consider the function

$h(x)=\frac{1}{k}\sum_{i=k}^{2k-1}u_{2^{i}\lambda}(x)\,,\qquad x\in X\,.$

We want to apply Lemma 3.3. In order to do so, observe that
$u_{2^{i}\lambda}|_{X\setminus A}=u|_{X\setminus A}$, where

$A=X\setminus(B\setminus U^{2^{i}\lambda})=X\setminus(B\setminus
U_{B}^{2^{i}\lambda})=(X\setminus B)\cup U^{2^{i}\lambda}_{B}\,.$

Therefore, by Lemma 3.3 and properties (D1)–(D2), we obtain that

$g_{h}=\frac{1}{k}\sum_{i=k}^{2k-1}\Bigl{(}\kappa_{i}\mathbf{1}_{(X\setminus
B)\cup U^{2^{i}\lambda}_{B}\color[rgb]{0,0,0}}+g\mathbf{1}_{B\setminus
U^{2^{i}\lambda}}\Bigr{)}\in\mathcal{D}_{H}^{\beta}(h)\,.$

Observe that $U^{2^{k}\lambda}_{B}\supset
U^{2^{(k+1)}\lambda}_{B}\supset\dotsb\supset U^{2^{(2k-1)}\lambda}_{B}\supset
U^{4^{k}\lambda}_{B}$. By using these inclusions it is straightforward to show
that the following pointwise estimates are valid in $X$,

$\begin{split}\mathbf{1}_{B}g_{h}^{p}&\leq\bigg{(}\frac{1}{k}\sum_{i=k}^{2k-1}\Bigl{(}\kappa_{i}\,\mathbf{1}_{U_{B}^{2^{i}\lambda}}+g\mathbf{1}_{B\setminus
U^{2^{i}\lambda}}\Bigr{)}\bigg{)}^{p}\\\ &\leq
2^{p}\bigg{(}\frac{1}{k}\sum_{i=k}^{2k-1}\kappa_{i}\,\mathbf{1}_{U_{B}^{2^{i}\lambda}}\bigg{)}^{p}+2^{p}g^{p}\mathbf{1}_{B\setminus
U^{4^{k}\lambda}}\\\
&\leq\frac{C(\beta,p,c_{\mu})}{k^{p}}\sum_{j=k}^{2k-1}\bigg{(}\sum_{i=k}^{j}2^{i}\lambda\bigg{)}^{p}\mathbf{1}_{U_{B}^{2^{j}\lambda}}+2^{p}g^{p}\mathbf{1}_{B\setminus
U^{4^{k}\lambda}}\\\
&\leq\frac{C(\beta,p,c_{\mu})}{k^{p}}\sum_{j=k}^{2k-1}(\lambda
2^{j})^{p}\mathbf{1}_{U_{B}^{2^{j}\lambda}}+2^{p}g^{p}\mathbf{1}_{B\setminus
U^{4^{k}\lambda}}\,.\end{split}$

Observe that $h\in\operatorname{Lip}_{\beta}(X)$ is zero in $E_{Q}$ and $h$
coincides with $u$ on $B\setminus U^{2^{k}\lambda}$, and recall that
$g_{h}\in\mathcal{D}_{H}^{\beta}(h)$. Notice also that $B\subset Q^{*}$ and
$x_{B}\in E_{Q}$. The local boundary Poincaré inequality in Lemma 6.6 implies
that

$\displaystyle\frac{\mathbf{1}_{E_{Q}}(x_{B})}{\mathrm{diam}(B)^{\beta
p}}\int_{B\setminus U^{2^{k}\lambda}}\lvert
u(x)\rvert^{p}\,d\mu(x)=\frac{\mathbf{1}_{E_{Q}}(x_{B})}{\mathrm{diam}(B)^{\beta
p}}\int_{B}\lvert h(x)\rvert^{p}\,d\mu(x)$ $\displaystyle\leq
K\int_{B}g_{h}(x)^{p}\,d\mu(x)$
$\displaystyle\leq\frac{C(\beta,p,c_{\mu})K}{k^{p}}\sum_{j=k}^{2k-1}(\lambda
2^{j})^{p}\mu(U_{B}^{2^{j}\lambda})+2^{p}K\int_{B\setminus
U^{4^{k}\lambda}}g(x)^{p}\,d\mu(x)\,.$

The desired local inequality (7.30) follows by combining the estimates above.
∎

###  7.4. Completing proof of Theorem 7.6

We complete the proof as in [19]. Recall that $u\colon X\to{\mathbb{R}}$ is a
$\beta$-Hölder function with $u=0$ in $E_{Q}$ and that

$M^{\sharp}u+M^{E_{Q}}u=M^{\sharp,p}_{\beta,\mathcal{B}_{0}}u+M^{E_{Q},p}_{\beta,\mathcal{B}_{0}}u\,.$

Let us fix a function $g\in\mathcal{D}_{H}^{\beta}(u)$. Observe that the left-
hand side of inequality (7.7) is finite. Without loss of generality, we may
further assume that it is nonzero. By Lemma 7.11,

$\displaystyle\int_{B_{0}}\big{(}M^{\sharp}u(x)+M^{E_{Q}}u(x)\big{)}^{p-\varepsilon}\,d\mu(x)$
$\displaystyle\qquad\leq
C(p,c_{\mu})\color[rgb]{0,0,0}\int_{B_{0}}\big{(}M^{E_{Q}}_{Q}u(x)\big{)}^{p-\varepsilon}\,d\mu(x)+C(\beta,p,c_{\mu})\color[rgb]{0,0,0}\int_{B_{0}}g(x)^{p-\varepsilon}(x)\,d\mu(x)\,.$

We have

$\displaystyle\int_{B_{0}}\big{(}M^{E_{Q}}_{Q}u(x)\big{)}^{p-\varepsilon}\,d\mu(x)=\int_{Q^{*}}\big{(}M^{E_{Q}}_{Q}u(x)\big{)}^{p-\varepsilon}\,d\mu(x)=(p-\varepsilon)\int_{0}^{\infty}\lambda^{p-\varepsilon}\mu(Q^{\lambda})\,\frac{d\lambda}{\lambda}\,.$

Since $Q^{\lambda}=Q^{*}=Q^{2\lambda}$ for every
$\lambda\in(0,\lambda_{Q}/2)$, we find that

$\displaystyle(p-\varepsilon)\int_{0}^{\lambda_{Q}/2}\lambda^{p-\varepsilon}\mu(Q^{\lambda})\,\frac{d\lambda}{\lambda}$
$\displaystyle=\frac{(p-\varepsilon)}{2^{p-\varepsilon}}\int_{0}^{\lambda_{Q}/2}(2\lambda)^{p-\varepsilon}\mu(Q^{2\lambda})\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq\frac{(p-\varepsilon)}{2^{p-\varepsilon}}\int_{0}^{\infty}\sigma^{p-\varepsilon}\mu(Q^{\sigma})\,\frac{d\sigma}{\sigma}$
$\displaystyle=\frac{1}{2^{p-\varepsilon}}\int_{Q^{*}}\big{(}M^{E_{Q}}_{Q}u(x)\big{)}^{p-\varepsilon}\,d\mu(x)\,.$

On the other hand, by Lemma 7.28, for each $\lambda>\lambda_{Q}/2$,

$\displaystyle\lambda^{p-\varepsilon}\mu(Q^{\lambda})\leq
C(\beta,p,c_{\mu})\lambda^{-\varepsilon}\biggl{[}\frac{(\lambda
2^{k})^{p}}{2^{k\alpha}}\mu(U^{2^{k}\lambda})+\frac{K}{k^{p}}\sum_{j=k}^{2k-1}(\lambda
2^{j})^{p}\mu(U^{2^{j}\lambda})+K\int_{U^{\lambda}\setminus
U^{4^{k}\lambda}}g^{p}\,d\mu\,\biggr{]}.$

Since $p-\varepsilon>1$, it follows that

$\displaystyle\int_{Q^{*}}\big{(}M^{E_{Q}}_{Q}u(x)\big{)}^{p-\varepsilon}\,d\mu(x)$
$\displaystyle\leq
2(p-\varepsilon)\int_{\lambda_{Q}/2}^{\infty}\lambda^{p-\varepsilon}\mu(Q^{\lambda})\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq C(\beta,p,c_{\mu})(I_{1}(Q)+I_{2}(Q)+I_{3}(Q))\,,$

where

$\displaystyle I_{1}(Q)$
$\displaystyle=\frac{2^{k\varepsilon}}{2^{k\alpha}}\int_{0}^{\infty}(\lambda
2^{k})^{p-\varepsilon}\mu(U^{2^{k}\lambda})\,\frac{d\lambda}{\lambda}\,,\qquad$
$\displaystyle I_{2}(Q)$
$\displaystyle=\frac{K}{k^{p}}\sum_{j=k}^{2k-1}2^{j\varepsilon}\int_{0}^{\infty}(2^{j}\lambda)^{p-\varepsilon}\mu(U^{2^{j}\lambda})\,\frac{d\lambda}{\lambda}\,,$
$\displaystyle I_{3}(Q)$
$\displaystyle=K\int_{0}^{\infty}\lambda^{-\varepsilon}\int_{U^{\lambda}\setminus
U^{4^{k}\lambda}}g(x)^{p}\,d\mu(x)\,\frac{d\lambda}{\lambda}\,.$

We estimate these three terms separately. First,

$\displaystyle I_{1}(Q)$
$\displaystyle\leq\frac{2^{k(\varepsilon-\alpha)}}{p-\varepsilon}\int_{B_{0}}\big{(}M^{\sharp}u(x)+M^{E_{Q}}u(x)\big{)}^{p-\varepsilon}\,d\mu(x)$
$\displaystyle\leq
2^{k(\varepsilon-\alpha)}\int_{B_{0}}\big{(}M^{\sharp}u(x)+M^{E_{Q}}u(x)\big{)}^{p-\varepsilon}\,d\mu(x)\,.$

Second,

$\displaystyle I_{2}(Q)$
$\displaystyle\leq\frac{K}{k^{p}}\sum_{j=k}^{2k-1}2^{j\varepsilon}\int_{0}^{\infty}(2^{j}\lambda)^{p-\varepsilon}\mu(U^{2^{j}\lambda})\,\frac{d\lambda}{\lambda}$
$\displaystyle\leq\frac{K}{k^{p}(p-\varepsilon)}\bigg{(}\sum_{j=k}^{2k-1}2^{j\varepsilon}\bigg{)}\int_{B_{0}}\big{(}M^{\sharp}u(x)+M^{E_{Q}}u(x)\big{)}^{p-\varepsilon}\,d\mu$
$\displaystyle\leq\frac{K4^{k\varepsilon}}{k^{p-1}}\int_{B_{0}}\big{(}M^{\sharp}u(x)+M^{E_{Q}}u(x)\big{)}^{p-\varepsilon}\,d\mu\,.$

Third, by Fubini’s theorem,

$\displaystyle I_{3}(Q)$ $\displaystyle\leq
K\int_{B_{0}\setminus\\{M^{\sharp}u+M^{E_{Q}}u=0\\}}\bigg{(}\int_{0}^{\infty}\lambda^{-\varepsilon}\mathbf{1}_{U^{\lambda}\setminus
U^{4^{k}\lambda}}(x)\frac{d\lambda}{\lambda}\bigg{)}g(x)^{p}\,d\mu(x)$
$\displaystyle\leq
C(k,\varepsilon)K\int_{B_{0}\setminus\\{M^{\sharp}u+M^{E_{Q}}u=0\\}}g(x)^{p}(M^{\sharp}u(x)+M^{E_{Q}}u(x))^{-\varepsilon}\,d\mu(x)\,.$

Combining the estimates above, we arrive at the desired conclusion. ∎

##  8\. Local Hardy inequalities

We apply Theorem 7.4 in order to obtain a local Hardy inequality, see (8.2) in
Theorem 8.1. This inequality is then shown to be self-improving, see Theorem
8.4, and in this respect we follow the strategy in [23]. However, we remark
that the easier Wannebo approach [30] for establishing local Hardy
inequalities as in [23] is not available to us, due to absence of pointwise
Leibniz and chain rules in the setting of Hajłasz gradients.

######  Theorem 8.1.

Let $X$ be a geodesic space. Let $1<p<\infty$ and $0<\beta\leq 1$. Let
$E\subset X$ be a closed set which satisfies the $(\beta,p)$-capacity density
condition with a constant $c_{0}$. Let $B_{0}=B(w,R)$ be a ball with $w\in E$
and $R<\operatorname{diam}(E)$. Let $E_{Q}$ be the truncation of $E$ to the
Whitney-type ball $Q$ as in Section 6. Then there exists a constant
$C=C(\beta,p,c_{\mu},c_{0})$ such that

$\int_{B(w,R)\setminus E_{Q}}\frac{\lvert u(x)\rvert^{p}}{d(x,E_{Q})^{\beta
p}}\,d\mu(x)\leq C\int_{B(w,R)}g(x)^{p}\,d\mu(x)$ (8.2)

holds whenever $u\in\mathrm{Lip}_{\beta}(X)$ is such that $u=0$ in $E_{Q}$ and
$g\in\mathcal{D}_{H}^{\beta}(u)$.

###### Proof.

Let $u\in\mathrm{Lip}_{\beta}(X)$ be such that $u=0$ in $E_{Q}$ and let
$g\in\mathcal{D}_{H}^{\beta}(u)$. Lemma 7.14 implies that

$\lvert u(x)\rvert\leq
C(\beta,c_{\mu})d(x,E_{Q})^{\beta}\left(M^{\sharp,p}_{\beta,\mathcal{B}_{0}}u(x)+M^{E_{Q},p}_{\beta,\mathcal{B}_{0}}u(x)\right)$

for all $x\in Q^{*}$. Therefore

$\int_{Q^{*}\setminus E_{Q}}\frac{\lvert u(x)\rvert^{p}}{d(x,E_{Q})^{\beta
p}}\,d\mu(x)\leq
C(\beta,p,c_{\mu})\int_{B(w,R)}\left(M^{\sharp,p}_{\beta,\mathcal{B}_{0}}u(x)+M^{E_{Q},p}_{\beta,\mathcal{B}_{0}}u(x)\right)^{p}\,d\mu(x)\,.$

By Theorem 7.4, we obtain

$\int_{Q^{*}\setminus E_{Q}}\frac{\lvert u(x)\rvert^{p}}{d(x,E_{Q})^{\beta
p}}\,d\mu(x)\leq C(\beta,p,c_{\mu},c_{0})\int_{B(w,R)}g(x)^{p}\,d\mu(x)\,.$
(8.3)

It remains to bound the integral over $B(w,R)\setminus Q^{*}$. Since
$E_{Q}\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle Q}\kern
0.0pt}$\hss}{Q}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle Q}\kern 0.0pt}$\hss}{Q}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle Q}\kern
0.0pt}$\hss}{Q}}}$ and $Q^{*}=4Q$, we have $d(x,E_{Q})\geq 3r_{Q}>R/64$ for
all $x\in B(w,R)\setminus Q^{*}$. Thus, we obtain

$\displaystyle\int_{B(w,R)\setminus Q^{*}}\frac{\lvert
u(x)\rvert^{p}}{d(x,E_{Q})^{\beta p}}\,d\mu(x)\leq\frac{64^{\beta p}}{R^{\beta
p}}\int_{B(w,R)}\lvert u(x)\rvert^{p}\,d\mu(x)$
$\displaystyle\leq\frac{3^{p}64^{\beta p}}{R^{\beta
p}}\left(\int_{B(w,R)}\lvert
u(x)-u_{B(w,R)}\rvert^{p}\,d\mu(x)+\mu(B(w,R))\lvert
u_{B(w,R)}-u_{Q^{*}}\rvert^{p}+\mu(B(w,R))\lvert
u_{Q^{*}}\rvert^{p}\right)\,.$

By the $(\beta,p,p)$-Poincaré inequality in Lemma 3.7,

$\displaystyle\int_{B(w,R)}\lvert u(x)-u_{B(w,R)}\rvert^{p}\,d\mu(x)$
$\displaystyle\qquad\leq 2^{p}\operatorname{diam}(B(w,R))^{\beta
p}\int_{B(w,R)}g(x)^{p}\,d\mu(x)\leq C(p)R^{\beta
p}\int_{B(w,R)}g(x)^{p}\,d\mu(x)\,.$

For the second term, we have

$\displaystyle\mu(B(w,R))\lvert u_{B(w,R)}-u_{Q^{*}}\rvert^{p}$
$\displaystyle\qquad\leq\mu(B(w,R))\mathchoice{\mathop{\kern 1.99997pt\vrule
width=6.00006pt,height=3.0pt,depth=-2.49997pt\kern-8.00003pt\intop}\nolimits_{\kern-2.79996ptQ^{*}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{Q^{*}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{Q^{*}}}{\mathop{\kern
1.00006pt\vrule
width=5.0pt,height=3.0pt,depth=-2.59996pt\kern-6.00006pt\intop}\nolimits_{Q^{*}}}\lvert
u(x)-u_{B(w,R)}\rvert^{p}\,d\mu(x)$ $\displaystyle\qquad\leq
C(c_{\mu})\int_{B(w,R)}\lvert u(x)-u_{B(w,R)}\rvert^{p}\,d\mu(x)\leq
C(p,c_{\mu})R^{\beta p}\int_{B(w,R)}g(x)^{p}\,d\mu(x)\,.$

For the third term, we have $d(x,E_{Q})\leq d(x,w)<4r_{Q}<R$ for every $x\in
Q^{*}$. Thus

$\displaystyle\mu(B(w,R))\lvert u_{Q^{*}}\rvert^{p}\leq
C(c_{\mu})\int_{Q^{*}\setminus E_{Q}}\lvert u(x)\rvert^{p}\,d\mu(x)\leq
R^{\beta p}\int_{Q^{*}\setminus E_{Q}}\frac{\lvert
u(x)\rvert^{p}}{d(x,E_{Q})^{\beta p}}\,d\mu(x)\,.$

Applying inequality (8.3), we get

$\mu(B(w,R))\lvert u_{Q^{*}}\rvert^{p}\leq C(\beta,p,c_{\mu},c_{0})R^{\beta
p}\int_{B(w,R)}g(x)^{p}\,d\mu(x)\,.$

The desired inequality follows by combining the estimates above. ∎

Next we improve the local Hardy inequality in Theorem 8.1. This is done by
adapting the Koskela–Zhong truncation argument from [21] to the setting of
Hajłasz gradients; see also [23] and [18, Theorem 7.32] whose proof we modify
to our purposes.

######  Theorem 8.4.

Let $X$ be a geodesic space. Let $1<p<\infty$ and $0<\beta\leq 1$. Let
$E\subset X$ be a closed set which satisfies the $(\beta,p)$-capacity density
condition with a constant $c_{0}$. Let $B_{0}=B(w,R)$ be a ball with $w\in E$
and $R<\operatorname{diam}(E)$. Let $E_{Q}$ be the truncation of $E$ to the
Whitney-type ball $Q$ as in Section 6, and let
$C_{1}=C_{1}(\beta,p,c_{\mu},c_{0})$ be the constant in (8.2), see Theorem
8.1. Then there exist $0<\varepsilon=\varepsilon(p,C_{1})<p-1$ and
$C=C(p,C_{1})$ such that inequality

$\int_{B(w,R)\setminus E_{Q}}\frac{\lvert
u(x)\rvert^{p-\varepsilon}}{d(x,E_{Q})^{\beta(p-\varepsilon)}}\,d\mu(x)\leq
C\int_{B(w,R)}g(x)^{p-\varepsilon}\,d\mu(x)$ (8.5)

holds whenever $u\in\mathrm{Lip}_{\beta}(X)$ is such that $u=0$ in $E_{Q}$ and
$g\in\mathcal{D}_{H}^{\beta}(u)$.

###### Proof.

Without loss of generality, we may assume that $C_{1}\geq 1$ in (8.2). Let
$u\in\mathrm{Lip}_{\beta}(X)$ be such that $u=0$ in $E_{Q}$ and let
$g\in\mathcal{D}_{H}^{\beta}(u)$. Let $\kappa\geq 0$ be the $\beta$-Hölder
constant of $u$ in $X$. By redefining $g=\kappa$ in the exceptional set
$N=N(g)$ of measure zero, we may assume that (3.2) holds for all $x,y\in X$.
Let $\lambda>0$ and define $F_{\lambda}=G_{\lambda}\cap H_{\lambda}$, where

$G_{\lambda}=\bigl{\\{}x\in B(w,R):g(x)\leq\lambda\bigr{\\}}$

and

$H_{\lambda}=\\{x\in B(w,R):\lvert u(x)\rvert\leq\lambda
d(x,E_{Q})^{\beta}\\}.$

We show that the restriction of $u$ to $F_{\lambda}\cup E_{Q}$ is a
$\beta$-Hölder function with a constant $2\lambda$. Assume that $x,y\in
F_{\lambda}$. Then (3.2) implies

$\lvert u(x)-u(y)\rvert\leq d(x,y)^{\beta}\left(g(x)+g(y)\right)\leq 2\lambda
d(x,y)^{\beta}\,.$

On the other hand, if $x\in F_{\lambda}$ and $y\in E_{Q}$, then

$\lvert u(x)-u(y)\rvert=\lvert u(x)\rvert\leq\lambda d(x,E_{Q})^{\beta}\leq
2\lambda d(x,y)^{\beta}\,.$

The case $x\in E_{Q}$ and $y\in F_{\lambda}$ is treated in the same way. If
$x,y\in E_{Q}$, then $\lvert u(x)-u(y)\rvert=0$. All in all, we see that $u$
is a $\beta$-Hölder function in $F_{\lambda}\cup E_{Q}$ with a constant
$2\lambda$.

We apply the McShane extension 2.7 and extend the restriction
$u|_{F_{\lambda}\cup E_{Q}}$ to a $\beta$-Hölder function function $v$ in $X$
with constant $2\lambda$. Then $v=u=0$ in $E_{Q}$ and $v=u$ in $F_{\lambda}$,
thus

$g_{v}=g\mathbf{1}_{F_{\lambda}}+2\lambda\mathbf{1}_{X\setminus
F_{\lambda}}\in\mathcal{D}^{\beta}_{H}(v)$

by Lemma 3.3.

By applying Theorem 8.1 to the function $v$ and its Hajłasz $\beta$-gradient
$g_{v}$, we obtain

$\displaystyle\int_{(B(w,R)\setminus E_{Q})\cap F_{\lambda}}\frac{\lvert
u(x)\rvert^{p}}{d(x,E_{Q})^{\beta p}}\,d\mu(x)$
$\displaystyle\leq\int_{B(w,R)\setminus E_{Q}}\frac{\lvert
v(x)\rvert^{p}}{d(x,E_{Q})^{\beta p}}\,d\mu(x)$ $\displaystyle\leq
C_{1}\int_{F_{\lambda}}g(x)^{p}\,d\mu(x)+C_{1}2^{p}\lambda^{p}\mu(B(w,R)\setminus
F_{\lambda})\,.$

Since $H_{\lambda}=F_{\lambda}\cup(H_{\lambda}\setminus G_{\lambda})$ and
$C_{1}\geq 1$, it follows that

$\begin{split}&\int_{(B(w,R)\setminus E_{Q})\cap H_{\lambda}}\frac{\lvert
u(x)\rvert^{p}}{d(x,E_{Q})^{\beta p}}\,d\mu(x)\\\ &\qquad\leq
C_{1}\int_{F_{\lambda}}g(x)^{p}\,d\mu(x)+C_{1}2^{p}\lambda^{p}\mu(B(w,R)\setminus
F_{\lambda})\\\ &\qquad\qquad+\int_{(H_{\lambda}\setminus E_{Q})\setminus
G_{\lambda}}\frac{\lvert u(x)\rvert^{p}}{d(x,E_{Q})^{\beta p}}\,d\mu(x)\\\
&\qquad\leq
C_{1}\int_{G_{\lambda}}g(x)^{p}\,d\mu(x)+C_{1}2^{p}\lambda^{p}\bigl{(}\mu(B(w,R)\setminus
F_{\lambda})+\mu(H_{\lambda}\setminus G_{\lambda})\bigr{)}\\\ &\qquad\leq
C_{1}\int_{G_{\lambda}}g(x)^{p}\,d\mu(x)+C_{1}2^{p+1}\lambda^{p}\bigl{(}\mu(B(w,R)\setminus
H_{\lambda})+\mu(B(w,R)\setminus G_{\lambda})\bigr{)}.\end{split}$ (8.6)

Here $\lambda>0$ was arbitrary, and thus we conclude that (8.6) holds for
every $\lambda>0$.

Next we multiply (8.6) by $\lambda^{-1-\varepsilon}$, where
$0<\varepsilon<p-1$, and integrate with respect to $\lambda$ over the set
$(0,\infty)$. With a change of the order of integration on the left-hand side,
this gives

$\displaystyle\frac{1}{{\varepsilon}}\int_{B(w,R)\setminus
E_{Q}}\biggl{(}\frac{\lvert
u(x)\rvert}{d(x,E_{Q})^{\beta}}\biggr{)}^{p-{\varepsilon}}\,d\mu(x)\leq
C_{1}\int_{0}^{\infty}\lambda^{-1-\varepsilon}\int_{G_{\lambda}}g(x)^{p}\,d\mu(x)\,d\lambda$
$\displaystyle\qquad+C_{1}2^{p+1}\int_{0}^{\infty}\lambda^{p-1-\varepsilon}\bigl{(}\mu(B(w,R)\setminus
H_{\lambda})+\mu(B(w,R)\setminus G_{\lambda})\bigr{)}\,d\lambda\,.$

By the definition of $G_{\lambda}$, we find that the first term on the right-
hand side is dominated by

$\frac{C_{1}}{{\varepsilon}}\int_{B(w,R)}g(x)^{p-\varepsilon}\,d\mu(x)\,.$

Using the definitions of $H_{\lambda}$ and $G_{\lambda}$, the second term on
the right-hand side can be estimated from above by

$\frac{C_{1}2^{p+1}}{p-\varepsilon}\biggl{(}\int_{B(w,R)\setminus
E_{Q}}\biggl{(}\frac{\lvert
u(x)\rvert}{d(x,E_{Q})^{\beta}}\biggr{)}^{p-{\varepsilon}}\,d\mu(x)+\int_{B(w,R)}g(x)^{p-\varepsilon}\,d\mu(x)\biggr{)}.$

By combining the estimates above, we obtain

$\begin{split}&\int_{B(w,R)\setminus E_{Q}}\biggl{(}\frac{\lvert
u(x)\rvert}{d(x,E_{Q})^{\beta}}\biggr{)}^{p-{\varepsilon}}\,d\mu(x)\\\
&\qquad\leq C_{2}\int_{B(w,R)\setminus E_{Q}}\biggl{(}\frac{\lvert
u(x)\rvert}{d(x,E_{Q})^{\beta}}\biggr{)}^{p-{\varepsilon}}\,d\mu(x)+C_{3}\int_{B(w,R)}g(x)^{p-\varepsilon}\,d\mu(x)\,,\end{split}$
(8.7)

where $C_{2}=C_{1}2^{p+1}\frac{\varepsilon}{p-\varepsilon}$ and
$C_{3}=C_{1}\bigl{(}1+2^{p+1}\frac{\varepsilon}{p-\varepsilon}\bigr{)}$. We
choose $0<\varepsilon=\varepsilon(C_{1},p)<p-1$ so small that

$C_{2}=C_{1}2^{p+1}\frac{\varepsilon}{p-\varepsilon}<\frac{1}{2}\,.$

This allows us to absorb the first term in the right-hand side of (8.7) to the
left-hand side. Observe that this term is finite, since $u$ is $\beta$-Hölder
in $X$ and $u=0$ in $E_{Q}$. ∎

##  9\. Self-improvement of the capacity density condition

As an application of Theorem 8.4, we strengthen Theorem 5.8 in complete
geodesic spaces. This leads to the conclusion that the Hajłasz capacity
density condition is self-improving or doubly open-ended in such spaces. In
fact, we characterize the Hajłasz capacity density condition in various
geometrical and analytical quantities, the latter of which are all shown to be
doubly open-ended.

######  Theorem 9.1.

Let $X$ be a geodesic space. Let $1<p<\infty$ and $0<\beta\leq 1$. Let
$E\subset X$ be a closed set which satisfies the $(\beta,p)$-capacity density
condition with a constant $c_{0}$. Then there exists $\varepsilon>0$,
depending on $\beta$, $p$, $c_{\mu}$ and $c_{0}$, such that
$\operatorname{\overline{co\,dim}_{A}}(E)\leq\beta(p-\varepsilon)$.

###### Proof.

Let $w\in E$ and $0<r<R<\operatorname{diam}(E)$. Let $E_{Q}$ be the truncation
of $E$ to the ball $Q\subset B_{0}=B(w,R)$ as in Section 6. Let
$\varepsilon>0$ be as in Theorem 8.4. Observe that

$E_{Q,r}=\\{x\in X\,:\,d(x,E_{Q})<r\\}\subset\\{x\in
X\,:\,d(x,E)<r\\}=E_{r}\,.$

Hence, it suffices to show that

$\frac{\mu(E_{Q,r}\cap B(w,R))}{\mu(B(w,R))}\geq
c\Bigl{(}\frac{r}{R}\Bigr{)}^{\beta(p-\varepsilon)},$ (9.2)

where the constant $c$ is independent of $w$, $r$ and $R$.

If $r\geq R/4$, then the claim is clear since
$\bigl{(}\frac{r}{R}\bigr{)}^{\beta(p-\varepsilon)}\leq 1$ and

$\mu(E_{Q,r}\cap B(w,R))\geq\mu(B(w,R/4))\geq C(\mu)\mu(B(w,R))\,.$

The claim is clear also if $\mu(E_{Q,r}\cap
B(w,R))\geq\frac{1}{2}\mu(B(w,R))$. Thus we may assume that $r<R/4$ and that
$\mu(E_{Q,r}\cap B(w,R))<\tfrac{1}{2}\mu(B(w,R))$, whence

$\mu(B(w,R)\setminus E_{Q,r})\geq\tfrac{1}{2}\mu(B(w,R))>0.$ (9.3)

Let us now consider the $\beta$-Hölder function $u\colon X\to{\mathbb{R}}$,

$u(x)=\min\\{1,r^{-\beta}d(x,E_{Q})^{\beta}\\}\,,\qquad x\in X.$

Then $u=0$ in $E_{Q}$, $u=1$ in $X\setminus E_{Q,r}$, and

$\lvert u(x)-u(y)\rvert\leq r^{-\beta}d(x,y)^{\beta}\quad\text{ for all
}x,y\in X.$

We aim to apply Theorem 8.4. Recall also that $w\in E_{Q}$. Thus we obtain

$\begin{split}\int_{B_{0}\setminus E_{Q}}\frac{\lvert
u(x)\rvert^{p-\varepsilon}}{d(x,E_{Q})^{\beta(p-\varepsilon)}}\,d\mu(x)&\geq
R^{-\beta(p-\varepsilon)}\int_{B_{0}\setminus E_{Q}}\lvert
u(x)\rvert^{p-\varepsilon}\,d\mu(x)\\\ &\geq
R^{-\beta(p-\varepsilon)}\int_{B_{0}\setminus E_{Q,r}}\lvert
u(x)\rvert^{p-\varepsilon}\,d\mu(x)\\\ &\geq
R^{-\beta(p-\varepsilon)}\mu(B(w,R)\setminus E_{Q,r})\\\ &\geq
2^{-1}R^{-\beta(p-\varepsilon)}\mu(B(w,R))\,,\end{split}$ (9.4)

where the last step follows from (9.3).

Since $u=1$ in $X\setminus E_{Q,r}$ and $u$ is a $\beta$-Hölder function with
a constant $r^{-\beta}$, Lemma 3.3 implies that
$g=r^{-\beta}\mathbf{1}_{E_{Q,r}}\in\mathcal{D}_{H}^{\beta}(u)$. Observe that

$\int_{B_{0}}g^{p-\varepsilon}\,d\mu\leq
r^{-\beta(p-\varepsilon)}\mu(E_{Q,r}\cap
B_{0})=r^{-\beta(p-\varepsilon)}\mu(E_{Q,r}\cap B(w,R))\,.$

Hence, the claim (9.2) follows from (9.4) and Theorem 8.4. ∎

The following theorem is a compilation of the results in this paper. It states
the equivalence of some geometrical conditions (1)–(2) and analytical
conditions (3)–(6), one of which is the capacity density condition. We
emphasize that the capacity density condition (3) is characterized in terms of
the upper Assouad codimension (1); in fact, this characterization follows
immediately from Theorem 5.7 and Theorem 9.1.

######  Theorem 9.5.

Let $X$ be a complete geodesic space. Let $1<p<\infty$ and $0<\beta\leq 1$.
Let $E\subset X$ be a closed set. Then the following conditions are
equivalent:

  * (1)

$\operatorname{\overline{co\,dim}_{A}}(E)<\beta p$.

  * (2)

$E$ satisfies the Hausdorff content density condition (5.4) for some
$0<q<\beta p$.

  * (3)

$E$ satisfies the $(\beta,p)$-capacity density condition.

  * (4)

$E$ satisfies the local $(\beta,p,p)$-boundary Poincaré inequality (6.7).

  * (5)

$E$ satisfies the maximal $(\beta,p,p)$-boundary Poincaré inequality (7.5).

  * (6)

$E$ satisfies the local $(\beta,p,p)$-Hardy inequality (8.2).

###### Proof.

The implication from (1) to (2) is a consequence of Lemma 5.3 with
$\operatorname{\overline{co\,dim}_{A}}(E)<q<\beta p$. The implication from (2)
to (3) follows by adapting the proof of Theorem 5.7 with $\eta=q/\beta$. The
implication from (3) to (4) follows from Theorem 6.6. The implication from (4)
to (5) follows from the proof of Theorem 7.4, which remains valid if we assume
(4) instead of the $(\beta,p)$-capacity density condition. The implication
from (5) to (6) follows from the proof of Theorem 8.1. Finally, condition (6)
implies the improved local Hardy inequality (8.5) and the proof of Theorem 9.1
then shows the remaining implication from (6) to (1). ∎

Finally, we state the main result of this paper, Theorem 9.6. It is the self-
improvement or double open-endedness property of the $(\beta,p)$-capacity
density condition. Namely, in addition to integrability exponent $p$, also the
order $\beta$ of fractional differentiability can be lowered. A similar
phenomenon is observed in [24] for Riesz capacities in ${\mathbb{R}}^{n}$.

######  Theorem 9.6.

Let $X$ be a complete geodesic space, and let $1<p<\infty$ and $0<\beta\leq
1$. Assume that a closed set $E\subset X$ satisfies the $(\beta,p)$-capacity
density condition. Then there exists $0<\delta<\min\\{\beta,p-1\\}$ such that
$E$ satisfies the $(\gamma,q)$-capacity density condition for all
$\beta-\delta<\gamma\leq 1$ and $p-\delta<q<\infty$.

###### Proof.

We have $\operatorname{\overline{co\,dim}_{A}}(E)<\beta p$ by Theorem 9.5.
Since $\lim_{\delta\to 0}(\beta-\delta)(p-\delta)=\beta p$, there exists
$0<\delta<\min\\{\beta,p-1\\}$ such that
$\operatorname{\overline{co\,dim}_{A}}(E)<(\beta-\delta)(p-\delta)$. Now if
$\beta-\delta<\gamma\leq 1$ and $p-\delta<q<\infty$, then

$\operatorname{\overline{co\,dim}_{A}}(E)<(\beta-\delta)(p-\delta)<\gamma
q\,.$

The claim follows from Theorem 9.5. ∎

A similar argument shows that the analytical conditions (4)–(6) in Theorem 9.5
are also doubly open ended. The geometrical conditions (1)–(2) are open-ended
by definition.

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# Parametric All-Optical Modulation on Chip

Zhan Li 1,2, Jiayang Chen 1,2, Zhaohui Ma 1,2, Chao Tang 1,2, Yong Meng Sua
1,2, Yu-Ping Huang 1,2,3,† 1Physics Department, Stevens Institute of
Technology, Hoboken, New Jersey, 07030, United States
2Center for Quantum Science and Engineering, Stevens Institute of Technology,
Hoboken, New Jersey, 07030, United States
3Quantum Computing Inc, Hoboken, NJ, 07030, United States
<EMAIL_ADDRESS>
###### Abstract

We demonstrate parametric all-optical modulation in a periodically-poled
lithium niobate microring resonator on chip. It employs quantum Zeno blockade
between two distinct waves, a signal and a pump, through their sum-frequency
generation at a large per-photon efficiency of $8.2$ MHz. With nanosecond pump
pulses at $6$ mW peak power, $85.7\%$ modulation extinction is observed,
achieving an efficiency improvement of over 30 times compared to previous
implementations. With only $2$ mW pump peak power, $43.0\%$ modulation
extinction is observed for a doubly-stronger signal at $4$ mW. This
demonstrates, for the first time, that optical transistors with cascadability
and fan-out are possible with just parametric nonlinear optics. These results,
together with inherent advantages in such photonic integrated circuits, open
the door to scalable technology for all-optical and quantum information
processing.

††preprint: APS/123-QED

Introduction— Quantum Zeno effect occurs when a coherently-evolving system is
coupled strongly to external degrees of freedom, with the result that the
evolution is suppressed or frozen [1, 2, 3, 4]. This counter-intuitive and
intriguing quantum phenomenon is implied in the postulates of quantum
mechanics, where performing measurement on a quantum object collapses its
wavefunction onto an eigenstate of the measuring apparatus. Here, the strong
coupling effectively implements fast repeated measurements, thereby forcing
the system to remain in the initial state. Since its first demonstration in
trapped ions in 1990 [5], the Zeno effect has been widely studied [6, 7, 8, 9,
10], and explored for exotic applications in quantum communications [11],
quantum computing [12], all-optical information processing [13, 14, 15], and
so on. Among them, quantum Zeno blockade (QZB) was analyzed and demonstrated
in nonlinear optical systems, including those of $\chi^{(2)}$ waveguides [16,
17], etalons with sandwiched crystals [18], whispering-gallary mode cavities
[14], a thin-film microdisk [19], and aluminum nitride microrings [20].

Optical QZB occurs in an open quantum system consisting of certain optical
modes coupled with external degrees of freedom through a nonlinear-optical
channel. When the channel is “fast” (i.e., the nonlinear interaction is
strong), the occupation of a mode can transiently alter the system condition
and suppress its coherent dynamics, including “block” additional photons from
coupling into a certain mode [21, 22, 13, 23]. Here, the mode can correspond
to a standing wave in an optical cavity, or a traveling wave in a waveguide or
optical fiber. The coherent dynamics can be the coupling from a bus waveguide
to the cavity, or evanescent coupling between two waveguides. In a cavity
setting, QZB can be implemented through sum-frequency generation (SFG) or
difference-frequency generation (DFG), where populating the cavity with pump
photons frustrates the cavity resonance condition for signal photons at a
different wavelength [22, 13, 18, 14]. As a result, the signal is deflected
from the cavity, and the frequency generation does not occur in the asymptotic
limit. During this process, the pump photons act effectively as a probe
dynamically monitoring if signal photons have entered the cavity, by coupling
the latter to another lossy mode [24]. When the monitoring is fast, the Zeno
effect prevents the signal from the cavity, thus the blockade effect [1, 2, 3,
5].

While QZB is a quantum Zeno effect, when the signal is a coherent light beam,
it can be intuitively understood in a classical-optics picture. Let’s consider
the case of a signal resonantly coupling to a cavity through a bus waveguide.
When there is no pump, the signal couples into the cavity by slowly building
up its field amplitude inside the cavity. Concurrent with this process, a
small portion of the cavity field leaks out to the bus waveguide and
destructively interferes with the signal remaining in the waveguide. It is
this destructive interference that will cause the majority of the signal to
enter the cavity. However, when there is a pump in the cavity to induce strong
SFG or DFG, any signal in the cavity will be quickly converted to a different
wavelength and effectively lost. This will prohibit its cavity-field building
up and thus disrupt the destructive interference. As a result, the majority of
the signal will continue to propagate in the bus waveguide and not enter the
cavity [22].

Hence, QZB provides a counter-intuitive implementation of the interaction
between optical signals: one to control another without them physically
overlapping to interact [7, 8]. Furthermore, unlike the photon blockade
mediated by single emitters, here no excitation of any kind is created, but
the interaction occurs due to the mere potential for a parametric nonlinear
optical process (which asymptotically does not occur) [16, 22]. These distinct
interaction-free and excitation-free features together eliminate the
occurrence of phase noise, photon dissipation, and quantum state decoherence
(e.g., via spontaneous emission) [25]. It thus promises to overcome the
fundamental challenges facing all-optical processing of quantum signals,
including realizing deterministic interaction between single photons for
scalable photonic quantum computing [24]. The same interaction-free and
excitation-free implementation of interaction makes QZB a strong candidate for
optical transistors [26], the backbone technology for of the future all-
optical information processing. Asides from the above practical advantages in
noise, loss, and decoherence, QZB naturally support superior cascadability,
fan out, and absence of critical biasing, as critical for practical optical
logic. This is because QZB effect transiently changes the cavity condition, so
that the blockade effect is independent of the signal’s power level and
detailed waveforms. It thus allows a weaker pump to control a strong signal,
or the output of one stage to drive multiple subsequent stages. Notably, as
QZB only utilizes nonlinear optics, there is no energy conversion from optics
to electronics or atomic, so that the otherwise detrimental heat sinking or
electrical interference issues are eliminated.

In view of those potentials, here we demonstrate, for the first time, a chip-
integrated device for QZB and efficient all-optical modulation with
cascadability and fan-out. Unlike previous QZB demonstrations, we use etched
and encapsulated microring cavities on thin-film lithium niobate [27, 28]. By
strong cavity confinement and quasi-phase matching, we observe high modulation
extinction ratio (MER) requiring only milliwatt peak power, pico-joule pulse
energy, corresponding to over orders of magnitude improvement over the state
of the art (see Table 1 below). For cascadability and fan out, we also
demonstrate that a weak pump can modulate a twice as strong signal. Together,
our results highlight QZB in integrated lithium niobate nanophotonics as an
enabling candidate for the future all-optical logic, photonic quantum
computing, and beyond.

Reference | Pump Peak Power | MER
---|---|---
[18] |  $17$ W | $\sim 80\%$
[19] |  $680$ mW | $\sim 51\%$
[20] |  $180$ mW | $\sim 80\%$
This work |  $6$ mW | $85.7\%$
Table 1: All-optical Zeno Effect Demonstrations. The MER of other works are
derived from their modulation results.

Experiments and simulations The periodically poled lithium niobate micro-ring
(PPLNMR), whose input-output is through a pulley bus waveguide, was fabricated
and characterized using procedures detailed in our previous works [29, 30, 31,
32]. We utilize the largest tensor $d_{33}$ of lithium niobate’s $\chi^{(2)}$
in 600 nm z-cut samples and fabricate PPLNMR chip with a photon-photon
coupling coefficient of $g=8.2$ MHz (angular frequency) and an overall
intrinsic Quality factor $Q_{i}=7\times 10^{5}$. It supports quasi-phase
matched sum-frequency generation among three quasi-transverse magnetic (quasi-
TM) modes, the signal at $\lambda_{s}=1545.73$ nm, the pump at
$\lambda_{p}=1558.01$ nm, and their sum-frequency at $\lambda_{f}=775.93$ nm.
Their intrinsic decay rates due to material absorption and scattering losses,
external decay rates due to coupling with the pulley waveguide, and the total
rates, are measured and listed in Table 2. As shown, for all three waves, the
external decay is faster than the intrinsic decay by at least four times,
which puts this chip in the significant over-coupling regime, as needed for
most optical information processing. In this experiment, the pump acts as the
control to frustrate the cavity resonance and modulate the signal, and the
sum-frequency light is nearly not generated when the pump is strong.

| Intrinsic Decay | External Decay | Total Decay
---|---|---|---
Pump |  $1.6$ GHz |  $8.45$ GHz |  $10.1$ GHz
Signal |  $1.8$ GHz |  $7.6$ GHz |  $9.4$ GHz
Sum-frequency |  $2.0$ GHz |  $12.1$ GHz |  $14.1$ GHz
Table 2: Decay rates of Pump mode, Signal mode and Sum-frequency mode in
angular frequency units. The intrinsic decay rates are derived by intrinsic Q
factor, which is $7\times 10^{5}$ at 1550 nm, corresponding to TM mode
propagation loss of 40 dB/m

Note that those measurements in Table 2 were performed at very low light
intensity level, when the cavity is “cold” (room temperature). During
experiment, the cavity may be heated up under high light intensity, and those
parameters may change modestly. Meanwhile, the phase matching condition will
change, which will require small change in chip temperature and the Pump and
Signal wavelength to restore.

The cavity dynamics can be modeled by the following equations [33]:

$\displaystyle\frac{dC_{p}}{dt}=(i\delta_{p}-\frac{\kappa_{t,p}}{2})C_{p}-ig^{*}C^{*}_{s}C_{f}-i\sqrt{\kappa_{e,p}}F_{p}$
(1)
$\displaystyle\frac{dC_{s}}{dt}=(i\delta_{s}-\frac{\kappa_{t,s}}{2})C_{s}-ig^{*}C^{*}_{p}C_{f}-i\sqrt{\kappa_{e,s}}F_{s}$
(2)
$\displaystyle\frac{dC_{f}}{dt}=(i\delta_{f}-\frac{\kappa_{t,f}}{2})C_{f}-igC_{p}C_{s}$
(3) $\displaystyle O_{p}=i\sqrt{\kappa_{e,p}}C_{p}+F_{p}$ (4) $\displaystyle
O_{s}=i\sqrt{\kappa_{e,s}}C_{s}+F_{s}$ (5) $\displaystyle
O_{f}=i\sqrt{\kappa_{e,f}}C_{f}$ (6)

In the above equations, the subscripts $\sigma=p,s,f$ denote the pump, signal,
and sum-frequency waves, respectively. $F_{\sigma}$ is the external wave at
the input. $C_{\sigma}$ is the amplitude of the cavity field normalized such
that $|C_{\sigma}|^{2}$ gives the photon number in the cavity. $O_{\sigma}$ is
the cavity output. The total cavity decay rate $\kappa_{t,\sigma}$ is the sum
of the intrinsic decay $\kappa_{i,\sigma}$ due to cavity losses and the
external decay $\kappa_{e,\sigma}$ through coupling with the bus waveguide.
$\delta_{\sigma}$ is the laser-cavity detuning, which is zero under ideal
phase matching. With cold cavity $\kappa_{\sigma}$, the simulation results are
calculated numerically.

Figure 1: Experiment schematic. Orange lines indicate optical connections
while blue lines are for electronic connections. Lensed fibers are used for
coupling lights in and out chip. TEC lies under the chip for temperature
controlling.

The experiment setup is depicted in the Fig. 1. It consists of two narrow-band
tunable lasers (Santec TSL-550/570), each followed by electronics and optics
for generating synchronized optical pulse. They include an arbitrary waveform
generator (AWG) generating synchronous electrical pulses to drive electric-
optical modulators (EOMs) at $1$ MHz for the pump light and $2$ MHz for the
signal light. The resulting optical pulse trains each undergo amplification by
an Erbium-doped fiber amplifier (EDFA), followed by wavelength division
multiplexers (WDMs) to filter out excessive ASE noise. Then, attenuators
(ATTs) are used to vary each optical power level, before combining via a
beamsplitter and directed jointly into the PPLNMR chip via a lensed fiber. The
chip sits on a thermoelectric cooler (TEC) that controls and stabilizes its
temperature with $0.01C^{\circ}$ precision. After the chip, another WDM is
used to separate the output pump and signal, before each is detected by a
photodetector (R2860D receiver module, 10 GHz) and an oscilloscope (HP Agilent
54750a, 20 GHz). The waveguide propagation loss is approximately 40 dB/m, as
inferred from our intrinsic Q factor of $7\times 10^{5}$. The bus waveguide is
4 mm long, which gives ignorable device insertion loss. For the lensed fiber
to chip edge coupling, the loss mainly comes from the mode mismatching between
fibers and the waveguide, which is 5 dB per facet.

Continuous-wave Signal and Pulsed Pump— We first demonstrate all-optical
modulation where a continuous-wave (CW) signal is modulated by a pulsed pump
(as the probe for QZB). To minimize thermal and photo-refractive (PR) effects
[34][35][36], the signal is prepared in a rectangular waveform with $10$-ns
quasi-Continuous Wave (quasi-CW) pulses at $2$ MHz repetition rate and $0.2$
mW peak power on chip (note that the power number to be quoted for the rest of
the paper always refers to the peak power in bus waveguide before the cavity).
The pump pulses, on the other hand, are in Gaussian form with $2$-ns full
width half maximum (FWHM), at $1$ MHz repetition rate, and are set in the
middle of the signal pulses.

The results are shown in Fig. 2(a-d). When there is no pump in cavity, the on-
resonance signal transmission loss through the chip is $63.5\%$ as compared to
the off resonance case, due to the signal mode over-coupling. Then, as shown
in Fig. 2(b), when pump pulses with low optical power ($0.25$ mW) are applied,
a dip is created in their overlapping regime, with the transmission dropping
to nearly zero (e.g., $\sim$ 100% loss). This is because the pump drives SFG
in the cavity, inducing additional losses and causing the cavity to transition
from overcoupling (in the absence of the pump) to critical coupling for the
signal.

Figure 2: Measured and simulated cavity transmission under various power
settings. The top panels plot the results for the case of quasi-CW signal and
pulsed pump, with the signal transmission under different conditions: (a)
without pump, (b) pump peak power at $0.25$ mW, (c) pump peak power at $2$ mW,
and (d) pump peak power at $6$ mW. The bottom panels plot the signal
transmission in the case of pulsed signal and pulsed pump, where, similar to
the top panels, (e) is the result without pump, and (f), (g), (h) are with
pump peak power at $0.25$ mW, $2$ mW, and $6$ mW, respectively. In all
figures, the solid curves represent the experimental results. The dashed
curves are the simulation results. The pink curves are the signal relative
transmission under off-resonance. The black curves are the signal relative
transmission on resonance with different pump peak powers. The orange curves
give the pump pulse profiles at the input.

In Figure. 2(c), as the pump power gradually increases to $2$ mW, the signal
after cavity transmission bounces back. At the peak of the pump pulses, the
signal mode transmission loss is reduced to $32.2\%$, which is about half of
the on-resonance case. This is the result of quantum Zeno blockade: when the
SFG-induced cavity loss is larger than the coupling decay, the signal is
effectively under coupled from the cavity, so that it mostly does not enter
the cavity [16, 18]. The MER, calculated as the percentage reduction of
transmission due to the pump in cavity, is $49.3\%$. As shown in Fig. 2(d),
further increasing the pump power to $6$ mW improves the MER to $74.5\%$ with
even lower transmission loss $14.8\%$. The trenches flanking the peak
indicates effective critical coupling for the signal in those regimes, where
the pump power induced cavity loses for the signal through SFG but not strong
enough to cause QZB. In both Figs. 2(c) and 2(d), the SFG at the pump peak is
suppressed compared to the weaker pump case in Fig. 2(b). Again, this is
because QZB blocks the signal from entering the cavity.

Also plotted in Figs. 2(a-d) are the simulation results by solving Eqs.(1-6)
with only the separately measured cold-cavity parameters in Table 2, without
using any fitting parameter. Here, the signal output power is 73 $\mu$W when
it is on resonance and there is no pump, which matches perfectly with the
simulation result. When the pump is 0.25 mW, the measured signal output power
is reduced to 5 $\mu$W (versus 0.3 $\mu$W as simulated). When the pump is 2
mW, the signal output power is measured to be 136 $\mu$W, while the simulation
gives 146 $\mu$W. Further increasing the pump to 6 mW, it becomes 170 $\mu$W
as measured and 180$\mu$W as simulated. As shown, the theoretical results
follow closely the experimental results, indicating a good modeling of
underlying dynamics. The disagreement in Figs. 2(c) and (d) may due to the an
over-estimate of the SFG efficiency $g$ and slight shift of the signal
resonance peak in a “hot” cavity with high pump power. Note that in Figs.
2(c-d), the transmitted signals have some oscillations around the peaks. This
could be due to the actual coupling condition for the SF wave is less over-
coupled, so that a strong SF field is trapped in the cavity, causing transient
parametric oscillation. More discussions can be found in the Supplemental
Materials.

Pulsed Signal and Pulsed Pump— Next we demonstrate the modulation of signal
pulses by pump pulses (the latter acting as the probe for QZB). The results
for Gaussian signal pulses at $2$-ns FWHM and with the same peak power are
shown in Fig.2(e-h). With a much lower average power for the signal, the chip
temperature and thus the cavity-coupling condition are changed, giving a
$53.9\%$ on-resonance transmission loss (which indicates higher over-
coupling). As seen, with the same pump applied, the modulation results follow
the trend with the above quasi-CW case. That’s when the signal overlapped with
0.25-mW pump pulses, the signal undergoes further suppression to critical
coupling. Increasing the pump power to $6$ mW results in the signal
transmission loss lowered to $7.6\%$ and MER reaching $85.7\%$.

The simulation results plotted in Fig.2(e-h), using the measured pulse
profiles and amplitudes and cold-cavity parameters in Table 2, give a signal
transmission dynamic mostly aligned with the experiment results. Here, without
the pump, the signal peak power after modulation is measured to be 92 $\mu$W
versus 73 $\mu$W as simulated. When the pump is 0.25 mW, the measured and
simulated power values at the output are 9 $\mu$W and 1 $\mu$W, respectively.
They increase to 132 $\mu$W and 146 $\mu$W for pump power at 2 mW, and 185
$\mu$W and 182 $\mu$W for 6 mW pump power.

Figure 3: Cavity transmission for the case of strong signal and weak pump.
Solid and dashed curves represent experimental and simulation results,
respectively. Black and pink curves are the transmitted pulse profiles of the
signal when it is on and off resonance, respectively. Orange curves are the
transmitted pump pulse profiles. Red curves show the transmitted signal pulse
profile modulated by the pump.

Strong Signal and Weak Pump— When the power of pump pulses reaches the
threshold to frustrate the cavity, they can modulate and switch signals of
much higher power. This unique feature, originating from QZB, is essential for
scalability, fan out, and absence of critical bias in practical optical
transistors. In Figure. 3, we demonstrate weaker pump pulses modulating
stronger signal pulses. The signal and pump pulse shapes are the same with
Fig. 2, but with the pump at $2$ mW and the signal at $4$ mW. As shown, the
signal on resonance transmission loss is $63.7\%$. By adjusting the phase on
AWG, the signal arrival time is tuned to achieve the best MER of $43.3\%$ at a
$600$ ps delay from the pump. Its transmission loss is reduced to $36.1\%$.
For a better understanding of the modulation as a function of the relative
delay time, we simulate the signal transmission as the relative arrival time
is swept. The results are plotted in Fig. 4, where the best MER of 35.5% is
obtained when the signal arrives 500 ps after the pump. More results can be
found in the Supplemental Materials.

Figure 4: Simulated modulation effects as a function of the signal arrival
time relative to the pump, where “+” and “-” in the legend indicate the signal
arriving before and after the pump, respectively.

Figure. 3 also plots the simulation results by using measured pulse profiles,
amplitudes, and the cold-cavity parameters, without any fitting parameters. As
seen, they are in good agreement with the experimental results. The on-
resonance transmission loss is $60.5\%$ and the MER is $35.5\%$, with $39.9\%$
transmission loss under the pump modulation. Compared with the experiment
results, the simulated MER is slightly lower. This is attributed to that the
actual signal coupling is weaker than the measured cold-cavity case. Also, the
simulated signal transmission shows a ringing bell oscillation around 0.5 ns
duration. This is because at the pump pulse tail, SFG is not strong enough to
prohibit the signal from entering the cavity via QZB, but rather causing
critical coupling for the signal. Further into the tail, the SFG is
diminished, because by then the pump has exited the cavity. Thus the signal
“sees” an overcoupling cavity, so that its transmission loss is less. Yet,
with the detector fast enough, this ringing is not observed in experiment,
which indicates other effects, such as those of photorefractive (PR), thermo-
optic, and second-harmonic generation (SHG), may need to be added to the
current model for a more accurate description of the system dynamics. This
will remain a task for the future studies.

Conclusion and Discussion— We have demonstrated parametric all-optical
modulation with high efficiency and presented clear evidences towards its uses
for practical optical transistor technology. It employs quantum Zeno blockade
in a periodic poled microring in thin-film lithium niobate, offering rich and
counter-intuitive nonlinear optical phenomena. Our simulation results agree
well with the experimental results, with subtle differences coming from side
effects not captured by our model, including those of PR, thermo-optic, and
SHG. They call for suppression of those effects for improved performance,
e.g., by addressing PR by MgO2 doping and addressing SHG by narrowing phase
matching. To reduce the thermo-optic effect, one can continuously monitor the
signal at an adjacent period to make sure it stays on resonance and fine
adjustment of the phase matching condition by tuning the tunable lasers and
the TEC.

From our results, the modulation performance is not sensitive to the pulse
length, as long as it exceeds the cavity life-time. In fact, in the “Pulsed
Signal and Pulsed Pump” case, a higher MER of $85.7\%W$ is seen for the same
pump power than in the case with a quasi-CW signal. Also, the MER can be
increased by operating in a less over-coupling regime. As of now, the power
consumption for modulation is 12 pJ. By using shorter pulses closer to the
cavity lifetime of $300$ ps, the modulation energy can be reducted to $\sim 1$
pJ, although we currently do not have this instrument to create such pulses.

We would like to note that the idea of light controlling light has gained
interest in the pursuit of all-optical information processing and beyond, as
an indispensable element and a powerful tool. Recent demonstrations have
employed two-photon absorption [37] and second-harmonic generation and its
reversion [38]. In comparison, QZB uniquely allows all-optical logic in an
interaction-free and excitation-free manner, promising chip-integrated optical
transistor technology that is robust and scalable. For example, in the
Supplemental Materials, we show how an all-optical switch is realized simply
by adding a drop port. For realistic device parameters and 1 ns pulses, a 6-mW
signal can be switched by a 2-mW pump with over 90% efficiency. Meanwhile, due
to interaction-free, $65\%$ of pump survives after the switch, so that it can
subsequently switch another signal pulse, providing the needed cascadability
for all-optical logic networks.

Another interesting feature with QZB is that, the roles of control and signal
are determined by which arrives at the cavity earlier. This gives rise to a
potentially powerful operation where the conditional logic is reversed by
controlling the relative time delay between the input pulses. For detailed
result figures please see the Supplemental Materials.

With the above features, the QZB devices provide fan-out for all-optical logic
operations. For example, two QZB switches can be cascaded to implement a
complex operation, with the signal output in one to act as the input pump for
the other, as long as the (quasi-)phase matching condition is still met. To
this end, one may apply a frequency converter to shift the signal wavelength,
or use a microcavity with multiple phase matching peaks.

Lastly, the interaction-free and excitation-free features make QZB
particularly suitable for quantum applications. Indeed, the same device shall
readily work for quantum signals at a single-photon level, except for
potential Raman scattering which can be suppressed by further detuning the
pump and signal. Furthermore, by improving the intrinsic Q-factor to $10^{8}$,
as demonstrated in a laser-written and chemical and mechanical polished
microring [39], the required pump power can be reduced to single-photon level
as well. This opens door to single-photon anharmonicity and quantum C-NOT
gates between single photons.

###### Acknowledgements.

The research was supported in part by the Office of Naval Research (Award No.
N00014-21-1-2898). Device fabrication was performed at Advanced Science
Research Center, City University of New York.

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