# Supplementary Information

###### Contents

  1. I SEOM device fabrication
  2. II RSFQ driver circuits
  3. III Quantum optics description of traveling-wave electro-optic modulation
    1. III.1 Hamiltonian for traveling-wave electro-optic modulation
    2. III.2 Dispersion and phase matching in traveling-wave electro-optics
    3. III.3 Dynamic equations and solutions
    4. III.4 Connection between $g_{0}$ and $V_{\pi}L$
  4. IV  $V_{\pi}$ change from room temperature to cryogenic temperature
  5. V Electro-optic bandwidth of SEOM
    1. V.1 Microwave dispersion and loss
    2. V.2 Electro-optic response
  6. VI  Eye diagram SNR analysis at low drive voltages
    1. VI.1 Theoretical model
    2. VI.2 Experimental results
  7. VII SEOM cryogenic packaging
  8. VIII Residual photorefractive effect and optical loss
    1. VIII.1 Photorefractive effect of HSQ cladded LNOI devices
    2. VIII.2 Electron beam induced material damage in LNOI
    3. VIII.3 The impact of photorefractive effect at cryogenic temperatures
  9. IX SEOM device performance projection

##  I SEOM device fabrication

The fabrication process flow for SEOM is depicted in Fig. S1. It begins with a
chip cleaved from an $x$-cut LNOI wafer (NanoLN), which has a 600 nm-thick
TFLN and 4.7 $\mathrm{\mu m}$ buried oxide on 500 $\mathrm{\mu m}$ high-
resistivity silicon substrate. Firstly, a hydrogen silsesquioxane (HSQ) resist
layer is spun on the chip to define optical waveguide using electron beam
lithography (EBL). Next, the LN film is etched by 350 nm using argon-base
reactive ion etching. After stripping the residual resist, a second layer of
900 nm-thick HSQ is spin-coated on the chip, followed by EBL to form HSQ
cladding. In this step, the dose of HSQ is gradually reduced at the edge to
give the HSQ cladding structure a sloped sidewall after development. The chip
is then annealed at 400 ∘C for 1 h. HSQ is chosen for LN waveguide cladding
instead of PECVD deposition to suppress the photorefractive effect Xu  _et
al._ (2021a). The Nb electrodes are defined through liftoff process using
polymethyl methacrylate (PMMA) as resist. Nb is evaporated using electron beam
in an ultra-high-vacuum chamber with $2\times 10^{-8}$ torr pressure during
the evaporation process. The deposition rate is approximately 10 Å/s.

Figure S1: SEOM fabrication process flow.

##  II RSFQ driver circuits

The RSFQ chip is designed by HYPRES, Inc., and fabricated at MIT Lincoln
Laboratory using SFQ5EE process Tolpygo  _et al._ (2016). It is capable of
both NRZ and RZ format signal generation, although only NRZ signal output is
utilized in our experiment. The circuit diagram NRZ signal generation is shown
in Fig. S2a. The operation of the RSFQ chip requires an external clock input
and 16 low frequency connections for current bias, control and monitoring
purposes. The logic circuit of the RSFQ chip has on-chip PRBS7 and PRBS15
signal generator, as described in Lehmann  _et al._ (2017). Additionally,
there is an external data input configured to accept customized data
generation.

In SFQ logic, information is stored in magnetic flux quanta and transferred
through voltage pulses. To interface with CMOS logic at room temperature, the
output circuit employs SQUID-stack drivers Inamdar  _et al._ (2009) called
Ostrich Gupta  _et al._ (2019), which generate approximately 5 mVpp digital
signal output. Within the data driver, an integrated SFQ/DC converter utilizes
a single SQUID driven by a flip-flop. When the flip-flop is in ’1’ state, it
generates a train of SFQ pulses; while in the ’0’ state, no output is
produced. This pulse train is directed to a stack of SQUIDs, and the output
signals from the SQUID stack are summed to generate the final output. By
carefully matching the excitation signal and the output signal from the SQUID
stack, high speed data rate can be achieved. The RSFQ circuit we use is
capable of data rates of up to 10 Gpbs. Two identical data drivers are
implemented in the RSFQ circuits, providing two identical outputs (output 1
and output 2 in Fig. S2a). While the RSFQ chip can operate at higher data
rates, the voltage waveform at the module’s interfaces may degrade due to the
limited interconnect bandwidth.

Figure S2: a, Circuit diagram of the RSFQ NRZ signal generation. The logic
circuit consists of PRBS7 and PRBS15 signal generator. The output circuit
employs a SQUID-stack driver. Operation of the RSFQ circuit takes a clock
input, external data input (optional) and 16 low frequency connection for
bias, control and monitors. The circuit has two identical outputs. b, Picture
of the packaged RSFQ stimulus module.

The RSFQ chip is enclosed in a cryogenic stimulus module, which incorporates a
magnetic shield and provides all necessary thermal connectivity and electrical
connections to the chip through pressure contact Dotsenko  _et al._ (2017).
The RSFQ stimulus module interfaces with external equipment through two types
of connections located at the base printed circuit board (PCB). The base PCB
is visible near the top of the Fig. S2b. Four high-speed signal connections
(clock, data input and two outputs) utilize female G3PO coaxial connectors by
Corning Gilbert. For the low frequency connections (current bias, monitoring
and control signals), a Min-E-Con 16-pin connector is used, as shown in Fig.
S2b. The low-frequency wiring employs a copper loom, while coaxial cables are
used for the high-speed signal. The complete stimulus module is mechanically
mounted onto the back side of 4 K plate of cryostat. Heaters and thermal
sensors are mounted on the module to tune the working temperature of the RSFQ
circuit for optimized signal quality.

##  III Quantum optics description of traveling-wave electro-optic modulation

With their remarkably low $V_{\pi}$ values, SEOMs can operate as broadband
microwave-to-optics transducers and offer competitive transduction efficiency
compared to cavity-based electro-optic quantum transducers. Here we present a
theoretical description of the traveling-wave electro-optic modulation process
within the quantum optics framework. Our approach follows the quantum
Hamiltonian treatment outlined in Sipe and Steel (2016). We use the theoretic
results derived here to fit our experimental results. Note that here we focus
more on the classical aspects of the electro-optic modulation process, and
vacuum fluctuation and quantum noise terms are omitted.

Figure S3: Dispersion consideration in traveling-wave electro-optic
modulators. A carrier (pump) mode $p$ at $\omega_{p}$ is scattered by a
microwave mode $m$ at frequency $\Omega$ into the sidebands $b$ at
$\omega_{p}+\Omega$ and $r$ at $\omega_{p}-\Omega$. The velocity difference
between microwave and optics leads to a wavenumber mismatch $\Delta k$.

###  III.1 Hamiltonian for traveling-wave electro-optic modulation

Consider an optical waveguide coupled with a co-propagating microwave
transmission line, the electrical and optic modal overlap leads to the
electro-optic coupling. We assume both optical waveguide and microwave
transmission line are invariant in the propagation direction ($z$ axis) and
have the same length $L$. With periodic boundary condition, the wave vector of
the microwave and optical modes must be an integer times $2\pi/L$. As
$L\rightarrow\infty$, the mode spectrum becomes continuous and mode branches
begin to form. Here we consider the dispersion of optical and microwave
branches of traveling-wave electro-optics, as depicted in Fig. S3. In this
case, the phase matching requirement is straightforward: the slope of the
optical mode dispersion curve (group velocity) must match that of the
microwave (phase velocity), known as velocity matching in traveling-wave
electro-optic modulator design. Note that if this phase matching requirement
is satisfied over a wide frequency range, high-order optical sidebands can be
generated in cascade. In the following analyses, we do not consider the higher
sidebands due to their low conversion efficiency. Hence, we have four modes of
interests in this model, optical pump $p$, blue and red sidebands $b$ and $r$,
and microwave mode $m$. The system Hamiltonian can then be expressed as
follows,

$H_{sys}=H_{p}+H_{b}+H_{r}+H_{m}+H_{V},$ (S1)

where $H_{p}$, $H_{b}$, $H_{r}$ and $H_{m}$ are the Hamiltonian for the four
modes of interests and $H_{V}$ is the electro-optic coupling term. In the
following, we derive the expression for each component of the system
Hamiltonian.

The electric field of the four modes shown in Fig. S3 can be expressed in
terms of the continuous mode creation and annihilation operators Blow  _et
al._ (1990),

$\displaystyle\mathbf{E}_{l}$ $\displaystyle=$
$\displaystyle\int\frac{l(k)\mathbf{u}_{lk}(x,y)e^{ikz}+H.c.}{\sqrt{4\pi\int\epsilon_{i}u_{lki}u^{*}_{lki}dxdy/\hbar\omega_{l}(k)}}dk,$
(S2)

where $l(k)$ is the annihilation operators of the continuous optical or
microwave electromagnetic modes of wavevector $k\mathbf{e}_{z}$, in which
$l=p,b,r$ or $m$; the electric field mode profile of each mode branch at wave
vector $k$: $\mathbf{u}_{lk}(x,y)=u_{lki}(x,y)\mathbf{e}_{i}$, where
$\mathbf{e}_{i}$ is the direction vector, $i=x,y$ or $z$ and Einstein
summation convention is applied. The denominators in the equation are for
normalization purpose and will disappear in the expression of Hamiltonian. The
commutation relations of the continuous annihilation and creation operators
indexed by $l$ and $l^{\prime}$ reads below Blow  _et al._ (1990)

$\displaystyle[l(k),l^{\prime}(k^{\prime})]=0,\;[l(k),l^{\prime\dagger}(k^{\prime})]=\delta_{ll^{\prime}}\delta(k-k^{\prime}).$
(S3)

With Eq. S2 and neglecting vacuum fluctuations, the Hamiltonian of each mode
branch is

$\displaystyle H_{l}$ $\displaystyle=$
$\displaystyle\int\epsilon_{i}E_{li}E_{li}^{*}dxdydz=\int\hbar\omega_{l}(k)l^{\dagger}(k)l(k)dk.$
(S4)

In Eq. S4, the $\frac{1}{2}$ factor of the electric field energy density is
multiplied by 2 to account for the magnetic field energy as the electric field
and the magnetic field have the same amount of energy.

With the continuous mode operators, a wave packet can be mathematically
expressed by the envelop operator, which is defined as

$\displaystyle\widetilde{L}_{\omega}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int l_{k}e^{i(k-k_{l}(\omega))z}dk,$ (S5)

where $\widetilde{L}_{\omega}(z)$ is the envelop of the wave packet with
central frequency $\omega$, $L=P,B,R$ or $M$ in our model. Through inverse
Fourier transformation, we have

$\displaystyle
l_{k}=\frac{1}{\sqrt{2\pi}}\int\widetilde{L}_{\omega}(z)e^{-i(k-k_{l}(\omega))z}dz.$
(S6)

The commutation relation of the envelop operators can be derived from the
commutation relation of the continuous creation and annihilation operators in
Eq. S3, which gives

$\displaystyle[\widetilde{L}_{\omega}(z),\widetilde{L}_{\omega^{\prime}}(z^{\prime})]$
$\displaystyle=$ $\displaystyle 0,$ (S7a)
$\displaystyle{[}\widetilde{L}_{\omega}(z),\widetilde{L}^{\dagger}_{\omega^{\prime}}(z^{\prime})]$
$\displaystyle=$ $\displaystyle
e^{i(k_{l}(\omega^{\prime})-k_{l}(\omega))z}\delta(z-z^{\prime}).$ (S7b)

Now we express the Hamiltonian using the envelop operators. Substituting Eq.
S6 to Eq. S4 and using the expansion expression for
$\omega_{l}(k)=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\partial^{n}\omega_{l}}{\partial
k^{n}}\mid_{\omega}(k-k(\omega))^{n}$ , the Hamiltonian for mode $l$ becomes

$\displaystyle
H_{l}=\int\hbar\widetilde{L}^{\dagger}(z,\omega)(\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\partial^{n}\omega_{l}}{\partial
k^{n}}\mid_{\omega}(-i\frac{\partial}{\partial
z})^{n}\widetilde{L}(z,\omega))dz.$ (S8)

To arrive at the expression above we used equation

$\displaystyle\int k^{n}f(x)e^{ikx}dkdx$ $\displaystyle=$
$\displaystyle(-i)^{n}\int f(x)\frac{\partial^{n}e^{ikx}}{\partial x^{n}}dkdx$
(S9) $\displaystyle=$
$\displaystyle(-i)^{n}\int\frac{\partial^{n}f(x)}{\partial x^{n}}e^{ikx}dkdx.$

Note that in Eq. S5, the integration is over all the wavenumber and physically
it means that the envelop operator represents the photons in all the frequency
range. In real experiments, what we encounter is narrow-band signals like the
laser. In this narrow bandwidth wavepacket assumption, we only consider the
first two terms of the expansion and the Hamiltonian can be simplified to

$\displaystyle H_{l}=\int($
$\displaystyle\hbar\omega\widetilde{L}^{\dagger}_{\omega}(z)\widetilde{L}_{\omega}(z)-i\hbar
v_{l}\widetilde{L}^{\dagger}_{\omega}(z)\frac{\partial\widetilde{L}_{\omega}(z)}{\partial
z})dz,$ (S10)

where $v_{l}=\frac{\partial\omega_{l}}{\partial k}\mid_{\omega}$ is the group
velocity at frequency $\omega$ for mode branch $l$. The first term assumes
that the signal is monochromatic and
$\widetilde{L}^{\dagger}_{\omega}(z)\widetilde{L}_{\omega}(z)$ is the photon
number density, while the second term is a correction to the monochromatic
assumption, because it is a wavepacket instead.

Next, we find the expression for $H_{V}$ in terms of the envelop operators. If
we take $p$ as the pump, there are two parts in $H_{V}$: parametric down
conversion between $p$, $r$ and $m$ and coherent electro-optic conversion (up
conversion) between $p$, $b$ and $m$. So we have

$\displaystyle H_{V}=\frac{1}{2}(\int
r_{ijk}\epsilon_{i}\epsilon_{j}E_{pi}E_{bj}E_{mk}dxdydz+\int
r_{ijk}\epsilon_{i}\epsilon_{j}E_{pi}E_{rj}E_{mk}dxdydz),$ (S11)

where $r_{ijk}$ is the electro-optic component and $(i,j,k)$ is the
permutation of $(x,y,z)$. In Eq. S2 the electric field mode profile
$u_{lk}(x,y)$ is a function of the wavelength, but within a narrow frequency
range, it is a good assumption that the mode profile does not change. Given
this, substitute Eq. S2 into $H_{V}$, we have

$\displaystyle H_{V}$ $\displaystyle=$ $\displaystyle\int(\frac{\hbar
g_{0}e^{i\Delta
kz}}{(2\pi)^{\frac{3}{2}}}p(k_{1})b^{\dagger}(k_{2})m(k_{3})+H.c.)dk_{1}dk_{2}dk_{3}dz+$
(S12) $\displaystyle\int(\frac{\hbar g^{{}^{\prime}}_{0}e^{i\Delta
kz}}{(2\pi)^{\frac{3}{2}}}r(k_{1})p^{\dagger}(k_{2})m(k_{3})+H.c.)dk_{1}dk_{2}dk_{3}dz$

$\displaystyle\hbar g_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{2}}\frac{\int
r_{ijk}\epsilon_{i}\epsilon_{j}u_{pi}u_{bj}^{*}u_{mk}dxdy}{\prod_{l=p,b,m}\sqrt{\int\epsilon_{li}u_{li}u_{li}^{*}dxdy/\hbar\omega_{l}}},\hbar
g^{{}^{\prime}}_{0}=\frac{1}{4\sqrt{2}}\frac{\int
r_{ijk}\epsilon_{i}\epsilon_{j}u_{ri}u_{pj}^{*}u_{mk}dxdy}{\prod_{l=r,p,m}\sqrt{\int\epsilon_{li}u_{li}u_{li}^{*}dxdy/\hbar\omega_{l}}}$
(S13)

where $\Delta k=k_{1}-k_{2}+k_{3}$ and $g_{0}$ is the vacuum electro-optic
coupling strength for the traveling-wave case. Suppose that the central
frequency of the wavepackets of mode $p$, $b$ , $r$ and $m$ are $\omega_{p}$,
$\omega_{b}$, $\omega_{r}$ and $\Omega$ respectively, then Eq. S12 can be
written in terms of envelop operators as

$\displaystyle H_{V}$ $\displaystyle=$ $\displaystyle\int(\hbar
g_{0}\widetilde{P}(z,\omega_{p})\widetilde{B}^{\dagger}(z,\omega_{s})\widetilde{M}(z,\Omega)e^{i\Delta
kz}+H.c.)dz+$ (S14) $\displaystyle\int(\hbar
g^{{}^{\prime}}_{0}\widetilde{R}(z,\omega_{p})\widetilde{P}^{\dagger}(z,\omega_{s})\widetilde{M}(z,\Omega)e^{i\Delta
k^{\prime}z}+H.c.)dz$

where $\Delta k=k_{p}(\omega_{p})-k_{b}(\omega_{b})+k_{c}(\Omega)$ and $\Delta
k^{\prime}=k_{r}(\omega_{r})-k_{p}(\omega_{p})+k_{c}(\Omega)$ are the phase
mismatch terms. Here we have derived the system Hamiltonian in terms of the
envelop operators, which can be applied to general electro-optic interaction
process. It will be simplified by the dispersion and phase matching conditions
when applied to traveling-wave electro-optic modulation process, as discussed
in the next section.

###  III.2 Dispersion and phase matching in traveling-wave electro-optics

With the optical pump at frequency $\omega_{p}$ and microwave signal at
$\Omega$, the blue and red sideband frequencies are
$\omega_{b}=\omega_{p}+\Omega$ and $\omega_{r}=\omega_{p}-\Omega$, as shown in
Fig. S3. In the traveling-wave electro-optic modulator scenario, there is only
one optical mode branch. Because the microwave frequency is far below that of
the optics, the optical dispersion can be simplified to a linear dispersion
relation, thus,

$\displaystyle k_{b}(\omega_{p}+\Omega)=k_{p}+v_{o}\Omega,$ (S15)
$\displaystyle k_{r}(\omega_{p}-\Omega)=k_{p}-v_{o}\Omega,$ (S16)

where $v_{o}$ is the optical group velocity at $\omega_{p}$. With this
dispersion relation, the $\Delta k$ and $\Delta k^{\prime}$ in Eq. S14 become

$\displaystyle\Delta k=\Delta k^{\prime}=(v_{m,p}-v_{o})\Omega,$ (S17)

where $v_{m,p}$ is the microwave phase velocity. Eq. S17 shows how the
wavenumber mismatch is related with the velocity mismatch. To realized a high-
bandwidth electro-optic modulation, $v_{m,p}$ needs to be a constant over that
frequency bandwidth and matches to $v_{o}$. As $n_{g}=n_{p}+f\frac{\partial
n_{p}}{\partial f}$, where $n_{g}$, $n_{p}$ and $f$ are the group index, phase
index and frequency, we have $n_{g}=n_{p}$ when the microwave is
dispersionless. So, the requirements for high bandwidth are that: the
microwave is dispersionless and its phase velocity (or group velocity) matches
to the group velocity of optics.

Because the optical modes $p,b$ and $r$ are in the same optical mode branch
and very close in frequency, the values $g_{0}$ and $g^{\prime}_{0}$ in Eq.
S13 are the same.

###  III.3 Dynamic equations and solutions

We are now prepared to derive the system dynamic equations. Using the
commutation relation Eq. S7, we obtain the Heisenberg equation in terms of the
envelop operators

$\displaystyle\frac{\partial\widetilde{P}}{\partial t}$ $\displaystyle=$
$\displaystyle-(i\omega_{p}+\frac{\kappa_{o}}{2})\widetilde{P}-v_{o}\frac{\partial\widetilde{P}}{\partial
z}-ig_{0}^{*}\widetilde{B}\widetilde{M}^{\dagger}e^{-i\Delta
kz}-ig_{0}\widetilde{R}\widetilde{M}e^{i\Delta kz},$ (S18a)
$\displaystyle\frac{\partial\widetilde{B}}{\partial t}$ $\displaystyle=$
$\displaystyle-(i\omega_{b}+\frac{\kappa_{o}}{2})\widetilde{B}-v_{o}\frac{\partial\widetilde{B}}{\partial
z}-ig_{0}\widetilde{P}\widetilde{M}e^{i\Delta kz},$ (S18b)
$\displaystyle\frac{\partial\widetilde{R}}{\partial t}$ $\displaystyle=$
$\displaystyle-(i\omega_{r}+\frac{\kappa_{o}}{2})\widetilde{R}-v_{o}\frac{\partial\widetilde{R}}{\partial
z}-ig_{0}^{*}\widetilde{P}\widetilde{M}^{\dagger}e^{-i\Delta kz},$ (S18c)
$\displaystyle\frac{\partial\widetilde{M}}{\partial t}$ $\displaystyle=$
$\displaystyle-(i\Omega+\frac{\kappa_{m}}{2})\widetilde{M}-v_{m}\frac{\partial\widetilde{M}}{\partial
z}-ig_{0}^{*}\widetilde{P}^{\dagger}\widetilde{B}e^{-i\Delta
kz}-ig_{0}^{*}\widetilde{R}^{\dagger}\widetilde{P}e^{-i\Delta kz}.$ (S18d)

where $\kappa_{o}$, $\kappa_{m}$ and $v_{o}$, $v_{m}$ are the intrinsic loss
rates and group velocities of the optical modes in the optical waveguide and
microwave mode in the transmission line respectively. We remove the fast
oscillating component in $\widetilde{P}$, $\widetilde{B}$, $\widetilde{R}$ and
$\widetilde{M}$ by introducing $P=\widetilde{P}e^{i\omega_{p}t}$,
$B=\widetilde{B}e^{i\omega_{b}t}$, $R=\widetilde{R}e^{i\omega_{r}t}$ and
$M=\widetilde{M}e^{i\Omega t}$. During the practical device operation, the
pump is a constant laser input with high intensity to boost the electro-optic
conversion efficiency. Because of its much stronger intensity than that of the
signal light and microwave, the back-action of electro-optic interaction on
the pump is negligible and the pump can be taken as a classical light source.
Therefore in the steady-state,

$\displaystyle P(z,t)=P_{0}e^{-\frac{\kappa_{o}}{2v_{o}}z}$ (S19)

where $P_{0}$ is a constant and the exponential decay term indicates the
dissipation in the optical waveguide of mode $p$. Suppose that at $z=0$, the
pump power is denoted as $P_{opt}=v_{o}\hbar\omega_{p}|P_{0}|^{2}$. If the
phase of the pump is set to be $0$,

$\displaystyle P_{0}=\sqrt{\frac{P_{opt}}{v_{o}\hbar\omega_{p}}}.$ (S20)

Combining this equation and Eq. S18, we can derive the temporal-spatial
dynamic equations of the system

$\displaystyle\frac{\partial B}{\partial t}+v_{o}\frac{\partial B}{\partial
z}$ $\displaystyle=$
$\displaystyle-\frac{\kappa_{o}}{2}B-ig_{0}P_{0}e^{(-\frac{\kappa_{o}}{2v_{o}}z+i\Delta
k)z}M,$ (S21a) $\displaystyle\frac{\partial R}{\partial t}+v_{o}\frac{\partial
R}{\partial z}$ $\displaystyle=$
$\displaystyle-\frac{\kappa_{o}}{2}R-ig_{0}^{*}P_{0}e^{(-\frac{\kappa_{o}}{2v_{o}}z-i\Delta
k)z}M^{\dagger},$ (S21b) $\displaystyle\frac{\partial M}{\partial
t}+v_{m}\frac{\partial M}{\partial z}$ $\displaystyle=$
$\displaystyle-\frac{\kappa_{m}}{2}M-ig_{0}^{*}P_{0}^{*}e^{(-\frac{\kappa_{o}}{2v_{o}}z-i\Delta
k)z}B-ig_{0}^{*}P_{0}e^{(-\frac{\kappa_{o}}{2v_{o}}z-i\Delta k)z}R^{\dagger}.$
(S21c)

With these dynamic equations, we solve for the intensity of the converted
optical sidebands and the microwave mode with microwave input $M(z)=M_{0}$ at
$z=0$. For the steady-state solution, where the time differential terms in Eq.
S21 are zero, the solution to Eq. S21 is

$\displaystyle B(z)$ $\displaystyle=$
$\displaystyle-i\frac{g_{0}P_{0}}{v_{o}}M_{0}e^{-(\alpha_{o}/2)z}\frac{1-e^{-(\alpha_{m}/2-i\Delta
k)z}}{\alpha_{m}/2-i\Delta k},$ (S22a) $\displaystyle R(z)$ $\displaystyle=$
$\displaystyle-i\frac{g_{0}P_{0}}{v_{o}}M_{0}e^{-(\alpha_{o}/2)z}\frac{1-e^{-(\alpha_{m}/2+i\Delta
k)z}}{\alpha_{m}/2+i\Delta k},$ (S22b) $\displaystyle M(z)$ $\displaystyle=$
$\displaystyle M_{0}e^{-(\alpha_{m}/2)z}.$ (S22c)

Here $\alpha_{o}=\frac{\kappa_{o}}{v_{o}}$ and
$\alpha_{m}=\frac{\kappa_{m}}{v_{m}}$ are the optical and microwave power
propagation loss. From this result we find that the microwave intensity is not
affected by the electro-optic interaction. This is because: in the parametric
down conversion process, one microwave photon is generated with one red
sideband optical photon; in the coherent electro-optic up conversion process,
one microwave is absorbed together with a pump photon to generate a blue
sideband optical photon. While for the traveling-wave electro-optic modulation
process, where the parametric down conversion and up conversion happen
simultaneously, the net effect on the microwave signal is that its photon
number remain the same if there is no propagation loss.

Under the condition that the modulation length is short (in the conventional
modulators’ case), $|(\alpha_{m}/2\pm i\Delta k)z|\ll 1$, the last terms in
solutions S22a and S22b reduce to $z$. The out-of-chip microwave-to-optical
transduction efficiency is then

$\displaystyle\eta=\frac{v_{o}|B(z)|^{2}}{v_{m}M_{0}^{2}}=\frac{(g_{0}P_{0})^{2}}{v_{o}v_{m}}e^{-\alpha_{o}z}z^{2}.$
(S23)

As $V_{\pi}$ is inversely proportional to modulation length, it implies that
the efficiency is inversely proportional to the square of $V_{\pi}$. In the
next section we point out the connection between $V_{\pi}$ and the vacuum
electro-optic coupling strength $g_{0}$, and Eq. S23 will be the same as the
result derived in Youssefi  _et al._ (2021).

When the modulation length is long, optical and microwave propagation loss and
the velocity mismatch $\Delta k$ come into play. In the last term of S22a and
S22b, if we ignore the propagation loss term and keep only the phase mismatch,
it becomes the sinc function, suggesting that if the velocity mismatch is
significant enough, there is null frequencies in the electro-optic response.
This will be illustrated in Section V where we use these equations to fit our
experimental results. If there is no velocity mismatch, the microwave and
optical loss become the fundamental limit on the conversion efficiency. Using
the results obtained here, we project in Section IX the ultimate achievable
performance of the SEOM devices.

###  III.4 Connection between $g_{0}$ and $V_{\pi}L$

In this section we show how the traveling-wave coupling strength $g_{0}$ used
in the quantum optics language is related to $V_{\pi}L$, the conventional
figure of merit to characterize the efficiency of electro-optic modulators.
Suppose that a voltage $V$ is applied across the electrodes, and the induced
optical mode index change is $\delta n(V)$, we have

$\displaystyle\frac{\delta n(V)}{n}=\frac{1}{2}\frac{\int
r_{ijk}\epsilon_{i}\epsilon_{j}u_{oi}u_{oj}^{*}u_{mk}(V)dxdy}{\int\epsilon_{i}u_{oi}u_{oi}^{*}dxdy/\hbar\omega}$
(S24)

where $\mathbf{u}_{o}(x,y)=u_{oi}(x,y)\mathbf{e}_{i}$, is the optical mode
profile, $i=x,y$ or $z$. $\mathbf{u}_{m}(V)(x,y)=u_{mi}(V)(x,y)\mathbf{e}_{i}$
is the microwave mode profile when voltage $V$ is applied. Here we see a
similarity between this equation and the definition of $g_{0}$ in Eq. S13.
Upon inserting $\frac{\delta n(V)}{n}$ into Eq. S13, we have

$\displaystyle g_{0}$ $\displaystyle=$
$\displaystyle\frac{\omega}{2\sqrt{2}}\frac{\delta
n(V)}{n}\sqrt{\frac{\hbar\Omega}{\int\epsilon_{i}u_{mi}(V)u_{mi}^{*}(V)dxdy}}$
(S25) $\displaystyle=$ $\displaystyle\frac{\sqrt{\hbar\Omega
v_{m}}v_{o}}{2}\frac{\omega\delta
n(V)z}{v_{o}n}\frac{1}{z\sqrt{2v_{m}\int\epsilon_{i}u_{mi}(V)u_{mi}^{*}(V)dxdy}}$
$\displaystyle=$ $\displaystyle\frac{\sqrt{\hbar\Omega v_{m}}v_{o}}{2}(k\delta
n(V)z)\frac{1}{z\sqrt{P_{m}(V)}}$

where $k$ is the optical wavenumber in vacuum and $k\delta n(V)z$ is the
optical phase change accumulated after an interaction length of $z$;
$P_{m}(v)$ is the microwave input power with voltage amplitude $V$. For a
given $z$, when $V=V_{\pi}$, $k\delta n(V_{\pi})z=\pi$ and
$P_{m}(V_{\pi})=V_{\pi}^{2}/(2Z_{0})$, $Z_{0}$ is the transmission line
impedance. Therefore we have

$\displaystyle g_{0}$ $\displaystyle=$ $\displaystyle\frac{\pi
v_{o}\sqrt{\hbar\Omega v_{m}Z_{0}}}{\sqrt{2}}\frac{1}{V_{\pi}L}$ (S26)

Substituting Eq. S26 into Eq. S23, we can express the conversion efficiency in
terms of $V_{\pi}$,

$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
P_{opt}\frac{\pi^{2}}{2}\frac{\Omega}{\omega}\frac{Z_{0}}{V_{\pi}^{2}}.$ (S27)

The $V_{\pi}L$ value is primarily determined by the gap between the modulation
electrodes, and the smallest possible gap distance is constrained by metal
induced optical loss when the electrodes are in close proximity to the
waveguide, as illustrated in the simulation shown in Fig. S4b. In order to
ensure that the metal loss is significantly lower than the intrinsic loss of
our fabricated optical waveguide (0.2 dB/cm), we aim for a metal loss of 0.02
dB/cm. Consequently, the value of $V_{\pi}L$ exceeds 2.1 V$\cdot$cm. In our
real device design, we choose the gap to be 5.2 $\mathrm{\mu m}$.

Figure S4:  Numerical simulation of metal induced opitcal loss and $V_{\pi}L$.

##  IV  $V_{\pi}$ change from room temperature to cryogenic temperature

Figure S5:  $V_{\pi}$ measurement at room temperature and 4 K. A 0.2 m-long
SEOM is measured at both room temperature and 4 K. Its $V_{\pi}$ increases
from 220 mV to 380 mV.

Cryogenic temperature dependence of the electro-optic coefficient of lithium
niobate has been studied Herzog  _et al._ (2008), and it shows that the
$r_{33}$ component (which is the component utilized in most electro-optic
modulators) decreases by 20% at 7 K. We also observe a $V_{\pi}$ increases by
70% from room temperature to cryogenic temperature in our SEOM. For a 0.2 m
SEOM, its $V_{\pi}$ at room temperature is 220 mV, and 380 mV at 4 K, as shown
in Fig S5. Compared with other studies, in Ti-diffusion waveguide electro-
optic modulators, $V_{\pi}$ decrease of 10% Youssefi  _et al._ (2021) and
increase of 74% Thiele  _et al._ (2022) are reported. On a ridge waveguide
based electro-optic modulator on LNOI, a $V_{\pi}$ increase of 15% is reported
Lomonte  _et al._ (2021a). In all the modulators mentioned above, $r_{33}$
component is commonly utilized in the modulator design. We think the variance
on the reported values might be due to different material growth methods and
fabrication conditions. With some of the reported results showing that the
$V_{\rm{\pi}}$ decreases or does not substantially change at cryogenic
temperature, it is possible that the 70% $V_{\rm{\pi}}$ increase could be
reduced or avoided, but this is subject to further investigations.

##  V Electro-optic bandwidth of SEOM

In the main text we present the measured electro-optic response along with
fitting. Here we provide the detailed data analysis and fitting procedure. We
use theoretical results Eq S22 obtained in section III.3 to model the electro-
optic frequency responses, which is captured by the last term of Eq S22a and
S22b. The parameters that depend on microwave frequency are $\alpha_{m}$, the
microwave loss and $\Delta k$, the wavenumber mismatch. We will discuss how we
determine the microwave loss and dispersion first and then use these
parameters to model the experimental data.

###  V.1 Microwave dispersion and loss

We first investigate the microwave dispersion through simulation. As shown in
Fig. S6a, the on-chip microwave transmission line mode is more dispersive at
low frequency than at high frequency. Above 10 GHz, the group and phase index
of microwave are close to each other, indicating that the microwave mode is
dispersionless in this frequency range. Exploiting this observation, we
approximately use the group index of high frequency to represent the microwave
phase index over the whole spectrum. The low frequency deviation is acceptable
because when the frequency $\Omega$ is small, $\Delta k=(v_{m,p}-v_{o})\Omega$
is more tolerant to the velocity mismatch.

The group index of the microwave transmission line is determined
experimentally using a VNA equiped with a time-domain option. By measuring the
time it takes a microwave pulse to travel through the transmission line, we
can calculate the microwave group velocity. The measured data is shown in Fig
S6b. Fig. S6a illustrates the simulated phase and group velocities without
considering the effect of kinetic inductance. When the kinetic inductance is
taken into account, the microwave group velocity is slowed down. The kinetic
inductance is dependent on temperature, and so is the microwave group index.
This temperature dependence is fitted by
$y=a\sqrt{1+b/\sqrt{1-(T/T_{c})^{4}}}$ Tinkham (2004), where $T$ is the
temperature, $T_{c}$ is the superconductive transition temperature 8.0 K
(measured), and $a,b$ are the parameters to be fitted. The kinetic inductance
provides us with the ability to tune the microwave velocity by adjusting the
temperature, thereby aiding in velocity matching.

The microwave propagation loss is quantified by directly measuring the S21
spectrum, as shown in Fig S6c. To extract the frequency dependent propagation
loss, a simple linear fitting is employed to determine the slope of the
envelope of the transmission line’s S21 response. In the linear fitting, the
frequency is in linear scale and the S21 response is in dB (logarithmic)
scale. In summary, we approximate the microwave phase index using its group
index, which has been validated through simulation. Both the microwave group
index and propagation loss are determined experimentally.

Figure S6: a, Numerical simulation of the microwave phase and group indices.
The cross-section of the microwave transmission line is as shown in Fig S4a.
The effective index (phase index) is directly simulated and the group index is
derived from the phase index through the equation $n_{g}=n_{p}+f\frac{\partial
n_{p}}{\partial f}$. Above 6 GHz, the phase and group index are close to each
other. b, Measured microwave group index of SEOM microwave transmission line
at different temperatures with fitting. The temperature dependence of the
kinetic inductance leads to the thermal tuning of group index. c, S21
measurement of SEOM microwave transmission line at different temperatures.
Solid lines are fitted envelop of the S21 response.

###  V.2 Electro-optic response

After extracting microwave group index and frequency-dependent microwave loss
from S21 response in Fig. S6, we next fit the electro-optic response shown in
Fig 3d in the main text. The only unknown parameter is the optical group index
that appears in $\Delta k$ as shown in Eq. S17. In principle, we can use the
simulated optical group index. However, the material’s properties like the
refractive index at cryogenic temperature is not well established from
literature. The optical group index is thus treated as a fitting parameter
inferred from the electro-optic response. In the scenario that the optical and
microwave index are very mismatched, there is null frequency in the electro-
optic response as predicted by Eq. S22. This null frequency in electro-optic
response provides a convenient signature to determine the optical index and
validate the model.

For this purpose, we fabricate and measure another SEOM device with the same
photonic waveguide structure as depicted Fig. 3d, but with a slightly
different transmission line geometry to create an intentional optical and
microwave index mismatch. The measured microwave index is shown in Fig. S7a,
where a large index mismatch is shown. The electro-optic response of this
device is shown in Fig S7b, and the bandwidth of the responses is only a few
GHz. Notably, the null frequencies appear at around 15, 17 and 20 GHz. By
fitting these electro-optic responses, we find that the best fit is achieved
when the optical group index $n_{o}$ is set to 2.28, while the simulated
optical group index using room temperature material coefficients is 2.25.
Employing this optical group index, along with the measured microwave index
and loss shown in Fig. S6, we obtain a good fit to the Fig. 3d in the main
text.

Figure S7: a, Measured microwave group index. In this device, the microwave
transmission line is intentionally designed to have a significantly lower
group index than the optical group index of 2.28. The optical group index is
obtained from the the fitting shown in the panel b. b Measured electro-optic
response of this SEOM device at 4.8 K, 5.6 K and 6.4 K, respectively. The
response is shifted by 24 dB for better visibility. Due to the index mismatch,
the 3 dB bandwidth of this device is only a few GHz and there is electro-optic
response dip at 15 GHz, 17 GHz and 20 GHz. The envelop of the curves are
fitted with $n_{o}=2.28$.

##  VI  Eye diagram SNR analysis at low drive voltages

In this section we study the fundamental limiting factors on the signal-to-
noise ratio (SNR) of eye diagrams in the low drive voltage region, where the
peak-to-peak voltage ($V_{\rm{pp}}$) is much smaller than the half-wave
voltage ($V_{\pi}$). The shot-noise-limited photoreceiver sensitivity has been
widely studied, and different receiver architectures and modulation formats
are employed to approach this limit HENRY (1989); Olsson (1989); Caplan  _et
al._ (2006); Lavery  _et al._ (2011). In this section we only focus on the
direct detection with on-off keying. One distinction here is that instead of
employing full-swing modulation ($V_{\rm{pp}}=V_{\pi}$), we utilize a small
drive voltage. We will examine how the shot noise limit is translated to the
optical carrier power, $V_{\pi}$ and working bias position in this specific
scenario.

###  VI.1 Theoretical model

Fig. S8a illustrate the eye diagram measurement setup. The MZI type SEOM is
biased near the transmission null point to enhance the measurement SNR, which
will be discussed further. As the transmission at this bias point is low, an
erbium-doped fiber amplifier (EDFA) is employed to overcome the electrical
noise in the photodetector and detection electronics. Here we assume a 100 %
quantum efficiency for the PD. Additionally, an optical filter is placed
between the EDFA and the PD to filter out the broadband amplified spontaneous
emission noise. After optical amplification, the optical power swing
encompasses a substantial portion of the photodetector’s output range, thus
overwhelming its thermal noise and dark current. Under this configuration, the
dominant noise source is from the optical shot noise, as shown in Fig S8b. The
optical output of an MZI type modulator is determined by phase difference
between the two interference arms. When biased at a specific point, the phase
modulation induces an optical output power change, encoding as digital bit 0
and 1. The photon number in each bit is $n_{1}$ and $n_{2}$, determined by the
optical power and data rate. The shot noise in these two bits can be expressed
as $\delta n_{1}=\sqrt{n_{1}}$ and $\delta n_{2}=\sqrt{n_{2}}$. We define the
SNR $=(n_{2}-n_{1})/(\delta n_{1}+\delta n_{2})$.

The eye diagram SNR at 1Gbps, with $V_{\rm{pp}}/V_{\pi}=0.01$, is plotted
versus different bias points with different optical output powers in Fig. S8c.
The optical power refers to the output power when the MZI is biased at its
peak transmission. Numerical simulation demonstrates that the highest SNR is
achieved when the MZI is biased near the transmission dip, owing to the steep
slope and consequent high modulation sensitivity. However, the SNR improvement
due to high extinction saturates as shot noise increases with decreasing
transmission. Experimentally, the SEOM is typically biased between -15 to -20
dB transmission. In our RSFQ circuit readout experiment, the SEOM’s output
power is approximately -3 dBm, roughly corresponding to the yellow curve in
Fig. S8c. Although the theoretical limit shows a SNR of about 40, out
experimentally result is about 10 dB away from this limit. Imperfections come
from noise figure of the EDFA, wide filtration band of the optical filter and
not deal quantum efficiency of the photodetector. SNR improvements can be made
from the device parameters point of view. If bit 0 is biased at the null
optical output, then $n_{1}=0$ and $\Delta
n=n_{2}\propto(V_{\rm{pp}}/V_{\pi})^{2}I_{o}/f$, where $I_{o}$ is the optical
output power and $f$ is the data rate. So, the SNR is

$\displaystyle\mathrm{SNR}\propto\sqrt{\frac{I_{0}}{f}}\frac{V_{\rm{pp}}}{V_{\pi}}.$
(S28)

The requirement for high SNR at a specific data rate speed aligns with the
need for high out-of-chip transduction efficiency shown in Eq. S27, which is
quadratic in $V_{\pi}$ and linear in optical power. With a fixed optical input
power, the output $I_{o}$ is determined by the propagation loss and modulation
length, while the $V_{\pi}$ is inversely proportional to the modulation
length. This tradeoff is captured by Eq. S23. In the following section, we
will discuss strategies to enhance the SNR (or equivalently electro-optic
transduction efficiency).

Figure S8:  a, Eye diagram measurement setup. EDFA: Erbium-Doped Fiber
Amplifier. PD: photodetector. b, Shot noise in a direct detection of on-off
keying with small driving voltage. The transmission of the MZI modulator is
determined by the phase difference of the two interference arms. The
transmission bias point and the modulation induced phase change determine the
photon number in bit 0 and 1. The two gray bars represent the photon number in
the digital 0 and 1 bits. The photon number difference is $\Delta
n=n_{2}-n_{1}$, shot noise $\delta n_{1}=\sqrt{n_{1}}$ and $\delta
n_{2}=\sqrt{n_{2}}$. SNR = $\Delta n/(\delta n_{1}+\delta n_{2})$. c, Eye
diagram SNR limit. In this calculation $V_{\rm{pp}}/V_{\pi}=0.01$ at 1 Gbps.

###  VI.2 Experimental results

Here we present experimental SNR results with different drive voltage and
optical power, and compare it with theoretical result Eq. S28. The results
shown in Fig. S9a is from the eye diagrams shown in the main text Fig 4, with
17 dBm optical input power at 1 Gbps data rate. When the drive voltage
increases from 5 mV to 10 mV, 20 mV, we see a relatively linear increase of
the SNR, consistent with Eq. S28. The SNR also improves with increasing
optical power as shown in Fig. S9b. The eye diagrams are taken with 50 mV
drive voltage at 3 Gbps data rate. The red curve is the square root function
fitting.

Figure S9:  a, Eye diagram SNR dependence with drive voltage. The eye diagrams
are taken with 17 dBm optical input power at 1 Gbps. The red curve is linear
fitting of the experimental data. b, Eye diagram SNR dependence with optical
power. The eye diagrams are taken with 50 mV $V_{\rm{pp}}$ at 3 Gbps. The red
curve is square root fitting of the experimental data.

##  VII SEOM cryogenic packaging

The photonic packaging employs on-chip optical grating couplers glued with
angled-facet bare fibers McKenna  _et al._ (2019). The grating couplers follow
the design principle outlined Lomonte  _et al._ (2021b). The SEOM chip is
first glued on a Si carrier chip. The optical fibers are the aligned to the
on-chip grating couplers and then glued to the SEOM chip surface using UV
glue. On each side, optical fibers are anchored to two small sapphire pieces
using stycast to improve the mechanical robustness, as shown in Fig. S10a. We
achieve a 6 dB/facet coupling efficiency after cooling to cryogenic
temperature.

The fiber-connectorized SEOM is then placed in an aluminum housing to
establish the interface with a PCB board. The PCB board features two 50
$\Omega$ microwave transmission lines. The SEOM chip is positioned alongside
the PCB board, and short wires are utilized for wire bonding (Fig. S10c). The
PCB board is soldered with two GPPO microwave connectors threaded within the
aluminum piece, as shown in Fig. S10b. There is another aluminum cover on the
package to ensure that the SEOM device is well shielded with robust mechanical
protection.

Figure S10:  SEOM device packaging scheme.

##  VIII Residual photorefractive effect and optical loss

It has been previously reported that PECVD silicon dioxide cladding on LNOI
optical waveguide causes significant photorefractive effect Xu  _et al._
(2021a). In our device fabrication process, we employ spin-on and thermally
annealed HSQ as cladding to mitigate the photorefractive effect. We find that
the photorefractive effect with this cladding approach is reduced and
comparable with that of air-cladded devices. Furthermore, we identify another
factor contributing to photorefractive effect and material damage in LNOI
optical devices, namely the electron beam dose during lithography. This dose
induces excessive optical loss, which emerges as the primary limiting factor
in our SEOM devices. Finally, we present the specific impact of
photorefractive effect at cryogenic temperatures for traveling-wave devices.

Figure S11:  Evaluation of photorefractive effect and optical loss in micro-
ring resonators. The wavelength is scanned from long wavelength to short
wavelength at 1 nm/s. The labeled optical input powers refer to the power in
the bus waveguide leading to the micro-ring resonators. Measured optical
quality factor and the corresponding propagation loss are labeled. a, The
original device transmission after TFLN etching and thermal annealing. b,
Device transmission after electron beam exposure. c, Remeasured device
transmission after a 2nd thermal annealing at 300 ∘C for 1 h.

###  VIII.1 Photorefractive effect of HSQ cladded LNOI devices

We fabricate micro-ring resonators to quantitatively assess the
photorefractive effect and process-induced optical loss. The waveguide etching
and HSQ cladding follow the procedures outlined in section I, with selected
devices omitting the HSQ cladding (thus air-cladded), serving as references.
After the device fabrication, thermal annealing is required to mitigate the
material absorption and restore device transmission Shams-Ansari  _et al._
(2022). Here we perform annealing at 400 ∘C for 1 h in a nitrogen atmosphere.
Immediately following the annealing, we measure the quality factor and
characterize the photorefractive effect in the ring resonator as shown in Fig.
S11a. The laser scan speed for the measurement is set at 1 nm/s, scanning from
long to short wavelength, to better reveal the photorefractive effect through
resonance shift Xu  _et al._ (2021a). The reference air-cladded device
exhibits no discernible resonance shift even at a high input power of 3 dBm,
maintaining a quality factor of 1.3 million. For the HSQ cladded device, the
optical quality factor is higher compared to the air-cladded resonators, as
the cladding layer reduces the index contrast of the sidewall roughness. The
measured resonance exhibits a quality factor close to 2 million. The
photorefractive effect in the HSQ-cladded device is not noticeable up to 0 dBm
optical power in the bus waveguide, indicating significant improvement over
PECVD cladded devices Xu  _et al._ (2021a). At 3 dBm optical power, the
observed resonance distortion suggests that thermal effect dominates over
photorefractive effect. Therefore, the HSQ cladding layer contributes to lower
optical loss and a weak photorefractive effect comparable with that of the
air-cladded devices.

###  VIII.2 Electron beam induced material damage in LNOI

In the process flow shown in section I, we have observed that the SEOM devices
after metal deposition experience a decrease in transmission and an increase
in propagation loss (from 0.2 dB/cm to 0.8 dB/cm). After ruling out the metal-
induced absorption loss as the cause, we pin down the actual cause of loss is
from the electron beam injected into TFLN during the electron beam
lithography. We confirm this by controlled ebeam exposure test on micro-ring
resonators. After the initial annealing, we use the electron beam writer to
expose selected areas near the optical waveguide (only electron beam is
injected, no resist spinning or any other following processes). The exposed
area is 2 $\mathrm{\mu m}$ away from the optical waveguide edge and the
electron beam dose is 600 $\mathrm{\mu C/cm^{2}}$, similar to the dosage of
the SEOM device fabrication process. Although the exposed area is not directly
on the optical waveguide, the scattered electrons still manage to expose the
waveguide regime through proximity effect.

After the electron beam exposure, a noticeable decrease of the quality factor
of the ring resonators is observed, as shown in Fig S11b. The air-cladded and
HSQ-cladded ring resonators possess quality factor of 1.3 million and 2.0
million, respectively, prior to the electron beam exposure. However, after the
exposure, the quality factor drops to around 600 thousand and 500 thousand,
respectively. The resonance shift in resonance scanning remain relatively weak
after the electron beam exposure. As a potential remedy, thermal annealing is
performed again in attempt to recover the quality factor, and the result is
shown in Fig. S11c. The device is annealed at 300 ∘C for 1 h. The air-cladded
device can recover its quality factor to its pre-ebeam value, while the HSQ-
cladded devices exhibit partial recovery. If annealed for a longer time or at
a higher temperature, the quality factor can also be fully recovered. However
this second thermal annealing presents two major challenges. Firstly, it is
incompatible with our SEOM fabrication flow, as the on-chip niobium is prone
to oxidation during the annealing process. Even at low temperature 200 ∘C
annealing under vacuum (1 $\times 10^{-5}$ torr at 200 ∘C), the
superconductive transition temperature decreases from 8 K to around 2 K.
Secondly, the photorefractive effect induced resonance shift is significantly
amplified after the 2nd annealing as indicated by the high-optical-power
resonance sweep in Fig, S11c. The underlying mechanism behind the enhanced
resonance shift after electron beam inject followed by thermal annealing is
subject to future studies. The impact of this residual photorefractive effect
on our SEOM device at cryogenic temperature will be discussed in the
subsequent section.

Given this incompatibility challenge between superconductor and thermal
annealing, our current SEOM devices experience an elevated optical loss (0.8
dB/cm) induced by electron beam exposure. This damage is solely caused by the
electron beam injection during the ebeam lithography process. Switching to
photolithography using a high resolution stepper will eliminate this problem
Luke  _et al._ (2020); Liu  _et al._ (2021). Additionally, alternative
solutions includes annealing in ultra-high vacuum chambers to protect the
superconductivity, or adopting alternative fabrication processes that allow
thermal annealing after electron beam lithography but before the
superconductor deposition. One example of such a process is the use of a high-
temperature tolerant liftoff resist. These approaches provide potential
strategies to mitigate the electron beam-induced optical loss and enhance the
performance of our SEOM devices.

###  VIII.3 The impact of photorefractive effect at cryogenic temperatures

In the previous section, we have shown that the photorefractive effect in LNOI
devices can vary significantly depending on the fabrication process. Here we
present how the photorefractive effect specifically impacts our SEOM device at
cryogenic temperatures. The SEOM device we use is based on a Mach-Zehnder
interferometer (MZI). In the presence of a strong photofrefractive effect, the
light induced material index change can lead to phase drift in the two arms of
the interferormeter, affecting the output power. To illustrate this, we
compare two devices processed differently: One device (Device 2) undergoes
waveguide etching, HSQ cladding and subsequent thermal annealing, and the
other device (Device 1) follows through the same process first and then is
deposited with metal through liftoff and further thermally annealed. As shown
in Fig. S11a and c, the photorefractive effect induced index change in Device
1 is considerably stronger than that in Device 2.

The optical transmission drift of these two devices at 4 K are shown in Fig.
S12. In the case of Device 2, with laser powers of 13 dBm and 3 dBm, and the
drift is relatively slow, within 5% over a few minutes. However, for Device 1,
strong optical transmission drift occurs even with 0 dBm laser input. The
drifting speed decreases as optical power decreases, and the transmission
remains relatively stable only when the input power intensity is down to -20
dBm. This transmission drift is of particular concern for applications that
require stable operation of the SEOM device, such as digital data links where
a stable transmission bias point is critical. With the slow transmission drift
observed in Device 2, the stability is sufficient for experimental
demonstrations. However, for long-term stability, active feedback on the SEOM
or laser wavelength is necessary. Another potential solution is to utilize
advanced modulation formats that do not rely on the absolute phase stability,
such as phase modulator in differential phase shift keying (DPSK). Despite the
relatively fast drift observed in Device 1, the drifting speed is well below
kHz, whose impact can be compensated for data transmission in the GHz speed
range.

Figure S12:  Optical transmission drift of two MZI-type SEOM devices at
cryogenic temperature caused by the photorefractive effect.

##  IX SEOM device performance projection

In sections III, V and VI, we present how the electro-optic transduction
efficiency, electro-optic bandwidth and the eye diagram SNR are related to the
SEOM device parameters. In this section we discuss the current limiting
factors and provide projections for future SEOM devices. We focus on two
essential figure of merits for electro-optic modulators: the transduction
efficiency (or modulation efficiency) and the bandwidth. According to Eq. S22,
the optical loss of the electro-optic materials sets the fundamental limit on
the transduction efficiency by determining the maximum achievable modulator
length. On the other hand, the microwave loss imposes the fundamental limit on
the electro-optic bandwidth under the velocity-matched condition. For the
completeness of the discussion, Fig. 5 in the main text is presented here
again.

As shown in Eq. S23, there exists an optimal modulation length
$l=2/\alpha_{o}$, that maximizes the transduction efficiency. Substituting
this value back into Eq. S23, we find that $\eta\propto(1/\alpha_{o})^{2}$.
However, as discussed in section VIII, our current fabrication process results
in elevated optical loss, which is the primary limiting factor as shown in
Fig. S13a. With the current propagation loss of 0.8 dB/cm, the optimal
modulation length is limited to 0.1 m (in each interference arm of the MZI
modulator). In section VIII we proposed solutions to reduce the propagation
loss back to 0.2 dB/cm. With this propagation loss, the optimal modulation
length would increase to 0.5 m and a transduction efficiency above 0.1 % could
be achieved. Also, even lower propagation loss down to 0.027 dB/cm has been
reported for TFLN Zhang  _et al._ (2017). With a 0.05 dB/cm propagation loss,
the transduction efficiency will approach 10% at meter-long modulation range.

The electro-optic frequency response is captured by the last term in Eq. S22a
and S22b. Assuming perfect velocity matching, the bandwidth limit arises from
the microwave loss. Fig. S13b illustrates the electro-optic response with
different index mismatch and microwave losses. In this analysis, we assume the
microwave propagation loss (in dB/m) is linearly proportional to the
frequency, by a linear coefficient $\alpha_{m}$ (dB/m/GHz). This assumption is
equivalent to considering a constant frequency-quality-factor product $fQ$ for
superconducting microwave resonators. For microwave losses of $\alpha_{m}=$0.2
and 0.1 dB/m/GHz, the corresponding $fQ$ products are $8\times 10^{11}$ and
$1.6\times 10^{12}$, respectively. It should be noted that much higher $fQ$
product than these on TFLN has been reported ($1\times 10^{13}$ Xu  _et al._
(2021b)). With this achievable microwave loss or $fQ$ product, the electro-
optic bandwidth would be limited by the optics and microwave index mismatch.
For a modulation length of 0.5 m, if the index difference is 0.01, the
bandwidth is estimated to be 20 GHz. By employing thermal fine tuning of the
velocity of the superconducting microwave, the index mismatch can be reduced
further to below 0.005. As shown in Fig. S13b, this can result in an electro-
optic bandwidth approaching 50 to 100 GHz.

Figure S13: a, Numerical simulation of the SEOM electro-optic transduction
efficiency with varying modulation length in each arm. The simulation
considers an optical input of 10 dBm and a microwave frequency of 10 GHz are
used. b, Numerical simulation of electro-optic response of an SEOM with a
modulation length of 0.5 m.

As stated in the main text, we use a 0.1 m-long each arm SEOM with 20 dB total
optical insertion loss for optical readout of an RSFQ circuit. In the
cryogenic environment, we applied a laser power of 17 dBm to the device and
performed the optical readout at 1 Gbps. If the limit on the optical loss is
lifted (0.05 dB/cm) and assuming a 2.5 dB/facet coupling loss, a 0.5 m-long
SEOM in each arm can have a total insertion loss of 10 dB and a $V_{\pi}$ that
is 10 times lower compared with the 0.1 m-long device. According to Eq. S28,
this would enable optical readout of the RSFQ circuit at 10 Gbps with an
optical input power of 0 dBm (100 fJ/bit). If we can enhance the receiver
sensitivity close to its theoretical limit (the current experimental SNR is 10
dB lower than that of the theoretical limit as shown in Sec. VI), the input
power requirement could be further reduced to -10 dBm (10 fJ/bit).

## References

  * Xu  _et al._ (2021a) Y. Xu, M. Shen, J. Lu, J. B. Surya, A. A. Sayem, and H. X. Tang, Optics Express 29, 5497 (2021a).
  * Tolpygo  _et al._ (2016) S. Tolpygo, V. Bolkhovsky, T. Weir, A. Wynn, D. Oates, L. Johnson, and M. Gouker, IEEE Transactions on Applied Superconductivity , 1 (2016).
  * Lehmann  _et al._ (2017) A. E. Lehmann, T. V. Filippov, S. M. Sarwana, D. E. Kirichenko, V. V. Dotsenko, A. Sahu, and D. Gupta, IEEE Transactions on Applied Superconductivity 27, 1 (2017).
  * Inamdar  _et al._ (2009) A. Inamdar, S. Rylov, S. Sarwana, and D. Gupta, IEEE Transactions on Applied Superconductivity 19, 1026 (2009).
  * Gupta  _et al._ (2019) D. Gupta, S. Sarwana, D. Kirichenko, V. Dotsenko, A. E. Lehmann, T. V. Filippov, W.-T. Wong, S.-W. Chang, P. Ravindran, and J. Bardin, IEEE Transactions on Applied Superconductivity 29, 1 (2019).
  * Dotsenko  _et al._ (2017) V. V. Dotsenko, A. Sahu, B. Chonigman, J. Tang, A. E. Lehmann, V. Gupta, A. Talalevskii, S. Ruotolo, S. Sarwana, R. J. Webber, and D. Gupta, IOP Conference Series: Materials Science and Engineering 171, 012145 (2017).
  * Sipe and Steel (2016) J. E. Sipe and M. J. Steel, New Journal of Physics 18, 045004 (2016).
  * Blow  _et al._ (1990) K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd,  _PHYSICAL REVIEW A_, Tech. Rep. (1990).
  * Youssefi  _et al._ (2021) A. Youssefi, I. Shomroni, Y. J. Joshi, N. R. Bernier, A. Lukashchuk, P. Uhrich, L. Qiu, and T. J. Kippenberg, Nature Electronics 4, 326 (2021).
  * Herzog  _et al._ (2008) C. Herzog, G. Poberaj, and P. Günter, Optics Communications 281, 793 (2008).
  * Thiele  _et al._ (2022) F. Thiele, F. vom Bruch, J. Brockmeier, M. Protte, T. Hummel, R. Ricken, V. Quiring, S. Lengeling, H. Herrmann, C. Eigner, C. Silberhorn, and T. J. Bartley, Journal of Physics: Photonics 4, 034004 (2022).
  * Lomonte  _et al._ (2021a) E. Lomonte, M. A. Wolff, F. Beutel, S. Ferrari, C. Schuck, W. H. P. Pernice, and F. Lenzini, Nature Communications 12, 6847 (2021a).
  * Tinkham (2004) M. Tinkham,  _Introduction to superconductivity_ (Courier Corporation, 2004).
  * HENRY (1989) P. S. HENRY, in  _Optical Fiber Communication Conference_, 3 (OSA, Washington, D.C., 1989) p. THK3.
  * Olsson (1989) N. Olsson, Journal of Lightwave Technology 7, 1071 (1989).
  * Caplan  _et al._ (2006) D. O. Caplan, M. L. Stevens, J. J. Carney, and R. J. Murphy, in  _2006 Conference on Lasers and Electro-Optics and 2006 Quantum Electronics and Laser Science Conference_ (IEEE, 2006) pp. 1–2.
  * Lavery  _et al._ (2011) D. Lavery, C. Behrens, and S. J. Savory, Optics Express 19, B836 (2011).
  * McKenna  _et al._ (2019) T. P. McKenna, R. N. Patel, J. D. Witmer, R. Van Laer, J. A. Valery, and A. H. Safavi-Naeini, Optics Express 27, 28782 (2019).
  * Lomonte  _et al._ (2021b) E. Lomonte, F. Lenzini, and W. H. P. Pernice, Optics Express 29, 20205 (2021b).
  * Shams-Ansari  _et al._ (2022) A. Shams-Ansari, G. Huang, L. He, Z. Li, J. Holzgrafe, M. Jankowski, M. Churaev, P. Kharel, R. Cheng, D. Zhu, N. Sinclair, B. Desiatov, M. Zhang, T. J. Kippenberg, and M. Lončar, APL Photonics 7, 081301 (2022).
  * Luke  _et al._ (2020) K. Luke, P. Kharel, C. Reimer, L. He, M. Loncar, and M. Zhang, Optics Express 28, 24452 (2020).
  * Liu  _et al._ (2021) Y. Liu, H. Li, J. Liu, S. Tan, Q. Lu, and W. Guo, Optics Express 29, 6320 (2021).
  * Zhang  _et al._ (2017) M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, and M. Lončar, Optica 4, 1536 (2017).
  * Xu  _et al._ (2021b) Y. Xu, A. A. Sayem, L. Fan, C.-L. Zou, S. Wang, R. Cheng, W. Fu, L. Yang, M. Xu, and H. X. Tang, Nature Communications 12, 4453 (2021b).