Supplementary
Materials
for
Few
-
cycle vacuum squeezing in nanophotonics
Rajveer Nehra
et al
.
Corresponding author
s
:
Rajveer Nehra, rnehra@caltech.edu; Alireza Marandi, marandi@caltech.edu
Sci
ence
37
7
,
1
3
33
(
202
2
)
DOI:
10.1126/sci
ence
.
abo6213
The
PDF
file
includes:
Materials and M
ethods
Figs. S1 to S13
Table S1
References
Supplementary information for, “Few-cycle vacuum
squeezing in nanophotonics”
Rajveer Nehra,
1
∗
, Ryoto Sekine
1
,
∗
, Luis Ledezma
1
,
2
,
Qiushi Guo
1
, Robert M. Gray
1
, Arkadev Roy
1
, and Alireza Marandi
1
,
†
1
Department of Electrical Engineering,
California Institute of Technology, Pasadena, California 91125, USA.,
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA.
∗
These authors contributed equally to this work.
†
E-mail: marandi@caltech.edu, rnehra@caltech.edu
1 Experimental Setup
The experimental setup for the generation and all-optical measurement of the squeezed state is
shown in Fig. S1. The Squeezer OPA and Measurement OPA are pumped by a mode-locked
Yb-fiber laser (Menlo Systems Orange A) generating
∼
75-fs-long pulses centered at
∼
1045
nm at a repetition rate of 250 MHz. The pump laser is first split into two paths, namely Pump
1 and Pump 2 in Fig. S1. The first beam (labeled as Pump 2) is sent to a delay stage with a
micrometer arm used for fine adjustments for temporal overlap; coarse adjustments are done by
tuning the position of the delay stage. Depending on the measurement at hand, the second beam
1
1045 nm
2090 nm
MLL
HWP
PBS
Delay Stage
BS
HWP
PZT
OPO
PD
LPF
VND
Pellicle
BS
FM
2
μ
m Source for device characterization
TEC
Chip
Obj.
Obj.
DM
1
μ
m OSA
FC
2
μ
m OSA
FM
Pump 2
Pump 1
Figure S1:
Experimental schematic for all-optical squeezing measurements
. The pump
laser is first split into two paths, namely Pump 1 and Pump 2, used for pumping the Squeezer
OPA and Measurement OPA, respectively. The relative phase between the pumps is modulated
by the PZT mounted on the delay stage in the Pump 2 arm. At the output of the nanophotonic
chip, the amplified squeezed signal and Measurement OPA pump are separated using a dichroic
mirror, and then are detected by optical spectrum analysers (OSAs) for spectral measurements.
PBS: Polarizing beamsplitter, HWP: Half-wave plate, DM: Dichroic mirror, Obj.: Reflective
objective, VND: Variable neutral-density filter, FC: Fiber Coupler, OSA: Optical spectrum an-
alyzer, PD: Photodetector, OPO: Optical parametric oscillator, FM: Flip Mirror.
2
(labeled as Pump 1) is either guided to the chip setup or to a synchronously pumped degener-
ate optical parametric oscillator (SPDOPO) used for generating pulses at 2090 nm (32 ). For
squeezing measurements, Pump 1 and Pump 2 are first combined at a 50:50 beamsplitter (BS),
and then focused on both Squeezer and Measurement OPA waveguides, respectively, using a
single high NA reflective objective (Newport: 50102-02). At the output of the nanophotonic
chip, we use another reflective objective (Newport: 50102-02) for collecting the light. Pump
powers are controlled using two variable neural density (VND) filters mounted on both the arms.
The relative phase between the Pump 1 and Pump 2 pulses is modulated by the piezoelectric
transducer (PZT) mounted on the delay stage. At the output of the chip, the amplified squeezed
vacuum and Measurement OPA pump are first separated using a dichroic mirror (DM) with
high transmission around 1045 nm and high reflectance around 2090 nm, and then are sent to
two different optical spectrum analysers (Yokogawa AQ6370D and Yokogawa AQ6375B) for
spectrum measurements at 2 nm resolution.
For linear characterization of our device and classical gain measurements of the OPAs, we
use the SPDOPO output centered at 2090 nm (32). Pump 1 is used for pumping the SPDOPO,
which is locked using a “dither and lock” technique, implemented with a Red Pitaya FPGA
board (33, 34). A variable ND filter is added to the output of the OPO to control the 2090
nm power. For OPA measurements, a well-attenuated SPDOPO signal is spatially overlapped
with Pump 2 at a dichroic mirror mounted in place of the 50:50 BS. A discussion on device
characterization and OPA gain measurements is provided in Sections 4 & 7.
2 Shot-noise calibration measurements
In this section, we discuss our shot-noise calibrations in greater detail. As discussed in the main
text, the design of our adiabatic coupler and its fabrication imperfections couples
∼
20% of
the Squeezer OPA pump to the high-gain Measurement OPA waveguide, which then interferes
3
with Pump 2, resulting in a gain modulation in the Measurement OPA. In order to accurately
determine the amount of squeezing, we measure this gain modulation by characterizing the
interference of the these two pumps in the Measurement OPA and removing its effect from
the squeezing measurements. Fig. S2A is an example of a Pump 1 and Pump 2 interference
fringe (green trace) acquired by an OSA in zero span mode at 1100 nm. While keeping the
Squeezer OPA Pump 1 blocked, we increase (decrease) the Measurement OPA Pump 2 power
to the maxima (minima) levels of the interference fringe, which corresponds to the blue (red)
trace in Fig. S2A. These two power levels of Pump 2 result in the “shot-noise maximum” and
“shot-noise minimum” shown in Fig. 2B in the main text. We measure the pump interference in
zero span mode at 1100 nm (the red arrow in the pump spectrum in Fig. S2C) to calibrate our
shot-noise measurements to ensure that our calibration is not affected by any nonlinear effects
in the Measurement OPA. This is confirmed by the linear transmission measurements, shown in
Fig. S2B, at 1100 nm for the Measurement O PA Pump 2 power levels varying in the range of
power levels of the recorded interference fringe.
To further verify our shot-noise calibrations, we recreate the interference pattern seen in
Fig. S2A off-chip and inject it into the Measurement OPA arm while keeping the Squeezer OPA
Pump 1 blocked. In Fig. S2D, we show the transmission measured in zero-span mode at 1100
nm. The interference fringe (green trace) is shown along with the maximum (blue) and mini-
mum (red) power levels of Pump 2. In Fig. S2E, we show the OSA output for these traces when
it was set to zero-span mode at 2090 nm. We can see that the maximum and minimum power
levels of the green trace in Fig. S2E stays within the maximum and minimum amplified shot-
noise levels. This is in stark contrast with the squeezing measurements (Fig. 2A and Fig. 3 in
the main text) where the green trace goes below/above the shot-noise minima/maxima, which is
due to the amplification of the squeezed and anti-squeezed quadratures of the original squeezed
state generated in the Squeezer OPA.
4
Zero span at 1100 nm
Zero span at 1100 nm
Pump spectrum
A
B
C
D
E
Zero span at 1100 nm
Zero span at 2090 nm
Figure S2:
Shot-noise calibration measurements
.
(A)
The green trace shows the interference
fringe between Pump 1 and Pump 2, measured from the Measurement OPA. The red and blue
traces correspond to the minimum and maximum power levels of Pump 2 when Pump 1 is
blocked.
(B)
Linear transmission measurements from the measurement amplifier in zero span
at 1100 nm.
(C)
Input pump spectrum normalized in linear units.
(D)
Measured fringes when
the Pump 1 and Pump 2 interference is created off-chip with the squeezer pump 1 blocked.
(E)
Measured amplified shot-noise levels for different Pump 2 power levels in Fig. D.
5
Next, we calibrate the shot-noise at various values of Pump 2 energy levels. In conventional
balanced homodyne detection (BHD), it is a common practice to verify the linearity of the
shot-noise by adding the linear losses or by increasing the local oscillator (LO) strength. This
verification test ensures that the BHD detection system does not add any measurement noise,
and the measured quadrature variances are truly due to the vacuum fluctuations. Mathemati-
cally, the measured quadrature variance of vacuum fields is
4
2
ˆ
X
φ
∝|
α
|
2
〈
ˆ
X
2
φ
〉
ρ
v
(S1)
where
ρ
v
is the vacuum state with
〈
ˆ
X
2
φ
〉
ρ
v
= 1
/
4
. As a result, by adding the losses (i.e.,
α
→
√
ηα
for
η <
1
) or by increasing the strength of the LO (i.e.,
α
→
√
kα
, for
k >
1
), one
can verify the linearity of the measured shot-noise, thereby confirming that the BHD does not
add any noise during measurements and the measured noise is the shot-noise limit.
Likewise, one can calibrate the shot-noise of an all-optical measurement in a similar manner
as BHD. In our all-optical measurements with high-gain OPAs, the roles of the beamsplitter
and LO in HD are played by the Measurement OPA and its pump (Pump 2). In this case, the
measured quadrature variance of the sufficiently amplified selected vacuum quadrature is given
by
4
2
ˆ
X
φ
∝
G
〈
ˆ
X
2
φ
〉
ρ
v
(S2)
where
G
=
e
2
Lη
√
P
is the gain of the Measurement OPA; L is the length of the Measurement
OPA,
η
is the nonlinear efficiency, and P is the Pump 2 strength. As a result, the amplified shot-
noise level (i.e., the noise of the high-gain OPA) grows exponentially with the Pump 2 (LO)
strength. We verified this by measuring the amplified shot-noise at various values of Pump
2 energy, as shown in Fig. S3. Our measured normalized shot-noise level (red points) agrees
well with the expected exponential behavior (black curve), as shown in the plot below. A good
agreement with the expected amplified shot-noise solely due to the phase-sensitive amplification
6
Figure S3: Shot-noise level measurements at various values of pump energy.
of the vacuum state confirms that the noise added by the Measurement OPA is negligible as
compared to the amplified shot-noise. We then ensured that the Measurement OPA has sufficient
gain. For a
direct
squeezing measurement, it is important for the Measurement OPA to amplify
the selected quadrature such that it entirely dominates the energy of the amplified field, and the
attenuated quadrature can be considered negligible. The desired and experimentally measured
gain levels are discussed in depth in Sec. S8.
3 Numerical Simulation Methods
In this section, we detail the methods used in our numerical simulations. To simulate the short-
pulse nonlinear dynamics in our devices, we solve a nonlinear envelope equation (NEE) in
the frequency domain using a split-step Fourier method (35), where the nonlinear step is inte-
grated using a fourth-order Runge-Kutta method. We obtained the NEE by ignoring counter-
propagating modes, which are usually phase mismatched, and assuming a constant nonlinear
7
coefficient across the entire simulation bandwidth. The NEE is given by:
∂A
∂z
=
−
i
[
β
(
ω
)
−
β
0
−
Ω
v
ref
−
i
α
2
]
A
−
iωε
0
X
0
8
d
(
z
)
F
Ω
{
a
2
e
jφ
(
z,t
)
+ 2
aa
∗
e
−
jφ
(
z,t
)
}
,
(S3)
where
A
(
z,ω
)
and
a
(
z,t
)
are the complex amplitude of the field during propagation in the
frequency and time domain,
φ
(
z,t
) =
ω
0
t
−
(
β
0
−
ω
0
/v
ref
)
z
,
β
0
is the waveguide propagation
constant at frequency
ω
0
,
Ω =
ω
−
ω
0
is the envelope frequency,
α
is the attenuation constant,
d
(
z
) =
±
1
is the sign of the nonlinear coefficient that varies along the waveguide due to quasi-
phase matching,
F
Ω
is the Fourier transform in
Ω
-space, and
X
0
is the effective nonlinear
coefficient. The pump and signal pulses were assumed to have a transform-limited, hyperbolic-
secant profile. We used a commercial software (Lumerical Inc.) to calculate the waveguide
modes and dispersion parameters used in our numerical simulations.
In Fig. S4A, we show the simulated squeezer OPA gain when it is seeded with a weak co-
herent pulse (
|
α
|
2
<<
1
) centered at 2090 nm and the relative phase between the pump and
weak signal is varied. For the pump pulse energy of
∼
1 pJ, the simulated OPA gain of
∼
10
dB agrees well with the measured gain (anti-squeezing) at the same pump energy levels. We
now determine the shortest signal pulse that can be amplified by our Squeezer OPA in a phase
sensitive manner. To determine that, we vary the pulse width of weak coherent pulse from 15 fs
to 100 fs while keeping the average pulse energy constant for a given pump pulse (
∼
75 fs) and
monitor the Squeezer OPA in the time domain. Fig. S4b shows the ratio,
R
=
τ
out
/τ
in
of the
FWHM widths of output and input pulses. Numerical simulations suggest that as we keep re-
ducing the pulse width of the input pulse from 100 fs, the amplified output pulse width remains
short until the input pulse width of
∼
30 fs, which corresponds to
∼
4 optical cycles. Beyond
that point the output pulse width starts to broaden in the time domain, as evident from
R >
1
in Fig. S4B. This may be attributed to stronger detrimental effects caused by a small group ve-
locity dispersion (GVD) and group velocity mismatch (GVM) in our devices for shorter pulses.
8
These effects can be minimized by a better management of GVD and GVM through dispersion
engineering. As a result, we find that the shortest pulse that our Squeezer OPA can amplify
while maintaining the pulse shape and offering high gain is
∼
30 fs. The temporal and spec-
tral evaluations of
∼
30 fs weak coherent pulse seeding the OPA are shown in Fig. S4C and
Fig. S4D, respectively. The input pulses are shown in blue traces and orange traces correspond
to output pulses. Dashed yellow and blue lines in Fig. S4C show the FWHM (
|
E
|
2
= 1
/
2
) for
determining the number of optical cycles. We see that the amplified pulse retains its shape and
contains
∼
4 cycles. Figure S4D shows the constant gain of
∼
18 dB over the entire bandwidth
for 4 pJ of pump energy. The simulated constant gain bandwidth agrees well with the measured
bandwidth of the optical parametric generation (OPG) from the squeezer OPA. The OPG sig-
nal is produced by amplifying (anti-squeezing) a quadrature while attenuating (squeezing) the
orthogonal quadrature of vacuum field. Therefore, our generated squeezed vacuum bandwidth
supports only a few optical cycles. Ideally, to measure the squeezing over the entire bandwidth
one needs to ensure that the Measurement OPA has the same gain bandwidth while having the
sufficient gain to amplify the microscopic squeezed vacuum to macroscopic levels, which can
be achieved by dispersion engineering in LN nanophotonics.
9
A
B
C
D
~4 cycles
Figure S4: Numerical simulations. (A), Simulated Squeezer OPA gain as the signal phase is
varied. (B), The ratio of FWHM bandwidths of output and input signal pulses for various values
of input pulse widths. (C), The electric fields of input (solid blue) and output (solid orange)
signal pulses. Dashed lines are at
|
E
|
2
= 1
/
2
for determining the pulse widths at FWHM. (D),
Spectrum of the weak coherent input and amplified output pulses.
10
4 Coupler design and characterization
In this section, we discuss our design for adiabatically tapered coupler. In our design, both
waveguides are linearly tapered while keeping the gap constant throughout the coupling length
as shown in Fig. S5A. Such a design offers a broadband coupling efficiency and is less suscep-
tible to fabrication imperfections. Due to the adiabatic nature of the coupling process, such a
coupler also relaxes the fixed beat length essential for a conventional non-tapered directional
coupler and prohibits the Rabi-like oscillations of optical field between the pair of waveguides
during propagation. The coupling efficiency of such a coupler given is by the Landau-Zener
formula (36):
η
c
= 1
−
exp
(
−
2
πg
2
λ
1
∂n
eff
/∂z.
)
(S4)
The coupling strength is
g
= (
n
e
−
n
o
)
/
2
, where
n
e
and
n
0
are the refractive indices for the
even and odd modes at the center of the coupler and
∂n
eff
/∂z
corresponds to the gradient in
the refractive index along the propagation length. From Eq. S4, one can see that the coupling
efficiency,
η
c
increases as the coupling strength,
g
increases. Due to the large difference in
the spatial mode sizes as seen in Fig. S5B, the fundamental transverse-electric (TE) modes
around 2090 nm of squeezed vacuum can be efficiently coupled from the squeezer waveguide
to measurement amplifier waveguide, while the fundamental TE mode around 1045 nm of the
squeezer pump mostly gets rejected. The numerically simulated coupling strengths around 1045
nm and 2090 nm are
O
(10
−
4
)
and
O
(10
−
3
)
, respectively. Figure S5C shows the simulated (solid
curves) and measured coupling efficiency and loss (dotted curves) over the measured squeezing
bandwidth. The simulated coupling efficiency (solid black) and loss (solid blue) are obtained
using using Eq. S4, where the effective refractive indices are calculated using the eigenmode
solver in Lumerical. We measure the coupling efficiency of
η
2090 nm
c
∼
0
.
70
using an auxiliary
signal centered at 2090 nm generated by our SPDOPO and
η
1045 nm
c
∼
0
.
20
around 1045 nm
11
W
2
W
1
Gap
L= 400 um
A
C
B
D
Figure S5: Design and characterization of adiabatically tapered coupler. (A), shows the design
in which we keep the gap constant and adiabatically vary the waveguide widths (
W
1
= 1668
nm and
W
2
= 1768 nm) along the propagation length. (B), Calculated fundamental TE modes
at 1045 nm and 2090 nm for the measured geometry. (C), Measured and simulated coupling
efficiency. Solid black and blue curves correspond to numerically simulated coupling efficiency
and losses, respectively. Shaded regions account for the fabrication uncertainty of
±
10
nm in
the etching depth. Black and blue dots show the measurements for coupling efficiency with an
auxiliary beam centered around 2090 nm. (D), An AFM image of the coupler region.
using the squeezer pump. The waveguide geometry for numerical simulations was obtained
using atomic force microscopy (AFM). An image for the coupler region is shown in Fig. S5D.
We measured the top widths of
W
1
∼
1668 nm and
W
2
∼
1768 nm, etching depth of
∼
380
nm,
thin-film thickness of
∼
713
nm, and sidewall angle of
∼
57
◦
.
In the coupling region of L = 400
μ
m, the measured bottom gap in the coupler region is
∼
400 nm. While our measured coupling efficiency around 2
μ
m is
∼
70%, it can be signifi-
cantly improved to
∼
100%, as suggested by our simulations. Figure S6 shows the numerically
simulated coupling efficiencies for the signal at 2090 nm and pump at 1045 nm. For these sim-
ulations, we use the thin-film thickness of 700 nm and etching depth of 350 nm. Top widths
12
A
B
Simulated coupling for the signal
Simulated coupling for the pump
Figure S6: Simulated coupling efficiencies, (A) for the signal, (B) for the pump field.
of
W
1
= 1750
nm and
W
2
= 1850 nm and the waveguide ridge angle is
∼
60
◦
. Three traces
correspond to the three different values of bottom gaps of 350 nm, 400 nm, and 450 nm. For
a coupler length of 300
μ
m, the coupling efficiencies for signal is
>
98% while
<
5
% of the
pump is getting coupled when the coupler gap is of 450 nm. Considering the ultra-low loss
propagation in LN nanophotonics and improved linear coupling of the coupler can lead to the
measurements of squeezing levels suitable for fault-tolerant quantum information processors.
5 Quantum formalism for the generation and all-optical mea-
surement of squeezed states
In this section, we derive the general formalism for the generation and all-optical measurement
of quadrature squeezed vacuum using ultrashort-pulse phase-sensitive optical parametric am-
plifiers (OPAs). In our formulation, we define the single-mode field quadrature operators as
ˆ
Q
:= (ˆ
a
+ ˆ
a
†
)
/
√
2
,
ˆ
P
:= (ˆ
a
−
ˆ
a
†
)
/
√
2
i
, and
̄
h
= 1
such that
4
Q
4
P
= 1
/
2
for the vac-
uum state. In the interaction picture, the effective Hamiltonian for a spontaneous parametric
13
down-conversion process in a
χ
(2)
waveguide OPA excited by a pulsed pump is given as (38,39)
ˆ
H
∝
∫ ∫
dω
s
dω
i
S
(
ω
s
,ω
i
)ˆ
a
†
s
(
ω
s
)ˆ
a
†
i
(
ω
i
) +
h.c.,
(S5)
where
S
(
ω
s
,ω
i
)
describes the spectral correlations in signal and idler modes described by the
photon creation operators
ˆ
a
†
s
(
ω
s
)
and
ˆ
a
†
i
(
ω
i
)
, respectively and H.C. stands for the hermitian
conjugate. By utilizing the Schmidt mode decomposition of the spectral correlation function
S
(
ω
s
,ω
i
) =
∑
m
c
m
φ
m
(
ω
s
)
ψ
m
(
ω
i
)
with
∑
m
|
c
m
|
2
= 1
, one can rewrite the Hamiltonian as
H
=
ξ
∑
m
(
c
m
ˆ
A
†
m
ˆ
B
†
m
+
c
∗
m
ˆ
A
m
ˆ
B
m
)
,
(S6)
where
ξ
determines the nonlinear interaction strength.
ˆ
A
†
m
and
ˆ
B
†
m
are broadband photon cre-
ation operators defined as (39)
ˆ
A
†
m
=
∫
dω
s
φ
m
(
ω
s
)ˆ
a
†
s
(
ω
s
)
(S7)
ˆ
B
†
m
=
∫
dω
i
ψ
m
(
ω
i
)ˆ
a
†
i
(
ω
i
)
,
(S8)
The resultant unitary evolution operator under the Hamiltonian in Eq. S6 is then given as
ˆ
U
=
exp
(
−
iξt
̄
h
∑
m
(
c
m
ˆ
A
†
m
ˆ
B
†
m
+
c
∗
m
ˆ
A
m
ˆ
B
m
)
)
,
(S9)
For brevity, we define
r
m
:=
−
iξtc
m
/
̄
h
=
|
r
m
|
e
iφ
m
, which determines the nonlinear interac-
tion strength of the
m
-th pairwise broadband spectral modes. The functions
{
φ
m
(
ω
s
)
}
and
{
ψ
m
(
ω
i
)
}
form a complete set of orthonormal functions, i.e.,
∫
φ
∗
m
(
ω
s
)
φ
l
(
ω
s
) =
δ
l,m
and
∫
ψ
∗
m
(
ω
i
)
ψ
l
(
ω
i
) =
δ
l,m
. As a result, the Hamiltonian in Eq. S6 can be considered as the effec-
tive Hamiltonian of an ensemble of independent broadband two-mode squeezers, which further
simplifies the unitary operator since we have
[
ˆ
A
l
,
ˆ
A
m
] = 0
and
[
ˆ
B
l
,
ˆ
B
m
] = 0
. Thus, we get
ˆ
U
=
⊗
m
exp
(
r
m
ˆ
A
†
m
ˆ
B
†
m
−
r
∗
m
ˆ
A
m
ˆ
B
m
)
.
(S10)
14
We now consider the
m
-th mode with signal and idler broadband modes described by the photon
annihilation operators
ˆ
A
m
and
ˆ
B
m
, respectively. In this case, the unitary operator is
ˆ
U
=
exp
(
r
m
ˆ
A
†
m
ˆ
B
†
m
−
r
∗
m
ˆ
A
m
ˆ
B
m
)
.
(S11)
In the Heisenberg picture, the evolution of the broadband operators is given by the Bogoliubov
transformation defined as (39)
ˆ
A
m
→
ˆ
A
m
cosh
r
m
+
e
iφ
m
ˆ
B
†
m
sinh
r
m
(S12)
ˆ
B
m
→
ˆ
B
m
cosh
r
m
+
e
iφ
m
ˆ
A
†
m
sinh
r
m
.
(S13)
In the quadrature representation, for
φ
m
= 0
, Eqs. S12 and S13 can be written as
ˆ
Q
A
m
±
ˆ
Q
B
m
→
(
ˆ
Q
A
m
±
ˆ
Q
B
m
)
e
±
r
m
(S14)
ˆ
P
A
m
±
ˆ
P
B
m
→
(
ˆ
P
A
m
±
ˆ
P
B
m
)
e
∓
r
m
,
(S15)
where we have used
ˆ
Q
= (ˆ
a
+ ˆ
a
†
)
/
√
2
and
ˆ
P
= (ˆ
a
−
ˆ
a
†
)
/
√
2
i
for the amplitude and phase
quadratures of the quantum field. From Eqs. S14 & S15, we see that the sum (difference) of
phase (amplitude) quadratures is squeezed and the difference (sum) of the phase (amplitude)
quadratures is anti-squeezed. We define
ˆ
Q
±
:=
ˆ
Q
A
m
±
ˆ
Q
B
m
and
ˆ
P
±
:=
ˆ
P
A
m
±
ˆ
P
B
m
. A simple
calculation shows that
[
ˆ
Q
±
,
ˆ
P
∓
] = 0
and hence, they can be diagonalized simultaneously. Their
shared eigenstate in the limit of
r
m
→∞
is the Einstein–Podolsky–Rosen (EPR) state with
ˆ
Q
−
|
EPR
〉
= 0
,
ˆ
P
+
|
EPR
〉
= 0
.
(S16)
In the finite squeezing limit, the EPR state serves as a crucial building block for many applica-
tions in continuous-variable quantum information processing (5-8). From Eq. S16, one can see
that the operators
ˆ
Q
−
and
ˆ
P
+
are the nullifiers for the EPR state and have been widely used for
15
their entanglement verification through the van Loock-Furusawa criterion in many frequency-
or time-multiplexed CV cluster state experiments (40). These nullifiers for each pair of modes
are typically measured using multimode balanced homodyne detectors (BHD) with local oscil-
lators (LO) tuned for each pair of frequency modes using electro-optic modulators (EOMs) (6-8,
10). While the multimode homodyne measurements have been successful in small-scale exper-
iments, it can be experimentally challenging when a large number of modes spanning over THz
of bandwidth are involved. Moreover, the nullifier measurements are inherently limited by the
electronic bandwidths of the EOMs, typically used for creating the LO sidebands to access the
individual frequency modes of the quantum optical frequency comb (8).
In the photon-number basis, the two-mode squeezed state can be expressed as
|
ψ
〉
m
A,B
=
∑
n
=0
C
nn
|
n
A
,n
B
〉
,
(S17)
where
c
nn
=
tanh
n
r
m
/
cosh
r
m
. As can be seen from Eq. S17, there are photon-number correla-
tions between the signal and idler modes. The average photon-number is
〈
N
A,B
m
〉
=
Tr
[
ρ
A,B
(
ˆ
N
A
m
+
ˆ
N
B
m
)] = 2
sinh
2
r
m
.
(S18)
So far, we have used single-mode operators for these derivations. We now adopt to frequency
agnostic two-mode complex quadrature formalism (41) where the two-mode complex quadra-
ture operators can be defined as
ˆ
X
A,B
:=
ˆ
A
+
ˆ
B
†
= (
ˆ
Q
+
+
i
ˆ
P
−
)
/
√
2
ˆ
Y
A,B
:=
i
ˆ
A
†
−
ˆ
B
= (
i
ˆ
Q
−
+
ˆ
P
+
)
/
√
2
,
(S19)
Here, we have omitted the mode index
m
because the formalism is equally valid for any num-
ber of correlated frequency modes (41). Similar to single-mode case, one can then define the
generalized two-mode quadrature operator as
ˆ
Z
A,B
=
cos
φ
ˆ
X
A,B
+
sin
φ
ˆ
Y
A,B
,
(S20)
16
where
ˆ
X
A,B
and
ˆ
Y
A,B
can be thought as the amplitude and phase quadratures of the two-mode
field. Using Eqs. S12, S13, S19, and S20, we can conclude that a two-mode OPA amplifies the
amplitude quadrature
ˆ
X
A,B
while attenuating the phase quadrature
ˆ
Y
A,B
without any additional
noise. Mathematically, we have
ˆ
X
A,B
→
ˆ
X
A,B
e
r
(S21)
ˆ
Y
A,B
→
ˆ
Y
A,B
e
−
r
.
(S22)
This is similar to a single-mode degenerate OPA where signal and idler photons are emitted
into the same optical mode. In the single-mode case, the Hamiltonian can be treated as an
effective Hamiltonian of an ensemble of independent broadband single-mode squeezers (39).
The resulting unitary evolution can then be simplified as
ˆ
U
=
⊗
m
=1
exp
[(
r
m
ˆ
A
†
m
2
−
r
∗
m
ˆ
A
2
m
)]
,
(S23)
In the Heisenberg picture, the evolution of the broadband operators is given by the Bogoliubov
transformation defined as (38)
ˆ
A
m
→
ˆ
A
m
cosh
r
m
+
e
iφ
ˆ
A
†
m
sinh
r
m
.
(S24)
Rewriting Eq. S24 in the quadrature representation for
φ
= 0
, we get
ˆ
Q
m
→
ˆ
Q
m
e
r
m
(S25)
ˆ
P
m
→
ˆ
P
m
e
−
r
m
(S26)
From Eqs. S21, S22, S25, and S26, one can see that the two-mode complex quadrature repre-
sentation puts the single-mode and two-mode OPAs at equal footing. In the two-mode complex
quadrature representation, the total average photon-number turns out to be
〈
ˆ
N
A,B
〉
ρ
∝
e
2
r
〈
ˆ
X
†
A,B
ˆ
X
A,B
〉
ρA,B
+
e
−
2
r
〈
ˆ
Y
†
A,B
ˆ
Y
A,B
〉
ρA,B
,
(S27)
17
where
ρ
A,B
is the input state to the OPA. In the high-parametric gain regime
e
2
r
>>
1
, the
average photon-number is entirely dominated by the amplified quadrature and the attenuated
quadrature can be considered negligible. The amplified quadrature power is in the macroscopic
regime, i.e., it’s sufficiently above the vacuum noise and can be directly measured with high-
tolerance to losses due to off-chip coupling, mode-mismatching, and imperfect detection. From
Eq. S27, we can see that the quadrature variance of the input state can be determined from a di-
rect measurement of the average photon-number of the amplified quadrature. For the generation
and all-optical measurement of squeezed vacuum states, one can cascade two such OPAs – the
first low-gain OPA can be pumped to generate squeezed vacuum, which can then be measured
by the second high-gain OPA, as shown in Fig. S7.
An ideal case is considered in Fig. S7a–OPA 1 is used to generated a squeezed vacuum state
which is then amplified with a high-gain OPA 2. From hereon, We will call OPA 1 and OPA 2 as
squeezer and measurement amplifiers, respectively. The squeezer phase,
φ
1
is kept fixed while
the measurement amplifier phase,
φ
2
is modulated to amplify each quadrature of the squeezed
vacuum state. The squeezer (measurement amplifier) gain is
G
1(2)
=
e
2
r
1(2)
. In the Heisenberg
picture, we find the evolution of annihilation operators after the first and second OPA. After the
OPA 1 in Fig. S7a, we get
ˆ
a
′
s
= ˆ
a
s
cosh
r
1
+
e
iφ
1
ˆ
a
†
i
sinh
r
1
ˆ
a
′
i
= ˆ
a
i
cosh
r
1
+
e
iφ
1
ˆ
a
†
s
sinh
r
1
.
(S28)
Likewise, after the OPA 2
ˆ
a
′′
s
= ˆ
a
′
s
cosh
r
2
+
e
iφ
2
ˆ
a
′†
i
sinh
r
2
ˆ
a
′′
i
= ˆ
a
′
i
cosh
r
2
+
e
iφ
2
ˆ
a
′†
s
sinh
r
2
.
(S29)
18
OPA 1
OPA 2
On
-
chip
Off
-
chip
Optical spectrum analyzer
OPA 1
OPA 2
On
-
chip
Off
-
chip
Linear
loss
Linear
loss
Microscopic
Macroscopic
A
B
Figure S7: Generation and all-optical measurement of a squeezed vacuum state. (A), An ideal
case where the squeezed vacuum is generated by OPA 1 and amplified by OPA 2 followed by
a total power measurement of the signal and idler modes. (B) Lossy case where linear losses
experienced by squeezed vacuum and amplified squeezed vacuum are considered. The green
and red arrows show the pump and signal-idler modes, respectively. Dashed arrows show the
initial vacuum for the signal and idler modes.
19