# Correlated twin-photon generation in a silicon nitrite loaded thin film PPLN
waveguide

Antoine Henry  1,2 David Barral  1 Isabelle Zaquine  2 Andreas Boes  3,4,5
Arnan Mitchell  3 Nadia Belabas  1 and Kamel Bencheikh1  1Centre for
Nanosciences and Nanotechnology, CNRS, Université Paris-Saclay, UMR 9001,10
Boulevard Thomas Gobert, 91120, Palaiseau, France
2LTCI, Télécom Paris, Institut Polytechnique de Paris, 19 place Marguerite
Perey, 91120, Palaiseau, France
3Integrated Photonics and Applications Centre (InPAC), School of Engineering,
RMIT University, Melbourne, VIC 3001, Australia
4Institute for Photonics and Advanced Sensing (IPAS), University of Adelaide,
SA 5005, Australia
5School of Electrical and Electronic Engineering, University of Adelaide, SA
5005, Australia<EMAIL_ADDRESS>
††journal: osajournal

Photon-pair sources based on thin film lithium niobate on insulator technology
have a great potential for integrated optical quantum information processing.
We report on such a source of correlated twin-photon pairs generated by
spontaneous parametric down conversion in a silicon nitride (SiN) rib loaded
thin film periodically poled lithium niobate (LN) waveguide. The generated
photon pairs have a wavelength centred at 1560 nm compatible with present
telecom infrastructure, a large bandwidth (21 THz) and a brightness of $\sim
2.5\times 10^{5}$ pairs/s/mW/GHz. The photons are correlated and exhibit a
cross correlation $g^{(2)}(0)$ of about 8000. Using the Hanbury Brown and
Twiss effect, we have also shown heralded single photon emission, achieving an
autocorrelation $g^{(2)}_{H}(0)\simeq 0.04$.

##  1 Introduction

Bright and wide-band twin-photon sources enable high rate emission of
entangled photons and heralded single photons, which are vital resources in
various quantum information processing protocols, such as quantum key
distribution, optical quantum computing, quantum simulation or quantum
metrology [1]. Twin photons are generally generated by spontaneous parametric
down conversion (SPDC), a second order nonlinear interaction process where a
single pump photon $\hbar\omega_{p}$ is annihilated, giving birth to a pair of
photons, signal and idler, with respective energies $\hbar\omega_{s}$ and
$\hbar\omega_{i}$, following the conservation energy relation
$\omega_{p}=\omega_{s}+\omega_{i}$. Different nonlinear materials have been
used to generated twin photons, however lithium niobate (LN) is a reference
and competitive material. It has indeed both large second order nonlinear
susceptibility ($d_{33}\sim 33$ pm/V) and electro-optic coefficient
($r_{33}\sim 31$ pm/V) and a wide transparency window ranging from 0.4 $\mu$m
to 4.5 $\mu$m.

Lithium niobate integrated devices are conventionally realized in either
proton-exchanged or titanium-indiffused waveguides [2, 3, 4, 5]. However, the
refraction index difference between the core waveguide and the surrounding
lithium niobate is weak for these waveguides, limiting the optical confinement
of the guided modes to a few $\mu$m2. Furthermore, the bending radius of such
waveguides is in the order of mm, hindering the integration of functional
elements on a compact chip. Recently, a new platform based on thin film
lithium niobate (TFLN) on insulator has emerged providing much tighter
confinement of the optical modes, allowing a more compact circuitry and hence
smaller footprint integrated compounds. This new TFLN platform has enabled
several breakthrough demonstrations over the last few years, including
enhanced nonlinear optical interactions with higher frequency conversion
efficiencies [6, 7, 8, 9, 10, 11, 12] and efficient and high-speed electro-
optic interaction [13, 14, 15]. Whereas second harmonic generation (SHG) has
been widely investigated on TFLN, it is only recently that twin photons have
been generated and their quantum properties studied on such platform [9, 16,
17, 18].

Among various waveguide geometries on the TFLN platform, we can distinguish
two main fabrication designs. The first one is based on the direct etching of
the TFLN to form the waveguides. This is the approach used for example in [9,
16, 17, 18]. However, the etching of lithium niobate can be challenging and
the phase matching wavelength is very sensitive to the waveguide dimensions.
It offers nevertheless a better confinement thanks to the high refractive
index contrast and thus allows for very high second-harmonic (SH) conversion
efficiencies, as demonstrated by C. Wang et al. [8] in 2018 achieving a SH
conversion efficiency of 2600 % W-1 cm-2 and by A. Rao et al. [11] in 2019,
measuring an efficiency of 4600 % W-1 cm-2. The second fabrication strategy
consists of depositing a layer of a transparent material with an appropriate
refractive index on top of the TFLN and whose processing technology is very
mature. The optical mode is confined horizontally by etching this material,
resulting in most of the light being confined in the TFLN beneath. This second
fabrication strategy is more accessible technologically and despite larger
modes, a still-high conversion efficiency can be obtained as it has been
reported by A. Boes et al. in 2019 [19], achieving about 1160 % W-1 cm-2 SH
conversion efficiency.

In this paper, we report to our knowledge the first generation of twin photons
via SPDC in periodically poled TFLN waveguides, using the second strategy,
depositing and partially etching a thin layer of silicon nitrate (SiN) on top
of the lithium niobate thin film. We have successfully generated about
$2.5\times 10^{5}$ pairs/s/mW/GHz at telecom wavelength with a bandwidth of
approximately 170 nm (21 THz), allowing for wavelength multiplexing and
quantum information processing in the frequency domain. Using the strong
emission correlation, we have been able to implement heralded single photon
emission, achieving a second order heralded autocorrelation function
$g^{(2)}_{H}(0)=0.04$, very close to the $g^{(2)}(0)=0$ of an ideal single
photon source.

In the next section, we will give a brief description of the fabrication
process of the periodically poled thin film lithium niobate waveguides (PP-
TFLN). Then, a classical characterisation through SHG is given in order to
determine the appropriate PP-TFLN waveguide for SPDC compatible with our
pumping laser source emitting at 780 nm. Section 3 and 4 are respectively
dedicated to the investigation of twin-photon generation properties and to the
heralded single photon emission.

##  2 PP-TFLN waveguide fabrication

The periodically poled lithium niobate waveguides were fabricated following
similar fabrication steps as described in [20]. We used commercially available
$X$-cut 300-nm thick thin-film lithium niobate wafer from NANOLN, on which we
patterned comb-like electrodes with a period of 4.87 $\mu$m, 4.93 $\mu$m and
4.98 $\mu$m, corresponding to the quasi-phase matching periods allowing twin-
photon generation at telecom wavelengths when the pump wavelength is set
approximately at 800 nm. We applied high voltage pulses to these electrodes to
periodically invert the spontaneous polarisation of the lithium niobate
crystal along the $Z$-axis, following the comb pattern of the electrodes, in
order to benefit from the highest nonlinear optical coefficient $d_{33}$.
Afterwards, we removed the electrodes by wet etching and deposited a 400 nm
thick film of silicon nitride (SiN) [21] on the lithium niobate surface. The
SiN film was then patterned by photolithography and reactive ion etching,
resulting in 2 $\mu$m-wide SiN stripes at the surface of the lithium niobate
thin film. The patterned SiN confines the optical mode horizontally along the
$Z$-axis, allowing TE modes for the fundamental (1560 nm) and second harmonic
(780 nm) wavelengths, with a good modal overlap for efficient nonlinear
interaction. Lastly, we coated the sample with a 1 $\mu$m thick silica layer
for protection and prepared the end facets of the waveguides by dicing and
polishing.

##  3 Classical characterisation of PP-TFLN waveguide

As a first step, we characterised the PP-TFLN waveguides in order to find the
waveguide that is suitable for efficient SPDC when using our pump laser source
with a wavelength of 780 nm. Since SHG is the reverse process of SPDC at
degeneracy ($\omega_{s}=\omega_{i}$), both phenomena fulfill the same quasi-
phase matching condition. Thus, we started by measuring the intensity of the
generated second harmonic field when the PP-TFLN waveguides were excited with
a laser source tunable around twice the SPDC pump wavelength, using the
experimental arrangement depicted in Fig. 1. For this purpose, the fundamental
field is delivered by a continuous-wave wavelength-tunable laser (TUNICS
T100R) emitting about 10 mW from 1490 nm to 1620 nm. To obtain higher powers,
we used an erbium-doped fibre amplifier seeded by the laser with a nominal
power of 500 $\mu$W. The amplifier gain is adjusted to achieve 200 mW at the
output which is sufficient for our measurements. The amplified fundamental
laser is then coupled into the PP-TFLN waveguides with a lensed fibre from OZ
Optics AR-coated at 1550 nm. To fulfill the phase matching condition, its
polarisation is adjusted to excite the TE mode with a fiber polarisation
controller (FPC). The generated second harmonic field, also in the TE mode, is
collected with a similar lensed fiber (OZ Optics) AR-coated at 780 nm. On the
other side, the lensed fiber is connected to a collimator and the output beam
is focused with a 100-mm lens into a free space Si photodiode (Newport
1801-FS), after spectral filtering with a dichroic mirror to remove the
fundamental pumping light that might have been collected by the output fibre.
In order to improve the detection sensitivity, we modulated the intensity of
the fundamental laser by driving its current with a sine-wave function at 80
kHz. The 30 %-depth intensity modulation induced on the fundamental field is
transferred to the second harmonic field and detected with the Si photodiode
connected to an SRS 100-kHz bandwidth locking-in amplifier, not shown in the
figure. The TFLN sample contains 30 periodically-poled waveguides having the
same physical length of about 5 mm and designed in three sets, each with a
different poling period: $4.87\,\mu$m, $4.93\,\mu$m and $4.98\,\mu$m. For each
set, the poling lengths of the PP-TFLN waveguides are 1.2 mm, 2.4 mm, 3.6 mm
or 4.8 mm. The sample is fixed on an aluminum holder and its temperature is
controlled and stabilised with a PID-controlled heater. The ensemble is
mounted on a 2 axis translation stage which allows to change easily the PPLN
waveguide under investigation with minor realignments.

Figure 1: Scheme of the experimental setup used to measure SHG from the PP-
TFLN waveguide. CW-TTL: continuous-wave telecom tunable laser, EDFA: erbium-
doped fiber amplifier, FPC: fibered polarisation controller, FC: fiber
collimator, DM: dichroic mirror, BD: beam dump, L: lens, PHD: photodiode, SH:
second-harmonic beam, F: fundamental beam

We used this setup to measure the SHG quasi-phase matching wavelength of the
waveguides at room temperature. As expected, we found that the second harmonic
wavelength is the same within a set of PP-TFLN waveguides and only depends on
the poling period. From our measurements, we deduce a phase matching
wavelength $\lambda=1553$ nm, $\lambda=1576$ nm and $\lambda=1600$ nm, with an
uncertainty of about $\pm$1 nm, for poling $\Lambda=4.87\,\mu$m,
$\Lambda=4.93\,\mu$m and $\Lambda=4.98\,\mu$m respectively. For the last two
poling periods, $\Lambda=4.93\,\mu$m and $\Lambda=4.98\,\mu$m, the SHG is
obtained for fundamental wavelengths larger than $1560$ nm. Unfortunately,
these waveguides are out of reach for SPDC with our pump laser source emitting
at 780 nm or would require cooling of the sample, which we want to avoid to
prevent condensation of ambient water vapour on the PP-TFLN waveguides.

Fortunately, the phase matching wavelength of waveguides with poling period
$\Lambda=4.87\,\mu$m is below $1560$ nm providing the opportunity to achieve
phase matching at $1560$ nm, by heating the sample up to 53∘C, as shown in
Figure 2. The red line is the theoretical phase matching
$\mathrm{sinc}^{2}(x)$ function where $x=(\Delta k-2\pi/\Lambda)L/2$, $L$
being the interaction length and $\Delta k=k_{p}-k_{s}-k_{i}$, with
$k_{m}=2\pi n_{m}/\lambda_{m}$ being the wavevector of mode $m=p,s,i$. The red
line is obtained by taking $L=4.8$ mm and after calculating the effective
indices $n_{m}$ of the TE modes associated with the geometry of the waveguides
at the working temperature and working wavelengths. No other extra parameter
is used. The oscillations in the measured spectra are due to interference
effects induced by the reflectivities on the input and output waveguide
facets. An oscillation period of about 0.11 nm is deduced, corresponding to a
length of 5.1 mm, in agreement with the physical waveguides length, which is
longer than the poled region. The maximum of the SH field is obtained at 1560
nm. This indicates that this temperature (53∘C) allows to satisfy the phase-
matching condition when pumping the PP-TFLN waveguide at 780 nm to generate by
SPDC twin photons having wavelengths around 1560 nm.

Figure 2: Second harmonic generation as a function the fundamental wavelength
in the waveguide with a poling period of $\Lambda=4.87\,\mu$m and poling
length $L=4.8$ mm. The temperature of the TFLN chip is at 53∘C.

##  4 SPDC and quantum characterisation of PP-TFLN waveguide

In order to implement spontaneous parametric down conversion and to measure
the twin photon emission, we use the experimental setup depicted in Fig. 3.
The source is a high-power cw fiber-coupled laser emitting up to 1 W at 780
nm. For our demonstrations we set the laser output power at 200 mW, which is
more than sufficient. The power is tuned with a fibre variable attenuator. The
monitoring of the pump power is achieved by sending the 1 $\%$ output port of
a 1:99 fiber coupler to a powermeter (NEWPORT). On the 99 % port, two cascaded
50:50 fibre coupled beamsplitters (not shown in the figure) are used to reduce
the power and a fibre polarisation controller is used to set the pump
polarisation, before coupling to the PP-TFLN waveguide with a lensed fibre
(AR-coated at 780 nm).

Figure 3: Scheme of the experimental setup used to characterize SPDC using
either a high sensitivity infrared (IR) spectrometer or single photon
detectors (SNSPDs) with a waveshaper to separate signal from idler photons.
Att: fibred optical attenuator, FBS: fibred beamsplitter, PM: powermeter,
WS4000A: waveshaper 400S, SNSPDs: superconducting nanowire single-ophoton
detectors.

As a first step, we estimated the coupling efficiency into the waveguide. This
is done by measuring the power at the output of the lensed fibre using a free
space optical powermeter. We then use this fibre to couple the 780 nm
wavelength beam into the PP-TFLN waveguide. A similar lensed fibre is used to
collect the guided pump laser. Using the same powermeter we measured the
transmitted pump and inferred the coupling efficiency to be $\sim$ 5 $\%$ at
780 nm, where we have made a reasonable assumption that the input and output
couplings are the same. Following the same procedure, we have deduced a
coupling efficiency of about $\sim\,29\,\%$ at 1560 nm by replacing the 780 nm
lensed fibers with 1560 nm lensed fibres. We have to emphasise here that no
particular design has been done in order to improve the coupling efficiencies.

Having estimated the coupling efficiencies, we set back the lensed fibre AR
coated at 780 nm to pump the PP-TFLN waveguides and kept the one AR coated at
1560 nm to collect twin photons. A fine determination of the SPDC phase
matching temperature is obtained by recording with a high-sensitivity infrared
spectrometer the wavelengths of the twin photons generated by the SPDC as a
function of the temperature. As the spectrometer is based on a 1D InGaAs array
with 532 pixels, we have only been able to measure the SPDC for the signal
photons, because the spectral sensitivity of the spectrometer drops
drastically above 1.6 $\mu$m at the low ($\sim$70 K) working temperature of
the spectrometer. The measured wavelengths are reported as red squares in Fig.
4 as a function of the temperature of the TFLN chip. The black squares are the
wavelengths of the corresponding idler photons deduced from the energy
conservation relation $1/\lambda_{p}=1/\lambda_{s}+\lambda_{i}$. The
background map is the theoretical $\mathrm{sinc}^{2}(x)$ function where
$x=\Delta k-2\pi/\Lambda$ and $L=4.8$ mm is the interaction length. It is
calculated using Lumerical finite-difference eigenmode solver to simulate the
effective mode indices for the full range of wavelengths and temperatures
ranging from 20∘C to 120∘C for our geometrical design of the PP-TFLN
waveguides. As in the case of SHG, we have a very good agreement between the
measured phase matching wavelengths and predicted ones, confirming the quality
of the PP-TFLN waveguides fabrication.

Figure 4: Left: Spectral SPDC bandwidth at degeneracy reached for a phase
matching temperature of 53∘C. Right: signal and idler wavelengths for
different phase matching temperatures. The red squares indicate the measured
signal wavelengths and black squares the deduced idler wavelengths. The
background map is the calculated phase matching using the poling period
$\Lambda=4.87\,\mu$m and interaction length of 4.8 mm. The green dashed
horizontal line highlights the degeneracy point.

When setting the temperature to 53∘C, the signal and idler are degenerate.
However due to the short interaction length, the bandwidth of the twin photons
is very large, extending over $\sim$ 21 THz. This large bandwidth allows easy
addressing of both degenerate and non-degenerate configurations using a
programmable optical filter WaveShaper 4000S. As depicted in Fig. 3, we
connect the lensed fibre collecting the twin photons to the programmable
filter. In the non-degenerate case, we program the filter such as the first
output has a 100 GHz bandwidth flat-top filter centered at $f_{0}+400$ GHz
away from the degeneracy at 1560 nm ($f_{0}$ = 192.113 THz), which corresponds
to the signal photons. A second identical band-pass filter is programmed on
the second output of the programmable filter, but centered at $f_{0}-400$ GHz
from degeneracy, which corresponds to the idler photons. Both outputs are then
directed to superconducting nanowire single-photon detectors (SNSPDs) from
Single Quantum to count signal and idler photons and to monitor the
coincidences using Time Tagger Ultra from Swabian Instruments. Both detectors
have a quantum efficiency of 70 %, a dead time of 20 ns and a jitter of about
30 ps.

Next, we measured a typical normalised temporal histogram of the coincidences
between signal and idler photons obtained with time bins of 10 ps, which is
shown in Fig. 5. The central peak at zero indicates that the signal and idler
photons are indeed twins created simultaneously in the PP-TFLN waveguide,
arriving at the same time to the detectors within a time resolution limited by
the detector jitters, the resolution of the time tagger and by the coherence
of the photons fixed by the bandwidth of the filters.

Figure 5: Signal and idler coincidence function $g^{(2)}(\tau)$ for different
delays between the signal and idler photons.

In order to determine the internal twin-photon generation rate $N_{t}$, we
have measured the signal $N_{s}$ and idler $N_{i}$ photons rates as well as
the coincidence rate $N_{c}$ in a time window ($\sim$600 ps) larger than the
width of the central peak shown in Fig. 5 for different pump powers coupled
into the PP-TFLN waveguide. In our case, the twin rate is simply given by
$N_{t}=(N_{s}\times N_{i})/N_{c}$ and thus does not require as a prerequisite
the determination of the different optical losses of our experiment setup,
including the detectors quantum efficiencies. Figure 6(a) shows the three
rates measured for different pump powers and in Fig. 6(b) the corresponding
internal twin-photon generation rates. A linear fit of the results in Fig. 6
(b), allows to extract a twin generation rate coefficient of 23 MHz/mW.
Moreover, as the signal and idler photons go through identical band-pass
filters of 100 GHz, we can deduce the brightness of our source to be
$2.3\times 10^{5}$ pairs/s/mW/GHz, a value close to the brightness of
$4.6\times 10^{5}$ pairs/s/mW/GHz reported by J. Zhao _et al._ [18] in PP-TFLN
fabricated by etched lithium niobate.

Figure 6: (a): Count rates of the signal $N_{s}$ and idler $N_{i}$ photons and
the coincidence rate $N_{c}$. (b): Evolution of the internal twin-photon
generation rate for different pumping powers. The signal and idler photons
have a bandwidth of 100 GHz.

Knowing the internal twin-photon generation rate, we can deduce the overall
optical losses on each channel. Indeed, the signal and idler measured rates
are given by $N_{k=s,i}=\eta_{k}\eta_{D}N_{t}$, where $\eta_{D}=(70\pm 2)\,\%$
is the quantum efficiency of the detectors and $\eta_{k}$ is the transmission
efficiency taken into account all optical losses from the waveguide until the
output of the final optical fibres connected to the detectors. From our
results we have been able to deduce $\eta_{s}=8\,\%$ and $\eta_{i}=9\,\%$ with
an uncertainty of about $\pm 2\,\%$ deduced statistically, repeating the
measurement several times. These efficiencies are coherent with the $29\,\%$
coupling efficiency at 1560 nm into the lensed fibre and taking into account
the $\sim$ 6 dB losses due to the waveshaper and the losses of $\sim$ 0.5 dB
at each fibre connection.

We also independently determined the SPDC brightness, namely by fixing the
pump power (180 $\mu$W) and tuning the bandwidth of the flat-top filters in
the signal and idler channels. In order to reach large bandwidths and avoid
any overlap between the signal and idler photons, we shifted the central
frequencies of the filters by $f_{0}\pm 800$ GHz away from the degeneracy. As
shown in Fig. 7, the measured internal twin-photon generation rate grows
linearly with the filter bandwidth. From the linear fit we deduce a slope of
0.058 MHz/GHz, which corresponds to a brightness of $\sim 3\times 10^{5}$
pairs/s/mW/GHz, close to the brightness we have found above.

Figure 7: Evolution of the twin-photon generation rate for different Signal
and Idler photons bandwidths. The pump power is 180 $\mu$W.

After having determined the brightness of our source, the generated twin
photons are further investigated by measuring the cross correlation between
the signal and idler photons, using the second order correlation function [22]

$g^{(2)}_{i,s}(\tau)=\frac{\langle\hat{a}^{\dagger}_{s}(t)\hat{a}^{\dagger}_{i}(t+\tau)\hat{a}_{i}(t+\tau)\hat{a}_{s}(t)\rangle}{\langle\hat{a}^{\dagger}_{s}\hat{a}_{s}\rangle\langle\hat{a}^{\dagger}_{i}\hat{a}_{i}\rangle}.$
(1)

Figure 5 is a typical signal-idler cross correlation function showing clearly
the bunching of the twin photons. For uncorrelated photons, such as those
emitted by a coherent laser, the cross correlation $g^{(2)}$ is unity, and is
independent of the delay between the arrival time of the photons on the
detectors. The cross correlation $g^{(2)}_{s,i}(\tau)$ is measured by sending
the signal and idler to the SNSPD detectors and collecting the time tags of
the generated electrical pulses with the time tagger. The true coincidences-
to-accidental ratio (CAR) is obtained by correcting
$\max(g^{(2)}_{s,i}(\tau=0))$ from the accidental coincidences for which
$g^{(2)}=1$. Figure 8 represents the CAR for different twin-photon rates,
hence different pump powers, for a 100-GHz bandwidth filters on the signal and
idler photons. As expected the CAR is proportional to the inverse of the
generation rate, following the theoretical analysis given in Ref. [23], with a
high value at low rates indicating a high level of photon bunching.

Figure 8: Coincidence-to-Accidental Ratio as a function of the twin-photon
generation rate $N_{t}$ measured using flat frequency filters of 100 GHz for
the signal and idler.

SPDC is a resource for producing bunched twin photons as shown in Fig. 8. This
is an extraordinary resource that is harnessed to produce heralded single
photon sources. Indeed, in the weak interaction regime, CAR $\gg 1$, the wave
function of the generated twin photons can be approximated by

$|\psi\rangle\simeq|0,0\rangle+\epsilon|1_{\omega_{s}},1_{\omega_{i}}\rangle,$
(2)

where the coefficient $\epsilon$ accounts for the nonlinear interaction
efficiency and the phase matching bandwidth. The bi-photon state Eq. (2)
indicates that a single pair of signal and idler photons is generated with a
probability $|\epsilon|^{2}/(1+|\epsilon|^{2})\ll 1$. By detecting for example
the signal photon emitted at frequency $\omega_{s}$, we project the state into
$|\psi\rangle=|1_{\omega_{i}}\rangle$ with one idler photon at frequency
$\omega_{i}=\omega_{p}-\omega_{s}$. Implementing such heralded single photon
sources on the compact and very efficient TFLNOI platform is a real asset for
integration with telecom C-band infrastructure. Another asset of such a source
is its large bandwidth enabling for wavelength multiplexing, broadband
tunability of the emitted single photon or fast repetition rates when SPCD is
excited with short pulses. In this work, the tunability is about 100 nm for
the PP-TFLN waveguide with $\Lambda=4.87\,\mu$m at $T=53\,^{\circ}$C, as shown
in Fig. 4. It can be extended to up to 300 nm by changing the working
temperature or shortening the interaction length.

In order to demonstrate that the idler photon is indeed a Fock state
$|n=1\rangle$, we measured its autocorrelation function which should be
$g^{(2)}(0)=0$. This is achieved through a Hanbury-Brown and Twiss experiment
[24] by connecting a 50:50 fibre beamsplitter at the idler output of the
waveshaper and sending the two beamsplitter outputs, called idler1 and idler2,
to the SNSPD detectors having a 70 % quantum efficiency. The signal photon is
detected with a third SNSPD detector having a quantum efficiency of 30 %. With
the time tagger card, we measured simultaneously the triple coincidences
signal-idler1-idler2 and all bi-coincidences signal-idler1, signal-idler2 and
idler1-idler2 in a time window of about 600 ps, slightly larger than the
cross-correlation temporal width. The count rate on each channel is also
recorded. To avoid long term drifts and have fair enough signal-to-noise
ratio, we collected the different coincidences and rates over one minute.
Finally we deduce the $g^{(2)}(0)$ of the idler photon heralded by the
detection of the signal photon [25, 26], which is given by

$g^{(2)}_{H}(0)=\frac{C_{s,i1,i2}\,\,N_{s}}{C_{s,i1}\,\,C_{s,i2}},$ (3)

where $C_{s,i1,i2}$ is the triple coincidence rate, $N_{s}$ the count rate of
the signal photons, and $C_{s,i1}$ ($C_{s,i2}$) is the coincidence rate
between the signal and idler1 (signal and idler2).

Figure 9: Autocorrelation $g^{(2)}(0)$ of the idler photon as a function of
the pump power.

Figure 9 shows the autocorrelation function $g^{(2)}(0)$ of the idler for
different pump powers, measured when heralding by the detection of the signal
photon (red) and without heralding (black). The light blue rectangle indicates
the area in which the source is considered to be in the single photon state
regime, whereas the light gray rectangle highlights the $g^{(2)}$ of Fock
states $|n\rangle$ with more than one photon. In Fig. 9, $g^{(2)}_{H}(0)\simeq
0$ indicates clearly single photon state emission. At a pump power of 385
$\mu$W, we measured $g^{(2)}_{H}(0)\simeq 0.04$, corresponding to a triple
coincidence rate $C_{s,i1,i2}=14$ Hz, a signal and idler detection rates of
about 428 kHz, and 987 kHz respectively and a coincidence rate between the
signal and idler around 48 kHz. The factor 2 between the signal and idler
detection rates is due to the unbalanced SNSPDs quantum efficiencies used for
this demonstration.

Figure. 9 also exhibits a slight increase of $g^{(2)}_{H}(0)$ as the pump
power is increased. This is due to the fact that the single photon character
of the source as described by Eq. (2) depends on the strength of $\epsilon$
which is pump-power dependent or equivalently dependent on the twin-photon
generation rate $N_{t}$. As the pump power increases, the probability of
generating more than a single pair within the measurement time window cannot
be neglected anymore. This gives rise to triple coincidence counts during the
measurements and thus to a higher value of $g^{(2)}_{H}(0)$.

When measuring the $g^{(2)}(0)$ without taking into account the detection of
the signal photons, we retrieve $g^{(2)}(0)\gtrsim 1$, indicating that our
source is emitting twin photons in a multimode regime with a statistic close
to the Poisson statistics and thus exhibits a coherent state behaviour. This
is due to the fact that the pumping source has a very long coherence time (kHz
bandwidth), whereas the photons are filtered with 100 GHz filters
corresponding to short coherence time, of the order of a picosecond. This
indicates that we are selecting much more than one signal or idler modes with
the band-pass filters. The slight higher values of $g^{(2)}(0)$ for pump
powers $<150\,\mu$W is attributed to statistical errors as the number of
coincidence counts is very low. A longer measurement time is required to infer
a $g^{(2)}(0)$ value with good uncertainty. In the case of an ideal single
mode detection, $g^{(2)}(0)=2$, a signature of thermal statistics
characterising twin photons emission [22]. This situation could be reached by
further narrowing for example the signal and idler filters down to the MHz
level, close the linewidth of the pumping source. However, the count rates,
and particularly the triple coincidence rate, will be very low. It is then
preferable to pulse the excitation source in order to increase its spectral
bandwidth to make it compatible with the twin photons detection bandwidth
which enables access to high rates.

##  5 Conclusion

Twin photon sources are important building blocks for quantum information
processing. We have reported here on an integrated source on the thin film LN
platform. Unlike traditional sources based on LN thin film where the LN is
etched, here the waveguide is defined by first depositing a thin layer of SiN
on top LN, followed by a shallow etching of SiN using well known and mastered
nano-fabrication techniques. Low loss, large transparency bandwidth SiN is a
also a promising platform when combined with efficient TFLNOI for optical
quantum technologies. A brightness of about $\sim 10^{5}$ pairs/s/mW/GHz is
achieved, allowing the demonstration a single photon emission. These results
pave the way to on-chip quantum applications based on twin photons and
advantage of their quantum correlations.

We acknowledge the support of the international Associated Laboratory in
Photonics between France and Australia (ALPhFA$+$). This work was supported by
the Australian Research Council’s Discovery Project scheme (DP190102773). It
has also been supported by Region Ile-de-France in the framework of DIM SIRTEQ
and by the Agence Nationale de la Recherche through Project INCQA.

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# An upper bound for the first positive eigenvalue of the Kohn Laplacian on
Reinhardt real hypersurfaces

Gian Maria Dall’Ara  Istituto Nazionale di Alta Matematica “F. Severi”
Research Unit Scuola Normale Superiore
Piazza dei Cavalieri, 7, 56126 Pisa (Italy<EMAIL_ADDRESS>and
Duong Ngoc Son  Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-
Platz 1, 1090 Wien (Austria<EMAIL_ADDRESS>
(Date: Oct 13, 2021)

###### Abstract.

A real hypersurface in $\mathbb{C}^{2}$ is said to be Reinhardt if it is
invariant under the standard $\mathbb{T}^{2}$-action on $\mathbb{C}^{2}$. Its
CR geometry can be described in terms of the curvature function of its
“generating curve”, i.e., the logarithmic image of the hypersurface in the
plane $\mathbb{R}^{2}$. We give a sharp upper bound for the first positive
eigenvalue of the Kohn Laplacian associated to a natural pseudohermitian
structure on a compact and strictly pseudoconvex Reinhardt real hypersurface
having closed generating curve (which amounts to the $\mathbb{T}^{2}$-action
being free). Our bound is expressed in terms of the $L^{2}$-norm of the
curvature function of the generating curve and is attained if and only if the
curve is a circle.

This project was begun while the first-named author was a Marie Skłodowska-
Curie Research Fellow at the University of Birmingham. He gratefully
acknowledges the support of the European Commission via the Marie Skłodowska-
Curie Individual Fellowship “Harmonic Analysis on Real Hypersurfaces in
Complex Space” (ID 841094). The second-named author was supported by the
Austrian Science Fund (FWF): Projekt I4557-N

##  1\. Introduction

Let $(M,\theta)$ be a compact strictly pseudoconvex pseudohermitian manifold.
The associated Kohn Laplacian $\Box_{b}$ is a nonnegative self-adjoint second-
order operator whose spectrum $\mathrm{Spec}(\Box_{b})$ in $(0,+\infty)$
consists of discrete eigenvalues of finite multiplicity, while zero is an
isolated eigenvalue if and only if $M$ is embeddable into some complex space.
The spectrum $\mathrm{Spec}(\Box_{b})$ in general, and estimates for the first
positive eigenvalue (in the embeddable case) in particular, have been studied
in several papers. For example, in [3] a Lichnerowicz-type estimate for the
first positive eigenvalue in terms of the Webster scalar curvature (or Ricci
curvature) was established, while the characterization of the equality case in
this estimate was treated in [6] and [2], for the higher-dimensional and the
three-dimensional cases respectively. In [5], several upper bounds for the
first positive eigenvalue of the Kohn Laplacian on the boundaries of certain
domains in $\mathbb{C}^{2}$ were obtained; the bounds are extrinsic, depending
on the realization of the CR manifolds as embedded compact real hypersurfaces
in $\mathbb{C}^{2}$. All of the aforementioned bounds are sharp, with equality
holding if $(M,\theta)$ is isomorphic to the standard CR sphere.

The main purpose of this paper is to provide an upper estimate for the first
positive eigenvalue of the Kohn Laplacian associated to a natural choice of
pseudohermitian structure on Reinhardt real hypersurfaces in $\mathbb{C}^{2}$.
A real hypersurface $M\subset\mathbb{C}^{2}_{z,w}$ is said to be Reinhardt if
it is invariant under the standard action of the $2$-dimensional torus
$\mathbb{T}^{2}$ on $\mathbb{C}^{2}$ (here
$\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}$). Under this assumption, the open
subset $M^{\ast}:=\\{(z,w)\in M\mid zw\neq 0\\}$ can be recovered from its
image $\gamma\subset\mathbb{R}^{2}$ under the mapping
$(z,w)\mapsto(\xi,\eta)=(\log|z|,\log|w|)$. We call $\gamma$ the “generating
curve” of $M$. (It may be of interest to remark that, by a result of Sunada
[11], two bounded Reinhardt domains in $\mathbb{C}^{2}$ are biholomorphically
equivalent if and only if the generating curves of their boundaries are rigid
equivalent.)

If $\gamma(s)=(\xi(s),\eta(s))$ is an arc-length parametrization of the
generating curve $\gamma$ (the symbol $s$ will always denote the arc-length
parameter), then $M^{\ast}$ is parametrized by

$(s,x,y)\mapsto(e^{\xi(s)+ix},e^{\eta(s)+iy})\in\mathbb{C}^{2},$ (1.1)

where $i=\sqrt{-1}$ and $x,y\in\mathbb{T}$. Notice that $M^{\ast}$ is
diffeomorphic to $\gamma\times\mathbb{T}^{2}$, and that $M=M^{\ast}$ if and
only if the induced $\mathbb{T}^{2}$-action on $M$ is free. In this paper, we
work under the following assumptions on the Reinhardt hypersurface
$M\subset\mathbb{C}^{2}$:

$M\ \text{is compact, connected, and}\ M=M^{\ast}.$ (1.2)

In this case, the generating curve is a simple closed curve, and hence $M$ is
diffeomorphic to a $3$-dimensional torus.

######  Definition 1.1.

We say that a Reinhardt hypersurface $M$ satisfying (1.2) has a “closed
generating curve”.

So, although the sphere is Reinhardt, the hypersurfaces of interest here are
not diffeomorphic to the sphere. By [6] and [2], the aforementioned estimates
for the first eigenvalue of the Kohn Laplacian are not sharp for manifolds
satisfying Definition 1.1, and this motivates the present paper.

Let $M$ be a strictly pseudoconvex Reinhardt hypersurface with closed
generating curve $\gamma(s)=(\xi(s),\eta(s))$, and let $\kappa=\kappa(s)$ be
the curvature function of $\gamma$. Using the parametrization (1.1), the
function $\kappa$ may be lifted to a $\mathbb{T}^{2}$-invariant function on
$M$. Since $M$ is strictly pseudoconvex, we can choose an orientation of
$\gamma$ such that $\kappa(s)>0$. A natural choice of a pseudohermitian
structure $\theta$ is then (following [1])

$\theta=\eta{{}^{\prime}}\,dx-\xi{{}^{\prime}}dy,$ (1.3)

whose associated volume form satisfies $dV:=\theta\wedge
d\theta=\kappa(s)ds\wedge dx\wedge dy$. We call (1.3) the “normalized
pseudohermitian structure” on $M$. As is well-known, $\theta$ allows to define
an associated Kohn Laplacian $\Box_{b}$ (acting on functions). Denote by
$\lambda_{1}(\Box_{b})$ its first positive eigenvalue, which is well-defined
since $M$ is strictly pseudoconvex and embedded (cf. [4]).

Our main result is

######  Theorem 1.2.

Let $M\subset\mathbb{C}^{2}$ be a strictly pseudoconvex Reinhardt hypersurface
with closed generating curve $\gamma$. Let $\kappa(s)>0$, $0\leq s<\ell$, be
the curvature of $\gamma$ in an arc-length parametrization $s$, where $\ell$
is the length of $\gamma$. Let $\lambda_{1}(\Box_{b})$ be the first positive
eigenvalue of the Kohn Laplacian associated to the normalized pseudohermitian
structure $\theta$ defined in (1.3). Then

$\lambda_{1}(\Box_{b})\leq\frac{1}{4\pi}\int_{0}^{\ell}\kappa^{2}ds.$ (1.4)

Equality occurs if and only if $\gamma$ is a circle, namely, if and only if

$M:=\left\\{\left(\log a|z|\right)^{2}+\left(\log
b|z|\right)^{2}=\frac{\ell^{2}}{4\pi^{2}}\right\\}$

for some $a,b>0$.

Since $\gamma$ is a simple closed curve, by the Hopf winding number theorem,

$\mathrm{Vol}(M):=\int_{M}\theta\wedge
d\theta=4\pi^{2}\int_{0}^{\ell}\kappa(s)ds=8\pi^{3}.$

Let $R$ be the Webster scalar curvature of $\theta$. It was shown in [1] that
$R=\kappa/2-(\log\kappa)^{\prime\prime}/(2\kappa)$ and thus

$\int_{M}R\,\theta\wedge
d\theta=4\pi^{2}\int_{0}^{\ell}\left(\kappa/2-(\log\kappa)^{\prime\prime}/(2\kappa)\right)\kappa
ds=2\pi^{2}\int_{0}^{\ell}\kappa^{2}ds.$

Estimate (1.4) is then equivalent to

$\lambda_{1}(\Box_{b})\leq\frac{1}{8\pi^{3}}\int_{M}R\theta\wedge
d\theta=\frac{1}{\mathrm{Vol}(M)}\int_{M}R\theta\wedge d\theta.$

In other words, $\lambda_{1}(\Box_{b})$ is not larger than the average value
of the Webster scalar curvature on $M$. For a comparison, the Chanillo-Chiu-
Yang’s lower bound [3] yields

$\lambda_{1}\geq\frac{1}{2}\min R,$

where the equality holds if only if $M$ is the sphere (which is never the case
in our setting, since $M$ is diffeomorphic to a 3-torus).

Let us conclude this introduction by remarking that our analysis uses the fact
that the generating curve is closed in an essential way. In fact, this
assumption allows to reduce the analysis of Kohn Laplacians on this class of
real hypersurfaces to that of a family of periodic differential equations. In
particular, the Kohn Laplacian on the “symmetric model” (i.e., when $\gamma$
is a circle) can be decomposed into a family of “quasi-exactly solvable”
periodic Schrödinger equations, the Whittaker–Hill equations, that have been
studied for a long time (cf. [13, 14]).

##  2\. Analysis on Reinhardt real hypersurfaces

As in the introduction, we have the following data:

  * •

a strictly pseudoconvex Reinhardt real hypersurface $M\subset\mathbb{C}^{2}$
with closed generating curve.

  * •

its generating curve $\gamma(s)=(\xi(s),\eta(s))$, parametrized by arc-length
$s$, so that $\mathbb{R}/\ell\mathbb{Z}\times\mathbb{T}^{2}\simeq M$ via
(1.1). Here $\ell$ is the length of $\gamma$, and we may naturally think of
$s$ as a variable in $\mathbb{R}/\ell\mathbb{Z}$.

  * •

$\kappa=\kappa(s)$, the positive curvature function of $\gamma$, also
expressed with respect to the arc-length parameter and thought of as a
$\mathbb{T}^{2}$-invariant function $\kappa(s,x,y)=\kappa(s)$ on $M$.

  * •

the normalized pseudohermitian structure
$\theta=\eta{{}^{\prime}}\,dx-\xi{{}^{\prime}}dy$, with attached volume form
$dV=\kappa(s)ds\wedge dx\wedge dy$.

  * •

the Kohn Laplacian $\Box_{b}$ associated with $\theta$, with first positive
eigenvalue $\lambda_{1}(\Box_{b})$.

To obtain a more concrete description of $\Box_{b}$, we exploit the
$\mathbb{T}^{2}$-invariance of our setting, that is, Fourier analysis. First
of all, a function $u$ on $M$ can be identified with a triple periodic
function of period $(\ell,2\pi,2\pi)$, i.e., as a function on
$\mathbb{R}/\ell\mathbb{Z}\times\mathbb{T}^{2}$. Therefore, $u$ admits a
partial Fourier expansion

$u(s,x,y)=\sum_{m,l\in\mathbb{Z}}u_{m,l}(s)\exp(imx+ily),$

where

$u_{m,l}(s)=\frac{1}{4\pi^{2}}\int_{\mathbb{T}^{2}}u(s,x,y)e^{-imx-ly}dx\,dy$
(2.1)

are the Fourier coefficient functions, which are $\ell$-periodic, and we have
the orthogonal decomposition

$L^{2}(M,dV)=\bigoplus_{l,m\in\mathbb{Z}}H_{m,l},$

where $H_{m,l}=\\{v(s)\exp(imx+ily)\colon\ v\in
L^{2}(\mathbb{R}/\ell\mathbb{Z},\kappa(s)ds)\\}$. Notice that
$L^{2}(\mathbb{R}/\ell\mathbb{Z},\kappa(s)ds)$ coincides with
$L^{2}(\mathbb{R}/\ell\mathbb{Z},ds)$ as a set, but the scalar products are
different.

######  Proposition 2.1.

Let $M$ be a strictly pseudoconvex Reinhardt real hypersurface in
$\mathbb{C}^{2}$ with closed generating curve, and let $\Box_{b}$ be the Kohn
Laplacian associated to the normalized pseudohermitian structure.
Then the spaces $H_{m,l}$ are invariant under $\Box_{b}$ for every
$l,m\in\mathbb{Z}$. More precisely, if $u(x,y,s)=v(s)\,e^{imx+ily}$ is smooth,
then

$\displaystyle\Box_{b}u=\left\\{\frac{1}{2\kappa}\left(-\frac{d^{2}}{ds^{2}}+V_{m,l}\right)v\right\\}e^{imx+ily}.$
(2.2)

Here,

$V_{m,l}=(lp+mq)^{2}+\kappa(lq-mp),$

where $q(s)=\xi^{\prime}(s)$, $p(s)=\eta^{\prime}(s)$, and $\kappa(s)$ is the
curvature function of $\gamma$.
In particular, $\Box_{b}u=0$ (for $u$ as above) if and only if

$v(s)=C\exp(l\eta(s)+m\xi(s)),$ (2.3)

where $C\in\mathbb{C}$ is a constant, that is, if and only if
$u(s,x,y)=C\exp\left(m(\xi+ix)+l(\eta+iy)\right)$.

Observe that the expression in curly brackets in formula (2.2) defines a
1-dimensional Schrödinger operator on $\mathbb{R}/\ell\mathbb{Z}$ with
potential $V_{m,l}(s)=(lp+mq)^{2}+\kappa(lq-mp)$. Thus, the analysis of
$\Box_{b}$ on $L^{2}(M,dV)$ is reduced to that of the two-parameter family of
operators
$\left\\{\frac{1}{2\kappa}\left(-\frac{d^{2}}{ds^{2}}+V_{m,l}\right)\right\\}_{m,l}$
on $L^{2}(\mathbb{R}/\ell\mathbb{Z},\kappa(s)ds)$.

We also remark that, by (2.3), $u(s,x,y)=v(s)e^{ilx+imy}$ is CR if and only if
it is the restriction to $M$ of a multiple of the holomorphic monomial
$z^{m}w^{l}$.

###### Proof.

We choose a frame $Z_{1}$ of $T^{(1,0)}M$ as follows (cf. [1]). Since $Z_{1}$
must satisfy

$Z_{1}\left(e^{\xi(s)-ix}\right)=Z_{1}\left(e^{\eta(s)-iy}\right)=0,$

we find that

$Z_{1}=\phi\left(-i\,\eta{{}^{\prime}}\,\frac{\partial}{\partial
y}-i\,\xi{{}^{\prime}}\,\frac{\partial}{\partial x}+\frac{\partial}{\partial
s}\right),$

for some function $\phi$ on $M$. Thus, if we require that the Levi “matrix” is
the identity, i.e., $h_{1\bar{1}}=-id\theta(Z_{1},Z_{\bar{1}})=1$, then we
must have $2\kappa|\phi|^{2}=1$. Hence, we take $\phi=1/\sqrt{2\kappa}$. With
this choice of $Z_{1}$, we have the dual coframe

$\theta^{1}:=\sqrt{\kappa/2}\,\left(i\,\xi{{}^{\prime}}\,dx+i\,\eta{{}^{\prime}}\,dy+ds\right).$

A computation (cf. [1]) shows that the connection and torsion forms are:

$\displaystyle\omega_{1}{}^{1}$
$\displaystyle=\kappa^{\prime}/\sqrt{8\kappa^{3}}\,(\theta^{1}-\theta^{\bar{1}})-i(\kappa/2)\theta$
(2.4) $\displaystyle\tau^{1}$
$\displaystyle=\frac{i\kappa}{2}\theta^{\bar{1}}.$ (2.5)

Thus,

$\displaystyle d\omega_{1}{}^{1}$
$\displaystyle=\left[\frac{\kappa}{2}-\frac{(\log\kappa)^{\prime\prime}}{2\kappa}\right]\theta^{1}\wedge\theta^{\bar{1}}-\frac{i\,\kappa^{\prime}}{\sqrt{2\kappa}}[\theta^{1}\wedge\theta+\theta^{\bar{1}}\wedge\theta].$
(2.6)

The Reeb field $T$ is determined by $d\theta(T,Z_{1})=0$, $\theta(T)=1$, and
its reality. Thus

$\displaystyle T=\eta{{}^{\prime}}\,\frac{\partial}{\partial
x}-\xi{{}^{\prime}}\,\frac{\partial}{\partial y}.$ (2.7)

If $u$ is a smooth function, then we have

$du=(Z_{1}u)\theta^{1}+(Z_{\bar{1}}u)\theta^{\bar{1}}+(Tu)\theta$ (2.8)

and $\partial_{b}u=(Z_{1}u)\theta^{1}$, which is a genuine one-form on $M$.
Integration by parts implies that

$\Box_{b}u=-u_{\bar{1},}{}^{\bar{1}}.$ (2.9)

Here, the indices preceded by a comma indicate the covariant derivative:

$u_{\bar{1},1}=Z_{1}Z_{\bar{1}}u-\omega_{\bar{1}}^{\bar{1}}(Z_{1})Z_{\bar{1}}u.$
(2.10)

In the chosen frame, raising and lowering indices of tensors only change their
“types” since $h_{1\bar{1}}=1$. Since $u(x,y,s)=v(s)\,\exp(imx+ily)$, where
$l$ and $m$ are constants, we have

$u_{\bar{1}}=Z_{\bar{1}}u=\frac{1}{\sqrt{2\kappa}}e^{i(ly+mx)}\left(v^{\prime}-(lp+mq)v\right).$

Since $\Box_{b}u=0$ if and only if $Z_{\bar{1}}u=0$, identity (2.3) follows
immediately. Moreover,

$\displaystyle-\Box_{b}u$
$\displaystyle=Z_{1}(u_{\bar{1}})-\omega_{\bar{1}}^{\bar{1}}(Z_{1})u_{\bar{1}}$
$\displaystyle=Z_{1}\left(\frac{1}{\sqrt{2\kappa}}e^{i(ly+mx)}\left(v^{\prime}-(lp+mq)v\right)\right)$
$\displaystyle\quad-\frac{1}{4\kappa}e^{i(ly+mx)}(\log\kappa)^{\prime}\left(v^{\prime}-(lp+mq)v\right)$
$\displaystyle=\frac{1}{2\kappa}e^{i(ly+mx)}\left(v^{\prime\prime}-(l^{2}p^{2}+2lmpq+lp^{\prime}+m^{2}q^{2}+mq^{\prime})v\right).$

Substituting $p^{\prime}=\kappa q$ and $q^{\prime}=-\kappa p$, we obtain
(2.2). ∎

Proposition 2.1 can be used to verify that $\Box_{b}$, initially defined as a
symmetric operator on $C^{\infty}(M)\subset L^{2}(M,dV)$, is essentially self-
adjoint. One way to see this is the following. By a well-known criterion [9,
Corollary to Theorem VIII.3], it is enough to check that
$\Box_{b}^{*}-\overline{\lambda}$ is injective when
$\lambda\in\mathbb{C}\setminus\mathbb{R}$. Here $\Box_{b}^{*}$ denotes the
Hilbert space adjoint. In other words, we have to show that a function $f\in
L^{2}(M,dV)$ such that

$\int_{M}f\,\overline{\left(\Box_{b}-\lambda\right)\varphi}\,dV=0\qquad\forall\varphi\in
C^{\infty}(M),$

must necessarily vanish. The choice $\varphi(s,x,y)=v(s)e^{imx+ily}$ for $v\in
C^{\infty}(\mathbb{R}/\ell\mathbb{Z})$ and $l,m\in\mathbb{Z}$ gives

$\begin{split}0&=\int_{M}f(s,x,y)\,\overline{\left(\left\\{\frac{1}{2\kappa}\left(-\frac{d^{2}}{ds^{2}}+V_{m,l}\right)v\right\\}-\lambda\right)v(s)}\,\kappa(s)ds\,e^{-imx-
ily}dx\,dy\\\
&=\int_{\mathbb{R}/\ell\mathbb{Z}}f_{ml}(s)\,\overline{\left(\left\\{\frac{1}{2\kappa}\left(-\frac{d^{2}}{ds^{2}}+V_{m,l}\right)v\right\\}-\lambda\right)v(s)}\,\kappa(s)ds,\end{split}$

where the functions $f_{ml}\in L^{2}(\mathbb{R}/\ell\mathbb{Z},\kappa(s)ds)$
are as in (2.1). In other words, $(B_{ml}^{*}-\overline{\lambda})f_{ml}=0$,
where

$B_{ml}=\frac{1}{2\kappa(s)}\left(-\frac{d^{2}}{ds^{2}}+V_{m,l}(s)\right),$

initially defined on $C^{\infty}(\mathbb{R}/\ell\mathbb{Z})$ and symmetric
with respect to the $L^{2}(\mathbb{R}/\ell\mathbb{Z},\kappa(s)ds)$ scalar
product. This operator is essentially self-adjoint (this is classical, see
e.g. [10, Theorem X.28]), and therefore, appealing again to the essential
self-adjointness criterion cited above, we must have $f_{ml}=0$ for every
$m,l$, and hence $f=0$, as we wanted.

It is also classical that $L^{2}(\mathbb{R}/\ell\mathbb{Z},\kappa(s)ds)$
admits an orthonormal basis consisting of smooth eigenfunctions
$\psi^{ml}_{n}$ ($n=0,1,\ldots$) of $B_{ml}$, corresponding to a sequence of
eigenvalues

$0=\lambda_{0}(B_{ml})<\lambda_{1}(B_{ml})\leq\cdots$

tending to $+\infty$. Notice that $\lambda_{0}(B_{m,l})$ vanishes and has
multiplicity one, because $B_{lm}$ is nonnegative (as a consequence of the
nonnegativity of $\Box_{b}$) and its kernel is one-dimensional.

Then, by Proposition 2.1, the collection
$\left\\{\psi^{ml}_{n}(s)e^{imx+ily}\right\\}_{m,l,n}$ is an orthonormal basis
of $L^{2}(M,dV)$ consisting of eigenfunctions of $\Box_{b}$ (they are
certainly in the domain of $\Box_{b}$, since they are smooth). The following
two observations follow (for $\mu>0$):

  1. (1)

To show that $\lambda_{1}(\Box_{b})\leq\mu$, it is enough to show that
$\lambda_{1}(B_{ml})\leq\mu$ for at least one pair $(m,l)\in\mathbb{Z}^{2}$.

  2. (2)

To prove that $\lambda_{1}(\Box_{b})\geq\mu$, we need to ensure that for every
pair $(m,l)\in\mathbb{Z}^{2}$, we have the inequality
$\lambda_{1}(B_{ml})\geq\mu$.

The first positive eigenvalues of the $B_{m,l}$’s are uniformly bounded below
by a positive constant, namely,

$\lambda_{1}(B_{ml})\geq\lambda_{1}(\Box_{b})>0.$ (2.11)

This uniform estimate seems to be nonobvious from the point of view of
Schrödinger operators. More precisely, since the Webster scalar curvature is
$R=(1/2)\left(\kappa-\kappa^{-1}(\log\kappa)^{\prime\prime}\right)$, the
Lichnerowicz-type estimate for $\lambda_{1}(\Box_{b})$ of [3] gives

$\lambda_{1}(B_{ml})\geq\frac{1}{4}\min_{s\in\mathbb{R}/\ell\mathbb{Z}}\left(\kappa(s)-\frac{(\log\kappa(s))^{\prime\prime}}{\kappa(s)}\right)\quad\forall
m,l.$ (2.12)

Unfortunately, this estimate is far from being sharp even when $\kappa$ is
constant.

##  3\. The upper bound for $\lambda_{1}(\Box_{b})$

In this section, we prove inequality (1.4) of Theorem 1.2. By the discussion
in the last section, this will follow from the next proposition.

######  Proposition 3.1.

Let $\gamma:\mathbb{R}/\ell\mathbb{Z}\rightarrow\mathbb{R}^{2}$ ($s:=$ arc-
length parametrization) be a smooth simple closed curve, and let $\kappa(s)$
be its curvature function. Assume that $\kappa(s)>0$ for every $s$. Let
$B:=B_{00}=-(2\kappa)^{-1}d^{2}/ds^{2}$, which defines a nonnegative and
essentially self-adjoint operator on $L^{2}(\mathbb{R}/\ell\mathbb{Z},\kappa
ds)$. Then its first positive eigenvalue satisfies the bound:

$\lambda_{1}(B)\leq\frac{1}{4\pi}\int_{\mathbb{R}/\ell\mathbb{Z}}\kappa^{2}ds,$

with equality if and only if $\kappa$ is constant, that is, if and only if
$\gamma$ is a circle of radius $\frac{1}{2\lambda_{1}(B)}$.

Notice that it also follows from this statement that, if equality is attained
in (1.4), then the generating curve is a circle of radius
$\frac{1}{2\lambda_{1}(\Box_{b})}$.

###### Proof.

Denote by $(u,v)$ the inner product in $L^{2}(\mathbb{R}/\ell\mathbb{Z},\kappa
ds)$, i.e.,

$(u,v):=\int_{\mathbb{R}/\ell\mathbb{Z}}u(s)\overline{v(s)}\kappa(s)\,ds.$
(3.1)

Notice that

$(Bu,v)=-\frac{1}{2}\int_{\mathbb{R}/\ell\mathbb{Z}}\kappa^{-1}u^{\prime\prime}(s)\overline{v}(s)\kappa
ds=\frac{1}{2}\int_{\mathbb{R}/\ell\mathbb{Z}}u^{\prime}(s)\overline{v^{\prime}(s)}ds,$

The first positive eigenvalue of $B$ has the following variational
characterization:

$\lambda_{1}:=\lambda_{1}(B)=\inf\left\\{\frac{(Bv,v)}{(v,v)}\colon v\in
W^{1,2}(\mathbb{T},\mathbb{R}),\int_{0}^{\ell}v\kappa
ds=0,v\neq\,\text{constant}\right\\}.$ (3.2)

Recall the notation $\gamma(s)=(\xi(s),\eta(s))$ and
$\gamma^{\prime}(s)=(q(s),p(s))$. We have
$\int_{\mathbb{R}/\ell\mathbb{Z}}p\kappa
ds=\int_{\mathbb{R}/\ell\mathbb{Z}}q^{\prime}\,ds=0$ and
$\int_{\mathbb{R}/\ell\mathbb{Z}}q\kappa
ds=-\int_{\mathbb{R}/\ell\mathbb{Z}}p^{\prime}\,ds=0$. Moreover,

$\displaystyle(Bp,p)$
$\displaystyle=-\frac{1}{2}\int_{\mathbb{R}/\ell\mathbb{Z}}\kappa^{-1}p^{\prime\prime}p\,\kappa
ds=-\frac{1}{2}\int_{\mathbb{R}/\ell\mathbb{Z}}p^{\prime\prime}p\,ds$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}/\ell\mathbb{Z}}(p^{\prime})^{2}ds=\frac{1}{2}\int_{\mathbb{R}/\ell\mathbb{Z}}\kappa^{2}q^{2}ds.$

Similarly, we have

$(Bq,q)=\frac{1}{2}\int_{\mathbb{R}/\ell\mathbb{Z}}\kappa^{2}q^{2}ds,$

and thus

$(Bp,p)+(Bq,q)=\frac{1}{2}\int_{\mathbb{R}/\ell\mathbb{Z}}\kappa^{2}ds,$

since $p^{2}+q^{2}=1$. On the other hand, by the Hopf winding number theorem,

$(p,p)+(q,q)=\int_{\mathbb{R}/\ell\mathbb{Z}}(p^{2}+q^{2})\,\kappa
ds=\int_{\mathbb{R}/\ell\mathbb{Z}}\kappa ds=2\pi.$

Hence, by the variational characterization (3.2), we have that

$\lambda_{1}\leq\frac{1}{4\pi}\int_{\mathbb{R}/\ell\mathbb{Z}}\kappa^{2}ds.$
(3.3)

Suppose that equality occurs in (3.3). Then necessarily $Bp=\lambda_{1}p$ and
$Bq=\lambda_{1}q$,

$\lambda_{1}p=Bp=-(2\kappa)^{-1}p^{\prime\prime}=(\kappa/2)p-\frac{1}{2}(\log\kappa)^{\prime}q,$
(3.4)

and thus

$\lambda_{1}p^{2}=(\kappa/2)p^{2}-\frac{1}{2}(\log\kappa)^{\prime}pq.$ (3.5)

Similarly,

$\lambda_{1}q^{2}=(\kappa/2)q^{2}+\frac{1}{2}(\log\kappa)^{\prime}pq.$ (3.6)

Adding these together and using $p^{2}+q^{2}=1$, we obtain
$\lambda_{1}\equiv\frac{\kappa}{2}$, i.e., $\kappa\equiv 2\lambda_{1}$ is
constant and $\gamma$ is a circle of radius $1/(2\lambda_{1})$. This completes
the proof. ∎

##  4\. Equality case

In this section we study the spectrum of $\Box_{b}$ (defined with respect to
the normalized pseudohermitian structure (1.3)) on the Reinhardt real
hypersurface having as closed generating curve $\gamma$ a circle of radius
$\kappa^{-1}$, that is, we assume that the curvature function
$\kappa(s)=\kappa$ is constant. In particular, we show that in this case
$\kappa/2$ is the lowest positive eigenvalue of $\Box_{b}$, thus completing
the proof of our main theorem.

Remark: Since in this case the Webster scalar curvature is $R\equiv\kappa/2$,
the inequality of Chanillo-Chiu-Yang [3] already gives the lower bound
$\lambda_{1}\geq\kappa/4$. In fact, the needed assumption that the CR Paneitz
operator is nonnegative has been shown to hold for embeddable manifolds by
Takeuchi [12].

Let then $\gamma(s)=(\xi(s),\eta(s))$, where $\xi(s)=\kappa^{-1}\cos(\kappa
s)$, $\eta(s)=\kappa^{-1}\sin(\kappa s)$, $q(s)=-\sin(\kappa s)$,
$p(s)=\cos(\kappa s)$, and $s\in\mathbb{R}/(2\pi\kappa^{-1}\mathbb{Z})$. We
note in passing that different values of $\kappa>0$ correspond to inequivalent
CR manifolds: see [1].

Recall that our task boils down to proving that
$\lambda_{1}(B_{ml})\geq\frac{\kappa}{2}$ for every $(m,l)\in\mathbb{Z}^{2}$,
where

$B_{ml}=\frac{1}{2\kappa}\left(-\frac{d^{2}}{ds^{2}}+V_{m,l}\right),\qquad
V_{m,l}=(lp+mq)^{2}+\kappa(lq-mp).$

Thus, if we make the change of variables $\tau=\kappa
s+\alpha\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$, $E=2\kappa^{-1}\lambda$, and
$a=\kappa^{-1}\sqrt{m^{2}+l^{2}}$, then the eigenvalue equation
$B_{ml}v=\lambda v$ becomes the well-known Whittaker–Hill equation

$\mathrm{WH}_{a}u(\tau):=-\frac{d^{2}u}{d\tau^{2}}+\left(a^{2}\sin^{2}(\tau)+a\cos(\tau)\right)u=E\,u.$
(4.1)

See, e.g., section 7.4 of [7] or [14]. Writing
$u(\tau)=w(\tau)\exp(-a\cos(\tau))$, (4.1) becomes

$\mathrm{I}_{a}w(\tau):=-\frac{d^{2}w}{d\tau^{2}}-2a\sin(\tau)\,\frac{dw}{d\tau}(\tau)=Ew(\tau),\qquad\tau\in\mathbb{R}/2\pi\mathbb{Z}.$
(4.2)

Notice that, while $\mathrm{WH}_{a}$ is self-adjoint with respect to the
$L^{2}(\mathbb{T},d\tau)$ scalar product (as a consequence of the self-
adjointness of $B_{ml}$ on $L^{2}(\mathbb{T},\kappa d\tau)$), the operator
$\mathrm{I}_{a}$ is self-adjoint with respect to
$L^{2}(\mathbb{T},e^{-2a\cos(\tau)}d\tau)$. Hence, in passing from (4.1) to
(4.2), we simplified the coefficients (passing from a quadratic to a linear
trigonometric polynomial) at the expense of losing self-adjointness with
respect to $L^{2}(\mathbb{T},d\tau)$.

At any rate, both $\mathrm{WH}_{a}$ and $\mathrm{I}_{a}$ have zero as a simple
(and smallest) eigenvalue, and preserve the parity of functions (as does the
transformation $w\mapsto we^{-a\cos(\tau)}$). Hence every positive eigenvalue
has multiplicity two and the corresponding eigenspace is spanned by an odd and
an even $2\pi$-periodic function. By restricting $\mathrm{I}_{a}$ to the
Hilbert space of $2\pi$-periodic odd functions, which is closure of the linear
span of $\\{\sin(k\tau)\\}_{k\geq 1}$, a simple computation shows that the
double eigenvalues of $\mathrm{I}_{a}$ coincide with the simple eigenvalues of
the infinite tridiagonal matrix
$\mathcal{I}(a)=\\{\mathcal{I}_{kk^{\prime}}(a)\\}_{k,k^{\prime}\geq 1}$ given
by

$\displaystyle\mathcal{I}_{kk^{\prime}}(a)=\begin{cases}k^{2}\quad&k^{\prime}=k\\\
(k^{\prime}-k)k^{\prime}a\quad&|k-k^{\prime}|=1\\\
0\quad&\text{otherwise}\end{cases}$

Notice that $\mathcal{I}(a)$ is not self-adjoint, because
$\\{\sin(k\tau)\\}_{k\geq 1}$ is not orthogonal with respect to the
$L^{2}(\mathbb{T},e^{-2a\cos(\tau)})$ scalar product. Denote by
$\mathcal{I}_{N}(a)$ the $N$-th principal minor of $\mathcal{I}(a)$. For
example,

$\mathcal{I}_{4}(a)=\begin{pmatrix}1&2a&0&0\\\ -a&4&3a&0\\\ 0&-2a&9&4a\\\
0&0&-3a&16\\\ \end{pmatrix}$ (4.3)

See [14] for a thoroughly discussion of this truncation. By the above
discussion, to established the desired inequality
$\lambda_{1}(B_{ml})\geq\kappa/2$, we have to show that if $E$ is an
eigenvalue of $\mathcal{I}(a)$, then necessarily $E\geq 1$, for every value of
$a$.

In order to achieve that, we need the following two lemmas. The first follows
from known results on Jacobi matrices.

######  Lemma 4.1.

Let $\mathcal{I}(a)$ and $\mathcal{I}_{N}(a)$ be the matrices defined above.
Then every eigenvalue of $\mathcal{I}(a)$ is a limit of eigenvalues of
$\mathcal{I}_{N}(a)$ as $N$ tends to $+\infty$.

###### Proof.

The result for complex Jacobi matrices is by now classical. For the non-
symmetric case, this follows from [14, Theorem 2] or [8, Theorem 2.1]. ∎

######  Lemma 4.2.

Assume that the real tri-diagonal matrix

$A=\begin{pmatrix}\delta_{1}&\mu_{1}&&&\\\ \nu_{1}&\delta_{2}&\mu_{2}&&\\\
&\nu_{2}&\delta_{3}&\ddots&\\\ &&\ddots&\ddots&\mu_{N-1}\\\
&&&\nu_{N-1}&\delta_{N}\\\ \end{pmatrix}$

satisfies the following properties:

  1. (1)

$\delta_{k}>0$ for every $k$;

  2. (2)

$\mu_{k}\nu_{k}\leq 0$ for every $k$.

Let $0<\delta\leq\frac{\pi}{2}\left(\sum_{k}\delta_{k}^{-1}\right)^{-1}$, and
$\mu:=\min_{k}\delta_{k}$. Then $A$ has no eigenvalues in the open sectorial
region

$T:=\left\\{z\in\mathbb{C}\colon-\infty<\operatorname{Re}(z)<\mu,\
|\operatorname{Im}(z)|<\delta(1-\mu^{-1}\operatorname{Re}(z))\right\\}.$

###### Proof.

Assume that $N\geq 2$ to avoid trivialities. Denote by $S(\theta)$
($\theta\in(0,\pi]$) the sector consisting of nonzero complex numbers whose
argument has modulus $<\theta$ (use the principal determination of the
argument). Let $\theta_{k}\in(0,\pi/2)$ be the unique angle such that

$\tan\theta_{k}=\frac{\delta}{\delta_{k}}.$

By the convexity of the tangent function, we have

$\sum_{k}\theta_{k}\leq\sum_{k}\tan\theta_{k}=\delta\sum_{k}\delta_{k}^{-1}\leq\frac{\pi}{2},$
(4.4)

by our choice of $\delta$.

Let now $A_{k}$ be the $k$-th principal minor of $A$, and let
$P_{k}(z)=\det(A_{k}-z\mathbb{I}_{k})$ be its characteristic polynomial
($\mathbb{I}_{k}$ is the $k\times k$ identity matrix). We are going to show
that

$P_{k}(T)\subset S(\theta_{1}+\cdots+\theta_{k})\qquad\forall k\leq N.$ (4.5)

The case $k=N$ says in particular that the characteristic polynomial of $A$
has no zeros in $T$, as we wanted.

We are going to prove (4.5) by induction. We use the following elementary
geometric facts:

  * •

The image of $T$ under the map $z\mapsto\delta_{k}-z$ is contained in the
sector $S(\theta_{k})$.

  * •

Any sector $S(\theta)$ ($\theta\leq\pi$) is invariant under right translations
$z\mapsto z+b$, where $b\geq 0$, and dilations $z\mapsto az$, where $a>0$.

  * •

If $z_{j}\in S(\theta_{j})$ ($j=1,2$) and $\theta_{1}+\theta_{2}\leq\pi$, then
$z_{1}z_{2}\in S(\theta_{1}+\theta_{2})$.

  * •

If $z_{j}\in S(\theta)$ ($j=1,2$) and $\theta\leq\frac{\pi}{2}$, then
$z_{1}+z_{2}\in S(\theta)$.

Computing explicitly, we see that

$P_{1}(z)=\delta_{1}-z,\quad
P_{2}(z)=(\delta_{1}-z)(\delta_{2}-z)-\mu_{1}\nu_{1}.$

Using the properties above and the assumption $\mu_{1}\nu_{1}\leq 0$, we see
that $P_{1}(T)\subset S(\theta_{1})$ and $P_{2}(T)\subset
S(\theta_{1}+\theta_{2})$, that is, (4.5) for $k=1$ and $2$. Expanding the
determinant defining $P_{k+2}$ along the last row, we find the recurrence
formula

$P_{k+2}(z)=(\lambda_{k+2}-z)P_{k+1}(z)-\mu_{k+1}\nu_{k+1}P_{k}(z).$

Assuming that (4.5) has been proved for $k$ and $k+1$ and using again the
elementary properties above, we see that if $z\in T$, then $P_{k+2}(z)$ is the
sum of an element of $S(\theta_{1}+\cdots+\theta_{k+2})$ and an element that
is either zero (if $\mu_{k+1}\nu_{k+1}=0$) or in
$S(\theta_{1}+\cdots+\theta_{k})$. Notice how we used the assumption on the
off-diagonal elements of $A$ and (4.4). The induction is now complete. ∎

The proof can now be easily concluded. By Lemma 4.1, every eigenvalue $E$ is
the limit of a sequence $\\{E_{N}\\}_{N\geq 1}$, where each $E_{N}$ is an
eigenvalue of $\mathcal{I}_{N}(a)$. By Lemma 4.2 applied to
$\mathcal{I}_{N}(a)$ and $\delta=\frac{3}{\pi}$ (all that really matters is
that this quantity is uniform in $N$), we see that $E_{N}\notin T$, where $T$
is as in the statement of Lemma 4.2. Since $T$ is open, we must necessarily
have $E\geq 1$. The proof of Theorem 1.2 is now complete.

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