# Path-entangled radiation from kinetic inductance amplifier

Abdul Mohamed and Shabir Barzanjeh<EMAIL_ADDRESS>Department of
Physics and Astronomy, University of Calgary, Calgary, AB T2N 1N4 Canada

###### Abstract

Continuous variable entangled radiation, known as Einstein-Podolsky-Rosen
(EPR) states, are spatially separated quantum states with applications ranging
from quantum teleportation and communication to quantum sensing. The ability
to efficiently generate and harness EPR states is vital for advancements of
quantum technologies, particularly in the microwave domain. Here, we introduce
a kinetic inductance quantum-limited amplifier that generates stationary path-
entangled microwave radiation. Unlike traditional Josephson junction circuits,
our design offers simplified fabrication and operational advantages. By
generating single-mode squeezed states and distributing them to different
ports of a microwave resonator, we deterministically create distributed
entangled states at the output of the resonator. In addition to the
experimental verification of entanglement, we present a simple theoretical
model using a beam-splitter picture to describe the generation of path-
entangled states in kinetic inductance superconducting circuits. This work
highlights the potential of kinetic inductance parametric amplifiers, as a
promising technology, for practical applications such as quantum
teleportation, distributed quantum computing, and enhanced quantum sensing.
Moreover, it can contribute to foundational tests of quantum mechanics and
advances in next-generation quantum information technologies.

Quantum entanglement, a fundamental aspect of quantum mechanics, has numerous
applications in quantum information processing [1, 2, 3], communication [4,
5], precision measurements, and sensing [6, 7]. Therefore, the efficient
generation of entangled states, such as spatially separated entangled photon
pairs known as Einstein-Podolsky-Rosen (EPR) states or two-mode squeezed
states, is key for further development of quantum technology and computing
[8]. These states have been used in several experiments to test the
foundations of quantum mechanics, such as demonstrating violations of Bell
inequalities [9, 10, 11, 12], and in practical applications like quantum
cryptography and quantum teleportation [13, 14]. The robustness of EPR states
against loss makes them a valuable asset for long-distance quantum
communication and quantum networks [15]. These states also offer unique
possibilities in remote quantum sensing and metrology [16, 17, 18, 19, 20],
imaging [21, 22], and tomography [23].

In the optical domain, EPR states are typically generated using nondegenerate
optical parametric amplifiers [24, 8, 25]. In this process, a pump photon
passing through a parametric amplifier down-converted into idler and signal
photons with lower energy, creating spatially separated two-mode squeezed
states [26, 27, 28, 29] or photon pairs [30, 31, 32]. In the microwave
frequencies, Josephson parametric amplifiers, operating as nonlinear media
[33, 34, 35, 36, 37, 38, 39, 40], can generate both single-mode [41, 42, 43,
44] and two-mode [45, 46, 47, 48, 49, 50] squeezed states. In addition to the
relatively complex fabrication process, a major challenge in utilizing
Josephson junction circuits for generating microwave entanglement is their
sensitivity to stray magnetic fields and the necessity for low-temperature
operation. Other methods, such as using electromechanical resonators [51], can
also produce entanglement between propagating microwave radiations, but they
face similar challenges.

An alternative approach to overcoming these limitations is to develop quantum-
limited amplifiers (as a source of entanglement), using kinetic inductance
superconducting thin films, such as Niobium Titanium Nitride (NbTiN). The
inherent nonlinearity of kinetic inductance materials allows much simpler
fabrication and chip design compared to Josephson junction amplifiers.
Recently, significant progress has been made in developing kinetic inductance
amplifiers that achieve very large gain and noise levels approaching the
quantum limit [52, 53, 54, 55, 56, 57, 58, 59]. Unlike Josephson junction
circuits, the fabrication of these amplifiers is straightforward.
Additionally, they demonstrate greater resilience to strong magnetic fields
[60, 61, 62, 63] and can operate at higher temperatures exceeding $5$ K [59].
While much of the research in this area has focused on amplification or very
recently on single-mode vacuum squeezing [63, 64] for weak signal
amplification, utilizing kinetic inductance quantum limited amplifiers to
deterministically generate spatially separated entangled states has not been
achieved so far.

Figure 1:  (a) A ring resonator with a diameter of $1.2$ mm and a width of $2$
$\mu m$ is capacitively coupled to two transmission lines on the left and
right sides. A separate line running through the top and bottom supplies the
dc-current ($I_{\text{dc}}$) and pump current ($I_{\text{p}}$) via a bias tee
to the system. By properly driving the system, we can generate a squeezed
state inside the resonator. (b) A two-sided Fabry-Perot cavity has a resonance
frequency $\omega_{a}$, extrinsic damping rates $\kappa_{\text{e,1}}$ and
$\kappa_{\text{e,2}}$, and a dissipation rate $\kappa_{\text{i}}$. The cavity
contains a $\chi^{(2)}$ nonlinear medium. By pumping the cavity at
$\omega_{p}=2\omega_{a}$, vacuum squeezing can be generated inside the cavity,
resulting in correlations and entanglement between the propagating signals at
the outputs of the resonator. (c) Circuit diagram of the measurement chain on
both sides of the resonator. A Vector Network Analyzer (VNA) is used for
measuring the resonator’s reflection properties. Squeezing is generated by
pumping the system at $\omega_{p}=2\omega_{a}$. Microwave signals emitted from
the system are amplified using High Electron Mobility Transistor amplifiers
(HEMT) at the 4K stage of the dilution refrigerator and room temperature
amplifiers, then down-converted using IQ mixers and a local oscillator (LO).
The down-converted signals are subsequently filtered and digitized using a
two-channel analog-to-digital converter (ADC). The device connects via two 50
cm long copper cables to radial switches at the 7mK stage, facilitating
switching between the sample and 50$\Omega$ loads for noise calibration
purposes. (c) The reflection parameters $S_{11}$ and $S_{22}$ of the resonator
measured at a fixed current $I_{\text{dc}}=1.575$ mA corresponding to a
frequency $\omega_{0}/2\pi=7.14$ GHz.

In this paper, we utilize a NbTiN-based Kinetic Inductance Parametric
Amplifier (KIPA) to generate stationary emission of path-entangled microwave
radiation. The entanglement is generated by the kinetic inductance
characteristic of the superconducting film, with the total inductance
described as $L_{k}(I)=L_{0}[1+(\frac{I}{I^{*}})^{2}]$ [65, 66, 67, 68], where
$L_{0}$ represents the kinetic inductance of the film in the absence of
current. The characteristic current, $I^{*}$, is proportional to the film’s
critical current and serves as a quantitative measure of the film’s
responsiveness to the applied current, $I$. As shown in Fig. 1a, our design
incorporates a nonlinear ring resonator with a resonance frequency
$\omega_{a}$, which is capacitively coupled to two transmission lines for
probing and reading out the circuit, thereby forming a two-sided nonlinear
resonator. This system is analogous to a two-sided Fabry-Perot cavity
containing a $\chi^{(2)}$ nonlinear medium, as illustrated schematically in
Fig. 1b. Our two-port resonator design provides a simple method for generating
squeezing. However, as we will explain below, this design restricts the
squeezing to no more than 3 dB at each port of the system.

By driving the system at $\omega_{p}=2\omega_{a}$, a squeezed state is created
inside the cavity and distributed to both ports, generating cross-correlation
and thus entanglement between the outputs of the resonators. In the KIPA
design, the $\chi^{(2)}$ nonlinearity arises from the kinetic inductance
property of the superconducting films. The nonlinearity’s strength can be
controlled by applying a dc-current ($I_{\text{dc}}$) to the circuit,
resulting in

$L_{k}(I)=L_{0}\Big{[}1+\Big{(}\frac{I_{\text{dc}}}{I_{*}}\Big{)}^{2}+\frac{2I_{\text{rf}}I_{\text{dc}}}{I_{*}^{2}}+\Big{(}\frac{I_{\text{rf}}}{I_{*}}\Big{)}^{2}\Big{]},$
(1)

where $I=I_{\text{dc}}+I_{\text{rf}}$ and $I_{\text{rf}}$ is the current of
the microwave signal. Combining the dc-current and a strong microwave pump
with current amplitude $I_{\text{p}}$ leads to the squeezing Hamiltonian [59]

$H=\Delta a^{\dagger}a+\frac{g}{2}(a^{2}e^{-i\phi_{p}}+a^{\dagger
2}e^{i\phi_{p}}),$ (2)

where $a$ is the annihilation operator of the ring resonator,
$g=g_{0}|\alpha_{p}|$ is the 3WM parameter, in which $|\alpha_{p}|$ represents
the amplitude of the pump with phase $\phi_{p}$. The Hamiltonian above has
been written in a reference frame rotating at
$\omega_{p}/2=\omega_{a}-\Delta$.

The emergence of squeezing and entanglement can be directly seen by solving
the quantum Langevin equations of motion along with using input-output theory
for a two-sided cavity. This gives the output operators
$a_{\mathrm{out,1(2)}}$ in terms of the input noise operators
$a_{\mathrm{e,1(2)}}$ at port 1 (2) of the resonator

$\displaystyle a_{\mathrm{out,1(2)}}$ $\displaystyle=$
$\displaystyle\Big{[}\mathcal{G}_{S,1(2)}a_{\text{e,1(2)}}+\mathcal{G}_{I,1(2)}a_{\text{e,1(2)}}^{\dagger}\Big{]}$
(3) $\displaystyle+$
$\displaystyle\sqrt{\frac{\kappa_{\text{e,2(1)}}}{\kappa_{\text{e,1(2)}}}}\Big{[}(\mathcal{G}_{S,1(2)}+1)a_{\text{e,2(1)}}+\mathcal{G}_{I,1(2)}a_{\text{e,2(1)}}^{\dagger}\Big{]}$
$\displaystyle+$
$\displaystyle\sqrt{\frac{\kappa_{\text{i}}}{\kappa_{\text{e,1(2)}}}}\Big{[}(\mathcal{G}_{S,1(2)}+1)a_{\text{i}}+\mathcal{G}_{I,1(2)}a_{\text{i}}^{\dagger}\Big{]},$

where $\kappa_{\text{e,1(2)}}$ is the extrinsic coupling rate between the
resonator and port $1(2)$ while $\kappa_{\mathrm{i}}$ shows the intrinsic
damping rate of the ring resonator with corresponding noise operator
$a_{\text{i}}$. In Eq. (3), the first bracket on the right-hand side of
$a_{\mathrm{out,1}}$ describes the single-mode squeezing of the resonator seen
at port 1. The second and third brackets represent the contributions of port 2
and internal loss, respectively. These terms act as noise terms with rates
$\frac{\kappa_{\text{e,2}}}{\kappa_{\text{e,1}}}$ and
$\frac{\kappa_{\text{i}}}{\kappa_{\text{e,1}}}$, which reduce the level of
measurable squeezing at port 1. The same interpretation also applies when
measuring the output of port 2, where port 1 serves as a loss channel. For
$\Delta=\omega=0$, the system’s gain parameters are given by
$\mathcal{G}_{S,1(2)}=\Big{(}\frac{2\eta_{1(2)}}{1-\mathcal{C}}\Big{)}-1,\,\,\mathcal{G}_{I,1(2)}=\Big{(}\frac{2\eta_{1(2)}\sqrt{\mathcal{C}}}{1-\mathcal{C}}\Big{)}e^{i(\phi_{p}-\frac{\pi}{2})}$,
where $\mathcal{C}=4g^{2}/\kappa^{2}$ is the cooperativity of the parametric
amplification and $\eta_{1(2)}=\frac{\kappa_{\text{e,1(2)}}}{\kappa}$ is the
ratio of the extrinsic damping rate to the total coupling rate
$\kappa=\kappa_{\text{e,1}}+\kappa_{\text{e,2}}+\kappa_{\text{i}}$. Note that
for a single port lossless resonator $\eta_{1}\approx 1$ and $\eta_{2}\approx
0$, thus
$a_{\mathrm{out}}=\mathcal{G}_{S}a_{\text{e}}+\mathcal{G}_{I}a_{\text{e}}^{\dagger}$
with $|\mathcal{G}_{S}|^{2}-|\mathcal{G}_{I}|^{2}=1$, as expected for an ideal
single-mode parametric amplifier [24, 25].

Eq. (3) possesses the typical structure of a squeezing operator, resulting in
quadrature squeezing below the vacuum noise. To demonstrate this, we introduce
the quadrature operators
$P_{\mathrm{out}}=(a_{\mathrm{out}}-a_{\mathrm{out}}^{\dagger})/\sqrt{2}i$ and
$X_{\mathrm{out}}=(a_{\mathrm{out}}+a_{\mathrm{out}}^{\dagger})/\sqrt{2}$
along with their variance by setting $\phi_{p}=-\pi/2$,

$\Delta
X_{\mathrm{out,1(2)}}^{2}=\frac{1}{2}\Big{(}1-\frac{2\eta_{1(2)}}{1+\sqrt{\mathcal{C}}}\Big{)}^{2}+n_{\text{noise,1(2)}},$
(4)

where
$n_{\text{noise,1(2)}}=\frac{2\eta_{1(2)}[1-\eta_{1(2)}]}{(1+\sqrt{\mathcal{C}})^{2}}$
is the added noise due to internal losses of the system and the presence of
port 2 (1). The first term of Eq. (4) shows single-mode squeezing for
$\mathcal{C}>0$. This becomes more evident when considering an ideal single-
port resonator where $\eta_{1}=1$ and $\eta_{2}=0$, resulting in single-mode
squeezing of the quadrature $X_{\mathrm{out}}$ with a variance $\Delta
X_{\mathrm{out}}^{2}=\frac{1}{2}\Big{(}\frac{1-\sqrt{\mathcal{C}}}{1+\sqrt{\mathcal{C}}}\Big{)}^{2}=\frac{e^{-2r}}{2}$,
see Supplementary Information. Here the squeezing parameter $r$ is defined as
$r=2\text{tanh}^{-1}(\sqrt{\mathcal{C}})$.

The squeezing distribution in the current two-port resonator system can
effectively be compared to a lossy beam splitter, with transmission and
reflection parameters proportional to $\eta_{1}$ and $\eta_{2}$. This analogy
becomes more clear for a two-port lossless ($\kappa_{\mathrm{i}}=0$) symmetric
resonator ($\eta_{1}=\eta_{2}=\frac{1}{2}$) in which Eq. (4) reduces to

$\Delta
X_{\mathrm{out,1(2)}}^{2}=\frac{1+\mathcal{C}}{2(1+\sqrt{\mathcal{C}})^{2}},$
(5)

At $\mathcal{C}=1$, the output of both ports reaches its maximum squeezing,
$\Delta X_{\mathrm{out,1}}^{2}=\Delta X_{\text{out,2}}^{2}=\frac{1}{4}$,
corresponding to $3$ dB squeezing below vacuum. The symmetric lossless
resonator effectively acts as an ideal balanced beam splitter, with a squeezed
state in one input port and a vacuum state in the other port. In this
scenario, the maximum achievable squeezing at the output ports of the beam
splitter is $3$ dB. Note that for $\mathcal{C}\gg 1$, $\Delta
X_{\mathrm{out,1(2)}}^{2}=\frac{1}{2}$.

The generation and distribution of squeezing in the two-sided resonator can
also lead to entanglement between the output fields of the system mainly due
to the generation of quadratures cross-correlation $\langle
X_{\text{out,1}}X_{\text{out,2}}\rangle=-\frac{2\sqrt{\mathcal{C}\eta_{1}\eta_{2}}}{({1+\sqrt{\mathcal{C}}})^{2}}$.
This behavior aligns with the predictions of the beam splitter model:
injecting a single-mode squeezed state into one port and a vacuum state into
the other port results in two correlated squeezed states at the output ports
of the beam splitter. The outputs of the resonator are entangled if the
parameter [69]

$\Delta_{\text{EPR}}:=\Delta X_{\text{+}}^{2}+\Delta P_{\text{--}}^{2}<1$ (6)

where
$X_{\text{+}}=\frac{1}{\sqrt{2}}(X_{\text{out,1}}+X_{\text{out,2}}),P_{\text{--}}=\frac{1}{\sqrt{2}}(P_{\text{out,1}}-P_{\text{out,2}})$
are the EPR quadratures. For a lossless symmetric two-sided resonator,
$\eta_{1}=\eta_{2}=\frac{1}{2}$, we obtain
$\Delta_{\text{EPR}}=\frac{1+\mathcal{C}}{(1+\sqrt{\mathcal{C}})^{2}}$. This
expression indicates that the output of the system is always entangled,
$\Delta_{\text{EPR}}<1$, for $\mathcal{C}>0$, and approaches the vacuum state,
$\Delta_{\text{EPR}}=1$, for $\mathcal{C}\gg 1$. The maximum entanglement,
however, is achieved at $\mathcal{C}=1$, corresponding to
$\Delta_{\text{EPR}}=\frac{1}{2}$ or again $3$ dB squeezing.

We experimentally demonstrate the generation of path-entangled emission for
split squeezed states at the outputs of a microwave resonator using a kinetic
inductance superconducting thin film. The optical image of the circuit and the
measurement setup are shown in Fig. 1a and b. The circuit includes a ring-
resonator with resonance frequency $\omega_{a}/2\pi=7.147$ GHz that is
capacitively coupled to the input transmission lines from the left and right
sides. The reflection parameters for both ports $S_{11}$ and $S_{22}$ are
shown in Fig 1d. Our design is very simple and doesn’t require a Bragg mirror
or impedance-matching step coupler [52, 53, 54, 55, 56]. The ring resonator
can be considered as a two-sided resonator with the extrinsic damping rates
$\kappa_{\text{e,1}}/2\pi=19.4$ MHz and $\kappa_{\text{e,2}}/2\pi=13.2$ MHz
for channels 1 and 2, respectively. The intrinsic damping rate of the
resonator is $\kappa_{\text{i}}/2\pi\approx 3$ MHz. Note that we determine the
sheet inductance, $L_{\square}$, by measuring the resonance frequency of the
ring resonator at zero bias current. By performing simulations in Sonnet and
adjusting the sheet inductance to match the experimental data, we deduce the
sheet inductance of the film to be $L_{\square}=30$ pH. This calculation
assumes that the film is uniformly evaporated across the entire chip.
Consequently, the total inductance corresponding to this sheet inductance is
$L_{0}=251$ nH.

Figure 1c shows the experimental setup used to measure squeezing and
entanglement from the chip. The output signals from the left and right sides
of the ring resonator are amplified using two identical cryogenic high
electron mobility transistor (HEMT) amplifiers. Subsequently, the signals are
further amplified at room temperature, filtered, down-converted to an
intermediate frequency $\omega_{\text{if}}/2\pi=20$ MHz, and digitized at a
sampling rate of 100 MHz by an 8-bit Alazar-tech analog-to-digital converters
(ADC).

The first step to measure non-classical radiation is to evaluate the mean
$\langle X\rangle$ and variance $(\Delta X)^{2}=\langle X^{2}\rangle-\langle
X\rangle^{2}$ of the quadrature operator $X$ from experimental data. Utilizing
the Fast Fourier transform (FFT) and subsequent post-processing of the
measured data, we find the quadrature voltages $I$ and $Q$ at channels 1 and 2
of the ADC. These voltages can be used to compute the scaled (unitless)
quadratures $X=\frac{I}{\sqrt{2\hbar\omega_{a}BRG_{\text{T}}}}$ and
$P=\frac{Q}{\sqrt{2\hbar\omega_{a}BRG_{\text{T}}}}$ of the generated signal at
the outputs of the ring-resonator. Here, $R=50\,\Omega$, and $B=200$ kHz
represent the measurement bandwidth set by a digital filter [70, 45, 51, 17].
The measured quadratures $X$ and $P$ have the same statistical characteristics
as the quadrature operators $X=\frac{\hat{a}+\hat{a}^{\dagger}}{\sqrt{2}}$ and
$P=\frac{\hat{a}-\hat{a}^{\dagger}}{\sqrt{2}i}$. We calibrate the noise
properties of both measurement chains and accurately determine the gains
$G_{\text{T,1}}=99.22\,(0.02)$ dB and $G_{\text{T,2}}=94.02\,(0.02)$ dB with
the number of noise quanta $n_{\mathrm{add,1}}=7.51\,(0.03)$,
$n_{\mathrm{add,2}}=14.80\,(0.09)$ added by the output chains, see the
Supplemental Information. In the gain and added noise calculations, we do not
consider the loss of the copper cables, $\mathcal{A}\approx 0.8$ dB,
connecting the ring-resonator sample to the cold switch. As a result, the
reported squeezing values reflect the signal propagation over a $0.5$-meter
lossy cable. After the cold switch, the initially created squeezed state is
affected by amplifier noise, additional losses, and exposure to a thermal bath
at elevated temperatures.

The squeezing of the propagating output states can be fully characterized from
the variances of the rotated quadrature $X(\phi)$ when pumps are on and off,
i.e., $\langle X^{2}(\phi)\rangle=\langle X^{2}\rangle_{\mathrm{on}}-\langle
X^{2}\rangle_{\mathrm{off}}+\xi$, where the parameter $\phi$ shows a rotation
of the detector phase, implemented during post-processing. Here,
$\xi=\frac{1}{2}\mathrm{coth}\,\hbar\omega_{a}/(k_{B}T)\approx 0.5$ shows the
input quantum noise at temperature $T$, and the brackets $\langle..\rangle$
indicate an average over the $576000$ measurements when pump were on or off.
Figure 2 shows the measured quadrature squeezing for outputs 1 and 2, $\langle
X_{1(2)}^{2}\rangle$, at the optimal angle $\phi$ as a function of drive power
$P_{p}$. The parameter $\langle X^{2}\rangle$ remains consistently below 0.5,
representing the squeezed nature of the ring resonator’s output, reaching
0.7(0.2) dB and 1.0(0.3) dB squeezing for output 1 and 2, respectively.
However, at higher pump powers, the squeezing gradually diminishes and
eventually disappears. This phenomenon is attributed to finite losses in the
superconducting film and the resonator, leading to pump power–dependent
heating and resulting in larger variances of the output field. The reported
errors represent the statistical errors of the measured means, which surpass
the statistical errors and the long-term variation observed in the subtracted
noise measurements (with the pump turned off), in addition to the error from
the calibration measurements.

Figure 2: Measured variance of quadratures $X_{\text{out,1}}(\phi)$ and
$X_{\text{out,2}}(\phi)$ (dots) for ports 1 and 2 of the resonator at an
optimal detector angle $\phi$ for different pump powers, fitted to the
theoretical model (solid lines). The blue area indicates the region of vacuum
squeezing. Error bars represent the standard deviation of the mean from 5
independent measurements of both quadratures, each comprising 115,200 values.

The appearance of squeezing at each port can indicate the presence of
entanglement between the spatially separated ports of the resonator. We
confirm this by computing the variance of the EPR quadratures $\langle
X_{\text{+}}(\phi)^{2}\rangle=\frac{1}{2}\langle(X_{\text{out,1}}(\phi)+X_{\text{out,2}})^{2}\rangle$
and $\langle
P_{\text{--}}(\phi)^{2}\rangle=\frac{1}{2}\langle(P_{\text{out,1}}(\phi)-P_{\text{out,2}})^{2}\rangle$,
and calculate the EPR parameter $\Delta_{\text{EPR}}:=\Delta
X_{\text{+}}^{2}+\Delta P_{\text{--}}^{2}$. In Fig. 3a, we measure parameter
$\Delta_{\text{EPR}}$ at an optimized angle $\phi$ for different pump powers.
This figure shows that the outputs of the ring resonator are entangled
reaching 0.9(0.2) dB squeezing.

We additionally quantify the amount of entanglement generated by our microwave
circuits using standard measures in quantum information theory. Specifically,
we are interested in the logarithmic-negativity [71, 72], which serves as an
upper limit on the amount of distillable entanglement bits produced by the
source. The log-negativity $E_{N}$ is given by

$E_{N}=\text{max}[0,-2\text{log}(\zeta^{-})]$ (7)

where $\zeta^{-}$ is the smallest partially-transposed symplectic eigenvalue
of the covariance matrix defined $V_{ij}=\frac{\langle
X_{i}X_{j}+X_{j}X_{i}\rangle}{2},$ with
$X_{ij}=\\{X_{\text{out,1}},P_{\text{out,1}},X_{\text{out,2}},P_{\text{out,2}}\\}$,
see Supplemental Information. Figure 3b shows logarithmic negativity versus
the pump power with the maximum value $E_{N}=0.24(0.05)$. The log-negativity
exhibits similar behavior to the EPR parameter; increasing the pump power
causes the entanglement to diminish.

In summary, we have demonstrated the generation of stationary microwave path-
entangled radiation using kinetic inductance superconducting quantum circuits
for the first time. Compared to Josephson junction amplifiers, our device
utilizes a single-layer NbTiN thin film, offering a high fabrication yield. By
generating single-mode squeezing and splitting it with a two-port microwave
resonator, we demonstrated entanglement generation and distribution between
two separate paths. Although this work did not explore resilience to magnetic
fields or operation at higher temperatures, high-kinetic inductance NbTiN
resonators exhibit exceptional magnetic field compatibility [60, 63, 58, 73].
This property makes the devices suitable for integration with spin systems and
defect centers [74], such as NV centers, enabling precise readout of quantum
nodes or entanglement distribution between remote spins in different
substrates.

Several improvements can further enhance the efficiency of this system. One
approach is to reduce loss and noise by refining the fabrication process or
using resonators with better impedance matching. The current design limits
squeezing to 3 dB per port. To achieve higher levels of squeezing and
consequently larger entanglement, designing two-mode resonance circuits to
generate two-mode squeezed states would be beneficial. This device could
potentially be used to generate microwave nonclassical emissions at higher
temperatures, paving the way for designing sensors with real-world
applications in quantum illumination and microwave quantum radars [19] or
accelerometry without the need for dilution refrigerators or operation at
milliKelvin temperatures.

Figure 3:  (a) Measured EPR parameters $\Delta_{\text{EPR}}:=\Delta
X_{\text{+}}^{2}+\Delta P_{\text{--}}^{2}$ at different pump powers for
optimized detector angles. (b) Logarithmic-negativity $E_{N}$ as a function of
pump power. Solid lines represent the theoretical model fitting the
experimental results (dots). Mean values and error bars are obtained in the
same manner as in Fig. (2).

Acknowledgments We thank Jarryd Pla for helpful comments and discussions. S.B.
acknowledges funding by the Natural Sciences and Engineering Research Council
of Canada (NSERC) through its Discovery Grant and Quantum Alliance Grant,
funding and advisory support provided by Alberta Innovates (AI) through the
Accelerating Innovations into CarE (AICE) – Concepts Program, support from
Alberta Innovates and NSERC through Advance Grant project, and Alliance
Quantum Consortium. This project is funded [in part] by the Government of
Canada. Ce projet est financé [en partie] par le gouvernement du Canada.

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Supplementary Materials

##  .1 Hamiltonian and theoretical model

The Hamiltonian describing a two-port Fabry–Pérot cavity containing
nonlinearity $\chi^{(2)}$ and resonance frequency $\omega_{a}$ is given by,

$H=\omega_{a}a^{\dagger}a+\frac{g_{0}}{2}(a^{2}a_{p}^{\dagger}+a^{\dagger
2}a_{p}),$ (8)

This Hamiltonian represents a Three-Wave-Mixing (3WM) process, where a pump
photon at frequency $\omega_{p}$ is annihilated, resulting in the creation of
two photons at the resonance frequency $\omega_{a}$. In a strong pump regime,
we can treat the pump operator as a classical variable,
$\alpha_{p}=|\alpha_{p}|e^{-i\phi_{p}}e^{-i\omega_{p}t}$, where $|\alpha_{p}|$
is the amplitude of the pump and $\phi_{p}$ is its phase. This approximation
along with moving to a rotating frame with respect to the frequency of the
pump allows us to rewrite the Hamiltonian (8)

$H=\Delta
a^{\dagger}a+\frac{g}{2}(a^{2}e^{i\phi_{p}}+a^{\dagger^{2}}e^{-i\phi_{p}})$
(9)

where $g=g_{0}|\alpha_{p}|$ and $\omega_{p}/2=\omega_{a}-\Delta$.

By using the quantum Langevin equations of motion, we derive the following set
of coupled equations that describe the system’s dynamics for a two-sided
nonlinear cavity

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-\Big{(}i\Delta+\frac{\kappa}{2}\Big{)}a-ige^{i\phi_{p}}a^{\dagger}+\sum_{j=1,2}\sqrt{\kappa_{\mathrm{e},j}}a_{\mathrm{e},j}+\sqrt{\kappa_{\mathrm{i}}}a_{\mathrm{i}},$
$\displaystyle\dot{a}^{\dagger}$ $\displaystyle=$
$\displaystyle\Big{(}i\Delta-\frac{\kappa}{2}\Big{)}a^{\dagger}+ige^{-i\phi_{p}}a+\sum_{j=1,2}\sqrt{\kappa_{\mathrm{e,j}}}a_{{\mathrm{e},j}}a_{\mathrm{e},j}^{\dagger}+\sqrt{\kappa_{\mathrm{i}}}a_{\mathrm{i}}^{\dagger},$
(10)

where $\kappa=\kappa_{\mathrm{e},1}+\kappa_{\mathrm{e},2}+\kappa_{\mathrm{i}}$
represents the total damping rate of the resonator. These equations
incorporate quantum noise affecting input fluctuations for the resonator
($a_{\mathrm{e,1(2)}}$ with extrinsic damping rate $\kappa_{\mathrm{e,1(2)}}$)
and the intrinsic losses in the resonator mode ($a_{\mathrm{i}}$ with
intrinsic damping rate $\kappa_{\mathrm{i}}$). The correlations for these
noises are defined as follows:

$\displaystyle\langle
a_{\mathrm{e(i)}}(t)a_{\mathrm{e(i)}}^{\dagger}(t^{\prime})\rangle$
$\displaystyle=$ $\displaystyle\langle
a_{\mathrm{e(i)}}^{\dagger}(t)a_{\mathrm{e(i)}}(t^{\prime})\rangle+\delta(t-t^{\prime})$
(11) $\displaystyle=$
$\displaystyle(\bar{n}_{\mathrm{e(i)}}+1)\delta(t-t^{\prime}),$

where $\delta(t)$ is the Dirac delta function and $\bar{n}_{\mathrm{e(i)}}$
are the Planck-law thermal occupancies of microwave mode (bath). By moving to
the Fourier domain, where $\dot{O}\rightarrow-i\omega$, and applying input-
output theory, we can find the output operators at ports 1 and 2 of the cavity
$a_{\text{out, 1(2)}}$

$\displaystyle a_{\mathrm{out,1(2)}}$ $\displaystyle=$
$\displaystyle\Big{[}\mathcal{G}_{S,1(2)}a_{\text{e,1(2)}}+\mathcal{G}_{I,1(2)}a_{\text{e,1(2)}}^{\dagger}\Big{]}+\sqrt{\frac{\kappa_{\text{e,2(1)}}}{\kappa_{\text{e,1(2)}}}}\Big{[}(\mathcal{G}_{S,1(2)}+1)a_{\text{e,2(1)}}+\mathcal{G}_{I,1(2)}a_{\text{e,2(1)}}^{\dagger}\Big{]}$
(12) $\displaystyle+$
$\displaystyle\sqrt{\frac{\kappa_{\text{i}}}{\kappa_{\text{e,1(2)}}}}\Big{[}(\mathcal{G}_{S,1(2)}+1)a_{\text{i}}+\mathcal{G}_{I,1(2)}a_{\text{i}}^{\dagger}\Big{]},$

with idler and signal gain parameters

$\displaystyle\mathcal{G}_{S,1(2)}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\eta_{1(2)}\kappa\Big{[}\frac{\kappa}{2}-i(\omega+\Delta)\Big{]}}{\Delta^{2}-g^{2}+(i\omega-\frac{\kappa}{2})^{2}}-1,$
$\displaystyle\mathcal{G}_{I,1(2)}(\omega)$ $\displaystyle=$
$\displaystyle\frac{-i\eta_{1(2)}\kappa
ge^{i\phi}}{\Delta^{2}-g^{2}+(i\omega-\frac{\kappa}{2})^{2}},$ (13)

where $\eta_{1(2)}=\frac{\kappa_{\text{e,1(2)}}}{\kappa}$ describes the
waveguide-resonator coupling. The Eq. (12) fulfills the bosonic commutation
relation for the output field operators
$[a_{\mathrm{out,j}}(\omega),a_{\mathrm{out,k}}^{\dagger}(\omega^{\prime})]=\delta(\omega-\omega^{\prime})\delta_{\text{jk}}$,
resulting in

$\frac{|\mathcal{G}_{I,1(2)}|^{2}}{{\eta_{1(2)}}}=|\mathcal{G}_{S,1(2)}|^{2}+\Big{(}\frac{1-\eta_{1(2)}}{\eta_{1(2)}}\Big{)}|\mathcal{G}_{S,1(2)}+1|^{2}-1,$
(14)

where for a lossless and single port resonator for example $\eta_{1}=1$ and
$\eta_{2}=0$, we get $|\mathcal{G}_{S,1}|^{2}-|\mathcal{G}_{I,1}|^{2}=1$.

##  .2 Squeezing and cross-correlation

By setting $\Delta=\omega=0$ and defining $\mathcal{C}=4g^{2}/\kappa^{2}$, we
can simplify the gain parameters to

$\displaystyle\mathcal{G}_{S,1(2)}$ $\displaystyle=$
$\displaystyle\frac{2\eta_{1(2)}}{1-\mathcal{C}}-1,$
$\displaystyle\mathcal{G}_{I,1(2)}$ $\displaystyle=$
$\displaystyle-\frac{2i\eta_{1(2)}\sqrt{\mathcal{C}}}{1-\mathcal{C}}e^{i\phi_{p}},$
(15)

as presented in the main text.

Using Eq. (12) we can calculate the variance of quadratures

$\displaystyle
X_{\mathrm{out,1(2)}}=\frac{a_{\mathrm{out,1(2)})}+a_{\mathrm{out,1(2)}}^{\dagger}}{\sqrt{2}}$
(16) $\displaystyle
P_{\mathrm{out,1(2)}}=\frac{a_{\mathrm{out,1(2)}}-a_{\mathrm{out,1(2)}}^{\dagger}}{\sqrt{2}i}$
(17)

For this we assume the input noise operators from the waveguides are all in
vacuum $\bar{n}_{\mathrm{e,1(2)}}=0$, thus the variances of quadratures are
given by

$\displaystyle\Delta X_{\mathrm{out,1(2)}}^{2}(\phi_{p})$ $\displaystyle=$
$\displaystyle\braket{X_{1(2)}^{2}}-\braket{X_{1(2)}}^{2}=\frac{1}{2}+\frac{|\mathcal{G}_{I,1(2)}|^{2}}{\eta_{1(2)}}+\frac{\kappa_{\text{i}}\bar{n}_{\mathrm{i}}}{\kappa_{\text{e,1(2)}}}\Big{[}|\mathcal{G}_{S,1(2)}+1|^{2}+|\mathcal{G}_{I,1(2)}|^{2}\Big{]}$
$\displaystyle+$
$\displaystyle\Big{(}|\mathcal{G}_{I,1(2)}||\mathcal{G}_{S,1(2)}|+\frac{\eta_{2(1)}}{\eta_{1(2)}}|\mathcal{G}_{I,1(2)}||\mathcal{G}_{S,1(2)}+1|+\frac{\kappa_{\text{i}}(2\bar{n}_{\mathrm{i}}+1)}{\kappa_{\text{e,1(2)}}}|\mathcal{G}_{I,1(2)}||\mathcal{G}_{S,1(2)}+1|\Big{)}\text{sin}(\phi_{p}),$

and

$\displaystyle\Delta P_{\mathrm{out,1(2)}}^{2}(\phi_{p})$ $\displaystyle=$
$\displaystyle\braket{P_{1(2)}^{2}}-\braket{P_{1(2)}}^{2}=\frac{1}{2}+\frac{|\mathcal{G}_{I,1(2)}|^{2}}{\eta_{1(2)}}+\frac{\kappa_{\text{i}}\bar{n}_{\mathrm{i}}}{\kappa_{\text{e,1(2)}}}\Big{[}|\mathcal{G}_{S,1(2)}+1|^{2}+|\mathcal{G}_{I,1(2)}|^{2}\Big{]}$
$\displaystyle-$
$\displaystyle\Big{(}|\mathcal{G}_{I,1(2)}||\mathcal{G}_{S,1(2)}|+\frac{\eta_{2(1)}}{\eta_{1(2)}}|\mathcal{G}_{I,1(2)}||\mathcal{G}_{S,1(2)}+1|+\frac{\kappa_{\text{i}}(2\bar{n}_{\mathrm{i}}+1)}{\kappa_{\text{e,1(2)}}}|\mathcal{G}_{I,1(2)}||\mathcal{G}_{S,1(2)}+1|\Big{)}\text{sin}(\phi_{p}),$

For $\Delta=\omega=0$ and $\bar{n}_{\mathrm{i}}=0$, the $X-$ and
$P$-quadrature variances (at optimum phases $\phi_{p}=\pm\pi/2$) reduce to

$\Delta X_{\mathrm{out,1(2)}}^{2}|_{\phi_{p}\rightarrow-\pi/2}=\Delta
P_{\mathrm{out,1(2)}}^{2}|_{\phi_{p}\rightarrow\pi/2}=\frac{1}{2}\Big{(}1-\frac{2\eta_{1(2)}}{1+\sqrt{\mathcal{C}}}\Big{)}^{2}+\frac{2\eta_{1(2)}(1-\eta_{1(2)})}{(1+\sqrt{\mathcal{C}})^{2}},$
(20)

as explained in the main text.

The generated squeezing creates a cross-correlation between the outputs of the
system, resulting in the following quadrature correlations

$\displaystyle\langle X_{\mathrm{out,1}}X_{\mathrm{out,2}}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\langle(a_{\mathrm{out,1}}a_{\mathrm{out,2}}+a_{\mathrm{out,1}}^{\dagger}a_{\mathrm{out,2}}^{\dagger}+a_{\mathrm{out,1}}a_{\mathrm{out,2}}^{\dagger}+a_{\mathrm{out,1}}^{\dagger}a_{\mathrm{out,2}})\rangle=-\frac{2\sqrt{\mathcal{C}\eta_{1}\eta_{2}}}{({1+\sqrt{\mathcal{C}}})^{2}},$
$\displaystyle\langle P_{\mathrm{out,1}}P_{\mathrm{out,2}}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2}\langle(a_{\mathrm{out,1}}a_{\mathrm{out,2}}+a_{\mathrm{out,1}}^{\dagger}a_{\mathrm{out,2}}^{\dagger}-a_{\mathrm{out,1}}a_{\mathrm{out,2}}^{\dagger}-a_{\mathrm{out,1}}^{\dagger}a_{\mathrm{out,2}})\rangle=\frac{2\sqrt{\mathcal{C}\eta_{1}\eta_{2}}}{({1-\sqrt{\mathcal{C}}})^{2}}.$
(21)

for $\phi_{p}=-\pi/2$, $\Delta=\omega=0$ and $\bar{n}_{\mathrm{i}}=0$. This
now allows us to calculate the entanglement parameter

$\displaystyle\Delta_{\text{EPR}}:=\Delta X_{\text{+}}^{2}+\Delta
P_{\text{--}}^{2},$ (22)

Using the definition of the quadrature
$X_{\text{+}}=\frac{1}{\sqrt{2}}(X_{\text{out,1}}+X_{\text{out,2}}),P_{\text{--}}=\frac{1}{\sqrt{2}}(P_{\text{out,1}}-P_{\text{out,2}})$,
we get

$\displaystyle\Delta_{\text{EPR}}=\frac{1}{2}\Big{(}\sum_{j=1,2}\big{[}\Delta
X_{\mathrm{out},j}+\Delta P_{\mathrm{out},j}\big{]}+\langle
X_{\mathrm{out,1}}X_{\mathrm{out,2}}\rangle+\langle
X_{\mathrm{out,2}}X_{\mathrm{out,1}}\rangle-\langle
P_{\mathrm{out,1}}P_{\mathrm{out,2}}\rangle-\langle
P_{\mathrm{out,2}}P_{\mathrm{out,1}}\rangle\Big{)},$ (23)

By substituting Eqs. (20) and (21) into Eq. (23), we find

$\displaystyle\Delta_{\text{EPR}}=\frac{\mathcal{C}^{2}+(1-4\sqrt{\eta_{1}\eta_{2}\mathcal{C}})-2\mathcal{C}(1+2\sqrt{\eta_{1}\eta_{2}\mathcal{C}}-2(\eta_{1}+\eta_{2}))}{(1-\mathcal{C})^{2}},$
(24)

For a lossless symmetric two-sided resonator, $\eta_{1}=\eta_{2}=\frac{1}{2}$,
we obtain
$\Delta_{\text{EPR}}=\frac{1+\mathcal{C}}{(1+\sqrt{\mathcal{C}})^{2}}$.

##  .3 Covariance matrix and logarithmic-negativity

We can reconstruct the covariance matrix (CM) between the two output channels
of the system, which can be expressed as,

$\displaystyle V_{ij}=\frac{\langle X_{i}X_{j}+X_{j}X_{i}\rangle}{2},$ (25)

with
$X_{ij}=\\{X_{\text{out,1}},P_{\text{out,1}},X_{\text{out,2}},P_{\text{out,2}}\\}$.
To verify the generation of entanglement in the system we compute the
logarithmic negativity $E_{N}=\text{max}[0,-2\text{log}(\zeta^{-})]$ where

$\displaystyle\zeta^{-}=\left(\frac{\Delta-\sqrt{\Delta^{2}-4\textbf{det}V}}{2}\right)^{1/2}$
(26)

is the smallest partially-transposed symplectic eigenvalue of the covariance
in which

$\Delta=\textbf{det}A+\textbf{det}B-2\textbf{det}C\quad,\quad
V=\begin{pmatrix}A&C&\\\ C^{T}&B\end{pmatrix}$ (27)

##  .4 Sample Fabrication

The fabrication process begins with a high-resistivity intrinsic silicon wafer
($R\geq 20\,k\,\Omega-\text{cm}$) that is 500 micrometers thick, coated with a
10 nm layer of NbTiN, and diced into smaller chips. The chips undergo a
cleaning procedure using acetone and IPA, each involving a 5-minute
sonication. Following this, the chip’s surface is coated with AZ 1529 resist
and patterned using optical lithography. The device is then etched using an
ICP Reactive Ion Etcher.

##  .5 Packaging and the PCB design

After the fabrication, the chip is attached to a gold-plated PCB using silver
paste. This PCB, made from Rogers AD 1000 laminate with a dielectric thickness
of 0.5 mm, is coated with oxygen-free copper on both sides and finished with
gold plating. The PCB includes 50-$\Omega$ coplanar waveguides that connect to
the external measurement line via a surface-mount mini-SMP microwave
connector. At the opposite end, aluminum bond wires connect the PCB to the
input and pump/DC ports of the KIPA resonator.

To mitigate unwanted parasitic modes, an array of vias connects the
transmission line on the PCB to the top and bottom ground planes. The entire
assembly is then housed in an oxygen-free enclosure, forming a 3D cavity with
a resonance frequency distant from the sample’s operational point. This cavity
shields the sample from parasitic noise and enhances its thermalization.

##  .6 Noise calibration

To accurately verify the entanglement between the two output channels, we
first determine the gain and added noise quanta of the measurement chain using
two temperature-controlled $50\,\Omega$ loads [51, 17, 75], see Fig. 4. By
adjusting and recording the temperature of the $50\,\Omega$ load, we can
produce predictable thermal noise. This noise is then used to calibrate the
measurement chain. The calibration devices are connected to the setup via two
5-cm-long superconducting coaxial cables and a thin copper braid, which
provides weak thermal anchoring to the mixing chamber plate, using a Radiall
R573423600 latching microwave switch. The noise generated for calibration
travels along the measurement chain, is down-converted to 20 MHz, filtered,
and then measured with our two-channel ADC. By fitting the noise density in
$V^{2}/\text{Hz}$ to the expected photon number at a given temperature

$N_{i}=G_{\text{T,i}}\hbar\omega
RB\left(\frac{1}{2}\text{coth}[\hbar\omega/2k_{b}T]\text{+}n_{\text{add,i}}\right),$
(28)

we can accurately determine the gains $G_{\text{T,1}}=99.22\,(0.02)$ dB and
$G_{\text{T,2}}=94.02\,(0.02)$ dB with the number of noise quanta
$n_{\mathrm{add,1}}=7.51\,(0.03)$, $n_{\mathrm{add,2}}=14.80\,(0.09)$ added by
the output chains. The confidence intervals are calculated from the standard
error of the fits. Here $R=50\,\Omega$ and $B=200$ kHz.

Figure 4:  (a) The schematic of the calibration setup. Two temperature-
controlled $50\,\Omega$ loads are used to generate known broadband thermal
noise at the 7 mK stage of the dilution refrigerator. This noise propagates
through the same measurement chain used for detecting squeezing and
entanglement. At each channel of the measurement chain, the generated noise is
amplified using high mobility electron transistors (HEMTs) and room
temperature amplifiers, down-converted using IQ mixers, and measured by a two-
channel analog-to-digital converter (ADC). (b) By knowing the temperature of
the noise calibration tool (the $50\,\Omega$ load) and fitting the data using
the Einstein distribution function, we can accurately determine the gain and
added noise for each channel separately. We utilize two cryogenic switches to
switch between the sample and the calibration tools. For down-conversion, we
use local oscillators (LO) to down converter signals to 20 MHz.


# On the degeneracy of whispering gallery modes in a high-Q sapphire microwave
resonator

Vincent Giordano, Samuel Margueron V. Giordano, S. Margueron are with the
Institute FEMTO-ST (Franche-Comté Electronique, Mécanique, Thermique et
Optique - Sciences et Technologies), CNRS (Centre National de la Recherche
Scientifique), Supmicrotech-ENSMM (Ecole Nationale Supérieure de Mécanique et
des Microtechniques), Université de Franche-Comté, 25000 Besançon France,
(e-mail: giordano@femto-st.fr)Manuscript created September 2023. ; This work
is partially funded by the Agence Nationale de Recherche (ANR) Programme
d’Investissement d’Avenir under the following grants: ANR-11-EQPX-0033-OSC-IMP
(Oscillator IMP project) ANR-10-LABX-48-01 (FIRST-TF network), and
ANR-17-EURE-00002 (EIPHI); by the French RENATECH network and its FEMTO-ST
technological facility, and by grants from the Région Bourgogne Franche Comté
intended to support the above.

###### Abstract

Cylindrical WGM resonators machined in high-quality sapphire monocrystal
cooled down to liquid helium temperature offer exceptionally-high Q-factors in
the microwave frequency domain. Such a resonator constitutes the core of an
ultra-stable oscillator featuring fractional frequency stability better than
$1\times 10^{-15}$ at short integration times. As in any cylindrical resonant
structure, the WGM resonator presents a two fold degeneracy. When a defect
breaks the cylindrical symmetry of the resonator, the WGMs split and appear as
doublets. In the high-quality sapphire resonator, the frequency separation of
these twin modes varies from one mode order to another with a maximum value of
a few tens of kHz. While the mode splitting for a given mode was considered
until now unpredictable and intrinsic to each resonator since resulting a
priori from randomly distributed defects. we show here, at the contrary, that
the observed mode splitting found on all the sapphire resonators whatever
their origin mainly comes from a perfectly determined defect resulting from
the manufacturing processes.

###### Index Terms:

Whispering gallery mode resonator, sapphire, ultra-stable oscillator.

##  I Introduction

Whispering gallery modes (WGM) low loss dielectric resonators cover a wide
range of applications in the fields of microwaves, millimetre waves and
optics. The strong confinement of the electromagnetic field inside the
dielectric medium obtained with such excitation modes provides exceptional
properties to the resonator such as: high quality factor, immunity to
environment perturbations, enlarged dimensions … In the microwave domain,
these resonators serve as frequency reference for the demonstration of ultra-
stable cryogenic oscillators [1, 2, 3, 4], and enable the accurate measurement
of material complex permittivity [5] or superconductors surface impedance [6].
Millimeter-wave filters, oscillators or multiple-port power dividers/combiners
have been developed using planar WGM dielectric resonators [7, 8]. While it is
certainly in the field of photonics that we find today the most innovative
applications of WGMs with a profusion of resonator shapes, sizes and operating
wavelengths [9, 10].

A wide variety of resonator shapes such as disks, rings, toroids or spheres,
can support the propagation of whispering gallery modes. The rotational
symmetry of these resonators induces a degeneration of the Helmhotz equation
solutions yielding the two-fold degeneracy of the WGMs. Any defect that breaks
the resonator symmetry lifts this degeneracy and a resonance line splitting is
generally observed [11, 12]. The defect can be a geometrical imperfection, a
particle sticked on the resonator surface or an inhomogeneity inside the
dielectric bulk [13, 14]. For many applications, this phenomenon is
detrimental. In the case of a microwave WGM resonator oscillator, frequency
jumps can occur between the two degenerate modes, and thus compromise the
oscillator stability [15, 16, 17]. At the opposite, the degeneracy and line
splitting phenomenon can be exploited in photonics WGM resonators for the
sensing of nanoparticles like contaminants or virus [18, 19]. In any case,
these respective applications would benefit from improved control of the mode
splitting and better understanding of its origin.

So far, it is generally accepted that the mode splitting observed in the
sapphire microwave WGM resonator arises from randomly located defects that
affect the resonator geometry or its homogeneity. In this paper, we refute
this common though idea and show that the mode splitting originates from a
perfectly determined geometrical defect. The latter is found to be observed on
all the sapphire resonators, whatever their origin. This defect follows the
6-fold symmetry of the sapphire crystal and is linked to the different
crystallographic planes, which react differently when machining the sapphire.

As a secondary benefit of our study, we bring here a modest answer to the
question Can One Hear the Shape of a Drum?  posed by Mr. Kac in 1966 in a
famous article [20]. This question took mathematicians about thirty years to
give in the general case a negative answer to this question [21]. It is also
noteworthy that the first experimental verification of the isospectrality of
two different geometries discovered by mathematicians was carried out using
flat microwave cavities as a drum [22]. In this article, we will obviously not
address this complex mathematical problem. We show however how the phenomenon
of ”mode-splitting” observed in a microwave whispering gallery mode sapphire
resonator enables to deduce its deviation from the ideal cylindrical form.

##  II The Cryogenic Sapphire Resonator

The mono-crystal of Al2O3 is the material presenting the lowest dielectric
losses in the microwave frequency range. With a loss tangent ($tg\delta$) of
about $5\times 10^{-6}$ at $300$ K and lower than $10^{-9}$ at liquid helium
temperature [23], it is an ideal dielectric to build a high Q-factor microwave
resonator in X-band. However, its relative permittivity is quite low, i.e.
$\epsilon_{r}\approx 10$. Modes of high azimuthal order, named whispering
gallery modes, are thus used to benefit fully from the sapphire’s low
dielectric loss tangent. A standard design consists of a large sapphire disk,
typically 30 to 50 mm diameter (for X-band operation), enclosed in a metallic
cylindrical cavity. For a whispering-gallery mode, total reflection occurs at
the curved air–dielectric interface, greatly limiting the power dissipation in
the metallic enclosure walls. The resonator unloaded Q-factor $Q_{0}$ is then
only limited by the sapphire dielectric losses:

$Q_{0}\approx\dfrac{1}{tg\delta}$ (1)

Since 2010, we have built and validated at the FEMTO-ST Institute a dozen of
ultra-stable Cryogenic Sapphire Oscillators (CSO) based on this technology.
Our most advanced design is currently offered as a commercial instrument under
the codename: ULISS-2G [24, 25, 26]. Describing the (in)stability as the Allan
deviation (ADEV) $\sigma_{y}(\tau)$ of the fractional frequency $y$ as a
function of the measurement time $\tau$ , ULISS-2G features
$\sigma_{y}(\tau)<3\times 10^{-15}$ for $1~{}\text{s}\leq\tau\leq 10^{4}$ s
and is limited by a drift of $\approx 10^{-14}$ at one day. It can run
unattended for years of continuous operation, with only simple maintenance
every 2nd year.

All the resonators we have integrated into our CSOs are manufactured according
to the same design shown in the figure 1 [27].

Figure 1: The ULISS-2G sapphire resonator.

Made in a high purity sapphire mono-crystal, it is 54-mm-diameter and 30-mm-
height with the cylindrical axis parallel to the Al2O3 crystal $c$-axis within
1∘. The $10$ mm diameter spindle machined from the bulk is used to attach the
resonator to the bottom flange of a gold plated standard Oxygen-free high-
thermal-conductivity (OFHC) copper cavity. The resonator operates on the
quasi-transverse magnetic whispering-gallery mode WGH15,0,0 resonating at
$\nu_{0}=9.99$ GHz [28]. Two diametrically opposed small magnetic loops
constitute the input and output resonator ports. The radial position of these
probes can be adjusted to tune the coupling coefficient at each resonator
port. The cavity is thermally linked to the $2^{\text{nd}}$-stage of a Pulse-
Tube cryocooler and stabilized near $6$ K.

##  III The mode degeneracy

Although whispering-gallery (WG) modes are strictly hybrid modes, they can be
classfied in quasi-TM (WGH) and quasi-TE (WGE) mode families. A WGH mode is
characterized by the electric field mainly directed in the axial direction
while its magnetic field is essentially transverse. The inverse situation
occurs for a WGE mode. Whispering gallery modes can be further characterized
by three integers, i.e. $m$,$n$ and $l$, representing the variation of the
electromagnetic field components in the azimuthal ($\phi$), radial ($r$) and
axial ($z$) directions of the cylindrical coordinate frame. In our
developments, we only concentrate on the WGHm,0,0 modes presenting the highest
Q-factor. For these modes, the axial electric field inside the sapphire can be
written as [29]:

$E_{z}(r,\phi,z)=E_{0}{J_{m}}(kr)\cos(\beta
z)\left\\{\begin{array}[]{r}\\!\\!\\!\\!\cos(m\phi)\\\
\\!\\!\\!\\!\sin(m\phi)\end{array}\right.\\\ $ (2)

where $J_{m}$ is the Bessel function of the first kind of order $m$, $\beta$
the axial propagation constant and $k$ the guided wave number. The brace
indicates that the resonator can support two orthogonal ”polarizations”, both
equivalent solutions of the Helmolzt equation. Figure 2 shows the map of the
transverse magnetic field in the resonator equatorial plane for the twin modes
WGH6,0,0.

Figure 2: Magnetic field in the resonator equatorial plane for the twin modes
WGH6,0,0.

In a perfect sapphire cylinder, the twin modes probe exactly the same medium,
and thus resonate at the same frequency. The position of the nodes of the
stationary wave pattern is set by the coupling structure. It has already been
shown how any defect affecting the resonator cylindrical symmetry lifts the
twin modes degeneracy [11]. The frequency splitting $\delta\nu_{m}$ between
the twin modes depends on the azimuthal number because $m$ conditions the
configuration of the electromagnetic field near the defect. It is worth noting
that in presence of the defect, the azimuthal position of the stationary wave
pattern is no longer set by the coupling structure, but stays locked to the
defect. One of the twin mode has a node at the defect location, the second one
a maximum.

In the CSO we exploit the mode WGH15,0,0 at $\nu_{15}=~{}9.99$ GHz for which
the twin resonances have always been observed. $\delta\nu_{15}$ is typically
of the order of $10$ kHz ($10^{-6}$ in relative value). Figure 3 shows the two
WGH15,0,0 modes of a typical sapphire resonator cooled near $6$ K. Their
frequency separation is $\delta\nu_{15}=8.6$ kHz
($\frac{\delta\nu_{15}}{\nu_{15}}\approx 0.9\times 10^{-6}$).

Figure 3: Typical modulus of the reflexion and transmission coefficients
$|S_{11}|$ and $|S_{21}|$ for the twin modes WGH15,0,0 observed during the
first cooling down to $6~{}$K. The insertion losses of the two modes are near
the same, which can lead to frequency instability for the oscillator.

The presence of the twin modes is an issue for our application. Indeed, the
oscillator will start on the mode having the lower insertion losses (higher
coupling). The latter will be hardly predictable before cooling as the
degeneracy lifting in not observable at room temperature. Under the worst
conditions, both modes may have equivalent insertion losses. In this case the
stability of the oscillator could be jeopardized by random frequency jumps.
This issue has been solved by rotating the sapphire resonator with respect to
the coupling probe position such that the mode with the lowest frequency is
suppressed, as demonstrated in the figure 4.

Figure 4: Modulus of the reflexion and transmission coefficients $|S_{11}|$
and $|S_{21}|$ for the twin modes WGH15,0,0 measured after a resonator
rotation and a second cooling down to $6~{}$K. The sapphire rotation favoured
the higher frequency mode, which is now the only mode that can oscillate.

Apart the WGH15,0,0 mode, which serves as frequency reference for our
oscillator, the experimental set-up enables the observation other WGH modes
ranging from about 4 GHz to 14 GHz corresponding to $5\leq m\leq 24$. Outside
this range, the limited bandwidth of the isolators placed at each resonator
port and the poor coupling of the remaining modes make usually their detection
impossible.

Until now we had not invested much on the origin of the mode splitting. It was
generally accepted, that it results from defects affecting the cylindrical
shape of the resonator or the morphology of the sapphire crystal. Since
disturbances should be randomly distributed, the different mode splittings
$\delta\nu_{m}$ are hardly predictable and should be specific to each
resonator. To our opinion this claim must be abandoned as demonstrated in the
figure 5.

Figure 5: Compilation of the relative mode splitting $\delta\nu_{m}/\nu_{m}$.
For a given azimuthal number $m$, each resonator is represented by a black
bar. The red bullets represent the calculation result obtained with $\Delta
r=1.3~{}\mu$m (see section IV).

Figure 5 compiles the relatives values of $\delta\nu_{m}/\nu_{m}$ measured on
more than fifteen resonators. We turn to manufacturers who manage all the
processing operations: from crystal growth to final polishing. Among the
tested resonators, some of them have been ordered more than ten years ago.
These resonators come from three distinct manufacturers located in three
different continents and using their own method of growth. Measurement results
obtained with a resonator of $50$ mm diameter and $30$ mm height and with
another operating on the WGE modes are also included in the data.

It appears obvious that the observed mode splitting distribution is not fully
random. The mode splitting is notably higher for modes with $m$ multiple of 3
($m=\text{mult}(3)$), regardless of the origin of the crystal, its dimensions
or the excited mode family. In the observable frequency range, a maximum
occurs for $m=6$ and then $\delta\nu_{m}$ decreases with $m$ following a curve
parallel to $1/Q_{L}$, $Q_{L}$ being the loaded Q-factor. This last
observation rather confirms the presence of a defect that affects the external
shape of the resonator. Modes of high order being more confined in the
dielectric are less affected by a defect on the resonator surface. For other
modes with $m\neq\text{mult}(3)$, the mode splitting is generally not observed
or stays of the order of the mode bandwidth. Figure 6 shows the resonator
transmission coefficient $|S_{21}|$ for the two modes WGH9,0,0 and WGH10,0,0
measured at $6~{}$K. The mode splitting $\delta\nu_{9}\approx 30~{}$ kHz is
here clearly resolved, whereas for $m=10$ the twin modes can be hardly
distinguished.

Figure 6: Modulus of the transmission coefficient $|S_{21}|$ for the WGH9,0,0
and WGH10,0,0 modes at $6~{}$K.

##  IV Deductive simplified model

It is unlikely that the same defects are found on all these resonators of very
different origin, unless these defects are related to an intrinsic property of
the sapphire crystal. Lu et al. showed for a photonics WG modes micro-cavity
that an intentionally modulated resonator diameter induces a controlled mode
splitting for a selected azimuthal mode number [30]. The fact that modes with
$m=\text{mult}(3)$ have the largest observed mode splitting suggests that
there may be a periodic cylindrical defect that would mostly interact with
this class of modes.

Let’s consider a resonator with a small geometrical defect as represented in
the figure 7.

Figure 7: Whispering Gallery Mode resonator with a small geometrical defect.

The resonance frequency shift resulting from this geometrical defect can be
evaluated with the perturbation method [31]. If the electric $\bf E$ and
magnetic $\bf H$ fields in the unperturbed resonator are known, the relative
frequency shift can be calculated by applying:

$\dfrac{\Delta\nu}{\nu_{0}}=\epsilon_{0}\dfrac{\iiint_{\delta
V}\bf{E}\rm\left([\epsilon]-1\right)\bf{E^{*}}\rm
dv}{\iiint_{V}\left(\mu_{0}\bf{H}\rm^{2}+\epsilon_{0}\bf{E}\rm[\epsilon]\bf{E^{*}}\rm\right)\rm
dv}$ (3)

where $\epsilon_{0}$ and $\mu_{0}$ are the vacuum permittivity and
permeability respectively, $[\epsilon]$ the medium permittivity tensor, $V$
the entire resonator volume and $\delta V$ is the volume affected by the
geometrical defect. $\delta V>0$ corresponds to a dielectric removal. We have
at our disposal a custom software based on the Mode Matching Method [29],
which enables to calculate the frequency and the electromagnetic field
components for any cylindrical WGM resonator.

The expression (3) can be simplified by considering that for a WGH mode, the
electric field is principally axial: $\bf{E}\rm\approx E_{z}(r,\phi,z)\bf z$.
Moreover, as the deformation is small with respect to the sapphire dimensions,
$J_{m}(kr)$ can be considered as a constant over $\delta V$. Taking in account
these approximations, it is can be shown from the expressions (2) and (3),
that for a given azimuthal number $m$, the twin modes will be differently
shifted and $\delta\nu_{m}$ is proportional to:

$I_{m}=\int_{0}^{2\pi}\cos(2m\phi)d\phi\int_{r(\phi)}^{r_{0}}rdr$ (4)

Let us consider a small modulation of the resonator radius such as:
$r(\phi)=r_{0}+\Delta r\cos(p\phi)$, with $p$ being an integer and $\Delta
r\ll r_{0}$. $I_{m}$ becomes:

$I_{m}=2r_{0}\Delta r\int_{0}^{2\pi}\cos(2m\phi)\cos(p\phi)d\phi$ (5)

$I_{m}$ takes a non null value only if $p=2m$. As a first consequence, $p$
should be an even integer to induce a mode-splitting. Thus the radius
modulation only affects the mode with $m=p/2$ , leaving the other WGH modes
unperturbed. To obtain a distribution like those of the figure 5, the radius
modulation should be less smooth. In other words, $r(\phi)$ should contain
harmonics of the initial modulation:

$r(\phi)=r_{0}+\Delta r\sum_{j\geq 1}a_{j}\cos(j\times p\phi)$ (6)

In that condition, $I_{m}$ becomes the pondered summation of terms
proportional to the Fourier coefficients $a_{j}$. The $j^{\text{th}}$ term
takes a non null value for a specific azimuthal number:

$m_{j}=j\dfrac{p}{2}$ (7)

Thus, for a given even integer $p$, the azimuthal numbers for which a mode
splitting appears follow an arithmetic progression with a reason $\Delta m$:

$\Delta m=m_{j}-m_{j-1}=\dfrac{p}{2}$ (8)

In our case, we have $m=\text{mult}(3)$, thus $\Delta m=3$ and:

$p=6$ (9)

From this simple observation, we deduce that the geometric defect must have a
periodicity of $\pi/3$ in the azimuthal direction to induce a mode-splitting
distribution compatible with the experimental observations.

The shape of the resonator is not yet known. We have now to determine the
coefficients of the development of $r(\phi)$. The non null terms in $I_{m}$ is
proportional to the Fourier coefficients $a_{j}$ such as $j=m/3$. The measured
$\delta\nu_{m}/\nu_{m}$ gives thus an estimation of the relative value of the
Fourier coefficients $a_{j}$. As previously mentioned, the mode splitting for
$m=3$ can not be experimentally determined. Thus, we assume for the doublet
WGH3,0,0 a fractional mode splitting $\delta\nu_{3}/\nu_{3}=3\times 10^{-5}$
corresponding to the typical mode half bandwidth. For the other $m$ values, we
take the average of the collected data. Setting $a_{1}=1$, we get for the
Fourier coefficients up to $j=6$:

TABLE I: Fourier coefficients of the expected radius deformation $r(\phi)$ $j$ | $m$ | $\delta\nu/\nu_{m}$ | Fourier Coef.
---|---|---|---
1 | 3 | $(3.0\times 10^{-5})$ | $a_{1}=1.000$
2 | 6 | $1.0\times 10^{-5}$ | $a_{2}=0.333$
3 | 9 | $4.5\times 10^{-6}$ | $a_{3}=0.150$
4 | 12 | $2.0\times 10^{-6}$ | $a_{4}=0.067$
5 | 15 | $1.0\times 10^{-6}$ | $a_{5}=0.034$
6 | 18 | $5.0\times 10^{-7}$ | $a_{6}=0.017$

The amplitude of the defect $\Delta r$ remains the only free parameter. We
computed for different values of $\Delta r$ and for each azimuthal number $m$,
the mode splitting using Equ. (3). Here, the previous approximations are not
used: we take into account of all electromagnetic fields components, their
variation with respect to the coordinates and the anisotropy of the dielectric
medium. In Fig. 5 the red points represent $\delta\nu_{m}/\nu_{m}$ calculated
with $\Delta r=1.3~{}\mu$m, which gives the best fit with the experimental
observations. The corresponding resonator shape is represented in Fig. 8 in
polar coordinates.

Figure 8: Resonator profile deduced for the measurement of
$\delta\nu_{m}/\nu_{m}$ assuming $\Delta r=1.3~{}\mu$m. The dashed line
represents the nominal resonator diameter.

The shape of the resonator deduced from our calculation is surprising at the
first glance. The calculated resonator profile looks like a flower with 6 well
marked petals. The peak-valley radius deviation is $\approx 3~{}\mu$m. We will
see in the next section that the real profile of the resonator is actually
similar to this one. We will also provide an explanation of this specific
shape.

##  V Actual resonator profile

The profile of two sapphire resonators from different manufacturers have been
measured using a metrological optical coordinate measuring machine $\mu$CMM
Bruker Alicona. With this instrument, the resonator contour was measured in
its equatorial plane with a resolution of $0.1~{}\mu$m. Fig. 9 shows the two
measured profiles compared to the modeled one (see Fig. 8).

Figure 9: Red plain line: Profile of two sapphire resonators measured with a
resolution of $0.1~{}\mu$m. Black plain line: Profile deduced from frequency
measurements. Dashed black line: Resonator nominal diameter.

It is remarkable that the calculated profile faithfully follows the real
contour of the resonator. This comparison fully validates the calculation
presented in the previous section. The radius of the two resonators is
modulated with a periodicity of $\pi/3$, and the peak-valley radius deviation
is about $3~{}\mu$m, and this same defect is found on all resonators whatever
their origin. It is therefore natural to think that this defect comes from an
intrinsic property of the sapphire crystal polishing.

##  VI Proposed explanation

In the following, a discussion on the intriguing contour of the sapphire
resonator is presented from a manufacturing and material perspective. All
$c$-oriented sapphire cylinder were prepared by chemical mechanical polishing
(CMP) composed of particles and chemical agent with rotation polishing machine
(whose exact production conditions may change from one manufacturer to
another). However, the contour presents systematically large curved surfaces
and concave engraving defects as shown in Fig. 9 that are schematically
redrawn in Fig. 10. Symmetry of order 6 indicates that one of the prismatic
planes $a~{}(11\overline{2}0)$ or $m~{}(10\overline{1}0)$ planes (the first
two densest planes perpendicular to the $c$-plane (1000)) is favoured over to
the other. Furthermore, looking carefully at Fig. 9, the curved surfaces show
a stepped shape indicating that it corresponds to a crystallographic plane
which tends to maintain its flat surface (low surface energy). This plane has
the slower CMP etching rate (noted $V_{slow}$) than any other prismatic plane
of sapphire. As for the second family of prismatic plane, located at $\pi/6$
to the first one, it presents a concave engraving. This family plane has the
fastest polishing rate (noted $V_{fast}$). The engraving phenomena can be
explained by the Rehbinder effect [32]. The polishing chemical agent with the
highest surface energy tends to accumulate the polishing particles to this
face under the polishing pressure. The particles create microcracks after
microcracks penetrating the crystal [33]. Figure 10 schematizes the polar
diagram of the crystal shape with two CMP etching rates. At this level of
discussion, the determination of faster and slower CMP etching rate surfaces
among the $a$\- or $m$-planes has not been determined by diffraction X-rays in
this study. [34]

Figure 10: Polar diagram representing the crystallographic planes, the etching
rates and the expected final shape (figure adapted from [33]).

The literature presents several conflicting results on the CMP polishing rate
of $a$\- and $m$-planes [35, 36], as well as crystal morphology, surface
energy, chemical dissolution and tribological/mechanical tests. Additionally,
CMP polishing rate may likely change behavior due to surface atomic
relaxation, crystal impurities, surfactant composition, water, wear rotation
speed, or temperature [33]. Meanwhile, the surface energy is generally
correlated to the friction but also to the hardness and etching rate [33]. As
a matter of fact, the two prismatic planes may have a very close CMP polishing
rate. Indeed, one can note that the final contour gives a very low crystal
roughness of $3~{}\mu$m for a crystal diameter of $54$ mm which may depend on
the polishing time. The relative tolerance of $5\times 10^{-6}$ in micro-
machining is in the state of the art for anisotropic crystal. Finally, another
argument in favor of the Rehbinder effect can be seen in figure 9. Almost all
stepped shape and engraving defects have an asymmetrical shape tilted in the
same direction, which is likely the direction of polishing rotation.

##  VII Summary

In this paper we have solved the problem of the origin of the mode splitting
effect in the cylindrical sapphire whispering gallery mode resonator. The same
mode splitting repartition as a function of the azimuthal mode order is
effectively observed in any sapphire resonator whatever their origin. Thus, we
deduced that the mode splitting occurs due to a radius 6 fold modulation of
the resonator contour resulting from the resonator polishing step. The
observed mode splittings enable to calculate an approximate resonator profile,
which is confirmed with high resolution resonator contour measurements.

##  VIII Acknowledgements

The authors would like to thank the team of the MIPHySTO (MIcrofabrication,
Hybridation de Systèmes Techniques et d’Outillage) platform who realized the
contour measurements and Rodolphe Boudot for its review.

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