illumination power and $C_s$ the heat capacity of the superconductor defined in the previous Sec. 5.2.1. $P_s$ is the power from the feedline dissipating into the MKID that can be expressed on resonance as [53, 86]:

$$P_s = P_r \frac{2Q^2}{Q_i Q_c} \qquad (5.8)$$

where $P_r$ is the MKID readout power. In the case of a phonon event, the superconductor is subject to a time-varying change of temperature $T_{ph}(t)$ such as:

$$T(t) = T_s + T_{ph}(t) \qquad (5.9)$$

where $T_s = (P_s/K + T_{bath}^n)^{1/n}$ is the superconductor temperature in the steady-state under no optical illumination. Subsequently, Eq. 5.7 can be rewritten to include the phonon perturbation as follows:

$$\frac{dT_{ph}}{dt} \approx -\frac{G}{C_s} \left[ 1 + \frac{(n-1)T_{ph}}{2T_s} \right] T_{ph} \qquad (5.10)$$

where $G = nKT_s^{n-1}$ is the thermal conductance. Finally, using the analytical solution of Eq. 5.10, we can model the inverse phase response $\Delta\phi$ as a function of phonon perturbation at $t > t_0$, when the normal Cooper pair breaking process is considered to be over:

$$\Delta\phi(T_s, t > t_0) \approx -\frac{4Q \partial f_0(t)}{f_0 \partial T} \frac{T_{ph}(t_0) \exp\left(-\frac{t-t_0}{\tau_{th}}\right)}{1 + \frac{(n-1)T_{ph}(t_0)}{2T_s} \left[1 - \exp\left(-\frac{t-t_0}{\tau_{th}}\right)\right]} \qquad (5.11)$$

where $\tau_{th} = C_s/G$ is the inverse pulse response characteristic time related to the thermal flow between the superconductor and substrate. This model has been successfully fitted to the inverse pulse evolution in Fig. 5.9a as a function of temperature and in Fig. 5.9b as a function of input power. The characteristic phonon time represented in the inset of Fig. 5.9a clearly shows the inverse pulse dependence to TLS strength as $\tau_{th}$ drops down by a factor of 3 between 100 and 150 mK and it becomes negligible at 200 mK. We can draw the same conclusion from the inset of Fig. 5.9b where $\tau_{th}$ drops down between -105 and -100 dBm of readout power. There is an abrupt temperature and power transition between
the TLS saturated and non-saturated states, which was not as obvious for the larger IDC
MKID size we measured where $\tau_{th}$ seemed to follow a more linear behavior [87].
<figure><img src="image_76.png" /><figcaption>Figure 5.9: Time-domain evolution of the pulse response on the reference sample following a 200 ns-long illumination with the 405 nm diode. The dashed lines represent the fitting from the model proposed in Eq. 5.11 and the insets show the fitted characteristic time $\tau_{\text{th}}$. Unlike for the normal quasiparticle phase response that is not represented here, the inverse pulse decay is TLS-dependant and varies as a function of (a) bath temperature or (b) readout power.</figcaption></figure>
## 5.3 Amplification of the inverse pulse response for a possible phonon-based detection

In this section, we will investigate whether the source of TLS regarded as noise for MKIDs can be amplified to reach single-photon sensitivity. It has already been demonstrated that thermometers based on MKIDs have had increased responsitivity by boosting TLS [88]. In our case, the detector would not probe the kinetic inductance change in the meander but rather the capacitance change in the IDC caused by TLS-phonon interactions. We will name this technology MTAD for *Microwave TLS Amplified Detector*. In contrast to MKIDs that need low-$T_c$ materials to count photons, a MTAD would be made from high-$T_c$ materials to be cooled down well below its TLS turning point. We will conduct this study with stoichiometric TiN resonators as we have already demonstrated its large phonon response at $T < T_c/20$. We propose to add the same 60 nm-thick $Al_2O_3$ dielectric layer used for the optical stack but this time deposited below the IDC. We expect to increase the TLS density in the vicinity of the capacitor responsible for the large inverse phase response. The dielectric layer surface is increased by a tenfold compared to the optical stack design as the IDC is much larger than the inductor which should also lead to an increase in phonon response.

### 5.3.1 In-dark measurements

Similarly to the optical stack study, we will first measure the different in-dark parameters before illumination. We have chosen to compare this new geometry (*Dielectric Below IDC*, DBI) to the same NDNR and YDNR samples presented in the optical stack chapter, starting with $Q_i$ as a function of temperature in Fig. 5.10.
<figure><img src="image_77.png" /><figcaption>Figure 5.10: Comparison of the internal quality factor of one resonator in each sample with similar $f_0$ as a function of bath temperature measured at -100 dBm.</figcaption></figure>

Surprisingly, the DBI sample has a similar $Q_i$ to the NDNR reference at low temperatures where TLS should be the most abundant and is confirmed in Fig. 5.11. However, we have to keep in mind that in DBI, there is no dielectric below the inductor where the Cooper pair density is the highest. We suspect that the dielectric in YDNR thermally isolates the inductor from the substrate and cannot reach temperatures low enough to reduce its quasiparticle population. In DBI and NDNR, the inductor is deposited directly on the substrate and may explain why we obtain this high level of $Q_i$.

<figure><img src="image_78.png" /><figcaption>Figure 5.11: Comparison of $\Delta f/f$ as a function of bath temperature measured at -100 dBm.</figcaption></figure>
The $\Delta f/f$ of the DBI sample is similar to YDYR in Fig. 4.27 as the resonance first shifts to higher frequencies due to TLS. However, the TLS turning point in DBI has doubled compared to YDYR and is now around 500 mK. This is consistent with the reported evolution of the fractional frequency shift as a function of the dielectric volume in contact with the resonator [78].

<figure><img src="image_79.png" /><figcaption>Figure 5.12: Normalized frequential noise of one pixel in each of the three samples mea-
sured at -100 dBm input power at (a) 50 mK where TLS is dominant and (b) at 400 mK
where TLS is saturated.</figcaption></figure>

In Fig. 5.12a, all three samples share the same $f^{-1/2}$ slope between $10^2$ and $10^4$ and the frequency noise in DBI caused by TLS is naturally higher than in YDNR where the dielectric volume is much smaller. In Fig. 5.12b, the slope of the DBI sample in the same spectral region has completely flattened out and has lowered to the same level as YDNR. This could be explained by the larger $Q_i$ of DBI at 40 mK compared to YDNR seen in Fig. 5.10 which is inversely proportional to the dielectric loss tangent in Eq. 2.42. Moreover, in YDNR, the dielectric is in contact with the inductor where the denser Cooper pair population responsible for detection directly interacts with TLS that may not be completely saturated at 400 mK.
### 5.3.2 Optical illumination response

In this section, we will present the detector's response to short light pulses with the setup detailed in Sec. 3.8. We illuminate the sample using the 405 nm laser diode with the same power of 4 pW as for previous samples. The largest inverse phase response at 50 mK and -100 dBm shown in Fig. 5.13 was measured on the resonator with the lowest $f_0 \approx 2.2$ GHz with a $Q_i = 3.5 \times 10^5$ and a total $Q = 1.3 \times 10^5$.

<figure><img src="image_80.png" /><figcaption>Figure 5.13: Averaged pulse phase response with different illumination width at 50 mK and -100 dBm. The inverse pulse response is much larger than for the reference sample in Fig. 5.1b</figcaption></figure>

The inverse pulse response at 50 mK has drastically increased compared to the reference sample without dielectric shown in Fig. 5.1b. For a 50 ns-long pulse, the phase change is around 80 degrees, a tenfold of what was recorded for the reference sample. However, if we consider the detector SNR, the frequential noise in the resonator bandwidth represented in Fig. 5.12a has also increased by a tenfold. For longer illumination pulse width, the phase seems to enter a non-linear response that may be caused by the resonator response time $\tau_{rsp} = 20$ µs computed from Eq. 3.4 not fast enough to register this large of a phase shift. The single-photon mode on this sample has not been reached but we have nonetheless demonstrated the amplification of the inverse pulse by cleverly placing a TLS-loaded material below the capacitor, where the fractional dielectric change has the most influence on the resonator.
Conclusion and outlook

As an effort to improve optical coupling in MKIDs, we have proposed a novel optical stack design below the absorber. We have conducted several optical characterization and modelization on our thin-film layers to optimize the coupling: an Al reflector is set at a quarter-wavelength of the absorber with the use of a transparent low dielectric tangent loss Al₂O₃ layer. In the first fabrication and measurement run, it was observed that high quality factors were not achievable with this geometry. The low-Tc Al layer has been suspected to have a large quasiparticle density at the 400 mK operating temperature which would increase loss. This was resolved in the second batch by placing a higher Tc Nb "booster" layer below Al so that it could share its superconducting properties through the proximity effect. While we obtained high-Q factors with this new optical stack sample, we have not reached single-photon counting due to a tenfold of frequential noise compared to the reference. This larger noise is probably distributed between the optical stack parasitic capacitance, the TLS inside the Al₂O₃ layer as well as the number of interface where nanometer-thick TLS-hosting layers prone to form has tripled. In order to counteract this larger noise, we can either reduce the optical stack noise or use a more sensitive material as the absorber. For the former, we are considering replacing the quarter-wavelength spacer Al₂O₃ layer with vaccum. Our group has already demonstrated in the past the feasability to suspend TiN using a sacrificial layer to fabricate vacuum MIM capacitors that allowed for high quality factors millimeter MKIDs [64]. The same principle could be reproduced for the optical stack so that the parasitic capacitance would be completely nullified. The amorphous layers forming at the surface of the reflector and absorber would not be avoided but it would help us determine their contribution to the total noise increase caused by the optical stack. For the latter, we have fabricated and characterized MKIDs with the more sensitive sub-stoichiometric TiNₓ. The results
were not discussed here as we obtained in both the reference and optical stack samples
mediocre quality factors that led to shallow resonances on which single-photon mode was
not possible. We suspect a mismatch between the TiNₓ and sapphire substrates supported
by other groups results [12]. Another solution would be to use multi-layer Ti/TiN: the
succession of pure Ti and stoichiometric TiN layers can act by proximity effect as a single
layer with a controllable T_c as a function of each layer's thickness. Once we will have
fully characterized multi-layer Ti/TiN, we plan to recreate the optical stack study with
a T_c ≈ 1 K to hopefully quantify the optical coupling efficiency. Moreover, both Ti
and TiN have better reproducibility and uniformity than TiNₓ which would favor multi-
layer Ti/TiN for kilopixels arrays such as the future SPIAKID instrument. Otherwise,
if we were to set aside Titanium, the most promising up-and-coming material would
be Hafnium for its high resistivity, high quality factors and uniformity largely superior
than for TiNₓ [62]. However, its very-low T_c ≈ 400 mK requires a high-end ADR to
operate.

In parallel, we have been the first to demonstrate with my reference sample the possibility
to use stoichiometric TiN for photon-counting at temperatures up to 500 mK. While the
energy resolution is still relatively low in this configuration compared to the theoretical
limit, it opens up new possibilities for the use of higher-T_c materials to count optical
photons without the need of a costly ADR to operate at 100 mK and below.

During this thesis work, a new anomalous response from stoichiometric TiN has been
observed on some of our samples in the form of a large transient inverse pulse. We have
attributed this phenomenon to phonon perturbations that excites the non-saturated TLS
in amorphous layers as well as the substrate bulk at temperatures below T_c/20 ≈ 300 mK.
This leads to a change of the substrate dielectric constant which directly affects the MKIDs
IDC and thus the resonance frequency. Since phonons are generated by the recombina-
tion of quasiparticles, we have confirmed their origin by covering the MKIDs absorber
with an opaque layer to prevent Cooper pair breaking. We have naturally observed a
large diminution in Cooper pair breaking during illumination followed by a proportional
diminution of the inverse phase response. This has confirmed that phonons are indeed
generated in the absorber after a photon event and not from the substrate direct heating.
With that in mind, we have deliberately added a TLS-hosting layer below the IDC to try
and amplify the phonon response. The detection would not be recorded by the change
in kinetic inductance inside the absorber like for normal MKIDs but rather through the
change of the substrate dielectric constant in what we called a MTAD. Such a technology
could be useful in other astronomy applications such as dark matter experiments where
phonons are probed [89]. We have successfully demonstrated the amplification of the
inverse response while also increasing the resonator frequential noise which has prevented
the recording of a photon event. We will further investigate this detection mechanism
by using more sensitive absorbing materials to increase the TLS-phonon interaction and
hopefully reach the single-photon response.
# Appendix A

## Acid cleaning parameters

<table><caption>Table A.1: Wet cleaning procedure</caption><thead><tr><td>Solution</td><td>Volume ratio</td><td>Temp</td><td>Time</td><td>Target</td></tr></thead><tbody><tr><td>Piranha</td><td>H<sub>2</sub>SO<sub>4</sub> (3) : H<sub>2</sub>O<sub>2</sub> (1)</td><td>95°C</td><td>600 s</td><td>Organic contaminants</td></tr><tr><td>SC1</td><td>NH<sub>4</sub>OH (1) : H<sub>2</sub>O<sub>2</sub> (1) : H<sub>2</sub>O (5)</td><td>70°C</td><td>600 s</td><td>Particles</td></tr><tr><td>Dilute HF</td><td>HF (1) : H<sub>2</sub>O (10)</td><td>20°C</td><td>30 s</td><td>Oxides and H-Passivation</td></tr></tbody></table>
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# RÉSUMÉ

Les détecteurs à inductance cinétique (MKIDs) sont des résonateurs LC à haut facteur de qualité faits à partir de couches minces supraconductrices. Ils peuvent simultanément déterminer le temps d'arrivée de chaque photon en mesurant leur énergie sans optique supplémentaire. Grâce au pouvoir de multiplexage des MKIDs qui permet de facilement réaliser des matrices de plusieurs kilopixels, cette technologie est au centre de l'attention pour les nouvelles applications en astronomie. Parmi eux se trouve l'instrument SpectroPhotometric Imaging for Astronomy with Kinetic Inductance Detectors (SPIAKID), projet du GEPI à l'Observatoire de Paris. Nous utilisons la technologie MKID pour construire un spectrophotomètre composé de 20 000 pixels qui sera déployé sur le New Technology Telescope (NTT) de 3.6m au Chili à l'horizon 2025. SPIAKID s'intéressera à la population stellaire des galaxies naines ultra faibles du Groupe Local pour mieux comprendre la formation des galaxies ainsi que la matière noire. Nous présenterons dans ce travail de thèse un design novateur pour améliorer le couplage optique entre les photons incidents et la partie sensible du détecteur ainsi qu'une réponse anormale de notre supraconducteur qui pourrait faire l'objet d'un nouveau moyen de détection.

# MOTS CLÉS

détecteur – supraconductivité – inductance cinétique – optique et infrarouge proche – couplage optique – interactions des phonons

# ABSTRACT

Microwave Kinetic Inductance Detectors (MKIDs) are superconductive thin films LC resonators with high quality factors. They can simultaneously record single-photon events and measure their energy without any added optics. Combined with the ease of multiplexing thousands of pixels into a large array, MKIDs are now at the heart of current and upcoming ground-based astronomy applications. Among them is the SpectroPhotometric Imaging for Astronomy with Kinetic Inductance Detectors (SPIAKID) project at Paris Observatory. We are using MKID technology to build a 20,000 pixels spectrophotometer that will be deployed in 2024 on the 3.58m New Technology Telescope (NTT) in Chile. SPIAKID aims to study the population and metallicity of stars in Ultra Faint Dwarf (UFD) galaxies in the Local group to have a better understanding of galaxy formation and evolution. We will present an original design intended to improve the optical coupling between incident photons and the absorber part of the detector as well as an anomalous response from our superconductor that could offer a new mean of detection.

# KEYWORDS

detector – superconductivity – kinetic inductance – optical and near-infrared – optical coupling – phonon interactions