Modelling the Performance of Single-Photon Counting Kinetic Inductance Detectors

Using conventional superconductor theory we discuss and validate a model that describes the energy-resolving performance of an aluminium LEKID to single-photon absorption events. While aluminium is not the optimum material for single-photon counting applications, this material is well understood and is used to understand the underlying device physics of these detectors. We also discuss data analysis techniques used to extract single-photon detections from noisy data.


Introduction
A principal goal of the next generation of space-based astronomy will be dedicated to the characterisation of extra-solar planets (exoplanets). Of the 1000s discovered only a handful have been characterised beyond their size and mass, thus there is increasing interest in new exoplanet missions aiming to carry out spectroscopy on very low intensity light and shallow transit light curves. Energy-resolving, single-photon counting detectors provide an elegant solution for carrying out spectroscopy without the need for gratings, prisms or combinations thereof. The lumped element kinetic inductance detector (LEKID) is a proven technology capable of counting and energy-resolving single-photon events at optical and near infra-red wavelengths [1,2]. We attempt to understand the underlying physics governing single-photon detection in LEKIDs.

Single-Photon Response Model
LEKIDs are thin-film, superconducting mirco-resonators. Photon absorption leads to the breaking of Cooper pairs to create quasiparticles (un-paired

Josie Dzifa Akua Parrianen • Andreas Papageorgiou • Simon Doyle •
Enzo Pascale electrons in the superconductor). The quasiparticle population determines the resonance frequency of the resonator, through its surface impedance. Changes to the surface impedance are governed by kinetic inductance.
The absorption of a photon of some energy ℎ creates a proportional number of excess quasiparticles. A 1 st order estimate is given by where = 0.4 and ∆ is the superconductor energy gap. Note that the typical value taken for = 0.57 [3] is actually only true for a bulk superconductor.
Recent studies suggests this value should be much smaller for thin films; as low as ~0.4 for thin-film Al [4], thus we take = 0.4. Any change in the quasiparticle population corresponds to a change in the detector's resonance frequency. Using conventional superconductivity theory we simulate the detector response as a function of change in quasiparticle population d to find the maximum resonance frequency shift, The expected response to single-photon absorption is a pulse with an exponential decay of the resonance frequency shift , The decay time is governed by the dominant time-constant of the detector. We assume this to be the quasiparticle lifetime , such that (3) Our current 1 st order model gives an expected pulse height of ~120 Hz.

Energy Resolution
The energy resolution of a photon detector can be given by The fundamental noise limit of a LEKID is governed by the generationrecombination (GR) noise, in which there is a continuous random fluctuation in the quasiparticle population at any given time: ± √ . Therefore, we take the √ term as our noise. When a photon is absorbed, an excess of quasiparticles is created following Eq. (1). We take this to be our signal, such that the fundamental energy resolution limit due to GR noise becomes

Modelling the Performance of Single-Photon Counting Kinetic Inductance Detectors
Note that this is the energy resolution integrated over the detector timeconstant: the quasiparticle lifetime. Whereas, the fundamental energy resolution for a detector capable of single-photon detection is governed by the Fano limit and given by where is the Fano factor [5]. For a LEKID, the Fano factor takes into account the variance in the number of quasiparticles generated due to photon absorption. It is generally accepted to take = 0.2.
It is reasonable to suggest, then, the ultimate energy resolution of a single-photon LEKID is a combination of the two fundamental limits. We therefore propose a maximum energy resolution: Calculating for our test device, which have meander volumes of ~1400 m 3 and measurements made at 100 mK, we get ∆ = 0.4 meV and thus ∆ = 21 meV. This suggests the detector is at the fundamental photon limit. However, it has been shown there is a limit to the responsivity of a LEKID at low temperatures [6]. The quasiparticle lifetime (and equivalently, the quasiparticle number) saturate at some temperature due to microwave heating [7]. We have measured this saturation in our device, shown in Fig. 2, showing a saturation temperature of approximately 190 mK. This adjusts our energy resolution to ∆ = 100 meV and thus from Eq. (7) we get ∆ = 103 meV. So we are now very firmly in the GR noise limit; which can be improved upon.

Experimental Set-up and Measurement
We used a LEKID array developed for part of the SpaceKIDs project [8], optimised as a 350 GHz narrowband Earth observation demonstrator.
Although not optimised for single-photon detection, it is a well-defined test device with good sensitivity. It is a 624 pixel array, formed of a 30 nm Al film (TC = 1.3 K) on a 320 m Si substrate. There is an additional TiAl bi-layer on the backside of the substrate which is used as a method to reduce cross talk

Josie Dzifa Akua Parrianen • Andreas Papageorgiou • Simon Doyle •
Enzo Pascale [9]. The pixel design follows standard LEKID architecture with the inductive meander patterned into a 3 rd order Hilbert fractal, shown in Fig. 1. The array was cooled down to a base temperature of 100 mK, in a miniature dilution refrigerator. A standard homodyne readout technique was used to measure the detector response. Measurements were made with a 1550 nm laser diode mounted on the 4 K stage. The incident photons are carried by a 9 m single-mode fibre optic cable up to the ultra-cold stage and are attenuated before entering the device holder. A modified plate, shown in Fig.  1, ensures roughly half of the pixels are directly illuminated. The device is flood illuminated at a constant DC power level. The change in resonant frequency is measured and compared to when the light source is off.

Results
The analysis of the data produced in this work has two main aims: i) to measure the impulse response decay time-constant τ and ii) to identify singlephoton absorption events and measuring the pulse amplitude of such events.
The impulse response time-constant was measured by illuminating the detector with square-wave pulses and fitting the pulse decay with an exponential. Stacking of several pulses was used to increase the signal-tonoise ratio (SNR). This procedure was particularly necessary for the significantly noisier data obtained at higher temperaturesin these cases, high accuracy time-alignment was achieved by cross-correlation. Above 200 mK the SNR became too low, at which point we use the traditional method: measuring the noise roll-off from the power spectral density (PSD) of the

Modelling the Performance of Single-Photon Counting Kinetic Inductance
Detectors detector response. Fig. 2 shows the combined detector time-constant fitting results, which we assume is dominated by the quasiparticle lifetime.
Another approach to measuring τ was in reconstructing the detector impulse response to single-photon absorption events. Potential absorption events were identified by a step-function match filter. Such events were stacked and averaged, as seen in Fig. 2, which has been fitted with an exponential of τ = 1.9 ms. This matches the same value found for square-wave measurements made at 100 mK. Having obtained an estimate of the impulse response, detector timestreams were Wiener filteredwith bandwidth ~2 kHzand potential photon absorption events were identified by match filter. The same procedure was applied to both illuminated and dark detector data; the extraction counts are shown in Fig. 3. The illuminated detector detections significantly outnumber the "false detections"limitations of the impulse extraction procedureof the dark detector, for impulses of amplitude >200 Hz. This surplus of detections is attributed to true photon absorption events. Given the current extraction method employed here, photon absorption events with amplitude <200 Hz are more difficult to distinguish between true and false detections. This uncertainty is depicted by the vertical error bars shown in Fig. 3.
The expected photon absorption distribution is then measured by subtracting the dark data detections (comprised of false detections only) from the detections of the illuminated data (false detections + photon absorption events), resulting in the plot seen in Fig. 3. Notice the photon absorption distribution peaks at an amplitude of ~120 Hz; matching our expected pulse

Josie Dzifa Akua Parrianen • Andreas Papageorgiou • Simon Doyle •
Enzo Pascale height. There may also be a peak at ~240 Hz with very small error bars, suggestive of 2-photon absorption events. As an initial estimate for the energy resolution of our measurements, we refer to Eq. (4); taking the full-width half-maximum (FWHM) as our noise and the peak value as our signal, such that ∆ = 660 meV. This is ~6.5 times larger than expected. However, the Weiner filtering process means we only integrate over ~4 ⁄ leading to a √4 increase in GR noise contributions: ∆ = 200 meV. This brings our estimated energy resolution to ~3 times larger than expected. Also note the FWHM is likely skewed by the apparent ~240 Hz peak and it is probable the distribution includes responses from photons absorbed in the capacitor; causing partial responses when only part of the quasiparticle diffusion occurs in the detecting element: the inductive meander. Therefore we consider this calculation to be an overestimate.

Conclusions and Future Work
We show confirmation of single-photon detection which corroborates our 1 st order model as well as the use of = 0.4 for thin-film Al. Our detection model will be developed further to include the response of photon absorption in nondetecting elements of the device. We will also continue this work using higher energy photonsto eliminate the false detectionsand develop optimised devices as a means of further investigating the mechanisms that enable energy-resolving detection in LEKIDs.

Modelling the Performance of Single-Photon Counting Kinetic Inductance
Detectors