Josephson photodetectors via temperature-to-phase conversion

# Josephson photodetectors via temperature-to-phase conversion

## Abstract

We theoretically investigate the temperature-to-phase conversion (TPC) process occurring in dc superconducting quantum interferometers based on superconductor–normal metal–superconductor (SNS) mesoscopic Josephson junctions. In particular, we predict the temperature-driven rearrangement of the phase gradients in the interferometer under the fixed constraints of fluxoid quantization and supercurrent conservation. This allows sizeable phase variations across the junctions for suitable structure parameters and temperatures. We show that the TPC can be a basis for sensitive single-photon sensors or bolometers. We propose a radiation detector realizable with conventional materials and state-of-the-art nanofabrication techniques. Integrated with a superconducting quantum interference proximity transistor (SQUIPT) as a readout set-up, an aluminum (Al)-based TPC calorimeter can provide a large signal to noise (S/N) ratio in the 10 GHz10 THz frequency range, and a resolving power larger than below 50 mK for THz photons. In the bolometric operation, electrical NEP of  W is predicted at 50 mK. This device can be attractive as a cryogenic single-photon sensor operating in the giga- and terahertz regime, with applications in dark matter searches.

## I Introduction

Detecting radiation in the microwave to terahertz regime has a range of applications from astrophysics to quantum devices. For applications where cooling to low temperatures is available, superconducting sensors provide a way to construct high-sensitivity detectors, both for bolometric detection of continuous power, [1; 2; 3] and also towards detection of few photons. [4; 5; 6; 7]

Thermal superconducting radiation detectors [8; 1; 2] generally make use of the sensitive dependence of superconducting electrical response on the local electronic temperature, which changes as photons are absorbed. The specific way this is utilized varies in different detectors. Kinetic inductance [9; 1; 10] and Josephson superconductor–normal–superconductor (SNS) [11; 7; 12; 13] based detectors make use of the temperature dependence of the supercurrent. The changes in the current response can be detected e.g. via resonant circuits [9] or inductively or directly [14; 1] coupled SQUID sensors. In the direct coupling scheme, variation of the critical current or inductance in the detector embedded as one junction of a dc SQUID affects the distribution of phase differences across the second junction and the current balance, which can be detected. Similarly, if the two junctions reside in a closed superconducting ring, the result is temperature to phase difference conversion (TPC), which is the basis of the device discussed below.

The phase difference across a superconducting junction can be generally determined via the current flowing through it. However, in SNS, it is also possible to determine the value via tunnel junction spectroscopy — this forms the basis of the superconducting quantum interference proximity transistor (SQUIPT). [15] In a SQUIPT, the phase difference is measured by observing the tunneling current through a tunnel junction, [2] connected to the middle of the weak link. Such measurements can also be made at high bandwidth, [16; 17] enabling fast measurement timescales that are required for calorimetry. SNS junctions also have an advantage for bolometry in that the superconductivity inhibits electronic heat conduction out of the detector region, [18] improving sensitivity.

In this work, we propose a mesoscopic superconducting bolometer/calorimeter for single-photon and continuous power detection in the GHz–THz frequency range. The operation is based on temperature-dependent kinetic inductance change of a superconducting weak link, resulting to temperature–to–phase conversion (TPC) in a superconducting ring, and its nonlocal readout via an integrated SQUIPT sensor (see Fig. 1). We predict temperature sensitivities of tens of in a temperature range tunable over with the choice of the magnetic flux. We provide an appropriate simplified theoretical model for the operation and thermal response, and analyze the main performance characteristics in the bolometric and calorimetric modes. In calorimetric operation, we predict signal-to-noise ratios up to in the range, and a resolving power larger than for single photons at detector bath temperatures of . In the bolometric mode, noise equivalent power (NEP) down to at is predicted, and mainly limited by the thermal fluctuation noise in the detector.

The structure of the manuscript is as follows. In Sec. II we outline the operation principle and the theoretical background for the temperature–to–phase conversion and its SQUIPT detection. In Sec. III we discuss the thermal model for the system, and the main performance characteristics of the calorimetric operation mode. Section IV discusses the performance characteristics in the bolometric mode, and Sec. V concludes with discussion.

## Ii Detector

### ii.1 Operating principle

We consider a dc superconducting quantum interference device (SQUID) based on two superconductor-normal metal-superconductor (SNS) Josephson junctions [see Fig. 1(a)]. Each junction consists of two superconducting leads coupled to a N wire of length through highly transmissive interfaces. The contact with S induces superconducting correlations in the N regions through proximity effect which is responsible for the supercurrent flow through the junctions as well as for the modification of the wires densities of states (DOSs). The transverse dimensions of the wires are assumed to be much smaller than so that they can be considered as quasi-one-dimensional.

Both SNS junctions are assumed to be short, i.e., they satisfy the condition , where is the S energy gap, and is the diffusion constant of N. The above choice is dictated by three main reasons: (i) Drastic reduction of the detector sensing element volume in order to achieve a substantial improvement of the sensor performance. (ii) The analytical solutions for both the supercurrent and the quasiparticle density of states (DOSs) in the weak-links are well known in the short-junction limit, which somewhat simplifies the whole analysis of the detector. (iii) For short junctions, the phase-dependent response of the DOSs in the read-out junction is maximized leading to enhanced readout transduction sensitivity.

The variables and denote the macroscopic quantum phase differences across the detector and readout junction, respectively. By neglecting the ring kinetic inductance it follows from fluxoid quantization that

 φd+φr=2πΦext/Φ0+2kπ, (1)

where is the applied magnetic flux, is an integer, and Wb is the flux quantum.

We suppose the S loop to be backed at a distance from the NS interfaces of the detector junction by Andreev mirrors, [18] superconductors S with a large energy gap in good electric contact, so that S blocks out-diffusion of the heat absorbed by the N wire and the S electrodes [see Fig. 1(b)], as quasiparticle energy cannot escape through the S/S interfaces due to Andreev reflections. The above design for the detector region allows thermal relaxation to occur predominantly via the slower electron-phonon mechanism in the N and S regions. In order to retain the larger kinetic inductance of the short junction link determined by instead of , we choose below the length where is the coherence length of the weaker superconductor.

The external incident radiation is coupled to the N region via a suitable antenna. If it is coupled as in Fig. 1, the absorbed power is distributed both to the detector and the readout junctions, and divided evenly if the junctions are identical. Different designs may also be able to concentrate the power dissipation mainly in the detector junction, between the Andreev mirrors. Here, we neglect the impedance of the ring itself, as it is small in the GHz–THz range compared to the junctions.

The antenna impedance matching is influenced by the impedance of the short detector junction, which is qualitatively similar to that of superconductors. [19; 20] At low temperatures and in the short-junction limit, the dissipative component has a frequency threshold determined by the DOS gap in the junction: . At frequencies above the gap, the inductive component is of order . As a consequence, we expect that good radiation coupling can be achieved at frequencies by matching the normal-state resistance of the junction to the antenna. At frequencies below the DOS gap, the ability of the proximized normal wire to absorb energy in linear response becomes suppressed at low  — which is a generic limitation of superconducting absorber elements. For large enough excitation amplitude — ie. phase oscillation induced by incoming radiation pulse being where is the voltage amplitude across the junction — multiphoton events can contribute to the absorption also at lower frequencies.

It is beneficial for the performance of the device if the readout junction remains at a low temperature compared to the detector junction. This condition is enforced by the presence of the superconducting tunnel probe and the lack of Andreev mirrors in the readout junction, resulting to electronic heat out-diffusion that is larger by a factor of . Depending on the parameters, it is possible to supplement this by an additional tunnel-coupled N cooling fin [21] in the readout junction, but as we discuss in Sec. IV.1, this is likely not necessary for the parameters we consider.

### ii.2 Model

In the short-junction limit, the Josephson current () flowing through the detector and readout weak-links at temperature can be written as [22; 23]

 Id,rc(T,φd,r)=πΔ(T)eRd,rNcos(φd,r2)∫Δ(T)Δ(T)cos(φd,r/2)dε ×1√ε2−Δ2(T)cos2(φd,r2)tanh(ε2kBT), (2)

where is the BCS temperature-dependent pairing potential in S, is the normal-state resistance of detector (readout) junction, is the Boltzmann constant, and is the electron charge. In the limit of zero temperature (), Eq. (2) reduces to [22]

 Id,rc(0,φd,r)=πΔ(0)eRd,rNcos(φd,r2)artanh[sin(φd,r2)]. (3)

In Eq. (2), , is the wire resistivity, the density of states at the Fermi level in N, and is the wire cross section of the detector (readout) Josephson junction. Moreover, in the following the electron temperature in the detector junction will be denoted with () whereas that in the read-out weak-link is supposed to coincide with the lattice temperature, (see discussion in Sec. IV.1).

The electromagnetic energy absorbed in the detector junction elevates the temperature in N and in the lateral portion of S thereby leading to a decrease of the dissipationless supercurrent circulating (for ) in the superconducting loop. The temperature dependence of the junctions critical current is shown in panel (a) of Fig. 2, being evaluated from Eq. (2) for by setting . Note that the critical supercurrent is almost saturated for ( is the critical temperature of S), and decreases linearly with temperature around .

Such a temperature-induced suppression of yields a finite variation of the phase drop across both SNS junctions owing to the following reasons: i) Conservation of the supercurrent circulating along the S loop, and ii) fluxoid quantization in the interferometer [Eq. (1)]. As a consequence, for a given , the phases and can be determined for any and from condition i), i.e., by solving the equation

 Idc(Te,φd)=Irc(Tbath,φr). (4)

This is at the origin of the temperature-to-phase conversion (TPC) process. By defining the parameter as the degree of asymmetry of the SQUID junctions, we can solve Eq. (4) for the phase existing across the readout junction in order to investigate in detail the full TPC process. We note that the asymmetry parameter can also be written as when both junctions are at the same temperature, where is the kinetic inductance of the detector (readout) junction, so that it immediately expresses which junction of the interferometer will mainly determine the phase biasing of the superconducting loop for a given external flux . In particular, for the phase drop along the ring will occur predominantly across the detector junction whereas in the opposite situation () the phase drop will occur mainly across the readout weak-link.

Top panels of Fig. 3 show the phase calculated from Eq. (4) as a function of and for [(a)] and [(b)] at . It turns out that the largest TPC effect occurs by setting around where the phase response can be very sharp [see panel (b)]. In particular, by choosing values for around or slightly smaller than yields sizable phase modulation amplitudes in the readout junction at higher temperatures . By contrast, for , the modulation amplitude occurs for smaller temperatures, although slightly reduced.

The parameter also controls the phase difference over the detector junction and therefore the energy gap . As discussed above, this provides a frequency threshold below which it can be difficult to impedance match the detector junction to an antenna. To reduce , a larger should be chosen — which however reduces the sensitivity of TPC. Below, we choose this tradeoff at (dashed lines in Fig. 3), corresponding to . If a higher frequency threshold is acceptable, better performance characteristics can be achieved by values closer to .

The temperature-induced suppression of the critical current may thereby yield a variation of phase drop () across the readout junction. By measuring with a suitable setup would enable to assess with accuracy the electronic temperature , and hence the radiation absorbed by the SNS weak-link of the detector. Yet, owing to proximity effect, affect as well the spectral characteristics of the corresponding N region, for instance, by determining the exact shape of the local quasiparticle density of states (DOSs) in the N wire. In the following we show that by probing the phase-induced variations of the DOSs with a superconducting quantum interference proximity transistor (SQUIPT) implemented in the readout junction is a simple and effective way to get direct and detailed information about -driven phase changes.

The SQUIPT [shown on the upper part of the scheme of Fig. 1(a)] consists of a superconducting tunnel junction with normal-state resistance coupled to the middle of the N wire of the readout junction [15]. So far, SQUIPTs have been implemented in a few geometrical configurations [15; 24; 25] and with different materials combinations [26; 15; 27]. For the sake of clarity, we assume here the probing electrode (of width ) to be made of the same superconducting material S as the SQUID ring. changes induced by temperature variations in the detector junction affect the readout N wire DOSs, and thereby the current vs voltage characteristic of the superconducting tunnel junction biased at voltage [see Fig. 1(a)].

Let us now analyze the N wires DOSs () in the SNS junctions. In the short-junction limit, which is the relevant one for the present case, we have the well-know result (cf. e.g. Refs. 23; 28):

 Nd,rN(x,φd,r,ε,T)=Re ⎷(ε+iΓ)2(ε+iΓ)2−Δ2(T)cos2φd,r2×cosh(2x−LLarcosh√(ε+iΓ)2−Δ2(T)cos2φd,r2(ε+iΓ)2−Δ2(T)), (5)

where is the energy relative to the chemical potential of the superconductors, is the spatial coordinate along the N wires, and is the Dynes parameter, accounting for broadening in S.

exhibits a minigap for whose amplitude depends on , and is spatially constant along the N wires. In particular, for and decreases by increasing the value of the phase, vanishing at . Therefore, the quasiparticle spectrum in the N region can vary from that of a gapped superconducting material (for ) to that of a gapless normal conductor (at ) just by changing the phase across the weak-link.

The impact of the electronic temperature of the detector on the DOSs in the readout N weak-link is displayed in the two bottom panels of Fig. 3 where the minigap amplitude is plotted as a function of and for [(c)] and [(d)], both calculated at . These results show that the minigap behavior is qualitatively similar to that of , and confirm that for it is possible to obtain sufficient -induced modulation of . The above value for will be set in all forthcoming calculations, unless differently stated, to evaluate the response and performance of the nanodetector.

### ii.4 SQUIPT response

Let us now turn on discussing the behavior of the SQUIPT current vs voltage characteristics which allow to understand how to exactly operate the superconducting interferometer for radiation detection in the present setup. The current flowing through the superconducting tunnel probe of the SQUIPT in the readout junction is dominated by quasiparticles, and can be written as [29]

 Ip(V)=1ewRp∫L+w2L−w2dx∫∞−∞dεNrN(x,ε,φr)NpS(~ε)F(ε,~ε), (6)

where is the BCS normalized DOS of the S probe electrode at temperature , , , is the Fermi-Dirac energy distribution function, and is the normal-state tunneling resistance of the probing junction. In the following calculations we assume for simplicity that the superconductor forming the probing electrode is identical to that realizing the loop so that . Moreover, we set and as characteristic parameters of the SQUIPT readout. For calorimetric operation, we also assume measurement of the current is possible on bandwidths comparable to the relevant inverse thermal relaxation time of the detector junction (see below) — possible fast readout schemes are discussed in Refs. 16; 17.

Figure 4(a) shows the low-temperature SQUIPT current vs voltage characteristics calculated at and in the absence of radiation (i.e., , ), and for a nonzero radiation (, ) heating the detector junction. In particular, the onset of large quasiparticle tunneling occurs at an energy corresponding to the sum of the gaps in the superconducting probe and the N proximity layer, as expected for a tunneling process through an SIS’ tunnel junction [30]. Therefore, in the absence of radiation this onset appears at whereas under the effect of radiation of energy the onset occurs at . Moreover, since as a consequence of heating in the detector junction originating from absorption of radiation [see Figs. 3(c,d)]. From this it follows that, by biasing the SQUIPT at voltage (with ), the absorption of a photon will yield a reduction () of the current flowing through the tunneling probe. As a consequence, a direct radiation readout can be performed with the SQUIPT by a simple measurement of its current at fixed bias voltage.

The impact of asymmetry of the two SNS Josephson junctions forming the SQUID on the SQUIPT characteristics is displayed in Fig. 4(b) which shows vs calculated for selected values of at , and . The figure shows that by reducing leads to a lowering of , as expected from the reduction of in the readout junction [see Figs. 3(c,d)]. Therefore, a precise tuning of the SQUIPT working voltage can, in principle, be achieved by setting the asymmetry of the junctions forming the SQUID.

### ii.5 Temperature-to-current conversion and noise analysis

Hereafter, we describe the behavior of the SQUIPT corresponding to a temperature-to-current transducer, i.e., the ability of the system to convert a temperature variation in the detector weak link into a current change in the readout tunnel junction.

Figure 5(a) shows the dependence of the current through the probing junction on the temperature for different values of by keeping fixed the external flux , for and . In particular, the current turns out to be strongly dependent on for specific values of the biasing voltage, in particular, we note that for and , can vary significantly for temperatures in the range . The above values stem from the chosen parameters of the structure, and will lead to a high sensitivity for radiation detection. By contrast, for other bias voltage values the current response is somewhat moderate in the whole range of temperatures.

A figure of merit which is useful to characterize the readout weak-link performance is represented by the temperature-to-current transfer function, , which is shown in Fig. 5(b) for the same parameters as in panel (a). For and obtains the largest values in the range , which is expected from the corresponding behavior of the current in the same temperature window.

Figure 5(c) displays the dependence of on for different values of the applied magnetic flux by keeping fixed the bias voltage , for and . Approaching the flux leads to a stronger response of tunneling current through the readout junction. On the other side, values of far away from correspond in general to a weaker dependence of on the temperature. The corresponding transfer functions are shown in Fig. 5(d), and are calculated for the same parameters as in panel (c).

The intrinsic temperature sensitivity (temperature noise) per unit bandwidth of the probe junction () is related to its current-noise spectral density () as

 sT=√SI|τI|, (7)

where is the temperature-to-current transfer function discussed before. The low-frequency current noise spectral density of the tunnel probe is [31]

 SI(V)=2eIp(V)coth(eV2kBTbath). (8)

We note that Eq. (8) describes both regimes of shot noise, i.e., , and thermal noise, i.e., , and holds in the tunneling limit.

In Fig. 6 (a) we show the temperature noise vs calculated for selected values of at . The maximum temperature sensitivity is obtained for where the temperature noise can be of order . Figure 6 (b) displays the temperature noise vs calculated for a few values of the external flux at . In particular, is minimized at different temperatures depending on the specific value of , and obtains values as low as a few tens of , at temperature range tunable over with the choice of .

## Iii Nanocalorimeter operation

### iii.1 Thermal model

The operation principle of single-photon detection (i.e., operation as a calorimeter) based on the TPC effect can be easily understood by inspecting the scheme displayed in Fig. 2(b) which shows a sketch of time evolution of the electronic temperature in the detector weak-link after the arrival of a photonic event. In particular, we assume that depending on the photon energy , is increased with respect to up to uniformly along the wire over a time scale set by the diffusion time, (see red full line). The latter, with the typical parameters chosen for our SNS junctions, is of the order of s. The electronic temperature then relaxes towards bath temperature over a time scale set by the electron-phonon relaxation time, . For instance, in the normal state, this time is given by where is the electron-phonon coupling constant. By setting for instance WmK typical of silver (Ag) we obtain s for K so that [see dashed line in Fig. 1(c)]. As we shall argue, in our system thanks to superconducting correlations induced in the N regions, can be longer than that in the normal state.

We describe now in detail the thermal behavior of the interferometer as a sensitive nanocalorimeter. After the absorption of a photon of energy at time , the electron temperature in the detector junction can be determined by solving the heat equation

 Cetot(φd,Te)∂Te∂t=hνδ(t), (9)

where is the Dirac function, is the total electronic heat capacity of detector junction comprising both the electronic heat capacity of the N wire and that of the lateral portions of the S electrodes between the Andreev mirrors. In particular, is the electronic heat capacity of the N wire (lateral S electrodes) with volume , and is the electronic entropy in N (S). Specifically we have [32]

 SeN(φd,Te)=−2νFkBL∫L0dx∫∞−∞dεNdN(x,φd,ε,Te) ×f0(ε,Te)ln[f0(ε,Te)], (10)

and

 SeS(Te)=−2νSFkB∫∞−∞dεNdS(ε,Te)f0(ε,Te)%ln[f0(ε,Te)], (11)

where is the normalized BCS DOSs of the S electrodes, and is the DOSs at the Fermi level in S. In writing Eq. (9) we assumed to neglect i) the spatial dependence of in the N wire, and ii) the electron-phonon interaction in the detector as it occurs on a time scale .

For the solution of Eq. (9) we choose a 10-nm-thick, 20-nm-wide Ag wire with nm (volume m), , , and ms. This corresponds to normal-state resistance of , but other material choices providing better impedance matching are also possible. In addition we set K as appropriate for aluminum (Al), , and m, corresponding thus to .

### iii.2 Temperature response of the detector weak-link

It is interesting to show first of all the behavior of the total electronic heat capacity of the detector junction, as it determines temperature response of the sensor. The total heat capacity, displayed in Fig. 7(a), is calculated vs temperature for two relevant values of the applied magnetic flux at mK. The general behavior of is the one typical of a BCS superconductor, i.e., it is characterized by an amplitude exponentially-suppressed with respect to that in the normal state at low (i.e., for ), and a sizable discontinuity at the critical temperature . The exponential suppression of at low is at the origin of high sensitivity of detection for microwave photons typical of our setup.

Figure 7(b) shows the final electronic temperature in the detector junction as a function of absorbed energy () of the incoming photon calculated from Eq. (9) for several values of at . We notice, first of all, the sizable enhancement of temperature, typically occurring below GHz, which can be achieved in the junction at low bath temperature, in particular, for mK. This enhancement stems predominantly both from the exponentially-suppressed amplitude of the junction electronic heat capacity and from the reduced volume of the N wire which is peculiar of the present setup. We emphasize that, for a bath temperature of 10 mK, obtains values as high as mK for 10 GHz photons, and up to mK at 100 GHz. For larger the increase of is less pronounced below 100 GHz owing to temperature-driven enhancement of the electronic heat capacity, and becomes sizable only at frequencies exceeding 1 THz.

The relative variation of temperature, versus for the same values of as in panel (b) and is displayed in Fig. 3(c). In the present setup, around for 10 GHz photons and around for 10 THz photons can be obtained at 10 mK of bath temperature. By increasing , gets reduced reaching about for 10 GHz photons, and for 10 THz radiation at mK. These results for substantial temperature variations in the weak-link suggest that large signal to noise ratio can be achieved with a TPC-based single-photon detector in the microwave frequency range.

The evolution of phase across the readout junction as a function of photon energy is shown in Fig. 7(d) for several bath temperatures at . Note that at low the phase across the weak-link starts to be reduced already for a few tens of GHz whereas at higher bath temperature a reduction of occurs only for larger frequencies. This behavior stems from the fact that at low bath temperature it is easy to enhance the electron temperature also at low photon energy due to suppressed electronic heat capacity, with the following reduction of circulating supercurrent and thereby of phase drop across the read-out junction [see Figs. 3(a,b)]. By contrast, at higher , the total heat capacity is larger so that higher photon energies are required to change appreciably the junction temperature. As a consequence, the reduction of is less pronounced.

### iii.3 Performance: Signal to noise ratio and resolving power

In the operation as a calorimeter, i.e., in the pulsed detection mode, we define the signal-to-noise (S/N) ratio of the detector as

 SN(Te(hν))=Ip(V,Tbath)−Ip(V,Te(hν))√SI(V,Tbath)√ω, (12)

where is current flowing through the SQUIPT biased at voltage in the idle state, is the current through the SQUIPT after the absorption of energy , is the SQUIPT current noise spectral density (shot noise) in the absence of radiation, and is the SQUIPT measurement bandwidth. Note that in the denominator of Eq. (12) the current noise spectral density of the tunnel junction is evaluated in the idle state (). The shot noise decreases for [see Fig. 4(a) and Eq. (8)], so that the above S/N expression is an underestimate. In the calorimeter operation mode, the expression is appropriate for measurement bandwidths , where is the characteristic time scale for the relaxation of the system back to equilibrium after absorption of a photon. In general, is determined by the relaxation processes occurring in the weak-link and, in the present setup, the predominant energy relaxation mechanism stems from electron-phonon interaction (see below). Moreover, we also note that so that a SQUIPT tunnel probe with lower resistance is, in general, beneficial in order to maximize the S/N ratio.

After energy absorption, the equation governing time evolution of the temperature in the sensor weak-link can be written as

 Cetot(φd,Te)∂Te∂t=−˙Qtote−ph(Te,Tbath), (13)

where is the total heat flow between electrons and lattice phonons in the detector region. Here, is heat exchanged in the N region whereas is the one exchanged in the lateral S portions of the detector. (with ) is given by [33; 34]

 ˙Qie−ph=−ΣiVi96ζ(5)k5B∫∞−∞dEE∫∞−∞dεME,E+ε(Δi(Te),Γi) ×ε|ε|[coth(ε2kBTbath)(fE−fE+ε)−fEfE+ε+1], (14)

where , , and is the electron-phonon coupling constant in the N(S) region. Here, are anomalous spectral densities, which for BCS superconductors have the form,

 FS(E) =ΔResgn(E)√(E+iΓ)2−Δ2. (15)

When the structure is in the normal state, Eq. (14) reduces to the well-known expression [2]. In our case, we approximate the normal-wire part of with its normal-state value, as for the minigap in the detector is almost suppressed. This choice leads to an underestimation of the S/N ratio.

Integration of Eq. (13) yields the electron-phonon relaxation half-time,

 τ1/2(ν,Tbath)=∫Tmaxe(ν)[Tmaxe(ν)+Tbath]/2dTeCetot(Te)˙Qtote−ph(Te,Tbath), (16)

which allows us to determine the relevant thermal time constant of the detector for any given energy of the incoming photon. In Eq. (16), is the maximum electron temperature reached in the weak-link after absorption of a photon of energy . For small temperature changes, the heat current can be linearized and written in terms of a thermal conductance, , and in this case is directly related to the linear thermal relaxation time .

The electron-phonon thermal conductances can be obtained by differentiating Eq. (14) vs. :

 Gith(T) Missing or unrecognized delimiter for \Bigr (17) gi(T) =1960ζ(5)∫∞−∞dEdεE|ε|3MiE,E−εk6BT6sinhε2kBTcoshE2kBTcoshE−ε2kBT.

Here, is a dimensionless function, obtaining the value in the normal state or at high temperature, and decreasing exponentially to at low temperatures in the superconducting state.

Finally, in the pulsed mode operation an important figure of merit is represented by the resolving power, ( is the energy resolution of full width at half maximum), [2]

 hνδE(ν)=hν2√2ln2NEPTFN√τ, (18)

where is the thermal fluctuation noise limited noise equivalent power of the sensor which stems from thermal fluctuations between the electron and the lattice phonon system in the detector region. The resolving power is calculated for , i.e., in the idle state of the detector in the absence of radiation. In this case, the fluctuation-dissipation theorem states that, at the equilibrium, is given by [2]

 NEPTFN=√4kBT2bathGtotth, (19)

where is the total thermal conductance of the detector. As a result, recalling that , the resolving power can also be written as:

 hνδE(ν,Tbath)=hν4√2ln2kBT2bathCetot(φd,Tbath), (20)

depending only on the electronic heat capacity.

Figure 8(a) shows the detector time constant calculated from Eq. (16) vs frequency for several values of at . From the figure it turns out that in the range .

The detector signal to noise ratio S/N at mK vs bias voltage across the SQUIPT is displayed in Fig. 8(b) for a few values of photon energy. Here we set , and assume the measurement bandwidth . We note, in particular, that the S/N ratio is maximized close to (compare to Fig. 4) and, depending on , it obtains values as large as . The maximum achievable S/N ratio (S/N) vs photon frequency is shown in Fig. 8(c) for the same bath temperatures as in panel (a). The S/N ratio is a non-monotonic function of frequency, decreasing at high frequency. Notably, S/N ratios of the order of can be obtained in the whole frequency range for a bath temperatures below 100 mK. Increasing leads to a general reduction of the S/N ratio.

The resolving power vs photon frequency calculated for the same as in panel (a) is displayed in Fig. 8(d). In particular, the figure shows that resolving power values can be achieved above 0.1 THz at 10 mK. At 100 mK these values are reduced by almost an order of magnitude. The resolving power makes the detector of potential use in microwave and far infrared single-photon detection.

The role of flux biasing of the interferometer is displayed in Fig. 9(a) where the S/N ratio vs frequency at 10 mK is shown for selected values of . In particular, moving away from leads to a reduction of the S/N ratio in the low-frequency end, and only significant deviations lead to suppression in the whole range.

The impact of the Dynes parameter on the signal to noise ratio is shown in Fig. 9(b), where S/N is calculated vs at 10 mK for a few values of . In particular, sufficiently small values of have no effect ot S/N. The resolving power is similarly insensitive to it (not shown).

## Iv Nanobolometer

### iv.1 Thermal model

The sensor operation in continuous power excitation (i.e., operation as a bolometer) can be described by considering those mechanisms that transport energy in the N and S parts of the detector. At low temperature, i.e., typically below 1 K, the main contribution stems from electron-phonon heat flux which can be modeled according to Eq. (14). In particular, the incoming radiation is first absorbed by electrons in the weak link while the lateral contacts with large superconducting gaps () prevent energy from escaping from the island. Then, the system can relax by releasing energy from electrons to the lattice phonons residing at . Under absorption of a continuous power , the steady-state temperature in the weak link is determined for any from the solution of the energy balance equation for incoming and outgoing power:

 Popt+˙Qtote−ph(Te,Tbath)=0. (21)

Thanks to the reduced amplitude of (and of the corresponding thermal conductance ) at sufficiently low temperatures, Eq. (21) predicts that a fairly large electronic temperature can be established in the sensor even for a quite moderate absorbed optical power. It is instructive to show first of all the behavior of obtained from Eq. (17) as a function of . The sensor thermal conductance is displayed in Fig. 10(a) (dash-dotted line). In particular, turns out to be somewhat suppressed with respect to that in the normal state (), and is reduced by an order of magnitude for . These results for suppressed thermal conductance indicate that reduced NEP