6.2 Modeling for optically bright condition
Under the optically bright condition, the main origin of the quasiparticles in the MKID is due to the optical loading. Therefore the number of quasiparticles in ab-sorb part increases and the internal quality factor and the resonance frequency de-creases comparing for the dark condition. Since the spectrum of the optical source is Planck’s law in both of the conventional measurement and the atmospheric ob-servation as mentioned in next section, we consider the optical loading is as thermal radiation in this section. The fluctuation of the photon number from the thermal radiation is added to the PSD model and NEP model of the dark condition.
6.2.1 Number of quasiparticles
For the optically bright condition, we consider the number of quasiparticles gener-ated by the optical loading and the thermal loading. The total quasiparticles,Nqp,tot, is the summation of the number of quasiparticles generated by the optical loading and the thermal loading given by
Nqp,tot= Nqp,th+Nqp,abs, (6.39)
where Nqp,th is the number of quasiparticles due to the thermal loading given by Eq. (6.24) and Nqp,abs is the number of quasiparticles due to the optical loading.
Since the quasiparticle lifetime lineally decreases with increasing the number density of quasiparticles, the relation between the quasiparticle lifetime and the number of quasiparticles [79] is given by
τqp,tot = τ0V Nqp,tot
N0(kBTc)3
2∆20 = X Nqp,tot
X=τ0VN0(kBTc)3 2∆20
. (6.40) To calculate the quasiparticles due to the optical loading, the relation between the number of quasiparticles due to the optical loading [78] is given by
Nqp,abs= ηpbτqp,tot
∆0
Pabs, (6.41)
wherePabsis the absorbed power in the MKID due to the optical loading. Using Eqs.
(6.39), (6.40), and (6.41), the number of quasiparticles due to the optical loading is obtained as
Nqp,abs=
−Nqp,th+qNqp,th2 +4ηpbXPabs/∆0
2 . (6.42)
The relation between the absorbed power and the quasiparticle lifetime of the proto-type MKID design is shown in Figure6.10. The quasiparticle lifetime decreases with increasing the absorbed power as expected.
6.2.2 The quality factors and the resonance frequency for the optically bright condition
The inverse of the total quality factorQi,totis the summation of the inverse of internal quality factors due to the optical loading and the thermal loading which is given by
1
Qi,tot = 1
Qi,th + 1
Qi,abs, (6.43)
2 4 6 8 10 Pabs [pW]
30 40 50 60 70
qp ,to t
FIGURE6.10: The quasiparticle lifetime of the prototype MKID de-sign in various absorbed power.
whereQi,th[85] is the internal quality factor due to the thermal loading given by Eq.
(6.15) andQi,absis the internal quality factor due to the optical loading given by 1
Qi,abs = δ(1/Qi)
δNqp Nqp,abs, (6.44)
where δ(δN1/Qi)
qp is the responsivity of the internal quality factor given by δ(1/Qi)
δNqp = αkβλ 2σ2V
δσ1
δnqp, (6.45)
where δnδσ1
qp is the responsvity of the complex conductivity given by Eq. 2.7. The in-ternal quality factor of the optically bright condition is lower than that of the dark condition due to the optical loading. The relation between the internal quality fac-tor and the absorbed power is shown in Figure 6.11. The internal quality factor decreases with increasing the optical loading.
Since the coupling quality factor does not depend on the optical loading, the total resonator quality factorQr,totis given by
1
Qr,tot = 1
Qi,tot + 1
Qc. (6.46)
The resonance frequency decreases with increasing the optical loading. The re-sponsivity of the resonance frequency is given by
δfr/fr0 δNqp
= αkβλ 4σ2V
δσ2 δnqp
. (6.47)
6.2. Modeling for optically bright condition 71
2 4 6 8 10
Pabs [pW]
40000 50000 60000 70000 80000 90000 100000 110000
Q i,t ot
FIGURE6.11: The internal quality factor of the prototype MKID de-sign in various absorbed power.
Using Eq. (6.14), the resonance frequency for the optically bright condition is given by
fr,tot =
1+ αkβλ 4σ2V
δσ2
δnqpNqp,tmp+ αkβλ 4σ2V
δσ2
δnqpNqp,abs
fr0, (6.48) where the second term of the right hand side is due to the thermal loading and the third term of the right hand side is due to the optical loading.
The complex transmissionS21as a function of readout frequency of the prototype MKID design in various absorbed power is shown in Figure6.12. The resonance fre-quency decreases with increasing the optical power. The resonance depth decreases with increasing the absorbed power.
5.695 5.696 5.697 5.698 5.699 frequency [GHz]
0.0 0.2 0.4 0.6 0.8 1.0
amplitude
1.0 pW 2.0 pW 3.0 pW 4.0 pW 5.0 pW 6.0 pW 7.0 pW 8.0 pW 9.0 pW 10.0 pW
5.695 5.696 5.697 5.698 5.699 frequency [GHz]
3 2 1 0 1 2 3
phase [rad]
1.0 pW 2.0 pW 3.0 pW 4.0 pW 5.0 pW 6.0 pW 7.0 pW 8.0 pW 9.0 pW 10.0 pW
FIGURE6.12: The amplitude (left figure) and the phase (right figure) of the complex transmission S21 of the prototype MKID design in
various absorbed power.
The resonator ring time for the optically bright conditionτres,totis given by τres,tot= Qr,tot
πfr,tot. (6.49)
It becomes shorter than the time of the dark condition due to decreasing the res-onator quality factor.
6.2.3 PSD model
The components of the PSD for the optically bright condition are the LNA noise, the TLS noise, the G-R noise, and the photon noise. The summation of the G-R noise and the photon noise is called the BLIP (Background LImited Performance) noise.
Therefore, the noise level of the optically bright condition is higher than that of the dark condition. The amplitude PSD for the optically bright condition is summation of the LNA noise and the BLIP noise given by
SA=SA,BLIP+SA,LNA, (6.50)
and the phase PSD for the optically bright condition is summation of the LNA noise and the TLS noise and the BLIP noise given by
Sθ =Sθ,TLS+Sθ,BLIP+Sθ,LNA. (6.51) For the photon noise observation, the LNA noise and the TLS noise needs to be less than the BLIP noise.
The BLIP noise is the summation of the G-R noiseSx,G−R and the photon noise Sx,photongiven by
Sx,BLIP= Sx,G−R+Sx,photon. (6.52)
The G-R noise due to the optical loading is given by Sx,G−R = 4∆
20Nqp,tot/ηpb2 τqp,tot
[1+ (2πfτqp,tot)2][1+ (2πfτres,tot)2] dx
dPabs 2
(x= A,θ), (6.53) where dx/dPabsis the responsivity for the optically bright condition[78] is given by
dx
dPabs = ηpbτqp,tot
∆0
dx
dNqp (x= A,θ). (6.54) The PSD of the photon noiseSx,photonis given by
Sx,photon= 2hνPabs(1+ηoptηemn¯) [1+ (2πfτqp,tot)2][1+ (2πfτres,tot)2]
dx dPabs
2
(x = A,θ), (6.55) where h is the Planck constant, ν is the optical frequency, ηem is the emissivity of the thermal radiation,ηoptis the optical efficiency which is the ratio of the absorbed power to radiation powerPrad, and ¯nis the photon occupation number given by
¯
n= 1
exp
hν kBT
−1
. (6.56)
6.2. Modeling for optically bright condition 73 Therefore, the PSD of the BLIP noise is given by
Sx,BLIP= 2hνPabs(1+ηoptηemn¯) +4∆20Nqp,tot/ηpb2 τqp,tot [1+ (2πfτqp,tot)2][1+ (2πfτres,tot)2]
dx dPabs
2
(x= A,θ)
= 2hνPrad(1+ηoptηemn¯) +4∆20Nqp,tot/ηpb2 τqp,tot ηopt[1+ (2πfτqp,tot)2][1+ (2πfτres,tot)2]
dx dPrad
2
(x = A,θ), (6.57) wherePabsis the absorbed power given byPabs=ηoptPrad.
The parameters included in the PSD of the TLS noise given by Eq. (6.35), and LNA noise given by Eq. (6.36) are converted from Qi, Qr, fr, τqp, andτres to Qi,tot, Qr,tot, fr,tot,τqp,tot, andτres,totin the optically bright condition.
6.2.4 NEP model for optically bright condition
The optical efficiency is ratio of the absorbed power to the radiation power as men-tioned in last subsection. When the NEP is dominated by the BLIP noise, we can obtain the optical efficiency by comparing the theoretical NEP and the measured NEP [110,109], because the difference between these NEP comes from the optical efficiency.
The theoretical NEPNEPtheory[57,113] is given by NEPtheory =
s2hνPrad(1+ηoptηemn¯) +4∆0Prad/ηpb
ηopt (6.58)
The measured NEP for the optically bright condition is given by NEP= pSx
dx dPrad
−1q
1+ (2πfτqp,tot)2 q
1+ (2πfτres,tot)2 (x= A,θ), (6.59) where dx/dPrad is the rate of change of the phase and amplitude response for the radiation power.
6.2.5 Summary of Reliability check
The comparison of the model results and the measurement results as mentioned in Chapter 3 is summarized in Table6.4. The difference of the model and the measure-ment is less than 70%. We estimate the tiny deference of the measured results and results by the forecaster is came from the production errors. It confirms that our forecaster provides reasonably good evaluation of the MKID performance.
We check our PSD model comparing previous results by S. Verheul in Figure 6.2 left figure in Ref [110]. He modeled aluminum and niobium titanium nitraide hybrid type MKID atT =270 mK. We obtained the same results.
We extended to this model to the optically bright condition. Since unfortunately the performance of the MKID installed in the GroundBIRD telescope is quite low, we could not compare the measurement results and results by the forecaster in the optically bright condition.