Quasiparticle density fluctuations δn qp under an optical loading p

When the sensor strip is under optical loading, the total quasiparticle density nqp is the sum of the thermal quasiparticle densitynth

qp (generated by thermal phonons) and the excess quasiparticle densitynex

qp (generated by optical photons).

nqp=nthqp+nexqp (6.1)

Three independent physical processes are involved in changingnqpand must be modeled: thermal quasiparticle generation, excess quasiparticle generation, and quasiparticle recombination. We can write out the following rate equation,

dnqp(t)

dt = [g

th_{(t) +g}ex_(t)

−r(t)] (6.2)

wheregth_(t),gex_{(t), andr(t) are the rates for the 3 processes.}

6.1.1.1 Quasiparticle recombinationr(t)

The average quasiparticle recombination rate only depends on the total quasiparticle density nqp and is calculated by[77]

hr(t)i=r(nqp) =Rn2qp. (6.3) With this definition, the quasiparticle lifetime is given by1

1

τqp(nqp) = 2Rnqp (6.4)

1

We usually express the quasiparticle lifetime asτ−1

qp =τ− 1

0 + 2Rnqp, to account a finite lifetimeτ0at lownqp. In

the regime that submm MKIDs operate,nqp from the background loading is usually large enough so that theRnqp

term will dominate over theτ−1

0 term. For this reason, we ignore theτ− 1

0 term throughout the calculations in this

whereR is the recombination constant. We write

r(t) =2R(t)

V =hr(t)i+δr(t) (6.5)

whereV is the volume of the sensor strip andR(t) represents the recombination events in the volume

V. R(t) is often modeled by a Poisson point process[78] and it can be shown that the auto-correlation function ofδr(t) is a delta function

< δr(t)δr(t′)> = 4 V2hR(t)iδ(t−t′) = 2 VRn 2 qpδ(t−t′) (6.6) and the power spectrum is white

Sδr( ˜f) = 2

VRn

2

qp. (6.7)

6.1.1.2 Thermal quasiparticle generationgth_(t)

The average thermal generation rate only depends on the bath temperature T and is in balance with the thermal recombination rate when the system is in thermal equilibrium and without excess quasiparticles

gth(t)=gth(T) =r(nthqp(T)) =Rnthqp(T)2 (6.8) wherenth

qp is the thermal quasiparticle density given by Eq. 2.93. We write gth(t) = 2G th_(t) V = gth(t)+δgth(t) (6.9)

where Gth_{(t) represents the thermal generation events, which is also modeled by a Poisson point} process. The power spectrum ofδgth(t) is

Sδgth( ˜f) = 2

VRn

th

qp(T)2. (6.10)

6.1.1.3 Excess quasiparticle generation gex_{(t)under optical loading}

We assume that the number of excess quasiparticles generated by each detected submm photon is given by ζ(ν,∆), which in general, depends both on the photon energyhν and the gap energy ∆ (binding energy of the Cooper pair). An empirical assumption aboutζ(ν,∆) often adopted for photon to quasiparticle conversion is that a fraction ofηe ≈60% of the photon energy goes to the

quasiparticles,

ηe= ζ∆

hν (6.11)

Thus, the excess quasiparticle generation rategex_{(t) and the photon detection rateG}ph_{(t) (num-} ber of photons detected per unit time) are related by

gex_{(t) =} ζ

VG

ph_{(t). (6.12)}

The statistical properties of Gex_{(t) can be found in photon counting theory. In addition, to} simplify the discussion, we make the following assumptions:

1. The optical loading is from the black body radiation with mean photon occupation number

nph= (e

hν

kT −1)−1;

2. The photon numbers and their fluctuations of different modes are independent; 3. The detector has a narrow band of response (R dν→∆ν);

4. The detector is single mode (AΩ =λ2_{, whereA is the area of the detector and Ω is diffraction} limited solid angle) and is only sensitive to one of the two polarizations;

5. The detector has a quantum efficiency of 1. (A reduced quantum efficiencyη can be introduced with the substitutionnph→ηnph.)

Under these assumptions, we can derive

< Gph(t)>= ∆νnph (6.13)

p=< Gph(t)> hν= ∆νnphhν (6.14)

< δGph(t)δGph(t′)>= ∆νnph(1 +nph)δ(t−t′) (6.15) wherepis the average optical power received by the detector. Therefore

hgex_(t)i=gex_{(p) =} ζ V∆νnph= ζp hνV (6.16) Sδgex( ˜f) = _ζ V 2 ∆νnph(1 +nph) = _ζ V 2 _p hν(1 + p hν∆ν). (6.17)

6.1.1.4 Steady state quasiparticle densitynqp

The steady state quasiparticle densitynqpcan be derived by solving

dnqp(t)

which leads to a quadratic equation

Rn2qp−Rnthqp(T)2−

ζp

hνV = 0. (6.19)

We usually operate at a low enough temperature so that the excess quasiparticle generation rate dominates over the thermal quasiparticle generation rate

gex_(p)≫gth_{(T). (6.20)}

Under this condition, the thermal generation terms can be neglected and the steady-state equation reduces to,

Rn2 qp=

ζp

hνV (6.21)

and the positive root is

nqp= r

ζp

hνRV (6.22)

6.1.1.5 Fluctuations in quasiparticle density δnqp

The fluctuations in the quasiparticle density δnqp(t) = nqp(t)−nqp can be shown to satisfy the following equation,

dδnqp(t)

dt =−2Rnqpδnqp(t) + [δg

th_{(t) +δg}ex_(t)

−δr(t)] (6.23) This allows us to calculate the power spectrum ofδnqp in the Fourier domain as,

Sδnqp( ˜f) = τ2 qp 1 + (2πf τ˜ qp)2[Sδgth( ˜f) +Sδgex( ˜f) +Sδr( ˜f)] = τ2 qp 1 + (2πf τ˜qp)2Sgr( ˜f) (6.24) where τqp =τqp(nqp) and we have used the fact that the 3 processes are independent. Under the condition that the excess quasiparticle generation dominates over thermal generation,

Sgr( ˜f) ≈ Sδr( ˜f) +Sδgex( ˜f) ≈ _hνV2ζp2 + ζ V 2 p hν(1 + p hν∆ν) (6.25)

where Eq. 6.14 and Eq. 6.21 have been applied. By applying Eq. 6.3 and Eq. 6.22 we can further derive the spectral density of the fractional quasiparticle density fluctuations,

Sδnqp nqp ( ˜f) = Sδnqp( ˜f) n2qp = 1/4 1 + (2πf τ˜qp)2 (2/ζ+ 1)hν p + 1 ∆ν . (6.26)

Note that the power spectra derived above are double-sided with−∞<f <˜ ∞.