The signal chain and the noise propagation

HEMT (4 K)

6.1 The signal chain and the noise propagation

Chapter 6

Sensitivity of submm kinetic inductance detector

In this chapter, we will discuss the sensitivity of submm MKIDs, as an example of applying the models and theories of superconducting resonators developed in the previous chapters.

As the first stage of the detector development, these submm MKIDs are to be deployed in the Caltech Submillimeter Observatory (CSO), a ground-based telescope. For ground-based observa- tions, it is inevitable that the detector will be exposed to the radiations from the atmosphere and have a background photon signal. Once the intrinsic noise of the detector is made smaller than the shot noise of this background photon signal, the sensitivity of the detector is adequate. This requirement is called background limited photon (BLIP) detection.

One of the important questions to be answered in this chapter is whether our submm MKIDs can achieve the background limited photon detection on the ground, or in other words, whether the intrinsic detector noise (g-r noise, HEMT amplifier noise, and TLS noise) is below the photon noise of the background radiation from the atmosphere.

( ) ( )

p( ) p t

p t S_G

G

Q

( ) ( )

qp( )

qp qp n

n t n t S_G G

Q

( ) ( )

L( )

L L z

z t z t S_G G

Q 2

2 2

( ) ( )

V ( ) V t

V t S_G G

Q

Figure 6.1: A diagram of the signal chain and noise propagation in a hybrid submm detector.

p: optical power; Zl: load impedance of the sensor strip nqp: quasiparticle density; V₂⁻: output microwave voltage seen at the input port of the HEMT

6.1.1 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 densityn^th_qp (generated by thermal phonons) and the excess quasiparticle densityn^ex_qp (generated by optical photons).

nqp=n^th_qp+n^ex_qp (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)

whereg^th(t),g^ex(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) =Rn²_qp. (6.3)

With this definition, the quasiparticle lifetime is given by¹ 1

τqp(nqp) = 2Rnqp (6.4)

1We usually express the quasiparticle lifetime asτqp⁻¹=τ0⁻¹+ 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 theτ0⁻¹ term. For this reason, we ignore theτ0⁻¹ term throughout the calculations in this chapter.

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

VRn²_qpδ(t−t^′) (6.6) and the power spectrum is white

Sδr( ˜f) = 2

VRn²_qp. (6.7)

6.1.1.2 Thermal quasiparticle generationg^th(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

g^th(t)

=g^th(T) =r(n^th_qp(T)) =Rn^th_qp(T)² (6.8) wheren^th_qp is the thermal quasiparticle density given by Eq. 2.93.

We write

g^th(t) = 2G^th(t)

V =

g^th(t)

+δg^th(t) (6.9)

where G^th(t) represents the thermal generation events, which is also modeled by a Poisson point process. The power spectrum ofδg^th(t) is

Sδg^th( ˜f) = 2

VRn^th_qp(T)². (6.10)

6.1.1.3 Excess quasiparticle generation g^ex(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 rateg^ex(t) and the photon detection rateG^ph(t) (num- ber of photons detected per unit time) are related by

g^ex(t) = ζ

VG^ph(t). (6.12)

The statistical properties of G^ex(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^kThν −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

< G^ph(t)>= ∆νnph (6.13)

p=< G^ph(t)> hν= ∆νnphhν (6.14)

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

hg^ex(t)i=g^ex(p) = ζ

V∆νnph= ζp

hνV (6.16)

Sδg^ex( ˜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)

dt = 0 (6.18)

which leads to a quadratic equation

Rn²_qp−Rn^th_qp(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

g^ex(p)≫g^th(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) = τ_qp²

1 + (2πf τ˜ qp)²[Sδg^th( ˜f) +Sδg^ex( ˜f) +Sδr( ˜f)] = τ_qp²

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) +Sδg^ex( ˜f)

≈ 2ζp hνV² +

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)

n²_qp = 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 <˜ ∞.