Resonant Circuit

2.2 Responsivity

2.2.5 Resonant Circuit

nonthermal quasi-particles.

Equations (2.59) and (2.60) give the desired relationship between the measurable quantities — the fre- quency and loss of the resonator — and the quasi-particle density of the superconducting thin film. The final step is to examine how the frequency and loss of the resonator are probed by the readout electronics.

DAC ADC

Figure 2.4: Diagram illustrating the basic principle behind the MUSIC readout electronics. Two digital-to- analog converters (DACs) output the real (in-phase or I) and imaginary (quadrature-phase or Q) components of a pre-programmed waveform that consists of a superposition of sinusoids (carriers or carrier tones) at base- band frequencies (−225 to 225 MHz). The absolute magnitude of the FFT of an example waveform is shown in the upper left panel; it has the appearance of a “frequency comb”. The FFT of this waveform is also shown in the complex plane in the lower left panel. Note that the phases of the carriers are randomized in order to prevent clipping and utilize as much of the dynamic range of the DACs as possible. The I and Q components are up-mixed with a local oscillator (LO) to microwave (RF) frequencies and sent through the cryostat on coaxial cable. The baseband frequencies are chosen so that each carrier is centered on the resonant frequency of an MKID after mixing. The MKIDs modulate both the amplitude and phase of the carriers. The carriers are then amplified by a cryogenic high-electron-mobility transistor (HEMT) and down-mixed with the same LO back to baseband frequencies. The I and Q components are digitized by two analog-to-digital converters (ADCs). A field-programmable gate array (FPGA) is used to FFT the raw ADC timestreams, select the bins corresponding to the carrier tones, and low-pass filter and decimate them to the desired sampling rate of 100 Hz. The final output is the I and Q components of all of the frequency bins that contain carrier tones, sampled at 100 Hz. An example of the output at a single time sample is shown in the right column. The difference between the left and right columns provides a measurement of the overall gain and phase delay through the system.

Figure 2.5:Top:Schematic of the MUSIC readout electronics board.Bottom: Picture of one of the MUSIC readout electronics board. Each board output 144 carrier tones and measures the complex amplitude of these tones at the output of the system at a rate of 100 Hz. Two boards are used to probe each detector array.

resonant frequency,S₂₁^res(fres) =1−Q/Q_c=Q/Q_iand it is maximally attenuated.

The squared magnitude of the transmission near resonance is given by

|S₂₁^res(f)|²=1− 1−(Q/Q_i)² 1+4Q²

f−f_res fres

2 . (2.64)

It is a Lorentzian dip from unity with depth 1−Q²/Q²_i and full width at half maximum FWHM=2∆fres= f_res/Q. We denote the half width at half maximum as∆f_resand also refer to it as the resonator bandwidth. The magnitude|S₂₁^res(f)|and argumentθ=arg(S₂₁^res(f))are shown in the left panel of Figure 2.6.

It is useful to introduce the variables

x= f−f_res fres

, y=Qx= f−f_res

2∆fres

, (2.65)

wherexis the fractional detuning of the carrier tone from the resonant frequency andyis the normalized

Figure 2.6: The magnitude (upper left), phase (lower left), and complex transmission (right) near resonance.

Stars denote the location of the resonant frequency. The cross symbols in the right panel are separated by

∆f =10 kHz with increasing f corresponding to clockwise motion along the circle. The resonance curve changes from the solid black line to the dashed gray line when the optical loading is increased. Note that we measure the phase with respect to the complex origin, whereas many others measure the phase with respect to the center of the resonance circle. All curves were calculated using Equation (2.62) with parameters typical of a MUSIC resonator under sky loading: f_res=3.2 GHz,Q=40,000, andQ/Q_c=0.75.

detuning in terms of the full width at half maximum of the resonance. Equation (2.62) then becomes

S₂₁^res(f) =1− Q/Q_c

1+2jy . (2.66)

In the previous section we showed that a change in the quasi-particle density results in a change in both the frequency and dissipation of the resonator. We now calculate how these perturbations^δ_f^fres

res andδ_Q¹

i affect the transmission of the carrier tone. This is given by

δS₂₁^res= ∂S₂₁^res

∂_Q¹

i

! δ ¹

Qi

+ ∂S₂₁^res

∂x

δx

= ∂S₂₁^res

∂_Q¹

i

! δ ¹

Qi

− ∂S₂₁^res

∂x δfres

fres

, (2.67)

where in the last line we have used the fact thatδx≈ −^δ_f^fres

res . The partial derivatives are evaluated using

Equation (2.62) to be

∂S₂₁^res

∂_Q¹

i

= Q²/Q_c

(1+2jy)² , ∂S₂₁^res

∂x = 2jQ²/Q_c

(1+2jy)² (2.68)

and therefore

δS₂₁^res= Q²/Q_c (1+2jy)²

δ ¹

Qi

−2jδfres

fres

. (2.69)

If the carrier is centered directly on the resonant frequency, theny=0 and the response is at a maximum. In this case perturbations to the resonator frequency (dissipation) will result in purely imaginary (real) changes in the transmission. The factor(1+2jy)⁻²encodes the effect of detuning on the small signal response. It is helpful to write it in the form(1+2jy)⁻²=χye^jφywith

χy= 1

1+4y² , φy=−2 arctan(2y). (2.70) If the carrier is mis-centered, then the response will be degraded by a factorχ_y. The frequency and dissipation response will still be orthogonal, and will still be oriented tangential and normal to the resonance curve.

However, this “frequency and dissipation basis” will be rotated with respect to the real and imaginary basis by an angleφ_y. It is traditional to define a coupling efficiency

χc= 4^Q_Qⁱ

c

1+_Q^Qi

c

2 , (2.71)

which takes a maximum value of 1 when the coupling quality factor is matched to the internal quality factor, orQc=Qi. Equation (2.69) can then be written as

δS₂₁^res=1 4Q_iχ_cχ_y

δ ¹

Q_i−2jδfres

fres

e^jφy . (2.72)

This equation suggest three steps should be taken in order to maximize response to a fixed change in frequency and dissipation. The sources of loss intrinsic to the resonator — including the loss sourced by optically generated quasi-particles due to the background loading — should be minimized so that Q_i is large. The coupling to the feedline should then be designed so that under typical loading conditionsQ_c≈Q_i ensuring thatχ_c≈1. Finally the carrier tone should be centered on the resonant frequency so thatχ_y≈1. The coupling efficiencyχ_cand tuning efficiencyχ_yare plotted as a function of their respective arguments in Figure 2.7.

Equation (2.72) holds for slow perturbations to the resonator frequency and dissipation. For fast pertur- bations we must account for the resonator ring-down response. This effect can be described in the Fourier

Figure 2.7: The coupling efficiency (left) and tuning efficiency (right). TheFWHMof the resonator is defined as fres/Q.

domain as [105]

δSe₂₁^res(ν) =1

4Q_iχcχyH_res(ν,f) δQe_i⁻¹(ν)−2jδefres

fres

(ν)

!

e^jφy. (2.73)

whereHres(ν,f)is the resonator transfer function given by

H_res(ν,f) =1−S₂₁^res(f+ν)

1−S₂₁^res(f) . (2.74)

If the carrier tone is centered on resonance this reduces to Hres(ν,fres) = 1

1+j_∆f^ν

res

(2.75)

and the resonator acts as a low-pass filter with bandwidth∆fres=_2Q^fres. We are interested in signals at temporal frequenciesν≤50 Hz (the readout electronics sample at a rateν_s=100 Hz). Since∆fres&20 kHz for the MUSIC resonators, the transfer function can safely be ignored for our purposes.

We now consider perturbations to the frequency and dissipation that are sourced by perturbations in the quasi-particle densityδnqp. In this case Equation (2.72) becomes

δS₂₁^res=1

4Q_iχ_cχ_yα[κ₁(T,ω,∆₀) +jκ₂(T,ω,∆₀)]e^jφyδnqp

=1

4Q_iχcχyα|κ(T,ω,∆0)|e^{j Ψ(T,ω,∆0)+φy}δn_qp. (2.76) This can easily be converted into changes in the amplitude of the carrier tone using Equation (2.61). This is the quantity that is digitized by the readout electronics, which means that we have followed the propagation

of signal through the entire instrument. We are now in a position to derive an expression for the response to an astronomical source. Before doing that, we should address several complications to the model just presented.