Nonlinear Kinetic Inductance

2.2 Responsivity

2.2.6 Nonlinear Kinetic Inductance

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.

We would like to determinexfor a given value of low-power detuningx0and the microwave carrier power P. We start with the standard definition for the quality factor of a resonator as the maximum energy stored divided by the energy dissipated in one cycle, or

Q_i= E Pdiss/2πfres

=⇒ E=PdissQ_i 2πfres

, (2.80)

wherePdissis the power dissipated by the resonator. We can derive an expression forPdissthrough conservation of energy considerations. The microwave probe signal can either be reflected, transmitted, or absorbed by the resonator. Therefore

P=|S₁₁^res|²P+|S₂₁^res|²P+Pdiss

Pdiss=P 1− |S₁₁^res|²− |S₂₁^res|²

. (2.81)

The reflection is related to the transmission via

S₁₁^res=S₂₁^res−1=− Q/Q_c

1+2jQx , (2.82)

where we have used Equation (2.62) for the forward transmission. Taking the squared magnitude of Equa- tion (2.62) and Equation (2.82) and inserting into Equation (2.81) yields

Pdiss= 2Q² Q_iQ_c

1

1+4Q²x²P. (2.83)

Therefore the energy stored in the resonator is E=2Q²

Qc

1 1+4Q²x²

P 2πfres

(2.84)

and the fractional shift due to the quadratic term is

∆x=−2Q² Q_c

1 1+4Q²x²

P

2πfresE_? . (2.85)

Inserting this into Equation (2.79) yields

x0=x−2Q² Q_c

1 1+4Q²x²

P 2πfresE?

, (2.86)

which is an implicit equation for the power-shifted detuningx. In order to clean up the presentation of this equation we define the nonlinearity parameter

a=2Q³ Q_c

P

2πf_resE_? (2.87)

Figure 2.8: Left: magnitude of the forward transmission for several values of the nonlinearity parameter a. Right: forward transmission in the complex plane fora=2.0. Stars denote the power-shifted resonant frequency. Regions of bifurcation are designated with dashed lines. In these cases the measured curve will depend on whether one is sweeping upward or downward in frequency and there will be a discontinuity at the location of the arrow. The dotted line is the third (unstable) root corresponding to intermediate stored energies. All curves were calculated using Equations (2.66) and (2.89) with parameters typical of a MUSIC resonator under sky loading: fres,0=3.2 GHz,Q=40,000, andQ/Q_c=0.75.

as well asy=Qxandy₀=Qx₀. Equation (2.86) can then be written as y₀=y− a

1+4y² , (2.88)

which is a cubic equation for the normalized detuningy. Rearranging gives

4y³−4y₀y²+y−(y₀+a) =0. (2.89)

Solving for the roots it is clear that there is a critical value of the nonlinearity parameter,acrit=4√

3/9≈0.77.

Fora≤acritthere is only one purely real solution for all values ofy₀. Fora>acritthere is range ofy₀values for which there are three purely real solutions, corresponding to three possible values for the resonant frequency and stored energy. This means that multiple internal states exist for a single value of the carrier frequency and power. Only two of these states, corresponding to the smallest and largest stored energies, are stable.

Therefore we say that the resonator has undergonebifurcation.

Figure 2.8 shows resonance curves for several values ofa. As the carrier power is increased, the Lorentzian shape becomes distorted and compressed towards lower frequencies. When the carrier power exceeds the value corresponding toa_crit the resonator undergoes bifurcation. Two different curves will be measured de- pending on whether one is sweeping upward or downward in frequency, and a discontinuity appears at the

Figure 2.9:Left:Prediction for ˜κ1and ˜κ2. The solid, long-dash, short-dash, and dash-dot lines correspond to a=0.0, 0.2, 0.4, and 0.6 respectively.Right:the ratio of frequency to dissipation response is shown in purple and the angle between the quasi-particle direction and the direction normal to the resonance circle is shown in green.

frequency where the resonator transitions from the bifurcated to nonbifurcated state.

In this model, the transmission near resonance is still given by Equation (2.66), but nowyis determined by Equation (2.88). This means that the resonance will still trace out a circle in the complex plane; although, if one is operating in a regime wherea>acrit then the full circle will be inaccessible due to bifurcation.

The model also predicts that the depth of the resonance is constant with carrier power. Therefore, if one measures a non-circular resonance curve or a power-dependent resonance depth, then it is likely that some other mechanism is causing a nonlinear dissipation (recall that in this model only the resonator frequency is nonlinear). One example of such a mechanism is heating of the quasi-particle population by the microwave readout power [100, 109].

We now examine how fractional perturbations to the frequencyδfres,0/fres,0 and dissipationδ_Q¹

i translate into perturbations to the complex transmission under the nonlinear model. We can write

δS₂₁^res= ∂S₂₁^res

∂_Q¹

i

! δ ¹

Q_i+ ∂S₂₁^res

∂x

δx

= ∂S₂₁^res

∂_Q¹

i

! δ ¹

Q_i+ ∂S₂₁^res

∂x "

∂x

∂_Q¹

i

! δ ¹

Q_i+ ∂x

∂x₀

δx₀

#

, (2.90)

where we now assume thatxdepends on 1/Q_i(first term in the square brackets), since a change in the quality factor will change the energy stored in the resonator, thereby changing the resonant frequency through the nonlinear kinetic inductance. Inserting the derivatives presented in Equation (2.68) and using the fact that

δx0=−δfres,0/fres,0, we have

δS₂₁^res=1

4Q_iχcχy δ ¹ Q_i+2j

"

∂x

∂_Q¹

i

! δ ¹

Q_i− ∂x

∂x0

δf_res,0 fres,0

#!

e^jφy. (2.91)

The derivatives∂x/∂_Q¹

i and∂x/∂x0are evaluated using Equation (2.88) to be

∂x

∂_Q¹

i

=− 2a

(1+4y²)²+8ay=− 2aχ_y²

1+8ayχ_y^{2 (2.92)}

∂x

∂x₀ = 1+4y²2

(1+4y²)²+8ay

= 1

1+8χ_y²ay . (2.93)

Inserting these into Equation (2.91) yields

δS₂₁^res=1 4Q_iχ_cχ_y

"

1−j 4aχ_y² 1+8ayχ_y²

! δ ¹

Q_i−j 2 1+8ayχ_y²

δfres,0

f_res,0

#

e^jφy. (2.94)

We see that it is no longer the case that the frequency and dissipation response are orthogonal. While the frequency response is still confined to the direction tangent to the resonance curve, the dissipation response now leaks into the direction tangent to the resonance curve as well. We also see that detuning the carrier tone from the power shifted resonant frequency has interesting, asymmetric effects on the response. The factor of(1+8ayχ_y²)⁻¹implies that for f> f_resthe frequency response will be attenuated, while for f <f_res the frequency response will be amplified. The further the resonance is driven into the nonlinear regime, the greater this attenuation and amplification.

We now consider changes in frequency and dissipation sourced by changes in the quasi-particle density.

This is given by

δS₂₁^res=1

4Q_iχcχyα

"

1−j 4aχ_y² 1+8ayχ_y²

!

κ1(T,ω,∆0) +j 1

1+8ayχ_y²κ2(T,ω,∆0)

# e^jφyδn_qp

=1

4Q_iχ_cχ_yα|κ˜(T,ω,∆0,a,y)|e^{jΨ(T,ω,∆˜ 0,a,y)+φy}δnqp, (2.95) where we have defined ˜κ=κ˜1+jκ˜2and ˜Ψ=arctan(κ˜2/κ˜1)with





κ˜1(T,ω,∆0,a,y) κ˜2(T,ω,∆0,a,y)



=





1 0

−4aχ²_y 1+8ayχ_y²

1 1+8ayχ_y²



×





κ1(T,ω,∆₀) κ2(T,ω,∆₀)



. (2.96)

If we are centered on the power-shifted resonant frequency so thaty=0, this simplifies to

δS₂₁^res=1

4Qiχcα|κ˜(T,ω,∆0,a,0)|e^{jΨ(T,ω,∆˜ 0,a,0)}δnqp (2.97)

with 



κ˜1(T,ω,∆0,a,0) κ˜₂(T,ω,∆0,a,0)



=





1 0

−4a 1



×





κ1(T,ω,∆0) κ₂(T,ω,∆0)



 . (2.98)

In Figure 2.9 we show the predicted values of ˜κ₁, ˜κ₂, their ratio, and the corresponding angle ˜Ψfor a typical MUSIC detector operated aty=0 anda= [0.0, 0.2,0.4, 0.6]. We see that as the carrier power is increased the quasi-particle response normal to the resonance curve remains fixed, while the quasi-particle response tangential to the resonance curve declines. This results in a decrease in the overall magnitude of the response and a clockwise rotation of the quasi-particle direction. The reduction in response should be taken into account in readout power optimization. Assuming that the carrier is tuned to the resonant frequency so that y=0 and Equation (2.98) applies, then whena>κ2/4κ1the dissipative frequency response will be larger than the reactive frequency response. Sinceκ2/κ1'4 for our resonators at base temperature, this corresponds to a value of the nonlinearity parametera'1, which is just past the point of bifurcation. As the carrier power is increased beyond this point, the quasi-particle direction rotates past the direction normal to the resonance curve. This means that the frequency response will occur in the opposite direction as it does in the low power regime, so that a small increase in optical loading results in an increase in the resonator frequency.