Frequency Fluctuations in Tunable and Nonlinear Microwave Cavities
B. L. Brock M. P. Blencowe A. J. Rimberg






Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA


Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA 


ajrimberg@dartmouth.edu
Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA 



We present a model for how frequency fluctuations comparable to the total cavity linewidth may arise in tunable and nonlinear microwave cavities, and how these fluctuations affect the measurement of scattering matrix elements.  Applying this model to the specific case of a two-sided cavity, we obtain closed-form expressions for the average scattering matrix elements in several important cases.  A key signature of our model is the subtle deformation of the trajectories swept out by scattering matrix elements in the complex plane.  Despite this signature, the fluctuating and non-fluctuating models are qualitatively similar enough to be mistaken for one another.  In the case of tunable cavities we show that if one fails to account for these fluctuations then one will find damping rates that appear to depend on the tuning parameter, which is a common observation in such systems.  In the case of a Kerr cavity, we show that there exists a fundamental lower bound to the scale of these frequency fluctuations in the steady state, imposed by quantum mechanical uncertainty, which can appreciably affect the apparent damping rates of the cavity as the strength of the nonlinearity approaches the cavity linewidth.  By using the model we present as a fitting function for experimental data, however, one can extract both the true damping rates of the cavity and the effective scale of these frequency fluctuations over the scattering measurement's bandwidth.  Lastly, we compare this new method for observing frequency fluctuations to other methods, one of which we extend beyond the regime of small fluctuations, and we discuss how ringdown measurements can be performed accurately in the presence of frequency fluctuations.


Frequency Fluctuations in Tunable and Nonlinear Microwave Cavities
    A. J. Rimberg
    December 30, 2023
==================================================================





§ INTRODUCTION


Tunable and nonlinear microwave cavities have become ubiquitous in the field of circuit quantum electrodynamics.  Tunable cavities have been used as parametric oscillators to achieve single-shot readout of a superconducting qubit <cit.>, as quantum buses for selective qubit coupling <cit.>, as broadband filters <cit.>, and as platforms for investigating the dynamical Casimir effect <cit.>.  Many such cavities are also nonlinear due to the presence of embedded Josephson junctions, but nonlinear microwave cavities on their own have found wide-ranging applications as well.  For example, they have been used as a platform for generating Schrodinger cat states <cit.> and as amplifiers operating near the quantum limit of added noise <cit.>. 

In order to benchmark these devices, however, one must first accurately characterize their basic properties.  Determining their damping rates (or equivalently their quality factors) is particularly important, since these affect the equations of motion governing the cavities such that inaccuracies will have far-reaching consequences.  Unfortunately, a common issue with tunable cavities is that measured damping rates appear to vary with the tuning parameter <cit.>.  Although many have attributed this to causes specific to their devices, here we argue that the tunability itself is at least partially responsible for these observations.  

The line of reasoning is as follows: fluctuations in the tuning parameter will necessarily induce fluctuations in the resonant frequency.  These frequency fluctuations will in turn affect the measurement of scattering matrix elements, which is a standard method for extracting the damping rates of microwave cavities <cit.>.  A similar line of reasoning holds for cavities with a Kerr nonlinearity: fluctuations in the amplitude of the intra-cavity field will lead to steady-state fluctuations in the resonant frequency, though this effect does not lead to an easily identifiable consequence in the literature as in the case of tunability.

Thus, to accurately determine the damping rates of these cavities from scattering measurements, one must account for the effect of frequency fluctuations.  Here we present a model for this effect, which in the case of tunable cavities both predicts the apparent dependence of the cavity's damping rates on the tuning parameter and allows one to extract the true damping rates in the presence of these frequency fluctuations.  Similarly, in the case of a cavity with a Kerr nonlinearity we analyze the quantum limit of steady-state frequency fluctuations and show that the apparent damping rates can deviate significantly from their true values as the nonlinearity approaches the cavity linewidth.  In both cases we provide bounds on the scale of the fluctuations necessary to yield greater than 10% inaccuracy in the apparent damping rates relative to their true values.  Finally, we analyze how these two sources of fluctuations come together in the case of a tunable cavity with a Kerr nonlinearity.  We expect the use of this model as a fitting function for experimental scattering measurements will allow for more accurate characterization of these cavities moving forward.

In addition to scattering measurements we also consider the effect of frequency fluctuations on ringdown measurements <cit.>, which use the decay of intracavity oscillations to extract the total damping rate of the cavity.  Although they are not sensitive to the individual contributions from multiple loss mechanisms, such measurements are nevertheless useful as an independent metric of damping that could help corroborate the presence of frequency fluctuations by comparison to the results of scattering measurements.  We find that in order to accurately perform a ringdown measurement in the presence of frequency fluctuations, one must directly measure the amplitude or power of the output signal as the cavity rings down, especially if an ensemble average of measurements is necessary to resolve the signal from the noise.  

While our primary goal is the determination of the true damping rates of these cavities, determining the properties of the frequency fluctuations themselves is also important, since these will lead to dephasing <cit.>.  Many recent studies have focused on characterizing the time dependence of frequency fluctuations in microwave cavities <cit.>, but current methods developed to this end have either assumed that these fluctuations are small compared to the total cavity linewidth or that the damping rates can be determined independently.  Here we show that in the case of tunable and nonlinear microwave cavities both of these assumptions may be broken, and we conclude by extending the most straightforward of these methods to the case of large frequency fluctuations.



§ SCATTERING MEASUREMENTS


Let S_jk(Δ) be the scattering matrix elements of the microwave network containing the cavity <cit.>, describing an experiment in which the cavity is driven by a sinusoidal signal on port k and measured on port j, where Δ = ω - ω_0 is the detuning of the drive from resonance.  In general, measuring these scattering matrix elements involves a process of averaging, whether implicitly through the time-scale associated with individual measurements or explicitly through the incorporation of multiple independent measurements.  The process of characterizing a cavity then consists of measuring this average value of S_jk(Δ) for a range of detunings around the resonant frequency and comparing these results with a theoretical model to extract the damping rates.

In the presence of fluctuations in the resonant frequency, however, ω_0→ω_0 + δω_0(t) such that the detuning Δ may vary over the course of this measurement.  Rather than attempting to take into account the time dependence of δω_0 explicitly, we treat these fluctuations as a random variable and model their effect on S_jk(Δ) as an ensemble average.  As a result, in the presence of a fluctuating resonant frequency what will actually be measured is the convolution

    S_jk(Δ) = ∫_-∞^∞S_jk(Δ-Ω)P(Ω)dΩ

where P(Ω) is the probability density function (PDF) associated with drawing the value Ω from the random variable δω_0.  

There are two generic features of this convolution worth discussing.  First, the average quantity S_jk(Δ) will be sensitive to those fluctuations occurring faster than the measurement time and insensitive to those occurring slower than it.  For example, if we can measure S_jk(Δ) faster than a given source of frequency noise then our measurement will not be influenced by this noise.  Thus, P(Ω) should depend on the power spectral density of the frequency fluctuations and the time-scale associated with the measurement.  In particular, since the power spectral density S_ω_0ω_0(f) is simply the variance per unit frequency, the variance σ_ω_0^2 of frequency fluctuations can be found by integrating over all frequencies f.  For a real measurement, however, the low frequency cutoff is set by the inverse of the time T spent measuring S_jk(Δ) at a fixed detuning and the high frequency cutoff is set by the total linewidth of the cavity κ_tot/2π, which determines the maximum rate of cavity response, such that

    σ_ω_0^2 = ∫_1/T^κ_tot/2πS_ω_0ω_0(f)df.

We note that it may be possible to extend this simple connection using the Allan variance <cit.>, which accounts for dead time between measurements and is well-behaved with respect to the 1/f power spectra that are ubiquitous in solid state devices <cit.>, but whose underlying formalism may not directly carry over to the scattering measurements we are considering. 

Second, these scattering matrix elements S_jk(Δ) generally take the form of linear fractional transformations that trace out circles in the complex plane <cit.>.  Since the convolution is essentially a convex combination of points around this circle, the shape traced out by S_jk(Δ) will be smaller by comparison, corresponding to apparent changes in the damping rates.  We operationalize the notion of `apparent damping rates' by considering the best fit of the non-fluctuating model S_jk(Δ) to data generated by the fluctuating model S_jk(Δ).  

The effect of this convolution can be thought of as a form of inhomogeneous broadening, in that we are effectively averaging over the response functions of an ensemble of cavities with different resonant frequencies.  Similar effects show up in a diverse range of fields, but there are two examples in the same vein as our work worth noting.  First is the Doppler broadening of Lorentzian spectral lines, in which case one obtains a Voigt Profile <cit.>.  Second is the broadening of cavity lineshapes due to a linear coupling with an ensemble of spins that serve as a noisy driving field <cit.>.  Here we apply similar methods to understand the effect of tunability and nonlinearity on the lineshapes of microwave cavities.



§ TWO-SIDED CAVITY



As a basis for our study we consider a generic two-sided cavity, depicted schematically in Figure <ref>, which we expect to be an appropriate model for a wide variety of systems.  Using input-output theory <cit.>, we can relate the fields at the ports of the network to the internal field of the cavity according to

    a_1^out(t) - a_1^in(t)    = -√(κ_1)a(t) 
    
    a_2^out(t) - a_2^in(t)    = -√(κ_2)a(t)

where κ_1 and κ_2 are the damping rates associated with their respective ports, such that the internal cavity field obeys the equation of motion

    ȧ(t) = i/ħ H_sys, a  - κ_1+κ_2/2 a(t)    + √(κ_1)a_1^in(t) 
       + √(κ_2)a_2^in(t).

We assume for the time being that the cavity is linear, such that H_sys = ħω_0a^†a and H_sys, a  = -ħω_0a.

In scattering experiments we are generally interested in the steady state response of the cavity as a function of drive frequency ω, to which end we Fourier transform Eq. (<ref>) and  solve for the cavity response algebraically

    ã(ω) = √(κ_1)ã_1^in(ω)  + √(κ_2)ã_2^in(ω)/i(ω_0 - ω) + (κ_1 + κ_2)/2

where a tilde denotes the Fourier transform of the given mode operator.  Plugging this result into the Fourier transform of Eq. (<ref>), we find a linear relationship between the input and output mode operators

    ã_1^out(ω)    = ã_1^in(ω) - κ_1ã_1^in(ω)  + √(κ_1κ_2)ã_2^in(ω)/i(ω_0 - ω) + (κ_1 + κ_2)/2
    ã_2^out(ω)    = ã_2^in(ω) - √(κ_1κ_2)ã_1^in(ω)  + κ_2ã_2^in(ω)/i(ω_0 - ω) + (κ_1 + κ_2)/2

that defines the scattering matrix according to

    [ ã_1^out(ω); ã_2^out(ω) ]
    =
    [ S_11^* S_12^*; S_21^* S_22^* ][ ã_1^in(ω); ã_2^in(ω) ]

where the complex conjugate of each matrix element is taken, by convention, so that its phase corresponds to a counter-clockwise rotation in phase space of the output quadrature operators relative to the inputs.  For ease of notation let κ_tot = κ_1+κ_2 be the total damping rate of the cavity and let ξ = κ_1/κ_2 be the coupling ratio, in terms of which the scattering matrix elements can be expressed

    S_21(Δ)    = -2√(ξ)/1+ξ1/1+2iΔ/κ_tot
    
    S_12(Δ)    = S_21(Δ) 
    
    S_11(Δ)    = 1 + √(ξ)S_21(Δ)
    
    S_22(Δ)    = 1 + 1/√(ξ)S_21(Δ)

where Δ = ω - ω_0 is the detuning, as before.  

As mentioned earlier, our analysis will center around the shape of the trajectories swept out by these scattering matrix elements in the complex plane as a function of detuning, and how these trajectories are deformed by convolving them with a probability distribution, as given by Eq. (<ref>).  To this end, we define the quantities

    r_12 = r_21 = √(ξ)/1+ξ
    
    r_11 = 1-r_22 = ξ/1+ξ

and re-express the scattering matrix elements in the form

    S_jk(Δ ; κ_tot, r_jk) = δ_jk - r_jk(1+e^-2iarctan(2Δ/κ_tot))

where δ_jk is the Kronecker delta. Written in this way, it is clear that each S_jk traces out a circular path with radius r_jk at a rate determined by κ_tot.  This is also a convenient expression for analyzing all of the scattering matrix elements at once, since they are equivalent up to translation and a rescaling of r_jk.  It is worth noting that in order to retrieve ξ from r_12 or r_21 one must additionally specify whether ξ>1 or ξ<1, since they are invariant under ξ→ 1/ξ.  



§ TUNABLE CAVITY


We now consider a cavity whose resonant frequency ω_0 is a function of tuning parameter x such that ω_0 = ω_0(x).  In general, this tuning parameter can never be perfectly fixed: it will always fluctuate around its mean value.  To model this effect, let x →x + δ x, where x is the mean value of x and δ x is a random variable describing fluctuations of x about x, which we assume to be Gaussian with mean zero and variance σ_x^2.  Expanding to lowest order in these fluctuations, we find 

    ω_0(x+δ x) ≈ω_0(x) + δω_0(x)

where the second term

    δω_0(x) = ∂ω_0(x)/∂ xδ x

is itself a Gaussian random variable with mean zero and variance σ_ω_0^2 given by

    σ_ω_0^2 = |∂ω_0(x)/∂ x|^2σ_x^2

which varies with x.  The PDF associated with drawing the value Ω from the random variable δω_0 is therefore given by

    P(Ω) = 1/√(2πσ_ω_0^2)e^-Ω^2/2σ_ω_0^2.


In the presence of these fluctuations, the average scattering matrix elements are found by plugging Eqs. (<ref>) and (<ref>) into Eq. (<ref>).  This convolution results in the closed form expression

    S_jk(Δ ; κ_tot, r_jk, σ_ω_0) = 
    δ_jk - r_jk√(π/2)κ_tot/σ_ω_0 w(iκ_tot-2Δ/2√(2)σ_ω_0)

where w(z) is the Faddeeva function, which can be written in terms of the complementary error function as w(z) = e^-z^2erfc(-iz) <cit.>.  As illustrated in Fig. <ref>(a), the effect of these fluctuations is a deformation of the trajectories of S_jk in the complex plane.  Compared to S_jk(Δ ; κ_tot, r_jk), S_jk(Δ ; κ_tot, r_jk, σ_ω_0) is slightly oblong, its apparent radius is smaller, and it traverses its path more slowly as a function of detuning (corresponding to an apparent increase in κ_tot).  Furthermore, although there is a systematic deviation between the closest fit of S_jk(Δ ; κ_tot^', r_jk^') to the trajectory generated by S_jk(Δ ; κ_tot, r_jk, σ_ω_0), this deviation is subtle.  It is even more subtle, in fact, if one considers only the squared magnitude of S_jk rather than its full complex trajectory as is sometimes done <cit.>.



Thus, presented with experimental data for which frequency fluctuations are significant, it is quite reasonable to believe it is well-modeled by S_jk(Δ ; κ_tot^', r_jk^').  If one tries to fit this model to the data, however, one will extract damping rates that appear to vary with σ_ω_0, as illustrated in Fig. <ref>(b).  In the case of a tunable cavity for which σ_ω_0 = |∂ω_0(x)/∂ x|σ_x, one will therefore find damping rates that seem to vary with x, which is precisely the syndrome we set out to explain.  By using this model S_jk(Δ ; κ_tot, r_jk, σ_ω_0) as a fitting function for experimental data, however, both the true damping rates and the scale of frequency fluctuations can be determined.  

We emphasize that these fluctuations will always couple into the system as given by Eqs. (<ref>) - (<ref>), but they will not always be significant enough to require the use of the fluctuating model S_jk(Δ ; σ_ω_0).  The relevant frequency scale for a two-sided cavity is κ_tot, as seen in Fig. <ref>(b): when σ_ω_0≪κ_tot, the apparent and actual damping rates coincide.  As a benchmark in the intermediate case, one must have σ_ω_0≲ 0.17 κ_tot in order for both the apparent κ_tot^' and r_jk^' to deviate from their true values κ_tot and r_jk by less than 10%.  

Even when σ_ω_0≪κ_tot, however, frequency fluctuations may still have a non-negligible effect on the apparent damping rates of the cavity.  At maxima and minima of ω_0(x), for example, σ_ω_0 will vanish and fluctuations in x will only affect the resonant frequency to second order such that

    δω_0(x) = 1/2∂^2ω_0(x)/∂ x^2δ x^2.

Making the same assumption that δ x is Gaussian distributed with mean zero and variance σ_x^2, these fluctuations will follow a chi squared distribution with one degree of freedom.  If we define the scale associated with these fluctuations to be

    D = σ_x^2/2∂^2ω_0(x)/∂ x^2

then the probability distribution associated with this random variable can be expressed as

    P(Ω) = 1/√(2π DΩ)e^-Ω/2DΘ(Ω/D)

where Θ is the Heaviside step function, such that P(Ω) has support on either the positive or negative real axis depending on the sign of D.  Performing the convolution given by Eq. (<ref>) for this case, we find

    S_jk(Δ ; κ_tot, r_jk, D) = 
    δ_jk - i r_jk√(π)/2κ_tot/Dw(i√((iκ_tot-2Δ)/4D))/√((iκ_tot-2Δ)/4D)

where the branch cut can be made on the negative real axis without any issue since κ_tot > 0.  



As can be seen in Fig. <ref>(a), the effect of these fluctuations is a deformation of S_jk that once again leads to an increase in the apparent total damping rate κ_tot and a decrease in the apparent radius r_jk.  Unlike the case of Gaussian fluctuations, however, S_jk(Δ ; κ_tot, r_jk, D) is now asymmetric with respect to the real axis as a result of the probability distribution only having support over the positive or negative reals, which also gives rise to an apparent shift in the resonant frequency.  We have only displayed the case of D>0 for simplicity; the corresponding trajectory for D<0 can be visualized using the relationship

    S_jk(Δ ; κ_tot, r_jk, -D) = S_jk(-Δ ; κ_tot, r_jk, D)^*

which amounts to reflecting the shape of the trajectory across the real axis and traversing it in the same clockwise orientation.  As before, we obtain the apparent cavity parameters as a function of D by fitting the model S_jk(Δ^' ; κ_tot^', r_jk^') to data generated by S_jk(Δ ; κ_tot, r_jk, D), the results of which are displayed in Fig. <ref>(b).  In this case, we find that in order for the apparent κ_tot^' and r_jk^' to both be within 10% of their true values κ_tot and r_jk, we must have |D|≲ 0.14κ_tot.

Thus, if we tune the cavity to a point x_0 where σ_ω_0(x_0)=0 and |D(x_0)|≪κ_tot, the effect of frequency fluctuations will be negligible and we can extract the correct damping rates of the cavity using the non-fluctuating model S_jk(Δ).  In certain cases it may be sufficient to characterize the cavity only at such points, but in many cases it may be either preferable or necessary to characterize the system over a wide range of the tuning parameter x.  For example, if the cavity is tunable over a range of frequencies far greater than its linewidth then the physical properties of both the system and its environment (such as characteristic impedances) are likely to vary appreciably with the operating frequency ω_0(x), leading to frequency-dependent damping rates.  This model for S_jk allows one to extract this dependence without it being obscured by the presence of frequency fluctuations.



§ KERR CAVITY


We next consider a cavity with a Kerr nonlinearity K, such that the system Hamiltonian takes the form

    H_sys = ħω_0a^†a + ħ K/2a^† 2a^2.

The quantum Langevin equation (<ref>) for the operator a then becomes

    ȧ(t) =    -i[ω_0 + K a^†(t)a(t) ]a(t) 
       - κ_1 + κ_2/2a(t) + √(κ_1)a_1^in(t) + √(κ_2)a_2^in(t)

from which we see that the resonant frequency of the cavity depends on the state of the cavity according to 

    ω_0(a,a^†) = ω_0 + K a^†a.

We will restrict our focus to the regime |K|<κ_tot so a semi-classical treatment is appropriate <cit.>, and further assume that kT≪ħω_0 such that the steady state response of the cavity is a coherent state to good approximation (rather than a displaced thermal state).   We also assume that characterization of this system will be performed with a sufficiently weak driving field such that |K|⟨ a^†a⟩≪κ_tot in the steady state <cit.>.  In this case the cavity response will be linear to a good approximation and the scattering matrix elements (in the absence of frequency fluctuations) will be given by Eq. (<ref>).  

Since the scattered signal carries information about the quadratures X_1 = a^†+a and X_2 = i(a^†-a) of the intracavity field <cit.>, the fluctuations in these operators are what will affect our measurement of the scattering matrix elements.  We therefore express the resonant frequency as

    ω_0(X_1, X_2) = ω_0 + K/4(X_1^2 + X_2^2)

where we've absorbed an overall constant into ω_0.  Fluctuations δ X_1,2 = X_1,2 - ⟨ X_1,2⟩ of these operators in the steady state will be independent and Gaussian, each with zero mean and unit variance.  The resulting frequency fluctuations δω_0 = ω_0(X_1, X_2) - ω_0 will follow a non-central chi squared distribution with two degrees of freedom, whose PDF is given by 

    P(Ω) = 2/|K|e^-2(n + Ω/K) I_0(4√(nΩ/K))Θ(Ω/K)

where n = ⟨ a^†a ⟩ = (⟨ X_1⟩^2 + ⟨ X_2⟩^2)/4 is the average number of photons in the cavity (assuming a coherent steady state) and I_0(z) is the zeroth modified Bessel function of the first kind.  Note that the number of photons n is related to the non-centrality parameter λ of the non-central chi squared distribution according to λ = 4n.  There are two key downsides to working with this probability distribution.  First, the convolution given by Eq. (<ref>) does not readily simplify into an expression in terms of special functions (for which efficient implementations exist in most programming languages), so the numerical integration must be implemented manually.  Second, if we try to fit this model (given by Eqs. (<ref>) and (<ref>)) to experimental data for the scattering matrix element S_jk, then we would have two parameters (K and n) that govern the subtle deviation of the fluctuating model S_jk(Δ) from the non-fluctuating model rather than one.  This may lead to overfitting problems unless either K or n can be determined independently.  

It is therefore convenient to additionally work in a regime where n ≪ 1, such that fluctuations in the resonant frequency can be approximated as

    δω_0≈K/4(δ X_1^2 + δ X_2^2).

which will follow a central chi-squared distribution with two degrees of freedom.  Since the variance of the non-central chi squared distribution increases with n and we've already taken the limit of kT/ħω_0→ 0, this limit as n→ 0 can be thought of as the quantum limit of steady-state frequency fluctuations in a Kerr cavity.  The probability of drawing the value Ω from the random variable δω_0 in this case is given by

    P(Ω) = 2/|K|e^-2Ω/K Θ(Ω/K)

where Θ is the Heaviside step function, such that P(Ω) has support on either the positive or negative real axis depending on the sign of K.  As seen in Fig. <ref>, this approximation works very well up to n ∼ 1/8 and accumulates significant errors by n ∼ 1/2.  For this probability distribution, Eq. (<ref>) simplifies to 

    S_jk(Δ ; κ_tot, r_jk, K) = 
    δ_jk - 2ir_jkκ_tot/Ke^(iκ_tot-2Δ)/KΓ(0,iκ_tot-2Δ/K)

where Γ(a, z) = ∫_z^∞t^a-1e^-tdt is the incomplete gamma function <cit.>.





As before, the effect of these fluctuations is an apparent increase in the total damping rate κ_tot and decrease in the radius r_jk of the resonance circle, as seen in Fig. <ref>(a).  Suprisingly, even as K approaches the cavity linewidth κ_tot the deviation of the resulting scattering matrix elements S_jk from the non-fluctuating model S_jk remains subtle.  As such, this systematic deviation is easily obscured by measurement noise and is therefore likely to be overlooked.  Just as in the case of a chi squared distribution with one degree of freedom P(Ω) only has support on either the positive or negative reals, which leads to both a shift in the apparent resonant frequency and asymmetry in the shape of the resonance circle, but these effects are slight.  The case of K>0 is displayed for simplicity, but the corresponding resonance circle for K<0 can again be visualized by using Eq. (<ref>) with the substitution D→ K.  By fitting the model S_jk(Δ^' ; κ_tot^', r_jk^') to data generated by S_jk(Δ ; κ_tot, r_jk, K) we have obtained the apparent values of κ_tot^' and r_jk^' as a function of |K|, which we present in Fig. <ref>(b).  We find that in order for the apparent κ_tot^' and r_jk^' to both be within 10% of their true values κ_tot and r_jk, we must have |K|≲ 0.35κ_tot.  Thus, this effect only becomes significant as |K| approaches the cavity linewidth.  

Another important case to consider is that of a cavity with a modest Kerr nonlinearity K driven at high enough powers such that the number of photons n in the cavity is non-negligible, a situation that is particularly important for experiments in which it is either necessary or useful to characterize the cavity using a range of input powers.  In this case there is not a closed form expression for the average scattering matrix elements; instead, they must be obtained by numerically evaluating the convolution

    S_jk(Δ ; κ_tot, r_jk, K, n) = 
    ∫_-∞^∞S_jk(Δ-Ω ; κ_tot, r_jk) P(Ω ; K, n)dΩ

where S_jk(Δ-Ω ; κ_tot, r_jk) is given by Eq. (<ref>) and P(Ω ; K, n) is given by Eq. (<ref>).  We analyze the deformation induced by this convolution for fixed Kerr nonlinearity K_0 = κ_tot/10, for which the deformation is negligible in the limit n→ 0 (see Fig. <ref>(b)).



As with the previous cases, this convolution yields an apparent increase in the total damping rate κ_tot and decrease in the radius r_jk of the resonance circle, as seen in Fig. <ref>(a).  Furthermore, the shape of the deformed scattering matrix element S_jk is qualitatively similar to that in the case of a stronger Kerr nonlinearity K and negligible cavity occupation n (compare with Fig. <ref>(a)).  And, once again, the resulting deviation of the scattering matrix elements S_jk from the non-fluctuating model S_jk is subtle to the point of being easily overlooked or obscured by measurement noise.  By fitting the model S_jk(Δ^' ; κ_tot^', r_jk^') to data generated by S_jk(Δ ; κ_tot, r_jk, K_0, n) we have obtained the apparent values of κ_tot^' and r_jk^' as a function of cavity occupation n = ⟨ a^†a ⟩, which we present in Fig. <ref>(b).  We find that in order for the apparent κ_tot^' and r_jk^' to both be within 10% of their true values κ_tot and r_jk when K = κ_tot/10, we must have n≲ 2.6.  It is important to note that this effect is not scale-free; it will become stronger (weaker) for increasing (decreasing) values of K/κ_tot, even if nK/κ_tot is held constant.  

Thus, for cavities with even modest Kerr nonlinearities K≳κ_tot/10 operated at the few-photon level, one will extract damping rates that differ appreciably from their true values if the effect of quantum fluctuations on the scattering matrix elements is not properly taken into account.  Numerous examples of Kerr cavities in this regime are described in the literature <cit.>; based on our work, we believe that there are likely discrepancies between the reported damping rates of such cavities and their true values.  The fact that these apparent damping rates vary with the cavity occupation n is particularly important to account for experimentally since it is well known that the true damping rates of microwave cavities vary with n as well, due to the two-level systems present in the dielectric substrates on which planar cavities are fabricated <cit.>.  As we've shown, however, these discrepancies will persist even in the limit of zero cavity occupation.  By using the model presented in this section, these systematic errors can be mitigated.



§ TUNABLE KERR CAVITY


It is useful to consider the combined effect of tunability and the Kerr nonlinearity on frequency fluctuations, since many tunable cavities contain Josephson junctions which introduce nonlinearity into the system.  Making the same assumptions as in Sections <ref> and <ref>, which led to Eqs. (<ref>) and (<ref>), the frequency fluctuations in this case take the form

    δω_0 = ∂ω_0(x)/∂ xδ x + K/4(δ X_1^2 + δ X_2^2)

which is the sum of a Gaussian-distributed and a chi squared-distributed random variable.  The resulting PDF is the convolution of a Gaussian and chi-squared distribution

    P(Ω)    = ∫_-∞^∞2e^-2Ω^'/K/|K|e^-(Ω-Ω^')^2/2σ_ω_0^2/√(2πσ_ω_0^2)Θ(Ω^'/K)dΩ^'
       = e^2σ_ω_0^2/K^2 - 2Ω/K[1/|K| - erf(√(2)σ_ω_0/K - Ω/√(2)σ_ω_0)/K]

where σ_ω_0 is again given by Eq. (<ref>).  Working with this probability distribution has the same downsides as the non-central chi squared distribution: the convolution of Eq. (<ref>) does not readily simplify in terms of special functions, and it depends on two extra free parameters rather than one.  When |K|/σ_ω_0 is either very small or very large, however, we can once again obtain simple expressions for the average scattering matrix elements.

When |K| ≪σ_ω_0, the resulting probability distribution will be well-approximated by a Gaussian.  This is most easily seen by expanding this distribution's cumulant generating function to second order in K

    C(t)    = 1/2σ_ω_0^2t^2 - log(1-Kt/2) 
       ≈1/2Kt + 1/2(σ_ω_0^2 + K^2/4)t^2

which is identical to that of a Gaussian with mean K/2 and variance σ_ω_0^2 + K^2/4.  More precisely, we can expand Eq. (<ref>) as an Edgeworth series <cit.> such that

    P(Ω) ≈exp[-(Ω - K/2)^22(σ_ω_0^2 + K^2/4)]/√(2π(σ_ω_0^2 + K^2/4))

to lowest order.  Since the higher order cumulants will be c_m = (K/2)^m for integers m>3, the leading order correction to this expression will go as K^3/(4σ_ω_0^2+K^2)^3/2.  Although this approximation is most accurate when |K| ≪σ_ω_0, it actually works very well up to |K| ≈ 2σ_ω_0 and only begins to significantly deviate from the exact result when |K| ≳ 4σ_ω_0, as can be seen in Fig. <ref>.  Thus, for |K| ≲ 2σ_ω_0, the resonant frequency of our cavity will be shifted by K/2 and the average scattering matrix elements can be approximated by Eq. (<ref>) with σ_ω_0→√(σ_ω_0^2 + K^2/4).  In the opposite regime, when |K|≫σ_ω_0, the cumulants of the chi squared distribution will dominate at all orders and the scattering matrix elements can be approximated by Eq. (<ref>) if n ≪ 1, and by Eq. (<ref>) otherwise.






§ RINGDOWN MEASUREMENTS


We now consider the effect of frequency fluctuations on ringdown measurements <cit.>.  This type of measurement is performed by driving the cavity on resonance to a steady state amplitude, turning off the drive, and observing the subsequent decay of the intracavity field, from which the total damping rate of the cavity can be determined.  Before the measurement, the average resonant frequency ω_0 must first be measured, which sets the drive frequency, after which the actual resonant frequency ω_0(t) may fluctuate from this average value by δω_0(t) = ω_0(t) - ω_0 during the ringdown measurement.  We assume that at t=0 the cavity has reached a steady state, at which time the drive is abruptly turned off.  Since the amplitude and phase at t=0 will depend on the detuning of the drive from the true resonant frequency while the cavity approached the steady state, for t>0 the average output voltage (on whichever port is to be measured) can be written

    ⟨ V_out(t>0) ⟩ = V_0(Δ_0) e^-κ_tott/2cos[ω_0(t)t + ϕ(Δ_0)]

where Δ_0 is a function of δω_0(t<0).  As we will see, the details of this dependence become irrelevant as long as care is taken in how the output signal is acquired and processed.  

Since microwave signals are generally too fast to be recorded directly, they are typically mixed down to an intermediate frequency before being recorded <cit.>.  Here we consider a homodyne IQ detection scheme, where the output signal is mixed down to DC using a local oscillator at the drive frequency ω_0, after which both the in-phase (I) and quadrature (Q) components of the signal are recorded <cit.>.  In the presence of frequency fluctuations, these ostensibly DC signals now take the form

    I(t)    = V_0(Δ_0) e^-κ_tott/2cos[δω_0(t)t + ϕ(Δ_0)] 
    
    Q(t)    = V_0(Δ_0) e^-κ_tott/2sin[δω_0(t)t + ϕ(Δ_0)]

which remain oscillatory due to δω_0(t).  However, this oscillatory time dependence is not present in the amplitude of the signal

    A(t) = √(I(t)^2+Q(t)^2) = V_0(Δ_0) e^-κ_tott/2

whose time dependence is a simple exponential decay from which the rate κ_tot/2 is easily determined.  Even if the noise associated with the output signal is sufficiently large that an ensemble of measurements must be performed and averaged, only the initial value of the amplitude will be affected by this process such that κ_tot/2 can still be determined from the rate of decay.  However, the ensemble-averaged amplitude will also acquire a DC offset due to the root mean square amplitude of the noise, which in practice could make it difficult to resolve the exponential decay.

To illustrate how the results of this measurement may be skewed by frequency fluctuations if the amplitude is not directly measured, we consider a single channel homodyne detection scheme in which only the in-phase part of the signal is recorded.  For simplicity, we assume that V_0 and ϕ are constants independent of Δ_0, that ϕ=0, and that the resonant frequency doesn't fluctuate appreciably over the course of a single measurement but does fluctuate over the course of repeated measurements.  In this case the in-phase part of the signal obtained from a single measurement takes the form

    I(t) = V_0 e^-κ_tott/2cos(δω_0t).

Assuming δω_0 is a Gaussian-distributed random variable with mean zero and variance σ_ω_0^2, then the average time series from an ensemble of such measurements will take the form

    ⟨ I(t) ⟩   = V_0 e^-κ_tott/2∫_-∞^∞e^-Ω^2/2σ_ω_0^2√(2πσ_ω_0^2)cos(Ω t)dΩ
       = V_0 e^-κ_tott/2 e^-σ_ω_0^2t^2/2.

Clearly, this signal is no longer ideal for extracting κ_tot; if one tries to fit it to a simple exponential decay, then one will extract a total damping rate skewed by σ_ω_0.  As before, this effect only becomes significant as σ_ω_0 approaches κ_tot.  

Thus, to accurately perform a ringdown measurement in the presence of frequency fluctuations, one should ideally detect the amplitude (or, equivalently, the power) of the signal directly.  As we've shown, this can be accomplished with an IQ measurement, but diode detection is another feasible approach.  How this is best accomplished, or if it can be accomplished at all, will depend on the specifics of the experimental device under consideration.  Regardless, if the signal to noise ratio of a single measurement is sufficiently small that an ensemble of measurements is required to resolve the signal, then it is absolutely necessary to compute the amplitude of the signal prior to averaging in order to avoid the results being skewed by frequency fluctuations.




§ TIME-DOMAIN MEASUREMENTS OF FREQUENCY FLUCTUATIONS


We now turn our attention to a common method for observing frequency fluctuations in the time domain <cit.>.  We again consider a continuous measurement of the scattering matrix elements, except instead of sweeping the detuning across the resonant frequency we keep it fixed at Δ =0 and measure the quadratures of the scattered signal as a function of time.  For small fluctuations δω_0(t) (relative to κ_tot) about this detuning, we can approximate Eq. (<ref>) as

    S_jk(t) ≈δ_jk - r_jk(1+e^4iδω_0(t)/κ_tot)

such that the amplitude and phase of the scattered signal are given by

    |S_jk(t)|    ≈√(δ_jk - 4δ_jkr_jk + 4r_jk^2)
    arg(S_jk(t))    ≈ -ilog[δ_jk-2r_jk/|δ_jk-2r_jk|] -  4r_jk/δ_jk-2r_jkδω_0(t)/κ_tot

to linear order in δω_0(t)/κ_tot, valid for δ_jk≠ 2r_jk.  We therefore see that frequency fluctuations are encoded as fluctuations in the scattered signal's phase to lowest order, while its amplitude remains constant.  Thus, if we can independently determine κ_tot and r_jk then we can easily measure frequency fluctuations in the time domain by sampling the phase of the scattered signal as a function of time and inverting the above relationship.

For frequency fluctuations comparable to the cavity linewidth κ_tot, however, the situation is more complicated for two key reasons.  First, we must use the techniques developed in the preceding sections to determine the parameters κ_tot and r_jk using an appropriate model for S_jk, otherwise our method for finding δω_0(t) from S_jk(t) will be inaccurate.  Second, fluctuations in the resonant frequency will now cause both the amplitude and phase of the scattered signal to vary, so we must measure both quantities as a function of time.  We can simplify the analysis of this time domain measurement of the scattered signal by translating our measurements S_jk(t) along the real axis

    S_jk^C(t)    = S_jk(t) - δ_jk + r_jk
       = -r_jke^-2iarctan(-2δω_0(t)/κ_tot)

so that the center of the resonance circle lies at the origin of the complex plane.  Now, as before, fluctuations in the resonant frequency encode themselves as fluctuations in the phase of the scattered signal according to

    arg(S_jk^C(t)) = -2arctan(-2δω_0(t)/κ_tot)

which can once again be inverted easily to find δω_0(t).  We note that this measurement can be used to corroborate the random-variable model for δω_0 by creating a histogram from the time series of δω_0(t) and comparing to the expected probability distribution P(Ω).  

The primary strength of this time domain measurement is that it allows one to measure the power spectral density of frequency fluctuations, rather than simply treating them as random variables with an effective probability distribution.  However, it also has a major downside: it is only sensitive to those fluctuations occurring on time-scales slower than the time it takes to sample the quadratures of the scattered signal, which is the exact opposite sensitivity range as that of the frequency-domain method we have presented.  Since this sampling rate is usually set by the bandwidth of the measurement, faster sampling means more measurement noise.  This must be reduced far below the level of the phase fluctuations of interest in order to resolve the effect of frequency fluctuations, which in practice means operating at either high powers or long time scales.  Often it is either desirable <cit.> or necessary <cit.> to perform this characterization at the single-photon level, which means this measurement will only be sensitive to very slow noise processes.  

This may suffice in many cases, most notably when the frequency fluctuations have a 1/f spectrum.  This has been shown to be the dominant spectral component of the intrinsic frequency fluctuations present in microwave cavities, which arise due to fluctuating two-level systems in the dielectric substrates on which planar cavities are fabricated <cit.>.  It is also likely to be dominant for cavities tuned by the flux threading SQUID loops <cit.> due to the 1/f flux noise ubiquitous in SQUIDs <cit.>, which is believed to arise from unpaired surface spins.  We do not expect to resolve the frequency fluctuations due to the Kerr nonlinearity using this method, however, since these fluctuations are likely to occur far faster than the measurement times required at the single photon level.  Further complicating this measurement when fast fluctuations are present is the fact that each sample of the scattering matrix element S_jk obtained will correspond to an average taken over all the faster fluctuations.  So, rather than sampling from S_jk one will actually be sampling from S_jk.  

Lastly, it is worth noting that some of these difficulties may be alleviated by using a Pound-locking loop <cit.>, but this technique has thus far only been applied in the case of small frequency fluctuations.  As such, it may have to be extended in order to study frequency fluctuations in tunable and nonlinear microwave cavities.  Furthermore, this technique requires specialized equipment, whereas one can measure the scattering matrix elements using standard equipment (typically a vector network analyzer) present in every lab studying microwave-frequency devices.





§ DISCUSSION


In conclusion, we have shown how frequency fluctuations comparable to the cavity linewidth may arise in tunable and nonlinear microwave cavities, and have presented a model for how these fluctuations deform the trajectories traced out by the scattering matrix elements in the complex plane.  For tunable cavities we have shown how fluctuations in the tuning parameter induce fluctuations in the resonant frequency, and that failing to account for these fluctuations may lead one to extract damping rates that appear to vary with the tuning parameter, which is a common observation in these systems.  For Kerr cavities we have shown that quantum fluctuations in the cavity quadratures induce frequency fluctuations, which can appreciably affect the apparent damping rates of the cavity as the strength of the nonlinearity approaches the cavity linewidth.  However, by using the model we have presented for the average scattering matrix elements as a fitting function for experimental data, one can extract the true damping rates even in the presence of these frequency fluctuations, allowing these cavities to be characterized more accurately moving forward.  Further, we have discussed how ringdown measurements can be performed in the presence of frequency fluctuations so as to accurately extract the total damping rate κ_tot, which can serve to corroborate the true damping rates of cavities.  Finally, we have extended a standard method for measuring frequency fluctuations in the time domain to the case of fluctuations comparable to the cavity linewidth, and compared these two measurement paradigms.  

It is straightforward to extend our model to many other systems that we have not considered explicitly, but for which the analysis will be similar to that presented here.  First and foremost, the convolution of Eq. (<ref>) can be used for any scattering matrix elements S_jk(Δ) and any probability distribution P(Ω).  Since most S_jk's are similar in form to those of the two-sided cavity, we expect similar closed-form expressions exist for this convolution in the case of the probability distributions we've considered.  For other probability distributions, this convolution can be implemented numerically.  Our treatment can also be extended easily to the case of impedance mismatches, the effect of which is generally a rotation of the resonance circle around the off-resonant point (S_jk=δ_jk for a two-sided cavity) and an expansion of its radius <cit.>, transformations which can be induced on S_jk by the addition and multiplication of complex numbers.  Since these operations commute with the convolution, they can be applied to the average scattering matrix elements S_jk in exactly the same way.

For the case of tunable cavities, our treatment is easily extended to the case of multiple tuning parameters.  If ω_0 = ω_0(x_1,...,x_N) and fluctuations in each of the x_i are independent, then Eq. (<ref>) becomes

    σ_ω_0^2 = ∑_i=1^N|∂ω_0(x_1,...,x_N)/∂ x_i|^2σ_x_i^2

to linear order in the fluctuations δ x_i, where σ_x_i^2 is the variance of fluctuations in δ x_i.  We may also apply the same techniques we've developed to any dispersive coupling to the cavity, not just the classical tuning parameter considered here.  Common examples are the dispersive Jaynes-Cummings interaction between the cavity and a qubit, the optomechanical interaction between the cavity and a mechanical resonator, and the cross-Kerr interaction between the cavity and either another cavity or another mode of the same cavity.  In these cases, fluctuations in the state of the coupled system will cause the resonant frequency of the cavity to fluctuate in the same manner discussed here, leading to analogous effects.

For the case of the Kerr cavity, it may be possible to extend our treatment into the regime of single-photon strength for which |K|≳κ_tot.  To do so, one would have to treat the steady-state dynamics beyond the semi-classical approximation, using either numeric <cit.> or analytic techniques <cit.>.  However, such systems may simply require the use of different characterization techniques than those considered here, rendering this extension moot.  A more straightforward extension is to consider a finite temperature, in which case the steady state will be a displaced thermal state.  Fluctuations in the quadratures will once again be Gaussian but they will now have a variance 1+2n_th, where

    n_th = 1e^ħω_0/kT-1

is the average thermal occupation of the cavity.  All the results obtained in Sections <ref> and <ref> can be extended to this case by scaling the Kerr nonlinearity by this new variance according to K→ K(1+2n_th).

Our model can also be applied to the intrinsic frequency fluctuations present in microwave cavities due to the fluctuating two-level systems in dielectric substrates <cit.>.  As improved fabrication techniques are developed to reduce damping rates in these cavities, the scale of intrinsic frequency fluctuations may become comparable to the cavity linewidths, yielding appreciable deformation of the scattering matrix elements.  However, the damping rates in these cavities are usually limited by the presence of the same two-level systems responsible for intrinsic frequency fluctuations <cit.>.  There is considerable experimental effort to understand, characterize, and reduce the effects of two-level systems in microwave cavities <cit.>, due in large part to the detrimental effect they have on the coherence times of superconducting qubits <cit.>.  It remains to be seen whether these efforts will have a significant effect on the ratio σ_ω_0/κ_tot for intrinsic frequency fluctuations.  Finally, our model can be extended to account for intrinsic fluctuations of the cavity damping rates <cit.> by treating these parameters as random variables and performing a second convolution with respect to them.  As in all previous cases, the scale of these fluctuations would have to be comparable to the total damping rate to yield an appreciable effect.



We would like to thank B. Thyagarajan, S. Kanhirathingal, W. Braasch, and C. Ramanathan for helpful comments and discussions.  B. L. B. and A. J. R. were supported by the National Science Foundation under Grant No. DMR-1807785.  M. P. B. was supported by the National Science Foundation under Grant No. DMR-1507383.














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