Analytical and Numerical Modeling and Analysis for Quantum Sensor-Based Experiments Connor Timm Fitzgerald

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Applied Mathematics Master's Theses Engineering Master's Theses

9-2019 Analytical and Numerical Modeling and Analysis for Quantum Sensor-Based Experiments Connor Timm Fitzgerald

Follow this and additional works at: https://scholarcommons.scu.edu/amth_mstr Analytical and Numerical Modeling and Analysis for Quantum Sensor-Based Experiments

Connor Timm FitzGerald

September 2019

Santa Clara University School of Engineering Department of Applied Mathematics Thesis submitted at Santa Clara University in partial fulﬁllment of the requirements for the degree of Master of Science in Applied Math- ematics.

Research Advisor

Thesis Advisor

Department Chair Contents

Abstract

1 Introduction1

2 Hysteresis Analysis for Asymmetric DC SQUID1 2.1 Background...... 1 2.2 Analytical Hysteresis Derivations...... 3 2.3 Simulation Results...... 6 2.4 Summary...... 8

3 Savitsky-Golay Filter Analysis9 3.1 Background...... 9 3.2 Modeling the Measured Signal...... 9 3.3 Savitzky-Golay Filter...... 12 3.4 Summary...... 19

4 Hidden Photon Simulation Analysis 20 4.1 Background...... 20 4.2 Combining Spectra...... 23 4.3 Simulation of a Hidden Photon Experiment...... 25 4.4 Hidden Photon Limit Plots...... 29 4.5 Summary...... 31

5 Conclusion 32

Acknowledgments 33

Bibliography 34 Abstract

The results presented in this applied mathematics Masters thesis are based on some of the research I performed while a Santa Cara University (SCU) student, working alongside experimental physicists Prof. Betty Young (SCU) and Dr. Arran Phipps, Prof. Kent Irwin and others at Stanford University. This thesis focuses on my analytical and numerical analyses, and mathematical modeling of experiments that use Superconducting Quantum Interference Devices (SQUIDs) to make precision measurements of modern physical phenomena. Using a variety of mathematical methods and analyses, I was able to understand and explain the origin of undesirable hysteresis observed in a key SQUID circuit and identify a theoretical criterion that can be used to avoid such instabilities. Next, I analyzed the Savitsky-Golay filter and produced a robust Matlab procedure that can be used to understand how filtering can be used to remove Gaussian noise from a Lorentzian distribution with minimum distortion. Lastly, I developed a simulation package for Dark Matter (DM) Radio, an innovative new physics experiment that uses a tunable superconducting lumped-element resonator to search for light-field dark matter. The package is capable of simulating an entire DM Ra- dio experiment and provides powerful insight into how the experiment can be optimized. I present here an overview of the primary mathematical analysis methods used for all of this work, and show results obtained from some of the calculations and simulations I developed as part of this thesis. 1 Introduction

Quantum sensor-based experiments provide ultrahigh sensitivity that allows physicists to answer fundamental questions and solve important problems that would otherwise seem im- possible. By exploiting purely quantum mechanical phenomena, these experiments are able to avoid issues that would typically hinder a classical experiment, and thus are able to push the experimental envelope. One example of a quantum sensor is a superconducting quantum interference device (SQUID). SQUIDs were developed in 1964 [1] and have been essential readout components for many sensitive low temperature experiments since their invention. The first part of this thesis will demonstrate an analytical and numerical technique that was developed in order to better understand SQUIDs and will place constraints on how to avoid SQUIDs exhibiting hysteretic behavior. In order to conduct a successful experiment, the data that is taken needs to be well understood and modeled. The second part of this thesis will use Monte Carlo simulations to explore signal modeling, filtering, and data analysis routines for Dark Matter Radio, a quantum sensor-based experiment searching for light-field dark matter at Stanford Univer- sity. First the filtering process will be analyzed in the context of using a Savitsky-Golay filter to reduce the noise of a Lorentzian signal. Although the filter analysis will be done specifically in the context of the DM Radio experiment, the analysis can be applied to many different fields such as physics, chemistry, and signal processing. After the filtering process has been defined, simulations of the entire DM Radio analysis procedure will be conducted. The simulations serve as a sanity check to see whether physical data that is taken by the experiment is consistent with current models, a way to check whether a “fake” dark matter signal injected will be detected, and as a technique to determine how the experiment can be optimally done in order to detect a hypothetical dark matter signal.

2 Hysteresis Analysis for Asymmetric DC SQUID

2.1 Background A superconducting quantum interference device (SQUID) uses both the quantum mechanical properties of ﬂux quantization and Josephson tunneling in order to measure changes in magnetic ﬂux at the quantum limit. Flux quantization occurs due to the fact that the superconducting electron pair wavefunction

Ψ(~r, t) = |Ψ(~r, t)|eiφ(~r,t) (1) must be single valued in going around a superconducting loop[2]. This forces the magnetic flux penetrating a superconducting loop to be an integer multiple of the magnetic flux h −15 quantum, Φ0 = 2e ≈ 2.07 · 10 Wb, and the change in phase around the loop will change by 2πn, where n is the number of flux quanta, when traversing a single cycle around the loop[2]. In 1962 Josephson predicted that if a thin normal metal (or oxide or other “weak link”) is placed in between two superconductors, what is now referred to as a tunnel junction, the Cooper pairs will tunnel through the junction[3]. Josephson also predicted that the voltage

1 and current across the junction should be given by: Φ dφ V = 0 (2) 2π dt

I = Icsin(φ) (3) where φ is the diﬀerence in phase between the two superconducting macroscopic wavefunc- tions and Ic is the critical current. The DC SQUID is a superconducting loop with two Josephson junctions and is used traditionally as a highly sensitive magnetic ﬂux to voltage converter.

(a) Circuit symbol for a DC SQUID (b) A simple DC SQUID circuit

Figure 1: On the left (a) is a superconducting loop containing two Josephson junctions, each represented by an “x”. This is the circuit symbol for the DC SQUID. On the right (b) is one example of the DC SQUID being used in a circuit. The voltage across the SQUID is read out by an amplifier. The inductor on the left is inductively coupled to the SQUID. A change in current through the inductor produces a change in flux through the superconducting loop which creates a voltage across the SQUID due to flux quantization and the Josephson effect.

One of the major advantages of using SQUIDs is the ability to read out a signal without ever having to directly probe the system, since the carrier of information is magnetic flux. Coupling to magnetic flux also allows for multiplexing capabilities with a low number of wires, which is essential for low temperature experiments. The circuit model in figure 1b is a very ideal case. In reality there will always be some stray level of inductance from geometric and kinetic factors. Hysteresis will occur if the inductance or critical current becomes large enough. Hysteresis is characterized by two different phase values, and therefore two different voltage values by equation2, satisfying all of the necessary constraints. If this occurs the SQUID readout is no longer reliable.

2 In this thesis, all of the inductance will be assumed to be located on a single side of the DC SQUID loop, making the circuit model asymmetric. This is not a necessary condition, but was the most physically relevant model to SQUIDs that were being tested.

2.2 Analytical Hysteresis Derivations Pictured below in ﬁgure2 is the relevant circuit model for the asymmetric DC SQUID. The SQUID is biased with a current that does not exceed the sum of the critical currents for each junction.

Ibias

I1 I2

J1 J2

Ibias

Figure 2: Circuit Model for DC SQUID with nonidentical Josephson junctions. Asymmetry is assumed and all of the inductance is place on one of the loop junction. Nominal values for the critical current of each junction are are 5 µA and the nominal value for L is 120 pH.

Voltage/Phase Condition : To obtain the voltage condition for the circuit in ﬁgure2 we use Kirchhoﬀ’s loop rule and the voltage equation given in equation2. Φ dφ dI Φ dφ 0 2 + L 2 − 0 1 = 0 (4) 2π dt dt 2π dt

The variables φ1 and φ2 are the respective Josephson phases across each junction, L is the inductance, and I2 is the current passing through the right side of the circuit. We then integrate this equation with respect to time in order to get a magnetic ﬂux condition. Z Φ dφ dI Φ dφ 0 2 + L 2 − 0 1 dt = C (5) 2π dt dt 2π dt Φ Φ 0 φ + LI − 0 φ = C (6) 2π 2 2 2π 1

3 Flux quantization due to the superconducting loop dictates Φ 0 (φ − φ ) + LI + Φ = Φ n (7) 2π 2 1 2 FB 0

where ΦFB is the feedback flux applied to the circuit externally like in figure 1b. Finally, by multiplying both sides by a factor of 2π , we obtain the phase condition. Φ0 2π (LI2) + φ2 − φ1 = −φFB + 2πn = φnet (8) Φ0 Current Condition : By using Kirchhoff’s junction rule and equation3, we get the following three equations.

I1 = Ic1sin(φ1) (9)

I2 = Ic2sin(φ2) (10)

Ibias = I1 + I2 (11)

Solving for the variable φ1 using equations9-11 gives −1 Ibias − Ic2sin(φ2) φ1 = sin (12) Ic1 and when we substitute equation 12 into equation8 we eliminate a variable and obtain the ﬁnal function. 2πLIc2 −1 Ibias − Ic2sin(φ2) φnet = sin(φ2) + φ2 − sin (13) Φ0 Ic1

For any given value of φnet, if multiple values for φ2 satisfy the equation then hysteresis can occur. Therefore non-hystertic behavior implies φnet as a function of φ2 is injective. φnet is a continuous function on the interval [0, 2π], and a continuous function on a closed interval is injective if and only if it is strictly monotonic. Thus, setting the derivative of φnet with respect to φ2 equal to zero is the necessary and suﬃcient criterion for hysteresis.

dφnet 2πLIc2 Ic2cos(φ2) 0 = = cos(φ2) + 1 + q (14) dφ2 Φ0 2 2 Ic1 − Ibias − Ic2sin(φ2)

As a sanity check, suppose Ibias = 0 and Ic2 = Ic1.

2πLIc2 Ic2cos(φ2) 0 = cos(φ2) + 1 + q (15) Φ0 2 2 Ic1 − − Ic2sin(φ2)

2πLI cos(φ ) 0 = c2 cos(φ ) + 1 + 2 (16) 2 p 2 Φ0 1 − sin (φ2)

2πLIc2 0 = cos(φ2) + 2 (17) Φ0

4 2πLIc2 = 2sec(φ2) (18) Φ0

LIc2 1 = βL2 = (19) Φ0 π

βL2 is known as the screening parameter, and equation 19 is a well known criterion for hysteresis in the most simple conﬁguration. Now suppose Ibias = 0 but Ic1 6= Ic2.

2πLIc2 Ic2cos(φ2) cos(φ2) = q + 1 (20) Φ0 2 2 Ic1 − Ic2sin(φ2)

Ic2 2πβL2 = q + |sec(φ2)| (21) 2 2 Ic1 − Ic2sin(φ2)

We are interested in the minimum βL2 that will satisfy this equation. Therefore we want to ﬁnd the value of φ2 that minimizes equation 21 . The value of φ2 that does so is φ2 = nπ. Plugging this value into equation 21 yields:

Ic2 2πβL2 = + 1 (22) Ic1 Which also implies the minimum value for the inductance in the loop, L, is: Φ 1 1 L = 0 + (23) 2π Ic1 Ic2

Finally, suppose Ibias 6= 0.

Ic2 2πβL2 = q + |sec(φ2)| (24) 2 2 Ic1 − Ibias − Ic2sin(φ2)

Again we are interested in the minimum value of βL2 that will satisfy this equation. Un- fortunately the value of φ2 that minimizes equation 24 can no longer be found analytically. However, we can ﬁnd the value numerically. Deﬁne the value of φ2 that minimizes equation 24 as φcrit. Then we get the following equations that constrain the screening parameter and the inductance: Ic2 2πβL2 = q + |sec(φcrit)| (25) 2 2 Ic1 − Ibias − Ic2sin(φcrit) ! Φ0 1 |sec(φcrit)| L = q + (26) 2π 2 2 Ic2 Ic1 − Ibias − Ic2sin(φcrit)

5 2.3 Simulation Results Numerical simulations were performed in order to verify the results above and to ﬁnd the conditions for hysteresis. Figure3 graphically demonstrates the way in which curves were determined to be hysteretic.

(a) Non-hysteretic (b) Hysteretic

Figure 3: A simulated example of both a non-hysteretic curve, (a), and hysteretic curve, (b). In both ﬁgures, plotted on the x-axis is φ2 which ranges from 0 to 2π. Plotted on the y-axis is φnet which is calculated using equation 13. The hysteretic behaviour is characterized by the function φnet not being an injective function as a function of φ2.

For the ﬁrst simulation, bias current was set equal to zero and the left critical current, Ic1 was ﬁxed at 5 µA. Both the inductance and Ic2 were varied and hysteretic behavior was evaluated.

Figure 4: Hysteresis simulation results for zero bias current and Ic1 ﬁxed at 5 µA. Green indicates non-hysteretic behaviour and red indicated hysteretic behavior.

The simulation results were then compared with the theoretical zero bias current results given in equation 23.

6 Figure 5: A plot of the curve between hysteretic and non hysteretic behavior from the simulation results in ﬁgure4 and the theoretical prediction from equation 23. The two agree exactly.

The exact same simulation procedure was carried out with setting bias current equal to zero and ﬁxing Ic2 = 5 µA, and agreement with equation 23 was demonstrated for that case as well. Finally, the non-zero bias current case was analyzed.

Figure 6: Plot of threshold curves between hysteretic behaviour and non-hysteretic behavior for Ic2 ﬁxed at 5 µA

Since hysteresis values occur “above” the bias curves, ﬁgure6 demonstrates that if hysteresis does not occur in the zero bias current case, then it will never occur. Intuitively this makes sense, because as bias current is increased the magnitude of the current circulating in the loop must decrease to stay below the critical current.

7 2.4 Summary Using both analytical and numerical techniques, hysteresis of the DC SQUID in the “superconducting” state was completely characterised. If there is no hysteresis present when bias current is zero, which can easily be veriﬁed by checking equation 23 is not met, then there will never be hysteresis as long as Ibias ≤ Ic1 + Ic2. A more complicated but useful characterization would be to analyze the hysteresis criteria for a SQUID in the “voltage” state. In this state, one now has to consider the resistively- and capacitively-shunted junction (RCSJ) model. Under the RCSJ model there are resistors and capacitors in series with each junction. This introduces four new parameters into the model and is outside the scope of this thesis, but would serve as a logical next step for entirely characterizing hysteresis in the DC SQUID.

8 3 Savitsky-Golay Filter Analysis

3.1 Background Dark Matter (DM) Radio is a innovative new experiment that uses a superconducting lumped element resonator circuit to search directly for light-field dark matter [4]. The idea behind the experiment is to use a LC resonator driven by Johnson noise in order to detect dark matter. In the absence of a dark matter signal, the data would ideally trace out a Lorentzian. In reality, the measured data traces out a Lorentzian with added Gaussian noise, since Johnson noise has a known Gaussian distribution. The distribution of interest to determine whether a dark matter signal is detected is normalized excess power, and in order to calculate normalized excess power we need an approximation of the baseline Lorentzian. To do this, a Savitsky-Golay (SG) filter is applied to the measured signal [5]. The goal of this section is to use simulated data from the DM Radio circuit model in order to find the parameter range of the SG filter that filters out the noise and well approximates the baseline distribution. In addition we want to better understand how the filtering process affects the standard deviation and shape of the normalized excess power distribution.

3.2 Modeling the Measured Signal

Figure 7: The simpliﬁed DM radio circuit model. From left to right is the resonator loop, the transformer loop, and the SQUID readout electronics. The resonator loop is a LC circuit with potential stray resistance. The transformer loop contains an inductor with stray resistance which is coupled to both the resonator loop and the SQUID readout electronics. The Johnson noise from either resistor couples to and rings up the resonator which is detected by the SQUID.

9 Parameter Value Description Lr 52.996µH Resonator inductor Cr 1.9722 nF Resonator capacitor Rr 0 Ω Resonator resistance Mrt 1.0740 µH Resonator-transformer mutual inductance Rn 35.4976 mΩ Transformer loop series resistance Lin + Lt 843.027 nH Total transformer loop inductance τ 10 Hours Sampling time ∆f 0.5 Hz Frequency bin size

Table 1: All circuit parameter used for SG ﬁlter analysis

Using the circuit model from ﬁgure7 and the variable substitution ω = 2πf, the current spectral density through the transformer coil is a Lorentzian given by:

2 4kT Rn |in| = 3 2 (27) 2 w MrtCr R + ω(L + L ) − 2 2 n in t (w LrCr−1) √ A constant ampliﬁer noise current spectral density of ew = 11.616 pA/ Hz is assumed. We deﬁne the total current spectral density:

2 2 Pmean = |in| + |ew| (28)

Figure 8: Plot of the current spectral density Pmean, as given in equation 28. The frequency range used is ± 1kHz from the resonant frequency, fr = 498.773 kHz. This corresponds to 4001 frequency bins.

For each frequency bin, the data is assumed to be taken from independent Gaussian

10 distributions. The equation that models the measured data is:

2 1 −(f−µmeas) 2σ Pmeas = e meas (29) p 2 2πσmeas The parameters for the distribution are:

µmeas=Pmean (30)

P σ =√mean (31) meas τ∆f For simplicity, we deﬁne the normalized standard deviation: 1 σ = √ (32) τ∆f We can then model the measured signal as:

Pmeas = Pmean(1 + σ · Z) (33)

2 1 −z Where Z is the standard normal distribution: Z=√ e 2 2π A rearrangement of equation 33 shows that the unﬁltered normalized excess power distribution is equal to the standard normal distribution:

Pmeas − 1 Z = Pmean (34) σ

Figure 9: Plot of Pmeas overlayed on top of Pmean. Pmeas becomes noisiest near the peak of the Lorentzian because the standard deviation is proportional to Pmean.

11 Pmeas −1 Pmean Figure 10: Plot of unﬁltered normalized excess power, σ . The unﬁltered normalized excess power is assumed to be white noise sampled from the standard normal distribution.

3.3 Savitzky-Golay Filter The SG filter can be thought of as a generalization of boxcar average [5][6]. The two parameters that determine a SG filter are the degree of the polynomial and the window size. For each data point, the SG filter calculates the best fit polynomial of a prescribed degree over a prescribed window range centered around the point. The filtered value is the best fit polynomial evaluated at the point that the fit is centered around. Only even degree polynomials need to be considered and the window size must be an odd number. We define the SG filtered data as SGPmeas. For any prescribed polynomial degree, as the window size increases the signal is smoothed out and the noise is reduced. However, the signal also becomes distorted when compared to the original Lorentzian. This can be seen as the shape of the filtered data tracing out a Lorentzian that is wider and has a reduced maximum value. To quantify the smoothing effect, first we consider the normalized excess power distribution:

SGPmeas − 1 X = Pmean (35) σ Note that equation 35 is identical to the equation for normalized excess power as given equation 34, only the ﬁltered data has been substituted in for Pmeas. For suﬃciently small window sizes the distribution in 35 can be approximated by a normal distribution with:

µX = 0 (36)

σX = σReduced (37)

Ideally we want σReduced to be as small as possible.

12 Likewise, we can consider the normalized excess power distribution: Pmeas − 1 Y = SGPmeas (38) σ Again, note that equation 38 is identical to the equation for normalized excess power as given equation 34, only now the ﬁltered data has been substituted in for Pmean. For suﬃ- ciently small window sizes the distribution in equation 38 can be approximated by a normal distribution with:

µY = 0 (39)

σY = σEst (40)

Ideally we want σEst to be equal to 1.

th th Figure 11: Plot of σReduced versus SG ﬁlter window size for 4 and 6 degree polynomials. th th The window size that gives the minimum value for σReduced is 263 (4 ) and 379 (6 ). The value for σReduced begins increasing past these window sizes due to the distortion near the peak of the Lorentzian.

th th Figure 12: Plot of σEst versus SG ﬁlter window size for 4 and 6 degree polynomials. The th th window size that returns a value closest to 1 for σEst is 321 (4 ) and 453 (6 ). The value for σEst begins exceeding 1 past these window sizes due to distortion near the peak of the Lorentzian.

13 It would appear that the ideal window sizes for 4th and 6th degree polynomials would therefore be values between 263 and 321 and 379 and 453 respectively, since this method would keep σReduced small and give σEst values close to 1. However, this method does not explicitly take into account distortion the SG ﬁlter has near the peak of the Lorentzian. To better understand the distortion, Figures 13-20 plot the normalized excess power distribution as given in equation 35 for 4th and 6th degree polynomials and various window sizes:

Figure 13: Left (blue): Plot of normalized excess power as given in equation (9) for a SG ﬁlter with polynomial degree 4 and window size 51. Right (red): Plot of Pmean for reference.

Figure 14: Left (blue): Plot of normalized excess power as given in equation (9) for a SG ﬁlter with polynomial degree 4 and window size 133. Right (red): Plot of Pmean for reference.

14 Figure 15: Left (blue): Plot of normalized excess power as given in equation (9) for a SG ﬁlter with polynomial degree 4 and window size 217. Right (red): Plot of Pmean for reference.

Figure 16: Left (blue): Plot of normalized excess power as given in equation (9) for a SG ﬁlter with polynomial degree 4 and window size 301. Right (red): Plot of Pmean for reference.

Figure 17: Left (blue): Plot of normalized excess power as given in equation (9) for a SG ﬁlter with polynomial degree 6 and window size 75. Right (red): Plot of Pmean for reference.

15 Figure 18: Left (blue): Plot of normalized excess power as given in equation (9) for a SG ﬁlter with polynomial degree 6 and window size 175. Right (red): Plot of Pmean for reference.

Figure 19: Left (blue): Plot of normalized excess power as given in equation (9) for a SG ﬁlter with polynomial degree 6 and window size 275. Right (red): Plot of Pmean for reference.

Figure 20: Left (blue): Plot of normalized excess power as given in equation (9) for a SG ﬁlter with polynomial degree 6 and window size 375. Right (red): Plot of Pmean for reference.

16 It looks as though the ideal window range is around 101 to 201 for the 4th degree polynomial and 175 to 275 for the 6th degree polynomial. Analyzing only the standard deviations as given in equations 37 and 40 did not work because as the window size increases, the distortion near resonance always get worse. However, there is almost no distortion far off resonance. Thus, for the standard deviation as defined in equation 37, there are two com- peting factors as a function of window size. The two factors are the smoothing off resonance, which reduces the standard deviation, and the smoothing near resonance, which initially reduces the standard deviation, but then increases the standard deviation as the window size increases and distortion gets worse. The periodic structure of the normalized excess power in figures 13-20 is due to the fact that a SG filter acts like a low pass filter that also passes certain high “frequencies”. This can be seen by calculating the Fourier transform of the the SG filter. In order to calculate the Fourier transform, we use the fact that SG filtering is a convolution process. Thus we can take the ratio of the Fourier transform of the filtered and unfiltered data to get the Fourier transform of the filter:

F (SGPmeas) F (SG Filter) = (41) F (Pmeas)

Figure 21: Transmission of a SG filter with polynomial degree 4 and window size 151. The x-axis has arbitrary units, but it can be seen that the the filter has a very flat pass band region and attenuates only certain “high frequency” values.

17 Figure 22: Transmission of a SG filter with polynomial degree 6 and window size 225. The x-axis has arbitrary units, but it can be seen that the the filter has a very flat pass band region and attenuates only certain “high frequency” values.

In order to choose the degree of the polynomial, we should consider the type of data that is expected. As the Q factor of the Lorentzian increases, we will need to use a higher degree polynomial. But as can be seen in figures 21 and 22, a higher degree polynomial also requires additional high “frequency” components. This leads to less of the “high frequency” noise being attenuated, and thus a larger window size is necessary for the same level of filtering. Another aspect of the SG filter to consider is how the filter affects the detection of 5σ events, since a 5σ event will indicate a potential dark matter candidate. To calculate this, 5,000,000 different signals are generated for each window size. The signals are normalized according to the following equation:

Pmeas − 1 SGPmeas ≈ Z (42) σ · σest where Z is the standard normal distribution. Note that equation 42 is very similar to equation 38. The extra factor of σest in the denominator is there to account for the fact that as shown in Figure 12, equation 38 does not always have a standard deviation of 1. Excess power that exceeds 5σ is ﬂagged. Each distribution has 4001 frequency bins, so the expected number of false positive events due to random ﬂuctuations for 5,000,000 independent normal distributions is:

N = 5,000,000 · 4001 · P (Z ≥ 5) = 5,000,000 · 4001 · P (Z ≤ 5) = 5734 (43)

18 Figure 23: Percent error of 5σ counts for a SG ﬁlter with polynomial degree 4 and window range from 51 to 351. Initially, the ﬁlter reduces the absolute percent error for the positive and negative 5σ counts equally. Distortion near the peak of the Lorentzian eventually causes the percent error to exceed 100 and the percent error of positive events to be larger than negative events.

Figure 24: Percent error of 5σ counts for a SG ﬁlter with polynomial degree 6 and window range from 51 to 501. Initially, the ﬁlter reduces the absolute percent error for the positive and negative 5σ counts equally. Distortion near the peak of the Lorentzian eventually causes the percent error to exceed 100 and the percent error of positive events to be larger than negative events.

3.4 Summary Although the results in this section only apply to a specific set of circuit parameters and are not general, the analysis procedure can be generalized to different circuit setups and will be very useful for determining SG filter parameters for the actual DM Radio experiment. Current questions that are being investigated are how to SG filter without distortion when the Q-factor of the Lorentzian becomes very large and the costs and benefits of performing multiple SG filter passes with a small window size as opposed to a single pass with a larger window size.

19 4 Hidden Photon Simulation Analysis

4.1 Background

It is estimated that dark matter constitutes ∼ 85% of the total mass in the Universe, yet it is still a mystery as to what comprises dark matter, and what fundamental physics underlies it[7]. One well motivated newer candidate is the hidden (or dark) photon [8][9]. The hidden photon is a hypothetical, massive spin-1 gauge boson that is expected to weakly interact with traditional photons through kinetic mixing. The basic physics that governs hidden photons will be introduced as it pertains to Dark Matter(DM) Radio[4]. In this section we will present results from Monte Carlo simulations of a hypothetical hidden photon search and walk through the data analysis procedure. The data analysis equations can then be inverted in order to produce theoretical limit plots and provide insight into how an optimal experiment can be done. Hidden photons can be thought of as “heavy” (i.e. non-zero mass) photons, which are proposed to interact with ordinary photons through kinetic mixing [10]. If one assumes hidden photons constitute all of the observed dark matter in the universe, it can be shown that this kinetic mixing leads to an “effective” current density filling all of space given by [10][11] 3 2 ~ mhpc p imhpc t Jeff = −ε0 2µ0ρDMexp zˆ (44) ~ ~

where ε is the kinetic mixing angle, mhp is the mass of the hidden photon, and ρDM = 7.21 · 10−5 J·m−3 is the local dark matter density[12]. The mass of the hidden photon can be 2 converted to a frequency using the energy relation hfhp = mhpc . In the quasi-static limit, the “eﬀective” current density in 44 leads to a real oscillating magnetic ﬁeld.

2 ~ mhpcp imhpc t r ˆ B = −ε 2µ0ρDMexp φ (45) ~ ~ 2

Using a rectangular geometry with inner radius y1, outer radius y2, and height h, the magnetic ﬂux through a single loop can be calculated.

2 2 2 mhpcp imhpc t h(y2 − y1) Φ = −ε 2µ0ρDMexp (46) ~ ~ 4 Finally, using Faraday’s law we can calculate the EMF induced from the oscillating ﬁeld

2 2 2 2 2 dΦ mhpc p imhpc t Nh(y2 − y1) V = −N = iε 2µ0ρDMexp (47) dt ~ ~ 4c where N is the total number of loops. The Fourier transform of a complex exponential is a delta function, thus the voltage spectral density of the hidden photon as seen by the resonator should be equal to:

2 √ 2 2  mhpc 2 Nh(y2 −y1 ) ε( ) 2µ0ρDM If f = fhp  ~ 4c∆f |v| = (48)  0 If f 6= fhp

20 Note that the entire expression in equation 48 is just a constant at the hidden photon frequency. Thus the hidden photon will trace out the exact same shape as the Johnson noise, which is also a constant in the frequency domain, and appear as excess power at the frequency of the hidden photon. Following the same procedure and using the same circuit model outlined in the SG ﬁlter section, circuit parameters from Table2 are used in order to generate Pmean, the signal seen by the transformer loop, from 450 to 550 kHz.

(a) Circuit model (b) The resonator

Figure 25: Figure (a) is the circuit model used for hidden photon analysis. The Johnson noise rings up the LC resonator at all frequencies and the hidden photon rings up the resonator only at a single frequency. Figure (b) is the actual lumped element resonator. Superconducting wire is wrapped 40 times around the Teﬂon support and connected to a sapphire wafer which acts as the capacitor.

Parameter Value Description Lr 52.996µH Resonator inductor Cr 1.9722 nF Resonator capacitor Rr .0052 Ω Resonator resistance Mrt 1.0740 µH Resonator-transformer mutual inductance Rn 0 mΩ Transformer loop series resistance Lin + Lt 843.027 nH Total transformer loop inductance τ 10 Hours Sampling time ∆f 0.5 Hz Frequency bin size y1 .117 cm Inductor inner radius y2 3.80 cm Inductor outer radius h 4.95 cm Inductor height N 40 Number of turns in inductor coil

Table 2: All circuit value parameters used to carry out single capacitance hidden photon analysis. If scanning is implemented, all values stay ﬁxed and the capacitance is tuned to hit desired frequency values.

Gaussian noise is added to Pmean in order to create Pmeas. The noise has mean equal to

21 zero and standard deviation equal to: 1 σ = √ (49) τ · ∆f

A frequency value in the 450-550 kHz range is chosen which generates PHP. The total power is then given by PTot = Pmeas + PHP (50)

By SG ﬁltering PTot we can construct an approximation to Pmean. Denote this power th spectrum as SGPTot. In our analysis a 14 degree polynomial and a window size of 65 were used. Therefore normalized excess power can be approximated by:

PTot − 1 R = SGPTot (51) σ · σCorrection(d, m)

Where σ is as defined in equation 49 and σCorrection is the correction factor used in order to account for the reduction in standard deviation due to SG filtering. For the SG parameters used in this analysis σCorrection = .9201. A typical normalized excess power spectrum can be seen in figures 26-27.

Figure 26: Normalized excess power as deﬁned in equation 51 with no hidden photon signal.

22 Figure 27: Histogram of normalized excess power as deﬁned in equation 51 with no hidden photon signal. A Gaussian ﬁt is plotted in red. The mean is approximately zero and the standard deviation is approximately one, as expected.

4.2 Combining Spectra Suppose we had multiple normalized excess power spectra as deﬁned in equation 51 with diﬀerent circuit parameters. In order to construct a single combined normalized excess power spectrum we take a weighted sum.

n X RFinal = wiRi (52) i=1 The optimal weights [11] for each normalized excess power spectrum is the signal to noise ratio (SNR). PHP,i wi = (53) SGPTot,i An example of a single weighted normalized excess power spectrum appears in ﬁgure 28

23 Figure 28: Rescaled excess power is plotted in black with arbitrary units on the left. On the right (in red) is a plot of the Lorentzian, Pmean. The SNR is largest at the peak of the Lorentzian and decreases substantially oﬀ resonance as ampliﬁer white noise becomes the dominant noise source.

For analysis purposes, we need to think about what we would expect a hypothetical hidden photon signal to look like. In the absence of a hidden photon, the mean of the normalized excess power spectrum would be zero. If there were a hidden photon signal in a single bin, then it would have mean: PHP,i µRi = (54) SGPTot,i · σ · σCorrection Therefore, after weighting we would expect the bin with the hidden photon to have a mean: 2 PHP,i µwiRi = 2 (55) SGPTot,i · σ · σCorrection And standard deviation: PHP,i σwiRi = (56) SGPTot,i Assuming each spectrum is independent we therefore have:  0 If no hidden photon  µRFinal = (57) P 2 Pn HP,i  i=1 2 If hidden photon SGPTot,i·σ·σCorrection v u n 2 uX PHP,i σR = t (58) Final SGP 2 i=1 Tot,i

And ﬁnally we normalize RFinal so that it has standard deviation equal to 1. R S = Final (59) σRFinal

24  0 If no hidden photon   µS = r (60) P 2  Pn HP,i  2 If hidden photon  i=1 (SGPTot,i·σ·σCorrection)

σY = 1 (61)

If a ﬁve sigma excess is detected for the distribution given in equation 60 at frequency f = fhp with excess S(fhp) = δ, then the estimated mixing angle can be calculated by backtracking through the circuit and solving for ε in equation 48. This leads to the equation √ 4cL δ · ∆f · σ · σ ε = T correction (62) √ q n 2 2 2 4 P 1 Mrt(2πfhp) 2µ0ρDMNh(y2 − y1) i=1 4 2 |Zi(fhp)| SGPTot,i(fhp) where Zi(fhp) is the impedance of the resonator loop for the ith capacitor value at the detection frequency.

4.3 Simulation of a Hidden Photon Experiment Using the procedures described above we simulated a scan from 450kHz to 550 kHz by varying the capacitance of the resonator. For this simulation values of fhp = 498.773 kHz and ε = 3 · 10−10 were used with scan times of 10 hours at each capacitance value. The ﬁrst interesting thing to notice is that the kinetic mixing angle is small enough so that even if one got really lucky and the resonance frequency was exactly the hidden photon frequency, the experimenter would still not see the signal in a single scan.

Figure 29: A signal scan with the resonant frequency equal to the hidden photon frequency. As can be seen in the normalized excess power, the hidden photon signal is indistinguishable from the noise because it does not exceed 5σ.

25 Now consider a scan with 10 equal steps in capacitance over the frequency range. We take the sum of spectra that look like the one given in Figure 28 in order to generate distribution given by equation 52.

Figure 30: Rescaled excess power for 10 diﬀerent capacitance values added together.

Finally we normalize in order to generate the distribution given in equation 59.

Figure 31: Rescaled excess power for 10 diﬀerent capacitance values added together, normalized according to equation 59.

26 Figure 32: Histogram of the total rescaled normalized excess power. A Gaussian ﬁt is plotted in red. Data appears to be just noise sampled from the standard normal.

There were no 5 sigma events detected, indicating our scan was not dense enough.

Next a scan with 1000 steps in capacitance was done. We take the sum of spectra that look like the one given in 28 in order to generate distribution given by equation 52.

Figure 33: Rescaled excess power for 1000 diﬀerent capacitance values added together. It looks like there may be a hidden photon signal, but the data has to still be normalized.

Finally we normalize in order to generate the distribution given in equation 59.

27 Figure 34: Rescaled excess power for 1000 diﬀerent capacitance values added together, normalized according to equation 59. There is an event that exceeds 5 times the standard deviation at precisely the hidden photon frequency

Figure 35: Histogram of the total rescaled normalized excess power. A Gaussian ﬁt is plotted in red. Data not at the hidden photon frequency appears to be just noise sampled from the standard normal.

The analysis code estimated the hidden photon frequency to be 498.773 kHz and ε= 2.8745· 10−10. The reduction in the estimated mixing angle ε can be understood from the fact that the SG ﬁlter does not fully remove the hidden photon power at the hidden photon frequency, and thus the signal is slightly lowered when dividing by the ﬁltered distribution during the normalization process.

28 4.4 Hidden Photon Limit Plots Ultimately, what will be essential to the DM Radio experiment is setting a limit plot for frequency (which can also be converted thought of as mass) versus kinetic mixing angle ε [13]. Equation 62 allows a theoretical limit plot to be made by entering in circuit value parameters and fixing δ, the average normalized excess power produced by the hidden photon at frequency fhp. Since we are looking for 5σ events, we will fix δ = 5. The hidden photon frequency will be varied and the kinetic mixing angle εThresh required to produce a 5σ event will be calculated at each frequency. Ideally, kinetic mixing angle values larger than εThresh would be detected and values smaller would not be detected since they do not produce a 5σ excess on average. Thus εThresh gives the theoretical line between kinetic mixing angles which should and should not be detected on average at a given frequency fhp. Being able to generate these hypothetical limit plots will greatly aid in determine an optimal scan strategy, as will be demonstrated in the following example. In this example we will consider an experiment which ranges over 450-550 kHz regime and scans by changing the capacitance value since that is what will be done in the actual experiment. The circuit value parameters will be the same as those in Table2. In the first case we will consider an experiment which does not scan and uses the same capacitance value the whole time.

Figure 36: A single capacitance scan, Cr = 1.97 nF, with red indicating values that the experiment should be sensitive to as given in equation 62. The experiment is highly sensitive at the resonance frequency of the circuit, fr = 499 khZ, but sensitivity decreases very quickly away from resonance.

An ideal experiment would maximize the amount of red area in a plot like figure 36. However we can see that in figure 36 the experiment is sensitive near resonance but there is still a lot of parameter space that is not excluded off resonance. An experiment with 10 linearly space capacitance value will be considered next to show the usefulness of scanning.

29 Figure 37: A scan with 10 linearly spaced capacitance values from 1.68-2.31 pF, with red col- ored indicating excluded values.The scan excluded much more area than a single capacitance scan, but there are still gaps.

Figure 37 demonstrates that much more exclusion area can be added by allowing the capacitance values to vary, however there are still clearly gaps that could have been excluded if more capacitance values were used.

Figure 38: A scan using 25 linearly spaced capacitance values from 1.68-2.31 pF, with red coloring indicating excluded values. Large gaps are mostly ﬁlled, but the scan is still not dense enough to have no gaps

Figure 38 eliminates the larger gaps that existed in ﬁgure 37, however the resonance values

30 do not overlap yet and there are still smaller gaps.

Figure 39: A scan using 75 linearly spaced capacitance values from 1.68-2.31 pF, with red coloring indicating excluded values. The scan is suﬃciently dense, as there are no longer gaps. Thus an experiment would not need to step any ﬁner and use more capacitance values, as it would do just as well using these 75 values.

Finally, figure 39 is dense enough that there are no longer any gaps, indicating that more capacitance values would no longer increase exclusion area and therefore the scan is sufficiently dense. This aids the DM Radio experiment because ideally the capacitance value is tuned as little as possible, and for a given set of circuit parameters this method allows one to find the theoretical minimum number of capacitance values necessary to eliminate gaps in the exclusion plot.

4.5 Summary By injecting a “fake” hidden photon signal blindly and successfully measuring it, we were able to demonstrate that the analysis procedure outlined in this section should work for the actual DM Radio experiment. Furthermore we veriﬁed that the simulation code is sensitive at the level that the theoretical limit plots predict. The theoretical limit plots will be used in order to decide scan strategies and how to ensure that exclusion area is maximized. Looking forward, the most relevant question is analyzing what happens when we move from distributing all of the hidden photon power into a single frequency bin to distributing the power over many bins with a lineshape given by the Standard Halo Model [11].

31 5 Conclusion

In this thesis, a wide array of general analytical and numerical methods were developed with specific applications to quantum sensor-based experiments. An analytical expression to determine whether a DC SQUID in the “superconducting” state would exhibit hysteresis was derived and then verified by simulations. Monte Carlo simulations were performed in order to understand how to appropriately choose Savitsky-Golay filter parameters when filtering a Lorentzian distribution with Gaussian noise. The analysis procedure for a hypothetical hidden photon search using normalized excess power was proven to work and agreed with theoretical predictions. In the future, we hope to answer some of the more open ended and complicated questions posed at the end of each section.

32 Acknowledgments

First I would like to acknowledge Professor Chiappari, Ph.D., who was extremely helpful and supported me as both thesis adviser and department chair. Second, I would like to acknowledge Professor Betty Young, Ph.D., of the Santa Clara University (SCU) physics department who acted as the research adviser for this thesis and assisted with editing and provided invaluable mentorship throughout my entire time at SCU. Third, I would like to acknowledge the entire Irwin Group at Stanford University for providing me with the op- portunity to conduct the research presented in this thesis, and speciﬁcally I would like to acknowledge Dr. Arran Phipps, Dr. Saptarshi Chadhuri, Carl Dawson, Professor Kent Ir- win, and Stephen Kuenstner for their insightful discussions. I would also like to acknowledge Professor Guy Ramon, Ph.D., of the SCU physics department for allowing the use of his computer cluster for some of the larger Monte Carlo simulations. Finally, I would like to acknowledge Dr. Geoﬀ and Josie Fox for generously providing summer funding which made this thesis possible.

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