Fundamental Limits of Electromagnetic Axion and Hidden-Photon Dark Matter Searches: Part I - The Quantum Limit
We discuss fundamental limits of electromagnetic searches for axion- and hidden-photon dark matter. We begin by showing the signal-to-noise advantage of scanned resonant detectors over purely resistive broadband detectors. Building on this calculation, we discuss why the optimal detector circuit must be driven by the dark-matter signal through a reactance (an equivalent inductance or capacitance); examples of such detectors include single-pole resonators, which are used broadly in axion and hidden-photon detection. Focusing thereafter on reactively coupled detectors, we develop a framework to optimize dark matter searches using prior information about the dark matter signal. Priors can arise, for example, from cosmological or astrophysical constraints, constraints from previous direct-detection searches, or preferred search ranges associated with the QCD axion. We define integrated sensitivity as a figure of merit in comparing searches over a wide frequency range and show that the Bode-Fano criterion sets a limit on integrated sensitivity. We show that when resonator thermal noise dominates amplifier noise, substantial sensitivity is available away from the resonator bandwidth. The optimization of this sensitivity is found to be closely related to noise mismatch with the amplifier and the concept of measurement backaction. Additionally, we show that the optimized one-pole resonator is close to the Bode-Fano limit. The Bode-Fano constraint establishes the single-pole resonator as a near-ideal method for single-moded dark-matter detection. We optimize the integrated sensitivity in a scanned tunable resonator search by optimizing time allocation using priors. We derive quantum limits on resonant search sensitivity. We show that, in contrast to some previous work, resonant searches benefit from quality factors above one million, which corresponds to the characteristic quality factor (inverse of fractional bandwidth) of the dark-matter signal. We also show that the optimized resonator is superior, in signal-to-noise ratio, to the optimized reactive broadband detector at all frequencies at which a resonator may practically be made. At low frequencies, the application of our optimization may enhance scan rates by a few orders of magnitude. Finally, we discuss prospects for evading the quantum limits using backaction evasion, photon counting, squeezing and other nonclassical approaches, as a prelude to Part II.
- I Introduction
- II A Comparison of Reactive Resonant and Resistive Broadband Coupling Schemes
- III Coupling to the Dark-Matter Signal
IV Sensitivity Calculation of a Search for Light-Field Dark Matter
- IV.1 Scattering Representation of Resonant Detection Circuit
- IV.2 Scattering Representation of A Single-Moded Reactive Detector
- IV.3 Signal-to-Noise Ratio of Search
V Search Optimization
V.1 Optimization of Matching Network
- V.1.1 Value functional for matching network optimization in a log-uniform search
- V.1.2 Matching network optimization for log-uniform search: the Bode-Fano constraint
- V.1.3 Matching network optimization for log-uniform search with single-pole resonator
- V.1.4 Discussion of single-pole resonator matching optimization for log-uniform search
- V.1.5 Evading the Bode-Fano Constraint
- V.1.6 Optimization of matching network for candidate signal
- V.1.7 Comparison to other calculations
- V.2 Optimization of Time Distribution
- V.1 Optimization of Matching Network
- VI Fundamental Limit: Resonant Detector Read Out by a Quantum-Limited Amplifier
- VII Conclusions
- A The Electromagnetic Properties of the Axion and Hidden-Photon Dark-Matter Signal
- B Amplitude, Direction, and Distribution Variation in the Dark-Matter Field
- C The Standard Halo Model
- D Noise Correlations for Quantum-Limited, Phase-Insensitive Amplifiers in the Scattering Mode Representation
- E Flux-to-Voltage Amplifiers
- F Scan Optimization for Tunable Resonator Searches with Flux-to-Voltage Amplifiers
- G Comparison of Resonant and Broadband Searches with Flux-to-Voltage Amplifiers
A significant body of astrophysical and cosmological evidence points to the existence of dark matter, which comprises 27% of the mass-energy in the universe Ade et al. (2014). Dark matter is a direct window to physics beyond the Standard Model. The vast majority of experimental efforts to directly detect cold dark matter have focused on weakly interacting massive particles (WIMPs) Agnese et al. (2016); Akerib et al. (2017); Aprile et al. (2012). These searches, conducted over the past few decades, have so far yielded no detections and have placed strong constraints on the existence of WIMP dark matter. The lack of a detection of weak-scale dark matter motivates a search for other dark-matter candidates.
Another possibility for cold dark matter is an ultralight boson with mass below 0.1 eV (in contrast to the GeV mass scale of WIMPs). The extremely low mass of an ultralight boson implies a large number density because the local dark-matter density is GeV/cm. As a result, these bosons are best described as classical fields oscillating at a frequency slightly greater than their rest frequency, ( is the rest mass of the dark matter, is the speed of light, and is Planck’s constant); the actual oscillation frequency is slightly higher than as a result of the small kinetic energy. Two prominent candidates in the class of ultralight bosons are axions and hidden photons. The “QCD axion” is a spin-0 pseudoscalar originally motivated as a solution to the strong CP problem Peccei and Quinn (1977a, b) that can also be dark matter. However, spin-0 pseudoscalar dark matter may exist even with parameters that do not solve the strong CP problem. Such particles are sometimes referred to as “axion-like particles.” In this work, we refer to both QCD axions and axion-like particles as “axions.”
Axions may be produced nonthermally (as would be required for a sub-eV particle to be cold, nonrelativistic dark matter) through the misalignment mechanism Dine and Fischler (1983); Preskill et al. (1983). One may search for axions via their coupling to the strong force Budker et al. (2014) or their coupling to electromagnetism Sikivie (1983, 1985). The latter interaction, described by the Lagrangian
will be discussed further in this paper. In particular, in the presence of a DC magnetic field, the axion is converted to a photon. We will focus on this detection technique, a decision that may be motivated by the above Lagrangian. In the limit of a stiff axion field–the relevant practical limit– equation (1) demonstrates that the amplitude of the photon signal grows with the amplitude of the background electromagnetic field. We would thus like the background field to be as large as possible. In laboratory searches, DC fields can be orders of magnitude larger than AC fields. As shown in Appendix A.2, the effect of DC electric fields is suppressed, relative to their magnetic counterparts, by the virial velocity of the dark matter , so a background DC magnetic field is optimal.
The hidden photon is a spin-1 vector. Such particles emerge generically from models for physics beyond the Standard Model, often from theories with new U(1) symmetries and light hidden sectors Holdom (1986). The hidden photon was initially described as a dark-matter candidate in Nelson and Scholtz (2011) and is further investigated in Arias et al. (2012). Like axions, hidden photons may be produced through the misalignment mechanism. They may also be produced during cosmic inflation. In fact, a vector particle in the 10 eV- 10 meV mass range produced from quantum fluctuations during inflation would naturally have the proper abundance to be a dominant component of the dark matter Graham et al. (2016). One may search for hidden-photon dark matter via its coupling to electromagnetism Chaudhuri et al. (2015), which arises from kinetic mixing:
A traditional WIMP detector registers energy deposition from the scattering of a single dark-matter particle with a nucleus or electron. For an ultralight boson, such a measurement scheme is not appropriate. The energy deposition would be too small to be measured. Instead, it is possible to search for the weak collective interactions of the dark-matter field. For example, as a result of their coupling to electromagnetism, the effect of the hidden-photon or axion field may be modeled as an effective electromagnetic current density. (See Appendix A.2 for more details.) For the hidden photon, the direction of this current density is set by the vector direction, whereas for the axion, it is set by the direction of the applied magnetic field required for axion-to-photon conversion. These current densities produce observable electromagnetic fields oscillating at frequency slightly greater than , which couple to a receiver circuit and may be read out with a sensitive magnetometer, amplifier, or photon detector.
In ADMX Sikivie (1983, 1985); Asztalos et al. (2010), HAYSTAC Brubaker et al. (2016), Axion LC search Sikivie et al. (2014a), and DM Radio Chaudhuri et al. (2015); Silva-Feaver et al. (2017), the receiver takes the form of a tunable high-Q resonant circuit. If dark matter exists at a frequency near the resonance frequency, the electromagnetic fields induced by the dark matter ring up the resonator. The signal in the resonator is read out by a Superconducting Quantum Interference Device (SQUID) amplifier or a near-quantum-limited parametric amplifier. By tuning the resonator across a wide frequency range, one may obtain strong limits on light-field dark matter. In this manner, a search for axion- or hidden-photon dark matter operates much like an AM radio.
The amplifiers in these searches couple to a single mode of the resonator. This is clearly not always the optimal receiver circuit. For instance, if the circuit has multiple resonant modes (at different frequencies) that do not interact parasitically with each other, and each couple to the electromagnetic fields from the dark matter, one may increase the scan rate, relative to a single-moded resonator. The principal purpose of this work is to determine the properties of the optimal receiver to search for the electromagnetic coupling of axion- and hidden-photon dark matter.
We start by asking: What are the characteristics of the optimal receiver? To answer this question, we must first understand the basic structure of a receiver, irreducible noise sources in such receivers, and the role of impedance matching and amplifier noise matching.
The elements of a dark-matter receiver circuit are shown as a schematic block diagram in Fig. 1. 111One may note that the readout of a receiver signal with phase-insensitive amplifier constitutes coherent detection, which has been studied extensively in the context of radio astronomy for the past several decades Zmuidzinas (2003). In fact, one way to model the detection circuit is as an astronomical receiver coupled very weakly to a photon source, i.e. the dark-matter field. This paper is organized around optimizing each of these blocks, and globally optimizing the blocks and their interactions across a full scan.
Central to every detection circuit is an element that couples to the dark-matter field. The element may be reactive, e.g. an inductive pickup coil that couples to the magnetic field produced by the dark matter or a capacitance that couples to the electric field. We may alternatively use a purely resistive detector. The resistive detector may be a phased antenna array presenting a real impedance to the electromagnetic field, or a simple resistive sheet. Electric fields induced by dark matter would drive current flow in a resistive sheet, resulting in a dissipated power. 222Whether it is better to couple to the electric field or the magnetic field is a subject of Section III. An important role is played by electromagnetic shielding. Over the next two sections, we will explain why a single-pole resonator is superior (in terms of signal-to-noise ratio (SNR)) to a broadband resistive sheet. (Similarly, we also show that a single-pole resonator is superior to a broadband reactive detector, such as that presently used in ABRACADABRA Kahn et al. (2016). See Appendix G.) We will in fact make the more general statement that the optimal detector has the property that it is driven by the dark-matter-induced electromagnetic fields through a reactance. The explanation relies on understanding the requirements, formulated in ref. Chaudhuri (tion), for coupling efficiently–or “impedance matching”– to power available in the dark-matter field. Thereafter, since a single-pole resonator is a prime example of a reactive detector, we will focus solely on reactive coupling. Every detection circuit has some loss (i.e. some resistance), which produces thermal noise. If the circuit is cold and , where is Boltzmann’s constant and is the physical temperature, one will observe the effects of the zero-point fluctuations in the detector. We refer to these three components (dark-matter signal, thermal noise from loss, and zero-point fluctuation noise) collectively as the “Signal Source” in Fig. 1. This is the first element in the receiver model.
The second element of every receiver is an impedance-matching network. The impedance matching network is used to provide a better match between the complex reactive impedance of the pickup element and the input impedance of an amplifier, which typically possesses a real part or is purely real. A single-pole LC resonator is an example of a matching network. It uses a capacitor or network of capacitors to transform the inductive pickup coil to a real impedance on resonance, as seen by the readout. In the high-frequency cavity limit (e.g. ADMX), each mode can be modeled as an equivalent RLC circuit serving as a matching network. Another matching network is a multi-pole resonator, which, for instance, could have many LC poles at the same frequency. One may also use broadband inductive coupling, where a pickup coil is wired directly to the input of a SQUID (e.g. ABRACADABRA). The signal source and matching network may contain mechanical elements, such as piezoelectrics and crystal oscillators which behave electrically as RLC circuits. We will restrict our attention to linear, passive circuits.
The third and final element is an amplifier or photon counter to read out the signal passed through the impedance matching network. Photon-counting and quantum-squeezing techniques are being developed for light field dark-matter detection Zheng et al. (2016), but we will focus here on readout with phase-insensitive amplifiers, which coherently amplify both quadratures of an incoming signal with the same gain Caves (1982). The application of photon counting and quantum squeezing in the context of this comprehensive consideration of axion- and hidden-photon detection through electromagnetic coupling will be left to future work. We will also assume that the output of the amplifier is not fed back to the detection circuit in an active feedback or feedback damping scheme, which can effectively improve the impedance matching between signal source and amplifier Rybka (2014); Seton et al. (1995). Feedback will also be addressed in future work.
In a readout with a phase-insensitive amplifier, both quadratures of the signal are amplified equally and analyzed for a signal. As dictated by the Heisenberg uncertainty principle, the experimental sensitivity is subject to a second source of effective noise, which is noise added by the amplifier. The limitation set by Heisenberg for phase-insensitive measurement is sometimes referred to as the Standard Quantum Limit (SQL). In the high-gain limit, the quantum limit on added noise corresponds to a minimum noise temperature of one-half photon, Caves (1982). 333The noise temperature of the amplifier is defined as follows. The resistor at temperature produces a thermal noise spectral density proportional to the Bose-Einstein thermal occupation number ; see equations (59), (60), and (271). We may define an added noise number as the increase in thermal occupation number such that the increase in thermal noise equals the amplifier noise. The noise temperature is then given by . This definition is equivalent to that used in Ref. Clerk et al. (2010), but differs from that in Ref. Caves (1982). We stress that this difference is simply one of convention, rather than one originating in physical principles.
In particular, every amplifier has two effective noise sources: imprecision noise and backaction noise. The imprecision noise simply acts to add some uncertainty to the output of the amplifier, independent of the input load. The backaction noise injects noise into the input circuit. This noise, having been filtered according to the impedance of the input circuit, is added to the amplifier input and appears as additional noise on the output. In contrast to imprecision noise, backaction noise (referred to the amplifier input) is inherently dependent on the source impedance. The spectral densities of both noise contributions can be frequency-dependent. Consider the simple case where the matching network possesses negligible loss, so that the only loss is that in the signal source (and the coupling to the input impedance of the amplifier). When the noise temperature is minimized with respect to the source impedance, it is said that the input circuit is noise matched to the amplifier. Clerk et al. (2010); Wedge (1991) If, in addition to noise matching, the imprecision and backaction noise modes obey certain relations Clerk et al. (2010) (see Appendices D and E.2 for more information), then the quantum limit on noise temperature of can be achieved.
The optimal receiver circuit therefore has the following characteristics:
The detection circuit should be reactively coupled and should present an impedance match to the electromagnetic signal from the dark matter. In other words, it should couple as efficiently as possible to the power available in the local dark-matter ”source.”
The energy coupled into the detector from the dark-matter signal increases with the detector volume, so the reactive detection element should possess as much coupled volume as is practical.
The loss in the detection circuit should be as low as possible and as cold as possible. Larger receivers may have more loss and be more difficult to get cold than smaller receivers, so this third characteristic must be optimized in combination with the second. Furthermore, some sources of loss are reduced at low temperatures (e.g. loss from electrons in superconductors), and some are increased (e.g. saturable two-level systems in a dielectric), so a practical optimization may be complicated.
The characteristics of the impedance-matching network (second block in Fig. 1) and the amplifier should permit the noise temperature (with respect to input load) to be minimized, subject to the quantum limit. This minimization should be achieved at as large a range of frequencies as possible to reduce scan time of the full search.
The first criterion is discussed in depth in Chaudhuri (tion), which presents a modified circuit model for understanding the dark-matter electromagnetic interaction. There, it is shown that a single-pole resonator, in principle, possesses the key attributes required for an efficient impedance match to the dark matter. Nevertheless, the intrinsic loss needed (in a resonant detector or otherwise) to optimally match to, or backact on, the dark-matter signal is far too low to be achieved in any practical experiment. The electromagnetic signal induced by dark matter can be assumed to be a stiff signal, thereby motivating the second and third criteria.
The second and third criteria are constrained by familiar experimental limitations. The detector volume will be limited by constraints on the fabrication of detector materials and the maximum possible size of the shield, which surrounds the detector and is required in order to block external electromagnetic interference. The loss in the circuit can be optimized by using superconductors, although their use can be precluded in axion searches if the detector and the applied DC magnetic field are not spatially separated Asztalos et al. (2010); Brubaker et al. (2016). The temperature is usually limited by the performance of the cryogenic refrigeration scheme; the use of a dilution refrigerator enables mK operating temperatures.
The last criterion, and more generally, the concepts of amplifier impedance matching and noise matching as they apply to dark-matter detection, will be the primary focus of the paper. Ideally, one would possess an amplifier that is quantum-limited (capable of reaching the half-photon noise temperature with appropriate noise match) and noise-matched to the input circuit at all frequencies. However, such a device has not been experimentally demonstrated. In fact, as we will show, if the input impedance of the quantum-limited amplifier is real-valued (which is the circumstance for practical quantum-limited microwave amplifiers, such as Josephson parametric amplifiers Castellanos-Beltran et al. (2008)), then the Bode-Fano criterion Bode (1945); Fano (1950) places a powerful constraint on the sensitivity of a single-moded detection circuit integrated across the search band. A single-pole resonator is approximately 75% of the Bode-Fano fundamental limit. Together with the circuit model framework of Chaudhuri (tion), the Bode-Fano limit motivates the single-pole resonator as a near-ideal technique to detect the dark-matter field.
Having produced this general framework for understanding circuits to detect electromagnetic coupling to ultralight dark-matter fields, we now return to single-pole resonant detectors. The thermal noise and the dark-matter signal are both filtered by the impedance-matching network. Therefore, if the filtered thermal noise from the loss in the signal source dominates the amplifier noise (i.e. the physical temperature of the resistor is larger than the minimum amplifier noise temperature), then the sensitivity is independent of the detuning from resonance. It is to our advantage to maximize the bandwidth over which thermal noise dominates amplifier noise, while maintaining high sensitivity. This bandwidth may be considerably larger than the resonator bandwidth. A unique aspect of light-field dark-matter searches is that a quantum-limited amplifier is desirable even when the frequency being probed satisfies . Some previous work only accounted for the information available within the resonator bandwidth and therefore underestimates the sensitivity of resonant searches. Chaudhuri et al. (2015); Kahn et al. (2016); Sikivie (1983, 1985)
As stated before, a fundamental feature of a resonant search is that one must scan the resonance frequency over a broad range. In this work, we discuss the optimization of scan strategies. We show that this optimization is closely related to the optimization of noise matching. When combined with the aforementioned constraints on impedance matching to the dark-matter signal and amplifier, this optimization results in a fundamental limit on the sensitivity of axion and hidden-photon searches read out with a phase-insensitive amplifier coupled to an induced electromagnetic signal.
The outline of this paper is as follows. In Section II, we focus on two simple examples of light-field dark-matter detectors: a resistive sheet and a tunable cavity resonator. In an apples-to-apples comparison, the cavity is found to be superior to the resistive broadband detector. We contrast this behavior with the absorption of vacuum electromagnetic waves that satisfy (unmodified) Maxwell’s equations. A similar result is found in comparison of a resonator and a reactive broadband search, such as ABRACADABRA Kahn et al. (2016); details are left for Appendix G. This appendix should be read after the full quantitative machinery for analyzing signal-to-noise ratio has been introduced; the SNR framework is constructed in Section IV. The comparison in Section II allows us to introduce many of the concepts which are mentioned in Fig. 1, and which will be of primary concern in this paper.
In Section III, we further examine and categorize the schemes for coupling to the dark-matter signal. Using the results from Chaudhuri (tion), we explain why the optimal detector must be driven by the electromagnetic fields induced by dark matter through an equivalent-circuit reactance, i.e. an inductor or capacitor. The advantage of resonators over broadband resistive sheets is explained in terms of the “drag” on electrons due to visible-photon radiation. We also elucidate the role played by electromagnetic shielding, which is a practical requirement in any dark-matter search. In the limit that the characteristic size of the shield is much smaller than the Compton wavelength , the dominant observable is a dark-matter-induced magnetic field. One should then couple to the signal using an inductor. In the limit that the shield size and Compton wavelength are comparable, the induced electric and magnetic fields are comparable. There is no advantage to a capacitive coupling over an inductive coupling. As such, for the remainder of the main text, we consider solely inductively-coupled detectors, while pointing out in various sections similar results for capacitively-coupled detectors. To provide a practical perspective for the analysis that follows, we discuss a variety of single-pole resonant detection schemes proposed or in use for ultralight field dark-matter detection. We discuss more specifically how such detectors are excited by electromagnetic fields induced by dark matter, and consider how such detectors are read out. Subsequent to this section, we will focus on a simple example, where the detector is coupled to a transmission line and read out by an amplifier operated in the scattering mode. Clerk et al. (2010) This is representative of the type of receiver used in ADMX and HAYSTAC. In Appendices E-G, we will consider a different example, directly related to DM Radio and the Axion LC search, where the resonator is read out with a flux-to-voltage amplifier (see Appendix E for a definition and description), such as a dc SQUID or dissipationless rf SQUID. The description is somewhat more complicated in this latter example, but the primary conclusions regarding resonator optimization and fundamental limits are the same as those found in the main text for the scattering-mode case.
In Section IV, we derive the signal-to-noise ratio (SNR) for a scanning search using a scattering matrix representation of the coupling (e.g. resonator) and amplification circuits. The analysis will apply not only to resonators, but to any single-moded reactive coupling scheme. This level of generality is critical in enabling us to set a limit on detection sensitivity with the Bode-Fano criterion. We give a brief quantitative discussion of the dark-matter signal as it relates to the scattering representation, leaving a more detailed treatment, including considerations of temporal and spatial coherence, for Appendix A. As will be discussed, the detection scheme is equivalent to a Dicke radiometer Dicke (1946) used in radio astronomy. We discuss the signal processing steps of a Dicke radiometer as they relate to the detection circuit. The SNR is calculated in two parts. First, given a fixed dark-matter search frequency, we calculate the SNR from a single instance of the scan, e.g. a single resonance frequency. In doing so, we derive an optimal filter for the signal processing which maximizes SNR Wiener (1949). Second, using the results for the single instance, we calculate the SNR for a scan, which is comprised of many resonant frequencies. We determine the weighting of the data from the various scan steps that yields the highest possible SNR. As an example, we then provide an explicit calculation of SNR (for both the single scan step and whole scan) for a quantum-limited scattering-mode amplifier with uncorrelated backaction and imprecision noise modes, for which the noise impedance and input impedance are real-valued and equal. This amplifier and its noise properties are discussed further in Appendix D . The results on SNR for a quantum-limited phase-insensitive amplifier sets a fundamental limit on the detection sensitivity of a search for electromagnetic coupling to ultralight dark matter. The optimization of a search using a quantum-limited phase-insensitive amplifier is the basis for the remainder of the main text. A similar SNR calculation for flux-to-voltage amplifiers is carried out in Appendix F.
In Section V, we discuss the optimization of the search. The optimization is carried out in two steps. In the first part, for a fixed scan step, we optimize the impedance-matching network. This network need not be resonant. We determine a value function for evaluating the merits of a matching network and optimize two types of searches. The first type is a “log uniform” search, to be defined in Section V, which makes the natural assumption of a logarithmically uniform probability for the mass and coupling of dark matter within the search band. It is here that we establish a limit on detection sensitivity with the Bode-Fano criterion, which constrains broadband matching between a complex impedance (the signal source) and a real impedance (the amplifier input). The second type is a candidate-signal search, where a signal has already been found in a previous search and one wishes to probe the signal as well as possible. In each of the two cases, we link our result to the concepts of noise matching, amplifier backaction, and sensitivity outside of the resonator bandwidth. We show that the optimized resonator is approximately of the Bode-Fano limit, thereby motivating the single-pole tunable resonator as a near-ideal single-moded detector for probing ultralight-field dark matter. Hereafter, we will only discuss resonators in the main text. We discuss the impact of these results on scan time. In the second step, given a fixed total search time, we determine the optimal allocation of time across resonant scan steps for the log-uniform search. We introduce the notion of a dense scan, where each search frequency is probed by multiple resonant frequencies. We also briefly consider other possible value functions for time allocation based on prior assumptions about the probability distribution of dark matter.
Similar optimizations are carried out in the appendix. In Appendix F.2, we derive a Bode-Fano constraint for an inductively-coupled detector read out with a quantum-limited flux-to-voltage amplifier possessing real-valued noise impedance. In Appendix F.3, we consider resonator optimization with a quantum-limited flux-to-voltage amplifier possessing uncorrelated imprecision and backaction noise (real noise impedance). The result is the same as that derived for the quantum-limited scattering mode amplifier. We again find that the optimized single-pole resonator is approximately 75% of the Bode-Fano limit. In Appendix F.4, we carry out the optimization for the log-uniform search after relaxing the assumptions of uncorrelated imprecision and backaction noise and minimum noise temperature equal to one-half photon.
The optimization of the resonant scan in Section V, in combination with the Bode-Fano criterion and the circuit model insights, yields a fundamental limit on the performance of axion and hidden-photon dark-matter searches read out by a phase-insensitive amplifier. We calculate this limit in Section VI. Owing to the identical results provided in the scan optimization, the limit is the same for scattering-mode and flux-to-voltage readouts. We analyze various features of this limit. In particular, we discuss the parametric dependence of the fundamental limit on quality factor and contrast it with previous works. Chaudhuri et al. (2015); Sikivie (1983, 1985) We show that use of the optimized scan strategy can increase the sensitivity to dark-matter coupling strength by as much as 1.25 orders of magnitude at low frequencies. This corresponds to an increase in scan rate of five orders of magnitude.
We conclude in Section VII. We provide directions for further investigation, with attention paid to topics that will be covered in Part II of this work. In particular, we discuss prospects for evading the Standard Quantum Limit of the dark-matter measurement using backaction evasion, squeezing, entanglement, and photon counting in the context of this comprehensive optimization framework.
Ii A Comparison of Reactive Resonant and Resistive Broadband Coupling Schemes
As shown in Fig. 1, one may couple to an electromagnetic signal induced by the dark-matter field using a reactive element or a purely resistive element. A purely resistive search can be modeled as a sheet of finite conductivity in free space. The resistive sheet is a model for a physical resistor or an array of antennas presenting a real impedance. An incident electromagnetic field induced by a dark-matter field drives currents in the sheet. Ohm’s Law dictates that these currents must dissipate power, which may be detected, for instance, with bolometric photon counting, or by antenna-coupled amplifiers.
Consider initially an electromagnetic plane wave governed by unmodified Maxwell’s equations, normally incident on a resistive sheet with vacuum on both sides. The absorption is maximized when the sheet impedance is set to . (Here, is the thickness of the sheet. For simplicity of analysis, is set to be much less than the skin depth at the frequency being absorbed. is the vacuum permeability.) This setup absorbs 50% of the power available in the field. With the addition of a quarter-wave backshort and a sheet impedance of , one may absorb 100% of the power. However, the addition of the backshort introduces an effective reactance in the problem, and limits the bandwidth over which the coupling is efficient.
In light of the fact that a purely resistive sheet is a relatively efficient detector of “normal” electromagnetic fields over wide bandwidth, one might ask why they are not used commonly to detect fields induced by dark-matter axions or hidden photons. Why does one choose a resonator over a broadband resistive absorber? After all, resistive absorbers are a critical piece of many millimeter-wave astronomy instruments Stahl (2005), which also seek to detect small photon signals from cosmic sources.
To answer this question, we perform an apples-to-apples comparison of the sensitivity of a broadband resistive sheet and a tunable cavity. We first describe the experimental setup and then calculate the signal size in terms of power dissipation. This is followed by a brief SNR analysis (a more rigorous version of which will appear in Section IV.3). We compare the SNRs at each dark-matter rest-mass frequency in our search band and show that the SNR for the cavity is always larger. In fact, we show that the advantage in SNR scales as the square root of the quality factor, a result that will be explored in far greater depth in Section VI.
See Fig. 2. In the introduction, we stated that the hidden-photon and axion fields can be modeled as effective current densities that produce electromagnetic fields. They may also be modeled more directly as electromagnetic fields themselves. For the hidden photon, representation as an effective current is known as the interaction basis, while representation as an effective electromagnetic field is known as the mass basis Graham et al. (2014). Relative to the electric field, the free-space magnetic field induced by dark matter is suppressed by the velocity of the dark matter . This velocity arises from the virial velocity, in combination with the detector velocity (Earth velocity) in the galactic rest frame (the frame in which the bulk motion of the dark matter is zero). As such, the dominant observable is an effective electric field. We denote this effective electric field by in the figure and assume that this field is stiff. We will revisit this assumption below.
Owing to the nonzero velocity from virialization and Earth’s motion, the electric field induced by dark matter is not monochromatic, but rather, possesses a nonzero bandwidth. The dark-matter velocity gives a dispersion in kinetic energy and therefore, a bandwidth
The dark-matter signal thus spans the frequency range . We have made explicit here the dependence of the dark-matter bandwidth on search frequency to indicate that bandwidth is not constant across the search range, but is expected to grow linearly. Nevertheless, in the preliminary calculation in this section, we assume that the cavity linewidth is larger than this bandwidth, so that for the purpose of calculating experimental sensitivity, we may treat the dark-matter signal as monochromatic.
We assume that the cavity separation is smaller than the de Broglie wavelength of the dark matter, which sets the coherence length of . This is an appropriate assumption, given that for dark-matter signals near the fundamental resonance frequency, where the response is strongest, the length satisfies , the Compton wavelength of the dark-matter signal, and is thus much less than the de Broglie coherence length . We also assume that the lateral extent of the cavity is much smaller than the de Broglie wavelength, so that we may treat as spatially uniform. Furthermore, we assume that the lateral extent of the cavity is much larger than , so that we may ignore fringe-field effects and effectively reduce the three-dimensional problem to a one-dimensional problem.
In Fig. 2a, we show a sheet of conductivity . For simplicity, we assume that the conductivity is frequency-independent. Frequency-dependence has no significant bearing on our conclusion that a tunable cavity is fundamentally superior to a broadband resistive sheet. The sheet lies in the x-z plane. Electric fields tangent to the surface dissipate power in the sheet.
In Fig. 2b, we show two sheets separated by length : one of frequency-independent conductivity and the other possessing perfect conductivity. These two sheets form a resonant cavity. If the frequency of the electric field satisfies (or nearly satisfies) the resonance condition
then the dark-matter fields ring up the cavity. The rung-up fields dissipate power in the left-side sheet. We assume that only a single detector mode is used: the fundamental resonance at . Although one might use multiple resonance modes or wider band information across the search range (the latter of which will be discussed in detail later in this paper), in this section we only consider signal sensitivity within the resonator bandwidth of this single mode. The cavity resonance frequency may be tuned by changing length .
We assume that the dark-matter electric field lies in the direction. For the axion, this may be arranged by applying a DC magnetic field in the direction. We assume that the magnetic field is uniform. For the hidden photon, the experimentalist does not control the direction of the electric field. However, this is not of consequence for the sensitivity comparison. Misalignment with the detector will result in the same multiplicative reduction in sensitivity for both the sheet and the cavity.
Under these assumptions, the dark-matter electric field can be approximated as
We may relate the complex amplitude of this electric field to the complex amplitude of the z-component of the mass-basis hidden-photon vector potential, denoted Chaudhuri (tion). The relation is
One may write a similar relationship for the axion pseudoscalar potential, whose complex amplitude is :
where is related to the more traditional axion-photon coupling by
( is the vacuum permittivity.), and is the magnitude of the applied magnetic field.
We calculate the steady-state power dissipated in the finite conductivity sheet in each of the two experiments. This will be the signal size that we use to compare experimental sensitivities. Note that, in the case of antenna-coupled amplifier readout, as compared to bolometric readout, we typically discuss signals in terms of power received rather than power dissipated. However, if the bolometer (sheet) and the antenna/amplifier setup present the same real impedance and possess the same receiving area, then the signal size is the same. Ohm’s Law at each sheet dictates that the volumetric current density in each sheet is related to the electric field by
where the subscript indicates whether the resistive sheet belongs to the broadband detector (“r”) or the cavity (“c”) and is the total electric field in the sheet. There is also a current in the perfect conductor–the right-hand sheet in the cavity–to screen electric fields from its interior and maintain the boundary condition that the electric field parallel to the surface vanish. We denote this current density as . The current in each of the three sheets arises both from the “incident” dark-matter electric field and the electric field radiated in response to this drive.
In general, the current density and the electric field will be position-dependent in the sheets. However, because we are ignoring fringing effects and because the thickness of each sheet is set to be much less than the skin depth at frequency , the current density and the electric field are considered to be uniform in each sheet. We may then turn the volumetric current density into an effective surface current density . Setting all sheets to have a common thickness , we find
where is now most readily interpreted as the surface field. We may similarly define an effective surface current density for the perfect conductor, , that keeps the surface field zero. We define the sheet impedances as
We also set the sheets to have a common surface area .
Maxwell’s equations yield a relationship between the electric field and the volume current density at any point in space:
For the broadband detector, reflects the current density for the sheet, while for the cavity, includes both the current densities on the finite conductivity and the perfect conductivity sheets: . We may solve for the current in each sheet, and consequently, the power dissipation using superposition. Because we ignore fringe-field effects, the electric field is only a function of , the coordinate normal to the sheets.
Power Dissipation in Resistive Broadband Detector
We find the steady-state solution for the current and fields. All quantities oscillate at frequency and the fields and currents point in the direction, so we may write
Setting the position of the sheet to be , the current density in eq. (12) is where is a Dirac delta function. The electric field produced by the sheet is then the plane wave
Equation (10) gives
The power dissipated by per unit area is then
The power dissipation is maximized for sheet impedance . At this sheet impedance, the power dissipation is
Interestingly, though the dark-matter electric field is not a wave solution to Maxwell’s equations in vacuum, the sheet impedance at maximum power dissipation is the same as the usual vacuum electromagnetic wave. We will explain this result further in terms of the dark-matter circuit model described in the next section.
Power Dissipation in Cavity Detector
Again, we have set the position of this sheet to . We may write the electric field produced by the perfectly conducting sheet as
Since the electric field must vanish at the conductor, we have
Ohm’s Law gives
We have four equations (21), (23), (24)-(25), and four unknowns, , , , and . The condition that the electric field must be continuous at each sheet, and zero at the perfect conductor, along with Ohm’s Law (10) yields four equations. Solving the system gives the complex current amplitude on the sheet at ,
and a power dissipation per unit area of
For dark-matter resonance frequencies close to resonance, , this equation may be expanded as
The response is a Lorentzian, as would be expected, and the quality factor of the cavity, as determined by the full width at half maximum, is . For an on-resonance dark-matter signal, the power dissipation increases linearly linearly in . Taking the limit (sheet impedance to zero), it seems that arbitrarily large amounts of power can be dissipated. Of course, this is unphysical. Two assumptions that we have made break down. First, for high quality factors giving cavity linewidths narrower than the dark-matter linewidth (), not all parts of the dark-matter spectrum are fully rung up. A more complicated calculation, in which the shape of the dark-matter spectrum is convolved with the resonator response, is needed to calculate the power dissipation. Such a calculation is carried out in Sec. IV. Second, power conservation dictates that at some quality factor, we must begin to backact on the dark-matter electric field, producing enough dark matter through the electromagnetic interaction to locally change the value of the dark-matter density and effectively modify the value of . Understanding the conditions for backaction requires us to understand the impedance properties of the dark-matter electromagnetic interaction. This topic is treated in ref. Chaudhuri (tion). There, it is proved rigorously that, for transverse dark-matter modes (hidden photons propagating in a direction transverse to their polarization, or axions propagating transverse to the applied field), that backaction can be neglected for typical resonator quality factors . The loss required to backact on the dark-matter drive field is far too low to be achieved in practice.
To perform the sensitivity comparison between the resistive broadband and cavity detectors, we assume that the rest-mass frequency is unknown. We set a search band between frequencies and and calculate the SNR from a dark-matter electric field causing power dissipation in the detector. ( may be dependent on the frequency of the dark matter. However, for the purpose of sensitivity comparison, this will not be relevant, so we do not explicitly include it.) While the broadband detector will not change during the search, always being set to the optimal sheet impedance , the cavity resonance frequency will be stepped between and .
We can set the total system noise power to be the same in the broadband search and on-resonance in the cavity for an apples-to-apples comparison of scan sensitivity. The impedance of the cavity in Fig. 2b is real on resonance, and the impedance of the resistance in Fig. 2a is real at all frequencies. We assume that the temperature of the resistance in both cases is the same. We further assume that the added noise power from the device reading out the power dissipation – that is by the bolometer or amplifier – is the same in both cases. In particular, for amplifier readout, we assume that the minimum noise temperature is the same and that the amplifier is noise-matched and impedance-matched (so as to prevent unwanted reflections) to the detector. The total system noise in both experiments, , is dominated by the sum of the thermal Johnson-Nyquist noise power and the noise power added by the device reading out the power dissipation, allowing us to set to be the same.
Additionally, we set the total experiment time, for each experiment, at . We assume that the total experiment time is large enough such that the cavity can ring up at every tuning step and so that the dark-matter signal can be resolved in a Fourier spectrum; the former requires time while the latter requires time . We also assume that the tuning time is negligible compared to the integration time at each step.
where (= or ) is the power dissipated in the experiment and is the integration time at dark-matter frequency . For the broadband search, this integration time is simply the total experiment time , while for the cavity detector, it can be taken as the amount of time during which the dark-matter frequency is within the bandwidth of the resonator, . By using signal information only within the bandwidth of the resonator, we discard valuable information outside of the resonator bandwidth. We will make optimal use of information at all frequencies in later sections, but not in this preliminary calculation. The ratio of the SNRs for the cavity and resistive broadband detectors is then
If, in the cavity search, we spend an equal time at each frequency and step at one part in , the total integration time can be taken as the times the number of frequency steps between and . For broad scans (), the number of frequency steps can be approximated as an integral, so
The SNR ratio in (30) then varies with as
This demonstrates that high-Q cavities are superior to resistive broadband searches for dark matter, even if information outside of the resonator bandwidth is not used. The advantage of high-Q searches will be even larger if information outside of the bandwidth is used, as will be shown later. The SNR in power varies as square root of quality factor. This, in turn, implies that the minimum dark-matter coupling ( for the axion and for the hidden photon) to which the cavity is sensitive scales as . The result may have been intuited from the form of the Dicke radiometer equation: an experiment gains sensitivity much faster with higher signal power (linear relationship) than with longer integration time (square root relationship). Interestingly, as we will show in Sec. VI, the scaling even applies when the quality factor is larger than the characteristic quality factor of the dark-matter signal, for which resonator response and dark-matter spectrum must be convolved.
Our calculation of the cavity response makes clear the following: the dark-matter electric field and the electric fields that are produced in response act on both the sheet of finite conductivity and the sheet of infinite conductivity. That is, they act on both the power detector and the “reflector.” In this sense, we can apply the insights made here to the dish antenna experiment in Horns et al. (2013). In that work, the authors propose to use a spherical reflecting dish to focus the electric field signal induced by dark matter onto a power detector. The authors consider reflection at the dish; radiation and absorption at the feed (the detector) is ignored. However, as we have shown, such a consideration can be critical to calculating accurate response. We must carefully consider the impedance properties of the feed and the feed’s direct interaction with the dark-matter electric field , in addition to the analogous characteristics of the dish. There are two electrical boundary conditions, one at the mirror and one at the feed, which sets intrinsic length and frequency scales for the experiment. Thus, the frequency response of a dish antenna experiment is not a broadband flat spectrum, but has significant frequency-dependence.
High-Q resonators are also superior to reactive broadband searches at all frequencies at which a resonator can practically be made. We show this explicitly in Appendix G where we compare a series RLC circuit read out by a flux-to-voltage amplifier (e.g. a SQUID) to a broadband LR circuit. We leave the quantitative details for later in the paper, after we have introduced the machinery needed to understand the comparison more completely. This will give greater depth to our result, allowing us to rigorously consider the effects of thermal noise, and also illustrate the full power of sensitivity outside of the resonator bandwidth. The result is in direct contrast to that in Kahn et al. (2016), which claims a frequency range in which a broadband search has superior sensitivity.
The comparison of resonant and broadband detection illustrates important differences between absorption of vacuum electric fields governed by the unmodified Maxwell’s equations and absorption of electric fields sourced by dark matter. One cause of this difference is that the electric field induced by dark matter is spatially uniform (and has a single phase) as long as the experimental dimensions are small as compared to the de Broglie wavelength. In contrast, consider a transverse vacuum electromagnetic wave of frequency (solving Maxwell’s equations) propagating in the direction linearly polarized in the direction. Suppose that at the sheet at , the complex electric field amplitude for the incident wave is . Then, the power dissipated in the cavity will be
The difference in the numerator of the right-hand side of (33), relative to (27), ( vs. ) arises from the phase shift of the incident visible electromagnetic field between the two sheets. When the cavity is a half-wavelength at frequency , the power dissipation vanishes. The phase shift results in destructive interference of the electromagnetic waves radiated by the sheets with the incident drive field, and consequently, no signal. The visible signal absorption is minimized at precisely the spacing that maximizes the dark-matter signal absorption. Furthermore, one can easily show that the absorption of a visible signal is maximized when the separation of the sheets is a quarter-wavelength at frequency and . The quarter-wavelength transforms the short (the perfect conductor) into an open, so that all power available in the wave will be absorbed in the sheet if impedance-matched to free space. However, this power per unit area is , which is only a factor of two larger than that absorbed by the purely resistive broadband detector (see eq. (19)). There is no large parametric improvement in the absorption of a visible signal due to the quarter-wave spacing, e.g. no dependence on a small value of as in (27), over the purely resistive broadband detector. This extra factor of two occurs only within a narrow bandwidth. While a quarter-wave transformer may seem like a natural choice for boosting sensitivity to dark matter, given the match to Maxwellian electromagnetic waves, that is not the case; the half-wave cavity is parametrically enhanced for the absorption of dark-matter signals.
A second disinction in the absorption comes from the assumption that the dark-matter electric field can be treated as stiff. This begs the question: what is required to optimally couple power from the dark-matter signal? How do we impedance match to it and how should this inform detection schemes? We now investigate more closely the optimal scheme for coupling to the dark matter–that is, the optimal coupling element in the Signal Source of Fig. 1.
Iii Coupling to the Dark-Matter Signal
In Sec. III.1, we show, based on the concept of impedance matching, that inductively-coupled detectors (or more generally, detectors that can be modeled as coupling to dark matter through an inductance) are optimal. We explain why single-pole resonators are a relatively efficient impedance match to the dark-matter source. In Sec. III.2, we provide examples of single-pole resonators being used for dark-matter searches. The presentation of these examples naturally leads to the scattering matrix formalism of Section IV, which is used to analyze SNR in inductively coupled detectors.
iii.1 Optimizing the Coupling Element
As discussed in the introduction, we may couple to the electric field induced by dark matter via a resistance or capacitance, or couple to the magnetic field induced by dark matter via an inductance. Here, we investigate which of these three couplings is fundamentally best for an experimental search.
A critical measure of a detector’s suitability for a sensitive search is its ability to extract power from the dark-matter field. Ideally, the detector would be able to absorb all of the available power. In traditional forms of electromagnetic detection, designing a detector to absorb maximal power is known as impedance matching, and the optimal impedance match is achieved by setting the detector impedance equal to the complex conjugate of the source impedance. To optimize a light-field dark-matter detector, one must then understand the source impedance properties of the dark-matter field.
Such an exploration is carried out in ref. Chaudhuri (tion). We refer the reader there for a detailed discussion of the dark-matter source. Here, we apply the results of that paper to the present work. Signals described by Maxwell’s equations are often characterized by developing circuit models for the detection process. However, by virtue of the interaction terms (1) and (2), dark-matter electrodynamics is governed by modifications to Maxwell’s equations and cannot be represented with traditional circuit models. In ref. Chaudhuri (tion), extended circuit models are developed that take into account these modifications to Maxwell’s equations. It is demonstrated that the effective source impedance of dark matter is far lower than the free-space impedance . Dark matter can be treated as a spectrum of dispersive waves (see Appendix A.1, as well as Chaudhuri et al. (2015)), following dispersion relation (193). For a hidden photon with transverse polarization, whose mixing angle is and whose frequency and wavenumber are and , respectively, ref. Chaudhuri (tion) shows that the effective source impedance is
where the approximation holds in the non-relativistic limit, and is the hidden photon velocity. The equivalent expression for the axion is given by replacing the hidden-photon mixing angle with an analogous “axion mixing angle”, quantifying axion-to-photon conversion in a background magnetic field of magnitude 444For an axion, an effective transverse mode is defined by the relative orientations of the uniform applied magnetic field and the direction of propagation. The “transverse” axion possesses a direction of propagation orthogonal to the applied field, while a “longitudinal” axion possesses a direction of propagation that is parallel. . The axion mixing angle is defined as
where is as defined in eq. (8). For a virialized QCD axion in a T magnetic field, , so the axion source impedance is 29 orders of magnitude lower than the free-space impedance! Such a low source impedance may have been expected from the weak coupling of dark matter to photons and in turn, to electronic charges and currents.
The low source impedance has important consequences for the sensitivity comparison in Section II. A broadband resistive sheet, like that shown in Fig. 2a, is an efficient radiator of visible photons, which possess wave impedance . We term these waves “visible waves” to distinguish them from the dark-matter waves. Because the visible waves possess much higher impedance than the dark-matter electromagnetic signal, they exert an enormous “drag” on the electrons in the sheet. This limits power absorption from the dark matter, and consequently, the experimental sensitivity. A similar argument may be used if the resistive sheet is coupled to an arbitrary network of lumped elements (assuming that none of these elements are also driven by the dark-matter field); the resistive sheet, driven by the dark matter, radiates visible waves, which prevents power absorption.
In a resistive coupling, one must thus decouple radiated visible waves from the detector using multiple sheet resistors. This is precisely what the half-wave cavity of Fig. 2b does. The phase shift that a visible wave radiated from one sheet accrues in propagating to the other sheet causes destructive interference of the two sheets’ radiated visible waves. The dark-matter signal then provides the only force on the electrons, and efficient power absorption can be obtained. Consequently, the half-wave resonator is superior to a broadband resistive sheet or any resistive sheet coupled to a network of lumped elements. Ref. Chaudhuri (tion) shows that the half-wave resonator can, in principle, couple to as much as 1/2 of the power in a monochromatic, transverse dark-matter wave.
We conclude that we require a distributed structure of multiple resistive sheets in order to achieve efficient matching, which introduces a large reactance into the equivalent circuit model.
One may alternatively couple to the dark-matter source using a lumped inductive or capacitive coupling. Unlike sheet resistors, ideal lumped elements do not radiate and therefore, radiated visible waves do not present a challenge to detection. However, the dark-matter source impedance, when referred as an equivalent resistor (with real-valued impedance) in series with the pickup inductor/capacitor, is very low; it is much lower than the self-impedance of the coupling element. The self-impedance of the inductor or capacitor acts as a similar drag on electrons, limiting power absorption. One must therefore null the self-impedance, which means that the net reactance of the detector must be zero. On resonance, lumped LC resonators possess zero reactance. Like the half-wave resonator, they can also couple to as much as 1/2 of the power in a monochromatic, transverse dark-matter wave. Resonators can thus be an efficient impedance match to the dark-matter signal, whether they are free-space resonators formed by multiple conductors, or lumped-element resonators.
Nevertheless, because of the extremely low dark-matter source impedance, the intrinsic detector loss required for any detector (resonant or otherwise) to match to, and thus backact on, the dark matter is far lower than may be practically achieved. For instance, a cavity resonator requires a quality factor of to achieve optimal match to the QCD axion in a 10 T magnetic field. We may thus treat the dark-matter source as stiff. Having justified the stiff-source approximation, it is useful to think of the dark-matter source as stiff electromagnetic charge and current densities (the interaction basis for the hidden photon), as mentioned in Section I. These charge and current densities source stiff electric and magnetic fields. The charge/current representation differs from the direct electromagnetic field representation used in Section II, but we stress that the two are physically equivalent. The effects of the charge density are suppressed by the virial velocity, so we focus only on the current density. See Appendix A.2 and refs. Sikivie et al. (2014b); Chaudhuri et al. (2015) for a discussion of the current density.
The resistively coupled distributed cavity structures and the lumped-element detectors discussed above are related in that they may be modeled as coupling to the dark-matter effective current through an equivalent circuit reactance (an inductance or capacitance). This is apparent for the inductively and capacitively coupled lumped-element detector, as well as any distributed-element counterpart.
The half-wave, resistively coupled cavity resonator may be modeled as an equivalent lumped RLC circuit. The electric and magnetic field energies contained in the cavity can be mapped onto electric and magnetic fields stored in capacitors and inductors. One may treat the effective current density as driving the capacitance, inductance, or both, as long as the excitation is treated appropriately Pozar (2012). More generally, for a resistive coupling with a distributed structure, one may treat the dark-matter effective current as driving the LC cavity modes of the structure. It is critical to note that the detector is not simply the resistive sheet, but the entire structure. Consider, for instance, a resistive sheet placed within an electromagnetic shield, where the signal read out by the observer is the power dissipation in the sheet. The shield acts as a cavity, and therefore, its internal electromagnetic fields can be decomposed into an orthonormal basis of modes. Each cavity mode can be treated as an equivalent RLC circuit, and the dark-matter effective current, on- or off-resonance, may be chosen to drive either the inductance or capacitance of each of these circuits. Hill (2009); Graham et al. (2014) The resistive sheet inside the cavity acts as the dissipation mechanism (the ”R” of the RLC) for each of these modes. For each mode, the excitation from dark matter results in some complex-valued electric field amplitude at the surface of the broadband sheet. The total electric field at the sheet surface is a coherent sum, over all modes, of these electric field amplitudes. In other words, the total electric field–and thus, the power dissipation signal–is determined by filtered summation of all LC cavity excitations. Thus, the detector may again be treated as being driven by dark matter through equivalent reactances.
We conclude that the optimal coupling element is thus inductive or capacitive. The same conclusion is reached more rigorously and quantitatively in ref. Chaudhuri (tion). Should we choose inductive or capacitive coupling? An important role is played by electromagnetic shielding, which is necessary to avoid unwanted interference and dissipation.
In the limit where the size of the detector and shield are much smaller than the Compton wavelength (the lumped-element limit), the current may be treated quasi-statically. The size of the magnetic field produced by the current is then much larger than the electric field (in terms of energy). Thus, the optimal detector in the lumped-element limit couples to dark matter via an inductance rather than a capacitance. The excitation of the capacitor is negligible.
In the limit where the size of the detector and shield are comparable to the Compton wavelength (the cavity limit), the magnetic field and electric field are comparable in size. If using lumped-element detectors, one could couple to either a capacitance or an inductance. There is no advantage either way. However, due to the challenge of significant stray impedances in this limit, it is common not to use lumped-element couplings, but rather mode couplings, e.g. a resonant free-space cavity. As discussed above, we can model its equivalent RLC circuit as being excited either electrically or magnetically (or both). Thus, without loss of generality, we can choose to model a cavity-limit circuit as coupling to an inductance, as in the case of the lumped-element limit.
For the remainder of the paper, we will focus on inductively coupled detectors, since they are appropriate for modeling both the lumped-element and cavity limit. Where useful, we will also point out equivalent results for capacitively coupled detectors.
We will now develop the SNR framework to optimize the impedance-matching network and readout, represented by the second and third boxes of Fig. 1. The analysis will culminate in a limit on detection sensitivity set by the Bode-Fano criterion. The limit will demonstrate the relatively high efficiency of searches with single-pole resonators. In other words, we will show that these resonators are near ideal for single-moded detection.
First, to enable a practical understanding of the broadly applicable SNR framework, we give some concrete examples of single-pole resonators that may be analyzed with the described framework. Our examples will span Sections III.2 and IV.1. Section IV.2 lays the foundations for the more general treatment of any inductively coupled detector.
iii.2 Single-Pole Resonant Receiver Circuits for Axion and Hidden-Photon Detection
Two implementations of resonant receiver circuits are shown in Fig. 3. For frequencies below 1 GHz, the signal may be amplified with a Superconducting Quantum Interference Device (SQUID). Chaudhuri et al. (2015) This detection scheme is displayed in the left panel of the figure. The dark-matter effective current, through its induced magnetic field, feeds a flux through the pickup inductor . Since the effective current densities in practical circuits are stiff, this flux can be treated as stiff. The flux produces a resonantly enhanced current in the input coil and a flux in the SQUID. The SQUID provides high gain with near-quantum-limited noise performance, which allows this flux signal to be read out at high signal-to-noise. Clarke and Braginski (2006) The SQUID may be a dc SQUID, or if a larger circuit is desired, a single junction dissipationless rf SQUID Mates et al. (2008). This detection scheme is used in DM Radio and ADMX LC. Chaudhuri et al. (2015); Sikivie et al. (2014a); Silva-Feaver et al. (2017) One could also remove the capacitor and do a broadband search Kahn et al. (2016).
For frequencies above 1 GHz, a system with controlled impedances must be used; otherwise, reflections in wires create challenges toward efficient dark-matter detection. In this case, the resonant RLC circuit, which may be a free-space cavity 555In terms of impedance, one mode of a free-space cavity can be represented as an equivalent lumped-LC circuit., is coupled (represented in the right panel with a coupling capacitance ) to a transmission line of characteristic impedance (typically 50 ohms). A voltage signal generated in the resonator from the external, dark-matter-induced flux propagates to the input of an RF amplifier, as represented by the wave . This wave is amplified and read out with further electronics. Such a scheme is presently utilized by ADMX and HAYSTAC, which use near-quantum-limited microstrip SQUID amplifiers Mück et al. (2001) and Josephson parametric amplifiers Castellanos-Beltran et al. (2008) to achieve exquisite sensitivity in the search for dark-matter axions. Asztalos et al. (2010); Brubaker et al. (2016) The resonant circuit is also a suitable description of a dielectric resonator, such as that in the MADMAX experiment, or a resonator with a mechanical element, such as a piezoelectric.
In both circuit models, the resonance frequency may be tuned by changing the capacitance . One may then search over large range of dark-matter mass by scanning the resonance frequency.
Here, we analyze the second circuit in detail. The case of a search using a generic quantum-limited flux-to-voltage amplifier is covered in the appendix; the key results are the same as for the near-quantum-limited, scattering-mode amplification scheme.
The detector, as drawn, has resonance frequency . The quality factor of the resonator is determined by two sources of loss. First, resonator energy is lost by dissipation in the resistance . This frequency-dependent resistance represents internal losses of the resonator due to loss in metals or loss in the quasiparticle system (if superconductors are used), loss in wire insulation (in the case of lumped-element inductor coils), loss by coupling to parasitic electromagnetic modes, and loss in dielectrics. Second, resonator energy is lost by power flow into the transmission line. The resonator quality factor is thus
where the internal quality factor is
and the coupled quality factor, representing losses to the transmission line, is
The resonator is driven by three voltage sources: a thermal noise voltage, a zero-point effective-fluctuation-noise voltage, and a dark-matter signal voltage. The two noise voltages, sourced by the resistor in the equivalent circuit model of Fig. 3, are lumped together in the calculations that follow. The dark-matter signal voltage may be represented by a voltage in series with the inductor. Dark matter of mass couples into the pickup loop through its induced magnetic field. The magnetic field, oscillating at frequency , produces a circuit voltage via Faraday’s Law . This stiff voltage can be treated classically, owing to the high number density of the light-field dark matter. Additionally, due to virialization, the dark-matter voltage signal possesses a nonzero bandwidth. This bandwidth was introduced in equation (3) and is . In Section IV, we discuss how the voltage spectrum scales with the dark-matter coupling to electromagnetism (quantified by in the case of the axion and in the case of the hidden photon); we also show how it is related to the dark-matter energy density distribution over frequency, which in turn, can be calculated from the more familiar dark-matter energy density distribution over velocity. (See Appendix D.) For the more quantitative treatments and in particular, the integral approximations, that follow in Sections IV and V, we will also need to define a “cutoff” bandwidth . We define this cutoff bandwidth such that, outside of the frequency band , the dark-matter signal power is small enough to be neglected. Because we are searching for nonrelativistic dark matter, we take this bandwidth to satisfy
As we will see, this cutoff bandwidth is needed to facilitate the calculations, but its exact value is unimportant, as long as this inequality is satisfied. For more information regarding dark-matter bandwidths and their relation to velocity, see Appendices A and C. Where convenient, we will suppress the bandwidth’s dependence on search frequency, with it implicitly understood.
The dark-matter signal at the amplifier input will be strongest when the transmission is largest: when lies within the resonator bandwidth. This observation suggests that a search should be conducted over a wide range of frequencies by scanning the resonance frequency (performed, e.g., via tuning of the capacitance) and amplifying the signal.
Iv Sensitivity Calculation of a Search for Light-Field Dark Matter
Before performing detailed calculations for the SNR of this detector, we qualitatively discuss the processing of the signal. This discussion applies not just to resonators, but to any detector; as such, our SNR treatment in this section will apply broadly. The dark-matter signal has a finite coherence time, set by the inverse of the dark-matter bandwidth, (see Appendix A.1)
On timescales much shorter than the dark-matter coherence time, the dark-matter signal behaves as a monochromatic wave with frequency . In such a situation, one may determine if a timestream contains a signal by using a time-domain Wiener optimal filter Wiener (1949). This filter gives an estimate for the amplitude of the dark-matter signal if one exists in the timestream. If this amplitude is much greater than the uncertainty, set by the noise, then a candidate signal has been detected.
However, stronger constraints on coupling parameter space will require integration longer than this time. We assume here integration longer than the coherence time. Once the integration time has exceeded the dark-matter coherence time, the dark matter no longer behaves as a coherent amplitude signal, but rather, an incoherent power signal. Conceptually, an incoherent power signal can be processed using a Dicke radiometer, first described in Dicke (1946). A schematic diagram of a Dicke radiometer, as it pertains to our dark-matter receiver circuit, is shown in Fig. 4.
As indicated in Fig. 1, the signals from the source, which, in this case are the dark-matter signal, thermal noise, and zero-point fluctuation noise, are fed into a matching network and read out with an amplifier. The output signal of the amplifier is sent to an optimal filter to maximize SNR and then to a square-law detector, whose output voltage is proportional to the input power. This voltage is sent to an integrator. The square-law detector and integrator average down fluctuations in the noise power (i.e. thermal, zero-point fluctuation, and amplifier noise power), which sets the variance (uncertainty) in the measurement scheme. When power is observed in excess of the mean noise power, then a candidate dark-matter signal has been detected, and follow-up is required to validate or reject this signal. In a typical experiment, the output of the amplifier is fed into a computer, and the optimal filtering, square-law detection, and integration steps are executed in software.
The well known expression for the SNR of a Dicke radiometer is given by
where is the signal bandwidth, t is the integration time, and are, respectively the signal and mean noise power within that bandwidth.
Our analysis of optimal signal processing does not directly use this formula. However, it uses signal manipulation parallel to that implemented in a Dicke radiometer (amplification, square-law detection, integration) in order to provide a rigorous framework for understanding sensitivity in these receiver circuits. In particular, this framework will allow us to derive the optimal filter. It will enable us to consider the effects on SNR from scanning the resonance frequency or more generally, using information from multiple detection circuits. The method is easily extended to nonclassical detection schemes (e.g. squeezing and photon counting) that will be considered in Part II. Nevertheless, we will use (41) to build intuition about our results.
We begin by producing a scattering-matrix representation of the resonant circuit in the right panel of Fig. 3 (Sec. IV.1). We then generalize this representation to any detection circuit with inductive coupling to the dark matter (Sec. IV.2).
iv.1 Scattering Representation of Resonant Detection Circuit
Expressing the resonant detection scheme in a scattering matrix representation gives rise to a simple method for analyzing the sensitivity. The detection schematic may be represented as a cascade of two separate two-port circuits, as shown in Fig. 5.
In the first of these two circuits (at the left), the resistance is replaced by a semi-infinite, lossless transmission line of characteristic impedance ; on this line, thermal noise and zero-point fluctuations are injected into the system. This is a typical representation in the quantum optics literature Yurke and Denker (1984). The dark-matter signal voltage in our model is in series with the inductor. The incoming and outgoing waves at each port in Fig. 5 are defined in the frequency domain by the complex phasors (in units )
where and are, respectively, input and output voltages and currents at the far ends of the two transmission lines. (Subscript 1 corresponds to the transmission line of impedance , and subscript 2 corresponds to the transmission line of impedance .) These wave amplitudes are related by
are the scattering parameters for the circuit. and are the length and phase velocity of the transmission line. In deriving eqs. (47)-(49), we assume ; we also assume , and , which are typical design parameters for resonant circuits. In equation (46), represents the frequency component of the dark-matter voltage signal at frequency . The voltage spectrum depends on the distribution of dark-matter energy density over frequency, denoted by , which in turn, depends on the rest mass . The factor is related to the dark-matter coupling to electromagnetism. We now define this factor and show how can be related to the coupling and the distribution.
We will assume that, while the resonator is scanned near the dark-matter frequency, the amplitude of the dark-matter field (proportional to the local density) does not change. In the case of a hidden photon (vector) field, we also assume that its direction also does not change during this period of time. In Appendix B, we discuss deviations from this behavior, and its implications for scan strategy. There we will also discuss strategies to mitigate inevitable detector misalignment with a vector dark-matter field. We will further assume that the distribution stays fixed during the scan. We discuss deviations from this assumption in the context of annual modulation in Appendix B. We also assume that the size of the resonant detector is much less than the coherence length , so that spatial variations of the dark-matter field are inconsequential.
Under these assumptions, we may relate the spectral density of the voltage signal to the energy coupled into the dark-matter circuit by
where is an energy spectral density (in units of energy per unit bandwidth). may be parametrized as
where is the volume of the pickup inductor 666When a free-space cavity, rather than a lumped-element inductor, is used to couple to the signal, represents the equivalent inductance and represents the coupled volume for the mode under consideration. , and is the effective dimensionless coupling of the DM field to the detector.
at frequencies for which the Compton wavelength is much longer than the characteristic size of the detector. For frequencies where the wavelength is comparable to the size of the detector,
We omit the explicit dependence of on dark-matter rest-mass frequency where convenient. is a geometrical factor relating to the pickup pattern and aspect ratio of the inductor and the shape of the drive field. In practice, it is determined by calibration and modeling of the detector geometry.
The energy-density distribution has units of energy per-unit-volume per-unit-bandwidth and must satisfy
where 0.3 GeV/cm is the total local energy density. can be calculated from the density distribution across velocity. As an example, in Appendix C, we calculate explicitly for the standard halo model. For more details on the determination of the signal strength, see references Sikivie (1985); Chaudhuri et al. (2015); Sikivie et al. (2014a).
It is shown in Chaudhuri (tion) that, at small velocities (or equivalently for the axion, ), backaction is significant at any quality factor. It may thus seem that our equations (50) and (51) are inaccurate. However, given that the dark-matter virial velocity is much greater than over all open parameter spaces, it is appropriate to assume that dark matter at these extremely small velocities makes up a very small component of the total dark-matter distribution. See, for instance, the standard halo model in Appendix C. Therefore, assuming the validity of (50) and (51) over all velocities, instead of only for those which satisfy , will result in a negligible change in calculated SNR.
However, for brevity, we will truncate the dependence to
Unlike and , which are parameters of the detection system, and are model-dependent parameters.
An amplifier is represented in the second of the two two-port circuits. We will assume that the amplifier has perfect input and output match to the transmission line, i.e. its input impedance is . Input and output match prevent the emergence of cavity modes on the transmission line, which complicate readout of the dark-matter signal. We will also assume that it possesses reverse isolation, so that waves incoming on the right port do not transmit to the resonant detector. Reverse isolation is an experimentally desired attribute for amplifiers, as it prevents heating of the resonator and interference due to spurious signals on the feedline. The amplifier is described by the following scattering relation for a signal at frequency :
where is the power gain of the amplifier. and are, respectively, the backward and forward-traveling noise waves generated by the amplifier. A phase-insensitive amplifier must necessarily add such noise to the detection scheme Caves (1982). The backward-traveling noise wave may be thought of as the backaction noise of the amplifier, while the forward-traveling noise wave is the intrinsic added noise, or in the language used in Section I, the imprecision noise. We will assume that the amplifier has high power gain over the entire search band such that we may ignore the noise introduced by follow-on electronics (e.g. further analog amplification and filtering, digitization). This condition may be realized by using different high power gain amplifiers in the different frequency regimes. A possible implementation of this is discussed in Chaudhuri et al. (2015).
The left-hand side of equation (58) is a single vector. The top term of this vector represents the wave that is dissipated in the resonator. The bottom term represents the wave that is amplified and read out with further electronics. The signal and noise content of determines the sensitivity of the detector.
On the right-hand side of equation (58), there are three terms. The first term represents the response of the cascaded circuit to waves injected at the ports. Note that the response is independent of the incoming wave at port 2 of circuit (2); this is a result of reverse isolation. As stated before, represents the thermal- and zero-point-fluctuation noise waves injected into the system. Assuming that the detector system is at temperature , the noise correlation for this wave is
is the thermal occupation number of the resonator. We will suppress the dependence on temperature where convenient. As stated before, we have lumped together the thermal and zero-point noise; in (59), the “” term represents the thermal noise, while the “” term represents the zero-point fluctuations.
The second vector in equation represents the response of the circuit to amplifier noise. The top term is the backaction noise that is dissipated in the resonator. The bottom term is the sum of the intrinsic added noise and the backaction noise which is reflected at the resonator and transmitted through the amplifier. The amplifier noise is typically quantified by use of a 22 noise correlation matrix with values given by
where the values and are either 1 or 2. Note that .
The third vector is the response of the circuit to the dark-matter signal. The top term is the portion of the signal that is dissipated in the resonator. The bottom term is the portion of the signal that is transmitted and amplified.
iv.2 Scattering Representation of A Single-Moded Reactive Detector
As yet, we have not determined whether a single-pole resonator is an optimal detection circuit. We may generalize our treatment for the resonator to a single-moded, inductively coupled detector read out by a scattering-mode amplifier. The prescription for doing so is displayed in Fig. 6.