Microwave Cavity Searches

Abstract

The axion “haloscope” technique is a well-established method to search for dark matter axions with a resonant microwave cavity and has excluded axion models over several frequency ranges with unparalleled sensitivity. This chapter describes the basics of microwave cavity searches, including overviews of the main experimental components and details on the figure of merit for these searches.

4.1 Historical Introduction

By the early 1980s it was realized that a low mass axion would be a compelling dark matter candidate; the excellent agreement of theory and data for the neutrino signal seen from the Type-II supernova SN1987A a few years later providing an upper mass bound of [1].¹ Problematically, however, the axion-photon coupling for the QCD axion associated with those masses was so extremely small as to preclude conventional accelerator- or reactor-based searches by many orders of magnitude.

This conundrum was potentially solved by Pierre Sikivie in a seminal paper in 1983, where he showed that axions constituting the galactic halo dark matter could be detected by their resonant conversion to photons in a microwave cavity permeated by a magnetic field [5]. While the signal expected was extraordinarily weak, sensitivity estimates based on the technology of large-volume, high-field superconducting magnets, high-quality-factor cavities and ultralow noise amplifiers of that time appeared to make detection of the QCD axion very nearly within reach. Two early pilot experiments were soon mounted, one at the University of Florida (UF) [6] and the other at Brookhaven National Laboratory (BNL) by a Rochester-BNL-Fermilab (RBF) collaboration [7] providing experimental validation for that optimism, and setting limits on the axion-photon coupling g_aγγ within a factor of 10–100 of the model band. A watershed moment for the field was a workshop convened at BNL by Adrian Melissinos of the University of Rochester on April 13–14, 1989 [8] that brought together forty scientists and engineers, including experts in low-noise receivers, microwave resonators and superconducting magnets, to study whether projections of those technologies supported the idea of actually reaching the QCD model band and whether planning for a large-scale experiment was warranted at that time. The answer was unequivocally yes, and an R&D collaboration was formed from among the participants, opening the path to what has become a three-decade, world-wide effort on microwave cavity experiments and variations on the theme. These in turn have not only been beneficiaries but also drivers of technology development, particularly in quantum metrology.

4.2 Detection Principles

Microwave cavity searches rely on the axion’s coupling to two photons through the inverse Primakoff effect (see Ref. [9] and Sect. 2.4.2). In a resonant microwave cavity immersed in a magnetic field, axions interact with the virtual photons of the magnetic field and convert to an oscillating electromagnetic field with a frequency ν_a corresponding to the axion mass m_a as ν_a ≈ m_ac²∕h. The resonant conversion condition is that the axion mass is within the bandwidth of the microwave cavity at its resonance frequency. Since the axion mass is unknown, the cavity resonance frequency must be tuned to access a range of axion masses. As the resonance frequency of the cavity is tuned, the electromagnetic field inside the cavity is measured by a small probe antenna inserted in the cavity, which is in turn coupled to an ultralow noise preamplifier. There is an ongoing effort to maximize the axion signal power while reducing the background system noise in order to maximize the frequency search rate.

A standard detection schematic is illustrated in Fig. 4.1. The axion field a interacts with the virtual photons γ^∗ of the magnetic field and converts into a measurable oscillating electromagnetic field γ when the axion mass m_a is within the bandwidth of the cavity resonance frequency ν_c. The width of the axion signal is expected to be Δν_a ≤ 10⁻⁶ ν_a, and the bandwidth of the cavity resonance frequency is determined by that resonance mode’s quality factor. To measure the power inside the cavity, an inserted coaxial antenna probes the longitudinal electric field. Ultralow noise amplifiers boost the signal to a level where it can be properly mixed down to a lower frequency with a local oscillator and then the phase and power information can be digitally recorded. Finally, a Fourier transform is applied to the time-dependent signal resulting in a frequency-dependent power spectrum. Axion candidate frequencies are identified as signals above a target threshold and are revisited during a rescanning process to confirm if they are a persistent signal or statistical noise.

Fig. 4.1

Simplified axion detection schematic of microwave cavity searches. Much like an AM-radio, the high frequency axion signal is mixed down to audio frequencies by mixing with a local oscillator maintained at a fixed offset frequency from the cavity frequency. This allows for much lower digitization rates. The cartoon power spectrum shows a sample axion signal above the noise

Several collaborations are implementing this axion detection method. These include, but are not limited to, the Axion Dark Matter eXperiment (ADMX), the Haloscope At Yale Sensitive To Axion Cold dark matter (HAYSTAC), the Center for Axion and Precision Physics (CAPP) Ultra Low Temperature Axion Search in Korea (CULTASK), and the CryOgenic Resonant Group Axion CoNverter (ORGAN). These modern experiments, derived from the early pilot experiments of the RBF and UF collaborations, use similar detection techniques but have unique designs and mostly operate over different frequency ranges.

4.2.1 Signal Power

The signal power in a microwave cavity search can be derived by solving the equations of motion for the electromagnetic field coupled to the axion in the case of a resonant microwave cavity permeated by a static magnetic field and the axion field [10]. It is determined by a combination of theoretical parameters describing axion physics and measurable parameters describing the experimental apparatus in the equation

$$\displaystyle \begin{aligned} P_{\text{sig}} = \left(g_{a\gamma \gamma}^2 \frac{\hbar^3c^3\,\rho}{m_a^2}\right) \times \left(\frac{1}{\mu_0}B_0^2 \omega_c VC_{mn\ell}Q_0 \frac{\beta}{\left(1+\beta\right)^2} \frac{1}{1+\left(2\Delta\nu_a/\Delta\nu_c\right)^2}\right), \end{aligned} $$

(4.1)

where the factors in the first set of parentheses involve theoretical parameters set by nature and the factors in the second set of parentheses are experimental parameters. Theoretical parameters include the model-dependent coupling constant g_aγγ, local dark matter density (commonly used in axion searches [11] and consistent with recent measurements [12]), and the axion mass m_a. The coupling constant itself has units of GeV−1 and can be further expressed as where the dimensionless g_γ changes between classes of models. Representative values are g_γ = −0.97 for the Kim–Shifman–Vainshtein–Zakharov (KSVZ) [13, 14] family of models and g_γ = 0.36 for the Dine–Fischler–Srednicki–Zhitnitsky (DFSZ) [15, 16] family of models. The relevant experimental parameters are external magnetic field strength B₀, cavity resonance frequency ω_c = 2πν_c, cavity volume V , mode-specific cavity form factor C_mnℓ (often, C₀₁₀), unloaded quality factor Q₀, cavity coupling parameter β (β = 1 corresponds to critical coupling, β < 1 is undercoupled, and β > 1 is overcoupled), and cavity linewidth Δν_c.

Typical values for the HAYSTAC detector are B₀ = 9 T, , , C₀₁₀ = 0.5, Q_L = 10⁴, β = 2, , and $\Delta \nu _c = \nu _c / Q_L = \omega _c / \left (2\pi Q_L\right )$, where Q_L is the cavity loaded quality factor defined by $Q_L = Q_0/\left (1+\beta \right )$. Altogether, the expected power for these parameters at the axion–photon coupling set by the KSVZ family of models is P_sig ≈ 10⁻²⁴ W.

4.2.2 Noise Considerations

For any phase-insensitive linear receiver the system noise temperature T_sys may be written

$$\displaystyle \begin{aligned} k_BT_{\text{sys}} = h\nu N_{\text{sys}} = h\nu\left(\frac{1}{e^{h\nu/k_BT} -1} + \frac{1}{2} + N_A\right), \end{aligned} $$

(4.2)

where the three additive contributions correspond, respectively, to a blackbody photon gas in equilibrium with the cavity at temperature T, the zero-point fluctuations of the photon field, and the input-referred added noise of the receiver. The latter two terms combine to form the standard quantum limit (SQL), with N_A ≥ 1∕2 [17].

The Dicke radiometer equation [18] combines the expected signal power with the system noise temperature to form the signal-to-noise ratio Σ:

$$\displaystyle \begin{aligned} \varSigma = \frac{P_{\text{sig}}}{k_BT_{\text{sys}}}\sqrt{\frac{\tau}{\Delta\nu_a}}, \end{aligned} $$

(4.3)

where τ is the integration time, and Δν_a is the expected linewidth of the axion.

There is an active effort in the microwave cavity search community working to increase expected signal power and decrease system noise temperature rather than integrating for longer to improve Σ. Increasing the applied magnetic field or improving cavity performance (volume, quality factor, or form factor) increases the expected signal power, as suggested by Eq. (4.1). To decrease system noise temperature, the experiments are cooled to temperatures as low as possible and state-of-the-art amplifier technologies are implemented.

4.2.3 Scan Rate

Because the mass, and hence the oscillation frequency, of the axion is unknown, resonant microwave cavity searches must scan over a wide range of frequencies. Therefore, the ultimate figure of merit is the scan rate, which incorporates the expected signal power and noise considerations, and quantifies how quickly searches can scan through different frequencies at a given sensitivity

$$\displaystyle \begin{aligned} R \equiv \frac{\mathrm{d}\nu}{\mathrm{d}t} \approx \frac{4}{5}\frac{Q_LQ_a}{\Sigma^2} \left(g_{a\gamma \gamma}^2\frac{\hbar^3c^3\rho_a}{m_a^2} \right)^2 \times \left(\frac{1}{\hbar\mu_0}\frac{\beta}{1+\beta}B_0^2VC_{mn\ell}\frac{1}{N_{\text{sys}}}\right)^2. \end{aligned} $$

(4.4)

Most of these terms are recognizable from the expression of the signal power in Eq. (4.1). Using typical values for HAYSTAC, to achieve the benchmark KSVZ sensitivity g_KSVZ, the scan rate of the first run would have been approximately . Since the scan rate scales as the fourth power of the coupling constant, the scan rate would have been to achieve twice the KSVZ sensitivity 2g_KSVZ.

Improving the scan rate allows us to search through a mass range more quickly, but we are still limited by the tuning range of the cavity and amplifier electronics. The frequency range we can probe depends on the resonance frequency of the microwave cavity. In general, higher-frequency cavities have a smaller volume and therefore suffer from a smaller expected signal power as well as an increase in operational complexity due to a higher resonance mode density. New cavity designs are being developed that expand the accessible frequency range while improving sensitivity. This requires investigating various geometries using electromagnetic simulations, prototypes, and microwave testing.

4.3 Resonant Microwave Cavities

A resonant microwave cavity supports many modes with various electric and magnetic field profiles. Microwave cavity searches generally focus on one resonant mode and use a cavity design that optimizes the mode of interest for the figure of merit within a frequency tuning range while preserving mode purity.

The figure of merit of a cavity resonant mode is determined by the scan rate R (Eq. 4.4), which is partially composed of cavity geometry and resonant mode characteristics. The components include the quality factor Q, which quantifies losses, the form factor C_mnℓ, which describes the alignment of the resonant mode electric field to the external magnetic field, and the cavity volume V . The scan rate depends on these quantities as

$$\displaystyle \begin{aligned} R \equiv \frac{d \nu}{dt} \propto Q\, C_{mn\ell}^2\, V^2. {} \end{aligned} $$

(4.5)

The volume in the cavity figure of merit involves the internal cavity space through which electric fields penetrate. The volume generally decreases with increasing frequency, which is one of the main challenges in designing higher-frequency cavities. The HAYSTAC cavity, shown in Fig. 4.2, is a closed cylindrical volume of 5.08 cm radius and 25.4 cm length with one tuning rod of 2.54 cm radius that can be rotated off center.

Fig. 4.2

HAYSTAC copper-coated stainless steel resonant microwave cavity tunable over the frequency range 3.6–5.8 GHz. The top cover is not shown

4.3.1 Resonant Cavity Modes

The resonant modes present in a cavity are characterized by their electromagnetic field profiles that obey the Maxwell equations and the usual boundary conditions. The Maxwell equations in vacuum, assuming that the electric and magnetic fields have time dependence e^−iωt, reduce to

(4.6)

where is the electric field and is the magnetic field. At a perfect electric conductor (PEC) surface, the tangential electric fields and normal magnetic fields vanish, the tangential component of magnetic field is related to the surface current density , and the normal component of electric field is related to the surface charge density ρ_s. These boundary conditions are, respectively, summarized in the following equations:

(4.7)

where $\hat {n}$ is the vector normal to the PEC surface s.

The solutions to these equations and boundary conditions describe the resonant modes present in a cavity. The modes include transverse magnetic (TM), transverse electric (TE), and transverse electric and magnetic (TEM) modes. TM modes are characterized by a transverse magnetic field and longitudinal electric field, TE modes are characterized by a transverse electric field and longitudinal magnetic field, and TEM modes are characterized by a transverse electric and magnetic field. TEM modes only exist in structures with a central conductor. Characteristics of the three types of resonant modes are summarized in Table 4.1.

Table 4.1 Resonant mode descriptions, assuming that the $\hat {z}$ direction is along the length of the cavity

Each TE and TM mode is classified by three numbers m, n, and ℓ that describe the variation in the azimuthal, radial, and longitudinal directions, respectively. For example, the TM₀₂₀ and TM₀₁₁ modes have one extra radial node and longitudinal node, respectively, in the electric field compared to the TM₀₁₀ mode. For an empty cylindrical cavity of radius r_cavity and height h_cavity, the resonant frequencies of the TM_mnℓ modes are given by

$$\displaystyle \begin{aligned} \omega\text{TM}_{mn\ell} = \frac{c}{\sqrt{\mu \epsilon}} \sqrt{\frac{x_{mn}^2}{r_{\text{cavity}}^2}+\frac{\ell^2\pi^2}{h_{\text{cavity}}^2}}, \end{aligned} $$

(4.8)

where m, ℓ = 0, 1, 2, …, n = 1, 2, 3, …, c is the speed of light, μ is the permeability, 𝜖 is the permittivity, and x_mn is the nth root of the Bessel function of the first kind $J_m\left (x\right )$. The resonant frequencies of the TE_mnℓ modes are given by

$$\displaystyle \begin{aligned} \omega\text{TE}_{mn\ell} = \frac{c}{\sqrt{\mu \epsilon}} \sqrt{\frac{(x^{\prime}_{mn})^2}{r_{\text{cavity}}^2}+\frac{\ell^2\pi^2}{h_{\text{cavity}}^2}}, \end{aligned} $$

(4.9)

where m = 0, 1, 2, …, n = 1, 2, 3, …, ℓ = 1, 2, 3, … and $x^{\prime }_{mn}$ is the nth root of the derivative of the Bessel function of the first kind $J^{\prime }_m\left (x\right )$.

Some of the resonant modes can be tuned in frequency by moving rods inside cylindrical cavities and some have approximately stationary resonance frequencies upon changing the position of the rod. Since the TE frequencies are primarily determined by the length of the cavity, they do not change significantly when the central conductor moves or changes in size.

4.3.2 Quality Factor

As suggested in Eq. (4.5), the cavity figure of merit scales as the quality factor Q. The quality factor is a property of each resonant mode in the cavity and is given by the ratio of the stored energy U to the dissipated power P_d in the cavity, multiplied by the resonant mode frequency ω:

$$\displaystyle \begin{aligned} Q = \omega \, \frac{U}{P_d}. {} \end{aligned} $$

(4.10)

The stored energy in the cavity is proportional to the square of the electric field integrated over the cavity volume filled with material of dielectric constant 𝜖:

(4.11)

The power loss in the cavity is proportional to the square of the magnetic field integrated over the metallic surfaces inside the cavity:

(4.12)

where μ is the magnetic permeability of the metallic cavity surfaces, and the skin depth δ is the distance that electric fields are allowed to penetrate into the metallic surfaces. The classical skin depth is given by

$$\displaystyle \begin{aligned} \delta = \sqrt{\frac{2}{\omega\, \mu \, \sigma}}, {} \end{aligned} $$

(4.13)

where σ is the conductivity of the metallic surface. Conductivity improves with decreasing temperature, and the classical skin depth is expected to improve as well. However, at sufficiently low temperatures, the skin depth reaches an asymptote. In HAYSTAC, cooling the cavities from room temperature to gives an improvement of the quality factor by approximately a factor of four at a frequency around . For comparison, the conductivity improves by a factor of over a hundred in that temperature range. The classical description becomes invalid when the skin depth decreases below the electron’s mean free path. In this regime, the skin depth depends on the electron density instead of the normal conductivity. This anomalous skin depth [19] is given by

$$\displaystyle \begin{aligned} \delta_a = \left( \frac{\sqrt{3}\, c^2 m_e v_F}{8\pi^2 \omega n e^2} \right)^{1/3}, \end{aligned} $$

(4.14)

where m_e is the electron mass, v_F is the Fermi velocity, n is the conduction electron density, and e is the electron charge [20].

The quality factor is determined primarily by the material and geometry of the cavity and can also vary greatly between resonant modes. Materials with a higher conductivity typically have smaller skin depths and therefore higher quality factors. The desire to have higher conductivity motivates making or plating the cavities with oxygen-free high-conductivity (OFHC) copper and annealing them for further conductivity improvement. Superconducting materials are appealing but the thickness of the superconductor must be kept small enough to prevent lossy vortices from forming in the strong magnetic field. This technology is being explored by the ADMX, CAPP, and QUAX experiments (the latter of which is further described in Chap. 8 [21, 22]),

4.3.3 Form Factor

The form factor is a measure of how well the electric field of a resonant mode aligns with the applied external magnetic field. It is given by

(4.15)

where V  is the cavity volume not occupied by a metallic object and filled with material of dielectric constant 𝜖 = 𝜖₀𝜖_r. Note that if the cavity is partially filled with a dielectric material, 𝜖_r varies in space.

The form factor is maximized when integrated over the volume is maximized. Since the applied magnetic field for microwave cavity searches is commonly in the $\hat {z}$ direction, all resonant modes without electric field components in the $\hat {z}$ direction have form factors that are identically zero, and many TM modes have portions of their field that cancel out, thereby lowering their integrated form factor to near zero.

4.3.4 Tuning and Mode Density

Quality factor, form factor, and volume quantitatively describe the behavior of interest of a single resonant mode at a given frequency. These quantities give a general sense of performance across a tuning range, but if the range is full of intruder modes, it will be interrupted. A TM mode resonance frequency decreases when a rod rotates away from the center of the cavity. In comparison, the TE mode resonance frequency does not change significantly. When the TE and TM mode frequencies approach each other, the two modes mix, producing two hybrid modes, in analogy with two-level mixing in quantum mechanics [23]. If the mode of interest hybridizes significantly, it will be difficult or impossible to interpret the results of the experiment, thus leading to a notch in frequency coverage of the experiment. Mode density is difficult to quantify, but it is a key consideration for cavity design. The problem of mode density worsens for cavities of too large an aspect ratio h_cavity∕r_cavity; practically one is constrained to stay with cavity designs of h_cavity∕r_cavity ∼ 5 or lower.

4.3.5 Multiple Cavity Systems

A cavity’s TM₀₁₀ frequency scales inversely to the radius and thus the volume, assuming a constant length-to-radius aspect ratio, decreases as V ∝ ν⁻³ in going to higher frequencies. For a fixed magnet solenoid volume one can simply increase the number of cavities N each with their own independent receiver chains which can then combine their powers statistically for a $\sqrt {N}$ improvement to the $ of a single cavity. However, one can also take advantage of the coherent nature of the axion signal to recover this volume more efficiently by co-adding the in-phase voltage signals of multiple frequency-locked cavities. The axion signal, though it has an unknown global phase, will generate the same, in-phase, signal in each cavity. The voltages from each cavity can thus be combined in phase to provide N × V_a output voltage. The noise power from each cavity would be added incoherently providing an added noise level of $\sqrt {N}\times V_{\text{noise}}$. Squaring these to get power we see that we can get a signal-to-noise enhancement of N × $_{single cavity} [24, 25]. The price that one pays in such a scheme is the added complexity of controlling all the cavity systems so that they are within a linewidth of the other cavities.

4.3.6 Testing Cavities

Before incorporating the resonant microwave cavity in the detector, it must be thoroughly studied and characterized. Changing the cavity geometry (for example, by moving a tuning rod) changes the mode frequencies. Mode maps track these changes by showing mode frequencies at each cavity geometry change.

A vector network analyzer measures reflection and transmission of microwave signals in the frequency range of interest. When the coaxial antennas couple to a resonant mode, more signal is transmitted. By measuring the scattering parameters between two ports either through transmission (S₁₂, S₂₁) or reflection (S₁₁, S₂₂), the frequencies and quality factors of resonant modes in the cavity can be measured. All measurements are done with weak coupling to the coaxial antennas to minimize perturbations due to the antenna presence in the cavity. Scattering parameter measurements give information on the frequencies and quality factors of resonant modes, but not on their electric field distribution. To get insight on the resonant mode electric field distribution, the cavity is probed by pulling a relatively (compared to the cavity size) small bead through the length of the cavity and measuring the resonance frequency at each step. An example setup and bead pull measurement are shown in Fig. 4.3. The presence of the bead inside the cavity perturbs the electromagnetic field and shifts the resonance frequency by a magnitude proportional to the square of the strength of the electric field at the bead location [26]. If the bead is only slightly perturbing the electric field, the expected frequency shift is given by

$$\displaystyle \begin{aligned} \frac{\Delta \omega}{\omega} = \frac{-\left(\epsilon -1 \right)}{2} \frac{V_{\text{bead}}}{V_{\text{cav}}} \frac{E(r)^2}{\left< E(r)^2 \right>_{\text{cav}}}, \end{aligned} $$

(4.17)

Fig. 4.3

Bead perturbation technique setup

where V_bead and V_cav are the volumes of the bead and cavity, respectively [27]. This bead perturbation technique, which is commonly used in the microwave engineering community, allows the resonant mode to be identified, to determine whether the mode is significantly hybridized, and to ensure that the cavity is properly aligned. This is essential to confirming that the cavity form factor corresponds to its calculated value.

4.4 Amplifiers

The current microwave cavity axion experiments are establishing limits on the axion that correspond to signals with powers on the order of 10⁻²⁴ W, or equivalently on the order of one axion-to-microwave-photon conversion per second. Achieving sensitivity to such small signals depends critically on the equivalent system noise temperature T_sys and thus requires the state-of-the-art ultralow noise detectors.

The first-generation RBF and UF experiments, and the ADMX experiment for its first several years of operation, utilized High Electron Mobility Transistor amplifiers (HEMTs). Noise added by HEMTs decreases with temperature down to a minimum of a few Kelvin when cooled to liquid helium temperatures, but plateaus before reaching the SQL. Since the scan rate in Eq. (4.4) is inversely proportional to T_sys², decreasing the system noise temperature would significantly decrease the amount of time it would take to scan through the axion parameter space accessible by resonant cavity searches.

Amplifiers presently in use are operating at or near a system noise temperature corresponding to the SQL, and recently a squeezed-vacuum state receiver has been employed to circumvent the SQL [28].

4.4.1 Quantum-Limited Amplifiers

Unlike the HEMT noise temperature that plateaus at a few Kelvin, the noise temperature of amplifiers based on DC superconducting quantum interference devices (SQUIDs) decreases roughly linearly as the physical system temperature decreases to around 0.1 K [29]. SQUIDs are naturally applied to low frequencies, but replacing the input coil with a tunable microstrip resonator enables operation of SQUIDs up to 1 GHz [30]. Using SQUIDs provides the ability to drive the system noise temperature close to the SQL while introducing the challenge of magnetic shielding. Since they are sensitive to magnetic flux, experiments must magnetically shield them from the high magnetic fields permeating the resonant microwave cavity.

SQUIDs can operate up to a few GHz, but the axion parameter space extends to higher frequencies. Josephson parametric amplifiers (JPAs) are naturally resonant devices designed to operate in the 2–12 GHz range. A JPA is a nonlinear LC circuit, with the inductance provided by an array of SQUIDs [31]. Like the SQUID, the JPA must be carefully magnetically shielded. To give the scale of shielding, in HAYSTAC, the magnetic shielding is composed of a second superconducting magnet coil to negate the field from the main magnetic in the region of the quantum amplifiers, passive persistent coils, and ferromagnetic and superconducting shields [32]. Ultimately successful operation of JPAs in HAYSTAC required that the remnant field be reduced to a level corresponding to much less than one flux quantum in the region of the device. Although these amplifiers offer improved noise performance at a range of frequencies, they are limited by the SQL.

4.4.2 Sub-quantum Limited Amplifiers

During initial operation, HAYSTAC operated the JPA as a low-noise phase-insensitive linear amplifier subject to the SQL. To overcome the SQL, experiments can borrow from developments in quantum measurement technology to manipulate the noise in the system. In microwave cavity experiments, an antenna measures a voltage that is proportional to the electric field inside the cavity and can be decomposed into components

$$\displaystyle \begin{aligned} \hat{V} = \hat{X}\cos{\omega_c t} + \hat{Y}\sin{\omega_c t}, \end{aligned} $$

(4.18)

where ω_c is the cavity resonance frequency, and $\hat {X}$ and $\hat {Y}$ are quadratures of the cavity field. The variances of the quadratures have a minimum uncertainty limit of $\sigma _{\hat {X}}^2 \sigma _{\hat {Y}}^2 \geq 1/4$. JPAs can squeeze the vacuum state to increase uncertainty in one quadrature while decreasing it in the other. To decrease the system noise temperature below the SQL, experiments can implement vacuum squeezing by operating two JPAs in a phase-sensitive mode (the amplifier applies different gains to the two quadratures). In this operation, one JPA prepares the microwave cavity in a squeezed-vacuum state before amplifying with a second JPA that is 90^∘ out of phase. Then, the noise is decreased in the measured quadrature increasing the signal-to-noise ratio [33]. Implementing a squeezed-state receiver (SSR) allowed HAYSTAC to enhance the scan rate by a factor of 1.9 [28]. The benefit of using an SSR is limited by the cable transmissivity and lossy microwave components, so future efforts will work to improve the connections in the system.

4.5 Operational Experiments

Since the initial experiments in the 1980s, several microwave cavity searches have excluded axion parameter space while updating their cavity and amplifier designs to probe various axion mass ranges. The parameter space that has been excluded by the various microwave cavity searches is shown in Fig. 4.4.

Fig. 4.4

Excluded parameter space by microwave cavity searches. The dashed lines represent the KSVZ and DFSZ models and the yellow band describes their uncertainty

Table 4.2 compares representative values of microwave searches. The ratio of system noise temperature to the SQL has improved with developments in cooling and amplifier technology over the years. Although the technological advances have been impressive, more innovation is needed to probe the vast axion parameter space available for microwave cavity searches.

Table 4.2 Summary of representative values from selected microwave cavity searches