Axion haloscope array with PT symmetry

We generalize the recently proposed PT -symmetric axion haloscope to a larger array with more PT - symmetric structures. By broadening the response bandwidth of the signal without increasing the readout noise, the optimized scan rate of the axion haloscope is signiﬁcantly enhanced, as well as is the signal power. Furthermore, we show that the robustness of the detector towards the variations of the array coupling is the strongest when a binary tree structure is introduced which contains a largely enhanced PT symmetry. The multiple allowed probing sensors can further increase the scan rate by a factor of the sensors’ number due to the correlation of the signals. This type of array can strongly boost the search for an axion compared to single-mode resonant detection. The enhancement to the scan rate becomes the most manifest when applied to the proposed detection using a superconducting radio-frequency cavity with an ac magnetic ﬁeld where most of the parameter space of the QCD axion above kHz can be probed.


I. INTRODUCTION
An axion, with the initial motivation to solve the strong CP problem in QCD [1], is a strongly motivated hypothetical particle beyond the standard model.Besides the QCD axion, axionlike particles (ALPs) also appear generically in theories with extra dimensions [2].These particles can be ideal candidates of cold dark matter [3][4][5], behaving as a coherent wave within the correlation time and distance.There are many strategies to search for axion dark matter, for example, using a resonant microwave cavity or superconducting circuit as a haloscope [6][7][8], an axion-induced birefringence effect for light propagation [9][10][11], and axion-induced nuclear magnetic precession [12][13][14].The main experimental platforms include ADMX [15], SN1987A [16], and CAST [17], which set the current limits to the axion parameters.
The strategy using a resonant microwave system is based on the axion-photon interaction (the so-called inverse Primakoff process) in strong background magnetic fields, which Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Since the microwave field generated by axion conversion is extremely weak, a microwave resonator with a high-quality factor (Q factor) is essential for the experiment, which at the same time narrows the bandwidth of the detector.Therefore, to search for the axion dark matter in a broad mass spectrum, we need to switch the detector among different central frequencies in the practical running of the detector.The figure of merit for the Sikivie-type axion haloscope with a tunable center frequency is the scan rate R a , defined as [26] where S α ( ) is the sensitivity (noise normalized by the signal amplitude) of the haloscope.Here, we assume the axion coherence time is short compared to the observation time.
The scan rate of resonant detection is limited by two kinds of noise, i.e., intrinsic fluctuations due to the dissipation of the experimental components and the quantum fluctuations from the readout channel.The resonant detector responds to the signal field in the same way as to the intrinsic fluctuations, where the strongest response occurs around the resonant frequency.
FIG. 1. Design concept of the detector with PT symmetry (when g = G), where two auxiliary degrees of freedom were introduced with beam-splitter and parametric types of interactions in red and blue, respectively.
Thus the frequency domain sensitivity depends on the range that intrinsic fluctuations dominate over the flat readout one, as discussed in detail in Refs.[23][24][25]27,28].
To improve the scan rate, one can either reduce the system noise level or broaden the detection bandwidth (at the same time not sacrificing the sensitivity), thus responding to a larger off-resonance frequency region for each scan.Several works targeted at reducing the system noise level [23][24][25][26][27][28][29][30][31] have been carried, for example, microwave squeezing technology is useful in improving the scan rate.Recently, a new design [32] which has the feature of PT symmetry is proposed to substantially broaden the detector bandwidth and thereby significantly reduce the switching time costs.The basic structure of this proposal is a Sikivie-type axion detector assisted with an auxiliary nondegenerate parametric interaction, which was inspired by the white light cavity concept used in laser interferometer gravitational wave detectors [33].
In this paper, we proposed different configurations to enhance the scan rate further through generalizations of the PT -symmetric design concept to an array of detectors.Since the coherent length of the axion dark matter signal is typically 10 3 times the Compton length (see, e.g., Refs.[27,[34][35][36]) that a resonant microwave cavity usually matches, one can upgrade it to a fully PT -symmetric setup with multiple probing sensors.On the other hand, variations of experimental parameters from the optimal values can potentially degrade the scan rate enhancement.It turns out that the fully PT -symmetric configuration is more robust against these variations.

II. HALOSCOPE WITH PT SYMMETRY
The detector design with a PT -symmetric feature in Ref. [32], shown in Fig. 1, can be described by the following Hamiltonian in the interaction picture where the free Hamiltonians of the modes are omitted, where â, b, ĉ are three cavity modes.The mode â is used to probe the axion signal with coupling constant α, while the mode b is connected to the readout port where we extract the detection result.There are two different kinds of interactions in this Hamiltonian, (1) the beam-splitter-type interaction with strength g between modes â and b, and (2) the nondegenerate parametric interaction [37,38] with strength G between modes b and ĉ.These modes couple with external continuous fields via and they take the same form for the b, ĉ modes.When g = G, the interaction Hamiltonian (2) reduces to and the PT symmetry emerges.In this case, P transformation switches â and ĉ and T transformation switches the creation and annihilation operators.Under the joint PT transformation, the Hamiltonian keeps its original form.Moreover, in the ideal case when â/ ĉ are lossless, the response of the â + ĉ † to the axion driving is inversely proportional to the frequency, thereby achieving a significant enhancement of the signal, thus sensitivity S α ( ) at low frequencies.The conceptual designs and how the system behaves under the nonideal situation where the â/ ĉ possess internal loss and g = G has been demonstrated and optimized in Ref. [32].
In the following sections, we show that the sensitivity can be further enhanced in systems with enlarged PT -symmetric structures.

III. A CHAIN OF PT -SYMMETRIC DETECTOR
The advantage of the PT -symmetric detector design comes from its high response to the signal at low frequencies.Such an advantage can be further enhanced by considering an extension of the model with more PT -symmetric structures which we named the chain detector design, as shown in Fig. 2. The interaction Hamiltonian of this chain detector design is In the case of g = G, we have In the lossless case when the external continuous electromagnetic bath ûr only couples to the readout b mode via strength √ 2γ r , we have the following equations of motion: In the frequency domain, these equations can be stacked in the following way, where is the frequency shift from the resonant frequency ω rf .Clearly, there is an enhanced amplification factor G n ( ) = (g/ ) n due to the chain structure.Thus the lowfrequency signal response scales as ∼ −n [39], while at the same time the noise is still at the shot-noise level since S u r u r = 1 (in principle, this shot-noise level can be further reduced via squeezing technology so that S u r u r = e −2r s , where r s is the squeezing degree).The signal power and noise spectrum in this ideal lossless case are given by

IV. IMPERFECTIONS AND ROBUSTNESS
In practice the lossless condition and the PT -symmetric condition cannot be achieved as ideal, and the robustness of the device's sensitivity towards varying the loss rate and PT -symmetric condition needs to be taken into account.Let us first consider the case when g = G, but there exists an internal loss γ a/c and we assume that the internal loss for each resonator is the same so that γ a = γ c = γ .The signal enhancement factor in this case would be G( ) = g 2 /(γ 2 + 2 ) and the signal power S CD sig ( ), so the noise spectrum due to the readout S CD r ( ) and the internal loss S CD int now can be written respectively as where the noise spectra S ua i , S uc i are the loss contributions from the âi and ĉi , including both vacuum and thermal fluctuations.Even at < γ , as long as g γ , there is still a significant amplification of the signal power.We can vary the value of γ r while fixing g = G to optimize the scan rate in Eq. ( 1) and the optimized spectrum for n = 2, 3 is shown in the Appendix.It turns out that increasing the resonant chain levels could improve the scan rate R a by a factor of (g/γ n occ ) 2n/(2n+1) , through broadening the range of S CD int ( ) S CD r ( ), where n occ is the thermal occupation number.
On the other hand, the mismatch between g and G, which breaks the PT symmetry, will affect the system response as well and it has been also discussed in Ref. [32].Therefore we also need to test the robustness of the optimized scan rate, and the result is shown in Fig. 3, where we calculate how the scan rate ratio R α (g = G)/R α (g = G) would change with respect to g 2 − G 2 at zero temperature.In this figure, the scan rate would drop by half if g 2 − G 2 ∼ 10 5 γ 2 when g/γ = 10 4 , which requires g − G 5γ .For the system with large g, it could be a fine-tuning problem and thereby the system is less robust towards the g − G mismatch.

V. BINARY TREE DESIGN
An alternative design, with the schematic diagram shown in Fig. 4 (in this paper, we call it the binary tree design), could relieve the problem of robustness shown in the previous section.The corresponding Hamiltonian can be written as FIG. 4.An example of a binary tree design with n = 3, with red lines denoting a beam-splitter-type interaction and blue lines denoting a parametric-type interaction.The probing modes to the dark matter are chosen to be the â1i modes in the lowest level.
It is important to note that in this design, we are allowed to have multiple resonant modes to probe the axions, while in the chain detector, we only have one.Since the typical correlation wavelength of the axion dark matter is much larger than the spatial scale between these resonators, there is a coherent enhancement of signal fields as well as the scan rate.However, we need a benchmark to compare different device designs [40], and such a benchmark is chosen to be the total signal amplitude at the input.This is to say that when we compare the chain resonator design and the binary tree design, we the total signal amplitude to be the same.Practically speaking, it means we can reduce the strength of the background magnetic field in the coupling cavities â1i (i = 1, . . ., N) by a factor of N. In this sense, there is no advantage of the binary tree design over the chain detector design when there is no loss and PT symmetry is strict.
The structure of the interaction between âij and ĉij renders the Hamiltonian (12) to have a largely enhanced PT symmetry.It turns out to demonstrate much higher robustness to the g − G mismatch than the resonant chain detector as we can see from Fig. 3, and increasing the level n of the binary tree design also enhances the robustness.
In the Appendix, we numerically optimize the scan rate in terms of both γ r and g 2 − G 2 , and the corresponding noise power spectral density (PSD) is in Fig. 5.The range where internal loss noise S int dominates over the readout noise S r is indeed broadened once the level n increases, contributing to the enhancement of the scan rate compared with the case of the single-mode resonator.Take the array extension of a typical Sikivie design concept (where a microwave cavity is embedded in a dc magnetic field for axion detection) as an example: Each scan can effectively probe the axion mass within ω rf ± ω sc , where ω sc (g 2n γ n occ ) 1/(2n+1) is the effective scan bandwidth in which S int ( ) S r ( ).Notice that the flat distribution of the noise PSD for the binary tree is another feature of robustness compared with the chain detector.FIG. 5.The corresponding noise PSD for n = 1 and 3 of a chain detector and binary tree at g/γ = 10 4 when both γ r and g 2 − G 2 are tuned to give the optimized scan rate.The PSDs of a binary tree are flatter compared to the one of the chain detector.In comparison, we also show the PSD for the single-mode resonator.Each scan can probe axion dark matter with mass ω rf ± .

VI. EXPERIMENTAL EXPECTATIONS
We now discuss the application of the PT -symmetric array to the electromagnetic resonant detectors.These include a microwave cavity embedded in a strong dc magnetic field with the coupling strength α = g γ ηB 0 √ m V , where g γ is the axion-electromagnetic-field coupling, η is the overlapping factor of the cavity mode with the background magnetic field B 0 , and V is the cavity volume [6,7].We also consider superconducting-LC circuits [8]/superconducting radio-frequency (SRF) cavity [23][24][25] embedded in the dc/ac magnetic field with corresponding coupling strengths to be α LC = g γ B 0 V 5/6 m 3/2 and α RF = g γ ηB 0 m √ V/ω rf , respectively, where ω rf is the resonant frequency of the SRF cavity.
We consider the binary tree design only due to its robustness, as discussed in the previous sections.The potential physics reaches based on the three different types of experiments are shown in Fig. 6 with the integration time distributed for each e-fold in the axion mass to be t e = 10 7 s and the signal-to-noise ratio (SNR) reaching 1.In the Appendix, we show that the SNR is proportional to the square root of the scan rate in Eq. ( 1).Since typically g cannot be larger than the resonant frequency ω, we take g = Q int γ .Thus the highquality factor as well as an almost fixed thermal occupation number of the SRF cavity leads to a much larger enhancement.The benchmark parameters for a cavity with a dc magnetic field is V = 1 m 3 , η = 1, B 0 = 4 T, T = 10 mK, Q int = 10 4 while the LC circuit only differs by Q int = 10 6 [27].For SRF, the differences with the traditional cavity are Q int = 10 12 , B 0 = 0.2 T, T = 1.8 K, and the resonant frequency is almost fixed to be ω rf = 2π GHz + m [23].Below kHz, we also include the contribution of the phase fluctuation noise that dominates over the thermal noise, with the overlapping factor between the pumping magnetic field and the signal electric field to be 1d = 10 −5 and the quality factor of the pumping field to be the same as Q int .We require the scan bandwidth ω sc to be no larger than the axion mass m , as discussed in FIG. 6. Physics reach for cavity, LC circuit with dc magnetic field, and SRF with ac magnetic field.Here, we choose the binary tree detector design as an example (because of its robustness to the g − G mismatch).The scaling with m changes at low frequency for SRF due to the phase fluctuation noise [23] and the scan bandwidth saturating m .
the Appendix, which makes the scaling of the physics reach change below 10 MHz.
The nondegenerate interactions for these three systems are already experimentally realized in Refs.[37,38].There are still several potential challenges to consider for the practical implementation.The first one comes from the compatibility between the strong magnetic field and the nondegenerate interaction.Since the realizations of such a type of interaction usually require superconducting ingredients, a spatial separation away from the magnetic background is necessary to maintain the superconductivity.A conducting wire from the cavity/circuit can solve issue, as mentioned by Ref. [41].Another potential obstacle comes from the phase fluctuation of the pumping mode to realize the nondegenerate interaction, leading to a time-dependent value of the coupling G [42].The binary tree is more robust against such variations as well.Finally, the transmission losses due to the interactions between different resonant modes will modify the intrinsic dissipation coefficients γ , thus requiring more precise calibration for the modes at the lowest level.

VII. CONCLUSIONS
In this work, we explore the generalization of the axion detector design assisted with a PT -symmetric quantum amplifier to be a multiple resonant system for the further enhancement of sensitivity.With comparable parametric coupling and beam-splitter coupling, a PT -invariant mode â + ĉ † is formed, which can transform the signal to the next connected mode without suppression of the signal response.Finally, the signal response is further enhanced by flowing through different PT -invariant pairs while the readout noise response function stays the same.Two generalized scheme configurations are discussed and compared, respectively: the chain detector configuration and the binary tree configuration in terms of the signal and noise response as well as the robustness towards the variation of the optimized system parameters.
We found that both of these detector configurations have the potential capability of a scan rate enhancement by broad-ening the bandwidth of the signal response without increasing the readout noise.Considering the variation of experimental parameters, we show that the chain detector configuration is less robust toward PT symmetry breaking than the binary tree configuration.
For an electromagnetic resonant system such as a cavity or LC circuit, these PT -symmetric couplings have been already achieved in Refs.[37,38], and thus can be directly applied to most current experiments/proposals.These improvements of detector capability for constraining the axion mass and the axion-photon coupling of the design concept in this work for three different types of electromagnetic resonant experiments (cavity, LC circuit with a static magnetic field, and SRF with an ac magnetic field) are also discussed.The enhancement to the scan rate can approach ∼Q int /n occ for static field experiments, or for SRF searches at sufficiently high axion masses.The high-quality factor Q int of the SRF thus can lead to a significant enhancement and probe most of the QCD axion parameter space above kHz.where S N ≡ S int + S r is the total noise power spectral density (PSD).It is assumed that the scan is performed uniformly in log m , with the same time t e distributed for each e-fold in axion mass.t int = t e ω sc /m is the integration time within one scan step, required to be larger than the cavity ring-up timescale and dark matter coherent time 2π Q /m .ω sc is defined as the single scan step within which the expected SNR is an O(1) of the maximum value and ω s is the bandwidth of (S sig /S N ) 2 .In the case where the dark matter bandwidth m /Q is smaller than the sensitivity width of the detector, i.e., the width of [S sig /(S N S )] 2 , one has ω s = m /Q and ω sc is the sensitivity width.Now it is clear that the quantity needed to be optimized is the integral in the last line of Eq. (A1), which is the scan rate in Eq. (1).We parametrize the SNR to be FIG. 8.The corresponding noise PSD for n = 1, 2, 3 at g/γ = 10 4 under the optimization condition in Fig. 7. S r for the three are almost the same.
To understand and generalize the optimized SNR analytically, one first takes γ r (g 2n γ ) 1/(2n+1) according to the optimization conditions in Eq. (A3).This makes S r in Eq. (10) to be a flat spectrum, also shown in Fig. 8.If the relevant bandwidth in Eq. (A1) is smaller than g, S int is dominated by the contribution from the lowest level, i.e., S u a 1 and S u c 1 terms in Eq. ( 11), and We can now see that the sensitivity bandwidth ω sc in Eq. (A1) is just the range that intrinsic fluctuations S int dominates over the readout noise S r , since [S sig /(S int S )] 2 remains a constant within the bandwidth.The inequality in Eq. (A5) leads to Taking it back to Eq. (A1), one has which matches well with our numerical optimization in Eq. (A3).For large n, this leads to an enhanced SNR by a factor approaching (g/γ ) 1/2 .
For the binary tree case, taking g = G and all the intrinsic dissipation coefficients to be universal, one can calculate the signal and the noise PSD, while the readout noise PSD remains the same as Eq. ( 10).
There is an enhancement of 2 2n−2 compared to the chain detector in Eq. ( 9) with a single sensor.On the other hand, each mode of the intrinsic noise is incoherent and the PSD at each layer i is counted by a factor of the number of the same type of the modes in that layer, 2 n+1−i , compared to Eq. ( 11).Similarly, one optimizes and fits the SNR numerically at zero temperature n = 2 : γ r = 1.15(g 4 γ ) 1/5 , SNR = 1.49(g/γ ) 2/5 , (A10) n = 3 : γ r = 1.21(g 6 γ ) 1/7 , SNR = 3.07(g/γ ) 3/7 , which are shown in the lower panel of Fig. 7.As expected, there is a 2 n−1 enhancement due to the multiprobing sensors: We further optimize SNR with both γ r and g 2 − G 2 taken as free parameters.The results are shown in Fig. 9, and the corresponding noise PSD is in Fig. 5.In the binary tree case, the optimized condition is g 2 − G 2 γ r (g 2n γ ) 1/(2n+1) .Thus a small deviation from the optimized value of g 2 − G 2 can lead to negligible impact on the SNR and is indeed more robust than the chain detector when g 2 − G 2 needs to be highly fine tuned.
In the presence of a large thermal occupation number n occ ≡ 1/2 + 1/(e ω/T − 1), the thermal noise is dominant for each intrinsic noise PSD S u = n occ T /ω.For SNR, the finite-temperature effect is equivalent to replacing g 2n → g 2n n occ with the optimized condition γ r (g 2n γ n occ ) 1/(2n+1) , and dividing the whole SNR by n occ .For the binary tree, Eq. (A11) becomes Notice that the thermal noise and the phase fluctuation noise for SRF both contain a frequency-dependent spectrum.Since our sensitivity bandwidth is largely broadened, one should consider this effect.The part with the frequency lower than the the center resonant frequency contributes more while the higher-frequency part contributes less.Thus evaluating the spectrum at the center resonant frequency serves as a viable approximation.
FIG. 9. Optimization conditions and SNR with both γ r and g 2 − G 2 being free parameters for both the chain detector and binary tree haloscope.
In the case of a microwave cavity with a dc magnetic field, the thermal noise is negligible with T = 10 mK and For the LC circuit and SRF case, the SNR in the thermal noise limit are given by respectively.
In the low frequency, the scan bandwidth ω sc (g 2n γ n occ ) 1/(2n+1) becomes larger than the axion mass m for SRF with g = ω rf .In such cases, the enhancement to the scan rate from the increasing bandwidth will be limited, otherwise the integration time defined as t int = t e ω sc /m will be much larger than the e-fold time t e .Thus we require the scan bandwidth ω sc to saturate m so that the integration time t int = t e .Thus the scaling with m changes below 10 MHz for Eq.(A14) and Fig. 6.

FIG. 2 .
FIG.2.Chain detector configuration, where the red lines represent the beam-splitter type of interaction while the blue lines represent the parametric type of interaction.Each circle represents a degree of freedom of the system and only â1 is driven by the axion signal field.The information of the axion field is detected at the b port.

FIG. 3 .
FIG. 3. The effect of PT symmetry breaking when g 2 − G 2 > 0 to the scan rate of the chain detector configuration and binary tree configuration at zero temperature for g/γ = 10 4 .

FIG. 7 .
FIG.7.Optimization conditions for γ r when g = G and corresponding SNR for chain detector and binary tree haloscope, shown also in Eqs.(A3) and (A10).