The works of Halpern & Laurent [2] and Ford [4] subsequently caught the attention of an Italian engineering researcher Giorgio Fontana at the University of Trento [3]. Fontana had been involved in the design of the Silicon Vertex Tracker (SVT) for the proposed SuperB experiment. SuperB was a proposed high‑luminosity electron–positron collider and B‑factory experiment, designed to study B mesons and search for CP violations. Cancelled in 2012, it would have been an asymmetric electron–positron collider (electrons at 7 GeV, positrons at 4 GeV) operating at the Υ(4S) resonance, where B-B̄ pairs are copiously produced.

The SVT for the proposed SuperB experiment was designed to meet extremely challenging requirements imposed by the accelerator’s target of achieving a luminosity 100 times greater than its predecessor, the BaBar experiment.

The SuperB SVT design was largely based on the successful BaBar 5 layer SVT but featured a crucial upgrade: the addition of a new, ultra-precise innermost layer, called Layer 0, to compensate for the reduced boost in the new collider’s asymmetric beam energies, the innermost closest to the Interaction Point, resulting in an SVT with a total of 6 layers.

Fontana’s insight was that a similar multilayer design might also be advantageously applied to the problems of GW detection and emission, including the GASER as described in reference [2]. This led Fontana to publish a series of papers on the topic, culminating the publication of “High Temperature Superconductors as Quantum Sources of Gravitational Waves: The HTSC GASER” in 2012 [3]. All quantum detectors and emitters operate via orbital energy transitions, and Fontana realized that certain spin 2 transition could help force graviton emissions. Quoting from the abstract to reference [3]:

“…it is known from theory that electronic transitions between s and d states may produce the emission of gravitational waves. In normal materials, s-d transitions are suppressed by competing electric and magnetic transitions that have much higher probabilities; this is not the case of Cooper pairs in HTSC, which cannot support states other than s-wave or d-wave. To give theoretical foundations to a potential technology capable of constructing a quantum source of gravitational waves, this paper will also discuss means for creating coherence and population inversion and means to increase [graviton] emission probability.”

Examples of photon and graviton transitions are depicted in Figure 1, where photon transitions are blue, and graviton transitions are depicted in red. For a GASER, the ideal emission junction would favor spin 2 graviton emitting transitions over spin 1 photon emitting transitions.

Figure 1. Examples of photon and graviton transmissions [3] (reprinted with permission).

For the task of maximizing the population of spin 2 graviton transitions Fontana designed an s-wave to d-wave Josephson Junction. If biased appropriately it should invert the graviton emissions over the photon emissions. See the situation at the emission junction as depicted in Figure 2.

There is a subtlety in Figure 2: coherent transitions from s-wave to d-wave states are the important factor in forcing spin 2 transitions. Here the “s-states” and “d-states” refer to orbital angular momentum symmetries in the superconducting order parameter, not the band structure.

What is important for this pairing is to pair a superconductor with s-wave state pairing with a superconductor with predominately d-wave state pairing, with some s-wave state paring being acceptable. It is the s-wave to d-wave dissimilarity that is the important design feature in DJJs. This distinction will be discussed further in Section 5.2.

In the case of spontaneous decay, comparing the decay rate of graviton transitions Γ g to those of electromagnetic dipole transitions Γ e , one can cite for instance the value listed in Kiefer’s 4^th ed. of Quantum Gravity, Equation (2.60) [6], reproduced here as Equation (15):

Figure 2. Graviton emissions at a s-wave to d-wave junction, adapted from [3].

Which is less than encouraging. However, turning to the graviton emission probability in superconducting junctions, Fontana [3] pointed out that the emission ratio for matrix elements of equal structures is more on the order of 8 π G m e 2 / e 2 , which leads one to an emission probability of gravitons is 4.8∙10⁻⁴³ times the emission probability of photons for the electron transition. Assuming Cooper pair density of N = 10²⁰/cm³, with 10 cm³ of material, wavefunction coherence can amplify the probability of gravitons emission to the level of typical spontaneous photon emission in a single atom, the “ideal GASER” case. Fontana observed that “There are indeed 5∙10²¹/cm³ elementary cells in YBCO” [5].

Coherence gain is n², where n is the number of emitters, because the individual emitters add together in phase, and the resulting intensity is proportional to the square of the resultant field emission.

While the upper emission bound will be limited by material saturation, Fontana [3] makes reference to Ford’s [4] discussion surrounding Equation (48) in [4], based in part on L. P. Grishchuck and M. V. Sazhin [5], to point out quantum gravity effects can be macroscopically large. See below for an extract from [4]:

“It is sometimes stated that quantum gravity effects should be characterized by Planck dimensions. Yet in [some] examples discussed it is possible for quantum gravity effects to be macroscopically large. The resolution of this apparent discrepancy can be found by examining results such as [Equation (14) above.] If we let ω ~ ℓ , then we can write this equation” [as (16)]:

where:

P = emitted GW power

P p = Planck power ~ 10⁻⁵g/10⁻⁴⁴sec - 10⁶⁰ ergs/sec

ω = emitted frequency

ω p = Planck frequency ~10⁺⁴⁴/sec

ℓ = characteristic cavity dimension

n = number of quantum emitters

“Thus it is true that the result is characterized by Planck dimensions, except for the factor of n², which can be arbitrarily large. It is the presence of such large dimensionless numbers in the theory arising from the choice of quantum state which can produce quantum gravity effects on scales much larger than the Planck scale.” [4]

Estimated GW emitted power out, and the estimated sensitivity to GW power inputs for Dissimilar Josephson Junctions, or DJJs, will be discussed in Sections 5.3 and 5.4, respectively.

5.1.1. Operating Frequency Selection

Designing a thin film HFGW emitter begs the questions of how to achieve an AC oscillating signal, and what operating frequency should be chosen. On the topic of creating an AC signal, if one is using Josephson Junctions, we may leverage the AC Josephson Effect as reproduced in Figure 5 [9].

Note that in the AC Josephson Effect the frequency at which the junction oscillates is related to the applied bias voltage V0. This is referred to the as the Josephson frequency, f_J. The higher the applied bias voltage, the higher the oscillation frequency.

Figure 5. The AC josephson effect [9].

The next question becomes what operating frequency should be selected for an HFGW emission (&/or detection) device? A few considerations here might include what frequencies permit the use of standard technology, what frequencies are authorized for use, and what frequencies permit reasonably phased arrays of emitter geometries to be masked onto standard wafer sizes. These three factors were combined to generate Table 1, with the frequency judged as currently optimum to be highlighted.

Table 1. US Amateur microwave bands selected for examination [10].

Table 1 was compiled based on amateur bands available for use in the USA [10]. In the case of this table 24 GHz was judged to be the optimum operating frequency because frequencies below this operating point had emitter features too large for a phased array to fit on a standard wafer size, and the use of frequencies above this operating point were judged to be too unwieldy to consider using standard state of the art microwave technology.

5.1.2. Pairing with Standard Wafer Sizes

We turn now to the problem of how to inscribe a thin film phased array of emitters, or receivers, onto a standard wafer size. The approach taken here is that the larger the wafer size, the larger and more effective the possible phased array. However, it is also the case that not every foundry can support every wafer size, so that there may be a practical limit to the largest size a particular foundry can accommodate. This situation is summarized in Table 2 [11].

Table 2. Standard electronics wafer sizes [11].

Wafer sizes are characterized in terms of their diameter, the most common of which are the 150 mm (6”) and the 200 mm (8”). In the next section possible phased array layouts for these two form factors are presented.

5.1.3. Phased Array Spacing Examples (150 mm, 200 mm Cases)

The layout approach taken for the HFGW phased array emitters (and detectors) is to create quarter wave emitter (& receiver) pads one quarter wavelength in diameter, for a nearly in phase wavefront, spaced by full wavelength increments to keep all emitters in phase. Example layouts are presented here for the 150 mm and 200 mm wafer sizes, see Figure 6(a) and Figure 6(b).

(a)

(b)

Figure 6. (a) Phased array layout example for 150 mm wafers; (b) Phased array layout example for 200 mm wafers.

Note that these example layouts create a flat wavefront, without a focal point. If a focal point is desired additional trace lengths may be added or subtracted for the appropriate delay lines to shape the wavefront.

Turning now to material consideration for DJJ device fabrication, we may recall here that in Fontana’s work he suggests Pb/YBCO as a possible material paring choice for s-wave/d-wave junctions [3][6]. While the development and production of Pb-curate Josephson junctions is considered settled art [12][13], subsequent work has demonstrated that “high-temperature cuprate superconductors are among the most complex materials ever explored for practical application” [14]. Among the manufacturing challenges for cuprates one may list the following [14]-[16]:

Cuprates are multicomponent oxides (e.g., Y–Ba–Cu–O, Bi–Sr–Ca–Cu–O) that must be synthesized at high temperatures (≈700˚C - 800˚C) with tight control of cation ratios and oxygen partial pressure to obtain the right superconducting phase.Some constituents (e.g., barium, bismuth, mercury, thallium) are volatile or reactive at processing temperatures, so loss, decomposition, or parasitic phases are common and can suppress Tc and current-carrying capability.High angle grain boundaries act as weak links that drastically reduce the critical current density in polycrystalline cuprates like YBCO and Bi 2212.Manufacturing must therefore create highly textured or near single crystal material (e.g., epitaxial films, biaxially textured substrates, melt textured bulks), which adds processing steps such as buffer layers, seed crystals, and careful thermal profiles.Superconductivity in cuprates is extremely sensitive to oxygen content; precise oxygen stoichiometry and doping require carefully controlled anneals in oxygen or mixed-gas atmospheres.For Bi‑2212, high-pressure overpressure heat treatments (≈50 bar Ar/O2) are needed to densify filaments and optimize connectivity, but these processes are technically demanding and can cause leakage or failure.

All of the above detail is just to point out the desirability of avoiding the use of HTSC cuprates entirely if possible, thereby avoiding the variations and unknowns introduced by such a demanding material. As was discussed in Section 4.3, what’s important at the Josephson Junction is creating an s-wave/d-wave junction. For a d-wave HTSC, only cuprates have exhibited this combination of properties. Therefore, with the current technology we have no choice but to select YBCO or a related cuprate for our HTSC material.

The s-wave trade space is examined in Table 3, where selected s-wave elemental superconductors are listed [17].

Table 3. List of elemental superconductors (selected s-wave choices) [17].

Although other choices and combinations are certainly possible, preliminary down selections for s-wave materials would be those highlighted here, most commonly lead (Pb), as suggested by Fontana [3], or one may substitute Tantalum (Ta), for those not willing to expose their deposition chamber to Pb.

In this section emission powers will be estimated, along with detection sensitivities, to determine whether these thin film devices may be sufficient for HFGW generation and detection in a laboratory setting. The experimental setup under analysis is depicted in Figure 7. We assume here the choice of Pb for the s-wave SC, and YBCO for the d-wave SC. The operating temperature will need to be sub 7 Kelvin to conform with Pb’s critical temperature T_c.

Figure 7. Josephson junction GW generation and detection lab setup.

The problem will be treated in two parts: emitted power estimates are made in 5.3.1, and detection sensitivity estimates are made in 5.3.2.

5.3.1. Emitted Power Estimates

For emitted power estimates we leverage the prior work done by Fontana [3], which estimated the power output for a Pb/BSCCO GASER. In the present paper, Pb/YBCO junctions are proposed. BSCCO and YBCO are both prominent d-wave high-temperature superconductors, with BSCCO offering higher critical temperatures around 90 - 110 K compared to YBCO’s ~93 K. However, YBCO generally excels in practical applications due to superior critical current density (J_c) and manufacturability [18]. The band gaps of the two material selections are nearly the same, so their power output performance for the present paper will be taken as nearly equivalent.

Leveraging Ford’s quantum approach [4] Fontana [3] derived the expected performances of a BSCCO HTSC GASER as summarized in Table 4. Fontana’s device under analysis was a high-power bulk design, whereas in the present paper we have a low power thin film device, so it will be necessary to scale Fontana’s results to fit the thin film constraints.

Table 4. Expected performance of bulk BSCCO HTSC GASER [3] (reprinted with permission).

We note here that for the bulk device described in [3] that a current input of 10 kA/cm² is predicted to produce 10 W/cm², i.e. 1 A/cm² is expected to produce 1 mW/cm². How are we to scale this to a thin film Pb-YBCO device? While the current-voltage characteristic ‘IVC’ of Josephson Junctions (JJ) is a matter of detailed research [18]-[21], it is generally accepted that the critical voltage V_c, at which the JJ loses its superconducting properties, will be related to the critical current times the normal state resistance: V_c = I_c × R_n.

Why normal state resistance? Because our JJ’s will be biased with the DC current required to create the AC Josephson effect frequencies prescribed in Table 1, a nearly zero superconducting resistance of the junctions will necessarily result in the critical current I_c being exceeded, thus triggering normal state resistance R_n. The Resistance-Area product RA is typically given as: RA = R_nA = 10² – 10³ Ω∙μm², so that for a emitter pad sized to a diameter of 3.1 mm, as shown in Figure 6(a) and Figure 6(b) one would then divide the RA product by the area A_e = π(3.1 mm)², or π(3100 μm)² = 30,190,000 μm². For the high end of RA this leads to a per emitter resistance estimate of 33 μΩ per (17):

With a bias voltage specified in Table 1 of roughly 50 μV, the expected bias current per emitter will be I_b = V_b/R_n = 50 μV/33 μΩ = 1.5 Amps. 1.5 Amps emitted from an area of 0.3019 cm² results in ~5.0 Amps/cm² of emitter current density, resulting in a gravitational wave output, based on the ratios of Table 4, of approximately 5.0 mW/cm² of GW output per emitter. Multiplying by the emitter area A_e = 0.3019 cm² yields 1.5 mW of GW power per emitter. Returning to Figure 6(a) and Figure 6(b), one readily observes that the 150 mm array will result in a GW output power of 1.5 mW × 81 = 121.5 mW, whereas the 200 mm array will result in a GW output power of 1.5 mW × 129 = 193.5 mW. See Table 5 for a summary.

Table 5. Expected performance of thin film YBCO HTSC GASER.

5.3.2. Detection Sensitivity Estimates

NEP of DJJs

In detection applications the sensitivity of a device is limited by its Noise Equivalent Power (NEP). In the context of detection, Dissimilar Josephson Junctions (DJJs) are more commonly known as ‘Superconducting Tunnel Junctions, or STJs. [22] For YBCO NEPs of 10⁻¹⁵ to 6 × 10⁻² W / Hz have been reported [23][24]. Holdengreber (2018) [25] goes on to observe that the energy gap of YBCO permits signal detection up into the THz range, therefore our target frequency of 24 GHz as identified in Table 1 should not pose any difficulty in the way of detection. At a bandwidth of 10,000 Hz NEP would be on the order of 600 pW, resulting in a signal to noise power ratio (SNR) of at least 202.5(10)⁶ for the 150 mm array, or better than a power ratio of 83 dB, assuming nearly perfect quantum conversion efficiency per (18):

Quantum Efficiency of DJJs

The d-wave to s-wave conversion efficiency needed for a graviton to photon(s) conversion process is not well understood but is likely a function of quantum measurement efficiency [26] and doping profiles [27]. A range of doping regimes is recommended for experimental use to validate the best approach to enabling d-wave to s-wave graviton to photon(s) conversion. Given the small NEP cited in the proceeding analysis, even a small fractional quantum efficiency should result in an easily measurable signal.

For DJJs, a range of doping regimes was recommended in 5.3.2 to optimize band matching, for enabling d-wave to s-wave graviton to photon(s) conversion. In the event a link budget is established successfully, such a test rig could be extended to testing for potentially gravitomagnetically active materials as described in Stephenson [28][29].

While not in the scope of this paper, other approaches to the generation and detection of gravitational radiation abound and bear mentioning here. The work of Gorlik et al. (2018) [30] is certainly at the forefront, outlining a method using intensive electromagnetic irradiation of high index dielectric materials.

An entirely unrelated technique using pairs of Josephson Junctions in an oscillatory regime has been proposed by Atanasov [31][32] and may provide a parallel path worth investigating.

Recent fundamental research into YBCO may also yield additional benefits. Tahir-Kheli et al. [27] investigated using unique doping profiles to stimulate an s-wave YBCO phase at T_c’s as high at 70 K, with a d-wave YBCO phase occurring in the same material at 100 K. If true, “then there is enormous value to current magnet technologies in uncovering this s-wave phase of YBCO” [27]. Relevant fundamental research into high-temperature cuprate superconductors is also evidenced in Yung-Yeh Chung [33], where operating near the quantum critical point may give rise to coherent behavior in quasi-particles potentially useful for spin-2 transition stimulation.

Finally, should the generation and detection of gravitational waves in a laboratory setting ever meet with success, the next logical question will be how to leverage gravitational wave technology for the application of propulsion. Concepts for GW propulsion are treated at a high level in Fontana’s 2012 JSE paper [34], and would bear revisiting, should success ever befall this field.