Stimulated Emission of Gravitational Waves via Dissimilar Superconducting Josephson Junctions

Stimulated Emission of Gravitational Waves via Dissimilar Superconducting Josephson Junctions

Gary V. Stephenson, Seculine Consulting, Cupertino, CA, USA.

Journal of High Energy Physics, Gravitation and Cosmology, 2026, 12(1), 585–605. Received 29 December 2025; accepted 26 January 2026; published 29 January 2026.

Abstract

An experiment is proposed to prove the stimulated emission of gravitational waves is experimentally possible. The proposal suggests that Josephson Junctions with dissimilar superconductors, an s-wave joined to a d-wave, can force spin 2 transitions, triggering the emission of gravitational waves.

Keywords: Gravitational Waves, GASER, Superconductors, Josephson Junctions.

1. Introduction

An approach to the quantum emission of gravitational waves is described, along with a suggested experiment.

Semi-classical approaches covered in this paper include details gleaned from these important historical references:

• Section 2.0, 1964 Halpern and Laurent, "On the Gravitational Radiation of Microscopic Systems".
• Section 3.0, 1982 Ford, "Gravitational Radiation by Quantum Systems".
• Section 4.0, 2012 Fontana, "High Temperature Superconductors as Quantum Sources of Gravitational Waves: The HTSC GASER".

After review of this historical supporting literature an experimental approach is proposed in Section 5.0.

2. Halpern and Laurent's "On the Gravitational Radiation of Microscopic Systems"

Leopold Ernst Halpern (1925-2006) was an Austrian-born theoretical physicist best known for his work on gravitation, especially attempts to generalize Einstein's general relativity by incorporating spin and the de Sitter group into the fundamental structure of spacetime. He was closely associated with both Erwin Schrödinger and Paul Dirac and later became a long-time member of the gravitational physics community in the United States, particularly at Florida State University and in connection with the Gravity Probe B experiment [1].

One of his earlier works was Halpern and Laurent (1964), "On the Gravitational Radiation of Microscopic Systems" [2], which later came to the attention of researcher Giorgio Fontana [3]. This paper forms the introductory material describing predicted radiation patterns and strengths for gravitational emissions triggered from spin 2 transitions in a suitable solid-state material. In it Halpern and his co-author Laurent:

• Developed the theory describing the emission of gravitational radiation from quantum systems.
• Applied the linearized theory of gravitational radiation to quantum systems.
• Calculated gravitational quadrupolar transitions permitted for the emission of gravitons.
• Compared the emission of gravitons to the emission of photons, showing that the spontaneous emission of gravitons from quantum systems is comparatively small due to the weakness of gravity.

Halpern's approach was largely semi-classical but was later extended to a more fully quantum approach by Ford in 1982 [4], and will be summarized in Section 2 of this paper.

Halpern's subsequent work also suggested the possibility of a "GASER" (gravitational counterpart of a laser) using stimulated emission. The concept of a GASER was expanded on by Fontana [3] and will be summarized in Section 3 of this paper.

For the purposes of the present section, the most relevant material regarding predicted gravitational radiation is drawn from [2], the angular distribution of electromagnetic and gravitational multipole radiation of the lowest orders, the gravitational transitions of which are reproduced below.

Gravitational radiation for J = 2, l = 0 (Equations 1 to 7 in the source). The angular dependence of the potentials is given by the spinor coupling spherical harmonic tensors acting on the spin potential. The angular dependence of the amplitudes is then listed for each pair of quantum indices M and μ: for M = 2 with μ = +2 the amplitude goes as the fourth power of the cosine of half the polar angle, for M = 2 with μ = −2 as the fourth power of the sine of half the polar angle, for M = 1 as a mixed third-power cosine and first-power sine combination, and for M = 0 as the product of squared cosine and squared sine of half the polar angle — each multiplied by the appropriate normalisation factor and by the azimuthal phase factor. The angular dependence of the total radiation for M = 2 goes as the eighth power of the cosine of half the polar angle multiplied by the eighth power of the sine of half the polar angle.

Gravitational radiation for J = 2, l = 1 (Equations 8 to 10 in the source). The angular dependence of the potentials for M = 2 combines a term in the cosine of the polar angle with a term in the sine of the polar angle carrying the azimuthal phase factor. The amplitudes for μ = +2 and μ = −2 again go as the fourth powers of the cosine and the sine of half the polar angle respectively. The angular dependence of the total radiation is the same as for J = 2, l = 0.

(The tabulated multipole expressions did not survive text extraction cleanly and are stated here as named results; the complete equations are in the source.)

See Section 6.0 for a summary of the nomenclature used here. As admitted by the authors themselves in the Appendix A of [2], these predicted radiation patterns are the result of a linearized theory of gravitational radiation, and "The linearized equations are therefore not consistent, but they may nevertheless serve to evaluate the radiation field approximately" [2].

3. Ford's "Gravitational Radiation by Quantum Systems"

Lawrence H. Ford is a professor emeritus of physics at Tufts University, associated with the Department of Physics and Astronomy and the Tufts Institute of Cosmology.

He joined Tufts as an assistant professor in 1980, was promoted to associate professor in 1985, and became a full professor in 1992; he was later elected a fellow of the American Physical Society in 2004 for work in quantum field theory in flat and curved spacetime. Early in his tenure at Tufts, in 1982, he published "Gravitational Radiation by Quantum Systems" [4], the topic of this section.

As introduced in the abstract of Ford 1982 [4]:

"The limits of validity of the semiclassical theory in which gravity is unquantized is discussed. This is done by comparing the emission of classical gravitational waves in the semiclassical theory with graviton emission in quantum gravity theory. It is shown that these can be quite different even for macroscopic systems. Thus, quantum gravitational effects can manifest themselves on a macroscopic scale."

This paper has an interesting structure: Section II "Gravitational Radiation by Quantum Systems" was divided into a Part A, which treated the subject semi-classically, and a Part B, where the author contrasts the prior approach with a purely quantum theory. Part A, from [4], (13), the semi-classical case, the metric perturbation is summarized here in Equation (11):

Equation (11). The metric perturbation equals the in-field metric component plus sixteen pi times the integral over spacetime of the retarded Green's function multiplied by the energy momentum tensor.

where the metric perturbation, the in-field metric component, the retarded Green's function and the energy momentum tensor are as listed in the nomenclature.

He concludes Section II Part B with an insightful comparison of the resultant integrated energy-momentum tensors from both approaches:

"The essential difference between the semiclassical and the quantum results is that the [semi-classical case] depends upon a product of expectation values whereas [quantum case] depends upon an expectation value of products."

The example of a scalar field source was then explored for comparative purposes in section III of this same reference [4] to explore what would happen if the two approaches were set equal. Reference [4] Equation (27) embodies the above quoted comparison by setting the two approaches equal, reproduced as Equation (12) here.

Equation (12). The expectation value of the product of the energy momentum tensor and the energy momentum tensor perturbation is set equal to the product of their separate expectation values.

But the author does not stop there; he goes on to compare the predicted gravitational wave (GW) power radiated in the semi-classical case P sub sc, with a predicted power in the quantum case, P sub q.

Equation (13). Radiated power in semi-classical theory is given in [4] [5]: it scales with the fourth power of the radiating frequency and with the mode function, and equivalently with the square of the average energy density in the cavity, the square of the radiating frequency and the sixth power of the characteristic cavity dimension, modulated by the same mode function.

where P approximately equal to P sub sc is the semi-classical radiated power, z is the resonant cavity dimension, ω is the radiating frequency, ℓ is the characteristic cavity dimension, ε is the average energy density in the cavity, and α is a scaling constant, where the mode function is the characteristic frequency-times-length product raised to the power minus α.

As contrasted with radiated gravitational wave power from quantum theory, given by [4] and reproduced here as Equation (14):

Equation (14). The quantum-case radiated power equals the square of the number of quantum radiators, multiplied by the fourth power of the radiating frequency, multiplied by the mode function.

where P sub q is the quantum case radiated power, ω is the radiating frequency, ℓ is the characteristic cavity dimension, β is the quantum scaling constant, and n is the number of quantum radiators.

Noting that "n" can be made arbitrarily large, Ford concluded that the difference should be measurable, were there a way to emit and measure GW power levels in a laboratory setting [4].

In Sections 3 and 4 we'll explore the proposals to do just that, which unfortunately very likely depend solely on the second approach being the correct one.

4. Fontana's "High Temperature Superconductors as Quantum Sources of Gravitational Waves"

4.1. Historical Background

The works of Halpern and 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 upsilon (4S) resonance, where B and anti-B 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.

4.2. The GASER Application

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).

4.3. Dissimilar Josephson Junctions (DJJs)

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.

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

4.4. Emission Power Estimation

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

Equation (15). The ratio of the graviton transition rate to the electromagnetic dipole transition rate is approximately 1.28 times ten to the minus forty-seven.

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 eight pi G times the square of the electron mass divided by the square of the electron charge, which leads one to an emission probability of gravitons is 4.8 times ten to the minus forty-three times the emission probability of photons for the electron transition. Assuming Cooper pair density of N equal to ten to the twentieth per cubic centimetre, with 10 cubic centimetres 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 times ten to the twenty-first per cubic centimetre elementary cells in YBCO" [5].

Coherence gain is n squared, 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 be of order the inverse characteristic cavity dimension, then we can write this equation" [as (16)]:

Equation (16). The emitted gravitational-wave power divided by the Planck power is approximately the square of the number of quantum emitters multiplied by the fourth power of the ratio of the emitted frequency to the Planck frequency.

where P is the emitted GW power, P sub p is the Planck power of order ten to the sixtieth ergs per second, ω is the emitted frequency, ω sub p is the Planck frequency of order ten to the plus forty-four per second, ℓ is the characteristic cavity dimension, and n is the number of quantum emitters.

"Thus it is true that the result is characterized by Planck dimensions, except for the factor of n squared, 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. Experiment Design Considerations

Thus far we've discussed only the generation and emission of High Frequency Gravitational Waves (HFGW) by DJJs, not their detection. However, as Fontana noted in [3], "It must be observed that reciprocity will permit the design and construction of HTSC GASERS pumped by HFGW instead of DC electric current." In other words, the same devices that can be biased to emit HFGW can also be used passively to detect them. This would be a necessary element of any laboratory test, for there is very little value in generating HFGW signals that cannot be detected. Whether the signals generated by DJJ technology are strong enough to be detected by that same technology is still an open question and will be treated in more detail in Section 4.3.

Then there is the matter of what device type should be chosen for an experimental setup. Fontana [3] suggested a bulk device, the conceptual design of which is depicted in Figure 3. In this particular diagram a capacitor bank 'C' supplies DC bias through a thyristor until the current drops below a holding level, at which point it switches off. There are two impracticalities associated with this design. The first is that since the DJJ includes LTSC material, the entire setup must be cooled to near absolute zero temperatures, which would require a very large and complex cryogenic cooling system. The second is that, while the LTSC could be lead (Pb), and may be vapor dispositioned, the device envisioned here would require bulk HTSC of a d-wave type, for instance YBCO, which is notoriously difficult and expensive to fabricate.

A more practical design might envision the use of thin films for both the s-wave superconductor (the LTSC) as well as the d-wave superconductor (the HTSC), ideally on a standard substrate such as silicon (Si). If one chose a standard size silicon wafer one could even envision testing such a thin film device in standard low temperature cryogenic probe stations, such as those used for existing quantum computing wafers. Leveraging existing quantum computing technology for the development of quantum gravitation devices would save an enormous amount of development costs. Design and material aspects of thin film device design considerations will be examined in Sections 5.1 and 5.2.

Figure 3. A bulk device experimental design for dissimilar Josephson junctions [3] (reprinted with permission).

Finally, it would be remiss to not mention in passing the notion of focusing HFGW energy, to boost its power cross section, thereby improving detectability. In a laboratory or engineering setting, it is well known that matter cannot be fashioned into a useful gravitational wave lens the way glass or mirrors are used for light, because the coupling of gravitational waves to any realistic material is too weak to produce appreciable focusing [7]. Only astronomical-scale mass distributions produce noticeable lensing effects, and those are fixed by astrophysics rather than engineered devices.

However, it would be possible to fashion a curved array of emitters such that the wavefronts constructively interfere at a certain focal point. Such a point was made by Fontana [3] and in [8], and this idea is reproduced in Figure 4.

Figure 4. A curved GASER to create an HFGW focal point, adapted from [3] [8].

In the technological landscape of flat two-dimensional thin films, such a device would be tedious to engineer and expensive to build. Better in practice one might judge to replace the delta radii reflected in the curvature of Figure 4 with delay times creating the same delay pattern as the curvature of the emitter array, as is commonly done in phased array communication antennae and radar arrays. Such will be the impetus for the phased array design choices presented in Section 4.1.

5.1. Phased Array Emitters and Receivers Design

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 V sub 0. This is referred to as the Josephson frequency, f sub 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 (and/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].

| Frequency (GHz) | Wavelength (cm) | Quarter-wave geometry (cm) | JJ operating voltage (µV) | |---|---|---|---| | 1.296 | 23.15 | 5.79 | 2.68 | | 2.32 | 12.93 | 3.23 | 4.80 | | 3.4 | 8.82 | 2.21 | 7.03 | | 5.76 | 5.21 | 1.30 | 11.91 | | 10.368 | 2.89 | 0.72 | 21.44 | | 24.048 | 1.25 | 0.31 | 49.73 | | 47.088 | 0.64 | 0.16 | 97.37 | | 76.233 | 0.39 | 0.10 | 157.64 | | 120 | 0.25 | 0.06 | 248.14 | | 249 | 0.12 | 0.03 | 514.89 | | 6250 | 0.0048 | 0.0012 | 12923.90 |

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 size | Typical thickness | Year introduced | |---|---|---| | 1-inch (25 mm) | — | 1960 | | 2-inch (51 mm) | 275 μm | 1969 | | 3-inch (76 mm) | 375 μm | 1972 | | 4-inch (100 mm) | 525 μm | 1976 | | 4.9-inch (125 mm) | 625 μm | 1981 | | 150 mm (5.9 inch, usually referred to as "6 inch") | 675 μm | 1983 | | 200 mm (7.9 inch, usually referred to as "8 inch") | 725 μm | 1992 | | 300 mm (11.8 inch, usually referred to as "12 inch") | 775 μm | 1999 | | 450 mm (17.7 inch) (proposed) | 925 μm | — | | 675 mm (26.6 inch) (theoretical) | unknown | — |

Wafer sizes are characterized in terms of their diameter, the most common of which are the 150 mm (6 inch) and the 200 mm (8 inch). 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 (and 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).

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.

5.2. Material Considerations

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 (approximately 700°C to 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 (approximately 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].

| Substance | Class | Tc (K) | Hc (T) | Type | BCS | |---|---|---|---|---|---| | Al | Element | 1.2 | 0.01 | I | yes | | Cd | Element | 0.52 | 0.0028 | I | yes | | Ga | Element | 1.083 | 0.0058 | I | yes | | α-Hg | Element | 4.15 | 0.04 | I | yes | | β-Hg | Element | 3.95 | 0.04 | I | yes | | In | Element | 3.4 | 0.03 | I | yes | | Ir | Element | 0.14 | 0.0016 | I | yes | | Mo | Element | 0.92 | 0.0096 | I | yes | | Os | Element | 0.65 | 0.007 | I | yes | | Pb | Element | 7.19 | 0.08 | I | yes | | Re | Element | 2.4 | 0.03 | I | yes | | Sn | Element | 3.72 | 0.03 | I | yes | | Ta | Element | 4.48 | 0.09 | I | yes | | α-Th | Element | 1.37 | 0.013 | I | yes | | Tl | Element | 2.39 | 0.02 | I | yes |

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.

5.3. Signal Power Approximations

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 Tc.

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 to 110 K compared to YBCO's approximately 93 K. However, YBCO generally excels in practical applications due to superior critical current density (Jc) 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).

| Parameter | Value | Units | |---|---|---| | Energy of s/d wave gap in BSCCO | approximately 1.3 | THz | | Power output with 10 kA per square centimetre | approximately 10 | W per square centimetre | | Power output for multilayer | approximately 10 to the third | W per square centimetre | | Volumetric power output | approximately 10 | MW per cubic metre |

We note here that for the bulk device described in [3] that a current input of 10 kA per square centimetre is predicted to produce 10 W per square centimetre, i.e. 1 A per square centimetre is expected to produce 1 mW per square centimetre. 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 Vc, at which the JJ loses its superconducting properties, will be related to the critical current times the normal state resistance: Vc equals Ic multiplied by Rn.

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 Ic being exceeded, thus triggering normal state resistance Rn. The Resistance-Area product RA is typically given as RA equals Rn multiplied by A, in the range of 100 to 1000 ohm micrometre squared, so that for an 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 Ae, which for a radius of 3100 μm is 30,190,000 μm squared. For the high end of RA this leads to a per emitter resistance estimate of 33 μΩ per Equation (17):

Equation (17). The emitter resistance equals the resistance-area product divided by the emitter area: 1000 ohm micrometre squared divided by 30,190,000 micrometre squared, giving 33 microohms.

With a bias voltage specified in Table 1 of roughly 50 μV, the expected bias current per emitter will be Ib equals Vb divided by Rn, that is 50 μV divided by 33 μΩ, giving 1.5 Amps. 1.5 Amps emitted from an area of 0.3019 square centimetres results in approximately 5.0 Amps per square centimetre of emitter current density, resulting in a gravitational wave output, based on the ratios of Table 4, of approximately 5.0 mW per square centimetre of GW output per emitter. Multiplying by the emitter area Ae equals 0.3019 square centimetres 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 multiplied by 81, which is 121.5 mW, whereas the 200 mm array will result in a GW output power of 1.5 mW multiplied by 129, which is 193.5 mW. See Table 5 for a summary.

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

| Parameter | Value | Units | |---|---|---| | Power emitted per unit area | approximately 5.0 | mW per square centimetre | | Power output of one emitter | approximately 1.5 | mW | | Power output for 150 mm array | approximately 121.5 | mW | | Power output for 200 mm array | approximately 193.5 | mW |

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 ten to the minus fifteen up to 6 times ten to the minus two watts per root hertz 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 million for the 150 mm array, or better than a power ratio of 83 dB, assuming nearly perfect quantum conversion efficiency per Equation (18):

Equation (18). The signal-to-noise ratio equals the power divided by the noise equivalent power: 121.5 mW divided by 600 pW, giving 202.5 million, or approximately 83 dB.

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.

5.4. Next Steps

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 Tc'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.

6. Conclusion

An approach for developing, building, and testing phased array emitters of gravitational wave energy has been outlined using specially prepared Josephson Junctions. The remaining work is for the described hardware to be built and tested.

Acknowledgments

The author gratefully acknowledges helpful discussions with Eric Davis, Andrew Beckwith, Christian Corda, Giorgio Fontana, William Rieken, David Rossi and Kyle Gustin of Phase Helix, and Gurken Sufi and Yusha Bey of Ravata Solutions.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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Nomenclature

Section 1.0 nomenclature. Spinor coupling spherical harmonic tensors; spin eigenvectors; spin potential; spin 2 amplitudes; quantum indices J, l, M and μ; angular indices θ and φ; EM radiation potential q.

Section 2.0 nomenclature. Metric perturbation; in-field metric component; retarded Green's function; energy momentum tensor; energy momentum tensor perturbation; semi-classical radiated power P equal to P sub sc; quantum case radiated power P sub q; resonant cavity dimension z; average energy density in the cavity ε; radiating frequency ω; characteristic cavity dimension ℓ; scaling constant α, where the mode function is the frequency-length product raised to minus α; quantum scaling constant β; number of quantum radiators n.

Section 3.0 nomenclature. Graviton transitions; electromagnetic dipole transitions; emitted GW power P; Planck power P sub p, of order ten to the sixtieth ergs per second; emitted frequency ω; Planck frequency ω sub p, of order ten to the forty-fourth per second; number of quantum emitters n.

Section 4.0 nomenclature. DC supply voltage V sub 0; AC Josephson effect voltage v; AC Josephson effect current i; DC level of critical current Ic; Josephson effect frequency f sub j; Josephson effect wavelength; AC phase and initial phase; ohms as units of resistance; critical temperature Tc; critical current density Jc; critical voltage Vc; emitter bias current Ib; emitter bias voltage Vb; normal state resistance Rn; emitter resistance Re; emitter area Ae; noise equivalent power NEP, in watts per root hertz; signal to noise ratio SNR.