For two ideal, perfectly conducting, infinite parallel plates of area A separated by d in vacuum at T = 0, Casimir’s original result [[1]] gives the attractive force (Equation (1)).

This is a pressure, not a stored, extractable energy. For a comprehensive monograph treatment of the Casimir effect across geometries, materials, and finite-temperature extensions, we direct the reader to Milton [[12]]. Following Jaffe [[13]], we emphasize that Equation (1) does not by itself constitute evidence that vacuum “zero-point energy” is a technologically extractable reservoir; the force can be derived without invoking zero-point energy at all.

Real cavities deviate from Equation (1) through finite conductivity, nonzero temperature, frequency-dependent dielectric response, surface roughness, PFA limitations, patch potentials, and mechanical instabilities. Each of these must be budgeted in the force-modulation and energy-balance calculations.

When a cavity boundary is modulated in time, mode frequencies become nonstationary; vacuum modes are parametrically driven, and real photons appear [[2], [3]]. In the standard Bogoliubov formulation, the output annihilation operators are related to the input operators by Equation (2).

so that the mean photon number produced from vacuum in mode k is Nₖ = |βₖ|². The DCE is therefore an energy-conversion process, not a free-energy process. Wilson et al. [[4]] verified this experimentally. The DCE sits in the same family of accelerated-observer/nonstationary-boundary vacuum phenomena as the Unruh effect [[14]].

Single-bubble sonoluminescence [[5], [6]] follows a Rayleigh-Plesset-type dynamics, generalized to compressible collapse via Keller-Miksis (Equation (3)).

Here R(t) is the bubble radius, c the speed of sound in the liquid, σ the surface tension, μ the viscosity, P_B the pressure at the bubble wall, and P_∞(t) the far-field acoustic drive. Reviews [[5]-[7]] treat SL as an acoustic-energy concentration process. This remains the baseline null hypothesis against which any DCE contribution is tested.

The idea of an SL-vacuum connection traces back at least to Schwinger [[15]]. Eberlein [[8]] developed it quantitatively; Milton and Ng [[9]] and Lambrechtet al. [[10]] challenged it on energetic and spectral grounds. The present manuscript takes this literature as its scholarly starting point: SL is not treated as an independent energy source, and the DCE-like interpretation of SL is treated as a hypothesis to be tested, not assumed.

Jiang and Wilczek [[16]] showed theoretically that chiral metamaterials and magneto-optical media can modify the sign and magnitude of the Casimir force. In this manuscript, such materials are considered only as devices to shape cavity mode structure and parametric coupling; they are not treated as an energy source. The energetic cost of external magnetic-field modulation and any material switching must be included in the global energy balance (Section 5).

Given the above, an experimentally decidable question remains: can a carefully engineered acoustic-Casimir cavity produce electromagnetic output with the specific spectral and photon-statistical signatures of a DCE process, at a level distinguishable from conventional SL and thermal channels, under a fully accounted global energy balance?

The proposed prototype has an outer inner-radius R_cav = 5 cm (“small-scale”). A medium-scale variant has R_cav = 25 cm. The earlier 2-m variant has been withdrawn.

M = 32 piezoelectric elements distributed over the outer wall operate at f₀ ∈ [[1], [5]] MHz with independent phase φ_i and amplitude A_i. P_acoustic is a measured quantity, not an assumed one.

The original phrase “billions of concentric plates” is withdrawn. For an available radial path Δr ≈ 5 cm and a target gap d ∈ [10 nm, 100 nm], the number of achievable shells is given by Equation (4).

Each “layer” is a segmented, acoustically transparent metallic-dielectric stack (Ag/SiO₂ or graphene/Al₂O₃ by ALD).

Because F_Cas ∝ d⁻⁴, gap tolerance is critical. A fractional gap error δd/d produces a fractional force error (Equation (5)).

so that a 5% gap error produces a 20% force error, which implies a fabrication tolerance δd/d ≲ 10⁻³ for any meaningful cavity-QED calculation.

Physically Realizable Multilayer Specification (R2.1)

A representative shell class S_i has: shell thicknesst_s = 50 nm realized as Ag(20 nm)/SiO₂ (30 nm) grown by ALD on a 15 nm SiNₓ carrier membrane; shell segmentation into N_seg = 6 geodesic tiles per shell (pentakis-icosahedral partition), no single tile spanning more than λ_acoustic/4 at f₀ = 2 MHz; mechanical supportsN_sup = 12 per shell, radial piezoceramic pillars of 5 μm diameter with integrated capacitive gap sensors; fluid access through hexagonal micro-perforations of pitch p = 2 μm, diameter 0.6 μm, open area φ ≈ 8%; acoustic impedance matching of the stack to water/glycerol within 10% across the operating band; bubble-induced pressure-to-gap coupling is given by Equation (6).

where G_i is the mechanical Green’s function of the segmented shell, k_sup ≈ 80 N/m per pillar and k_mem(ω) is the frequency-dependent membrane stiffness. For a bubble collapse depositing δP_bub ∼ 10⁵ Pa over an effective area A_eff ∼ 100 μm² within 100 ns of the wall, Equation (6) yields peak gap modulation δd ∼ 0.4 nm, i.e., a fractional modulation δd/d ∼ 4 × 10⁻³ for a 100 nm nominal gap. This is the mechanism by which SL bubble collapse modulates the Casimir gap rather than merely perturbing the surrounding fluid.

The revised manuscript adopts a water/glycerol+xenon-saturated fluid as the primary experimental platform. This is the well-characterized SL regime [[5], [6]] and permits comparison against decades of published data. Liquid mercury is demoted from the primary design to a later-stage option.

The instrumentation array includes PMT, SPAD coincidence pairs, streak camera, UV/X-ray scintillators, calorimeter, hydrophones, and near-field microwave probes for DCE-band emission. All channels are calibrated independently against traceable standards.

Inside the fluid, the linear-acoustic pressure p(r, t) satisfies Equation (7).

where S is the transducer source term and S_bub the coupling from bubble oscillations. Spherical-harmonic decomposition (Equation (8))

gives radial equations that make the “spherical-harmonic alignment” language mathematically meaningful.

Rather than tracking ∼10¹² bubbles individually, we introduce the bubble distribution function (Equation (9)) obeying a Boltzmann-like transport equation.

The controller acts on f’s moments, not on individual collapses.

Closure of the Bubble-Population Model (R2.4)

The collision operator in Equation (9) is closed as Equation (10): nucleation + coalescence + dissipative loss.

Nucleation follows classical theory [Blander & Katz, AIChE J. 21, 833 (1975)]; coalescence uses the Smoluchowski kernel with sticking α_c ∈ [10⁻³, 10⁻¹] from Chesters (1991); dissipative lossγ_diss combines acoustic radiation damping [Prosperetti, J. Acoust. Soc. Am. 61, 17 (1977)] and rectified diffusion at the bubble wall. Initial condition: equilibrium size distribution f₀(R) ∝ R^−3.5exp(−R/R_max). Boundary conditions:f = 0 on outer wall; zero-flux at R_min and R_max. Parameter sources:σ, μ,ρ, c from NIST tables; α_c constrained experimentally. Constraining observables: (i) ns-resolved SL flash rate; (ii) broadband MHz acoustic noise floor; (iii) optical holography of baseline f₀(R); (iv) DCE-band microwave noise floor.

Between two adjacent Casimir shells modulated by the passage of collapsing bubbles, the cavity mode obeys Equation (11).

with d_i(t) the instantaneous inter-shell gap and ε, μ the dispersive material responses.

Given ω_k(t) from Equation (11), we solve for the Bogoliubov coefficients α_k, β_k using the standard adiabatic-basis matching [[3], [17]] (Equation (12)).

The mean DCE photon number is N_k = |β_k|² and E_DCE = Σ_kħω_kN_k. The parametric resonance condition verified by [[4]] is that ω_k(t) oscillate at Ω_mod ≈ 2ω_k; DCE photons appear in pairs at ω₁ + ω₂ ≈ Ω_mod.

The previous manuscript wrote E_SL ≈ 10⁻³ - 10⁻¹ MeV per collapse and, with N = 10¹² collapses/s, obtained P ∼ 10⁸ W. This is arithmetically inconsistent: because 1 MeV = 1.602 × 10⁻¹³ J, the correct value is P_SL ≈ 1.6 × 10⁻⁴ - 1.6 × 10⁻² W. The original 10⁸ W figure was in error by 10-12 orders of magnitude.

For a single acoustic cycle, E_acoustic,in distributes among SL, thermal, chemical, mechanical, EM, and dissipative channels. The device-level, time-averaged balance is Equation (13).

Single System Boundary and Channel Assignment (R2.5)

The energy ledger is anchored on one closed control volume Σ: the outer surface of the cryostat vacuum jacket. Every channel appearing in Equation (13) is assigned exactly once with respect to Σ: P_electrical, P_control, P_cryo are external inputs; P_acoustic is an internal transfer between the transducer subsystem and the fluid subsystem and does not appear separately in the external-input sum, which prevents double-counting; P_EM,out and P_mech,out are outputs; P_thermal and P_losses are loss outputs. The Casimir energy is a configurational quantity strictly inside Σ and is never an external channel.

Scenario A—Null DCE (baseline): P_DCE ≈ 0, P_net ≤ 0, null hypothesis. Scenario B—Detectable DCE (target): 0 < P_DCE ≪ P_acoustic with statistically significant sideband, correlated-pair, and squeezing signatures; this is the primary scientific target. Scenario C—Anomalous energy balance: P_out > P_in; this scenario is not assumed anywhere in this manuscript.

The controller operates on modal statistics, not individual sub-nanosecond bubble events. The state vector contains the low-ℓ acoustic modal amplitudes and rates, mean bubble radius and dispersion, temperature, bubble number density, optical intensity, and per-layer gap monitors. The scalar objective is J = −w₁P_acoustic + w₂P_{optical,DCE-band} – w₃P_instability – w₄P_thermal.

Three control layers operate at three disjoint timescales (see Table 1). Inner loop (PLL, 10 - 100 kHz) keeps each transducer locked to the resonator’s first-harmonic radial mode. Mid loop (MPC, horizon ∼10 ms) adjusts amplitudes, phases, and frequencies. Outer loop (RL/Bayesian, s-hour) updates supervisory parameters. No controller is proposed to individually observe or control the ∼10¹² collapse events per second.

Table 1.Control-timescale ladder.

The coupled system is integrated with an operator-splitting scheme: Equation (7) on a spherical FEM mesh with radial refinement in the Casimir stack; Equation (9) on a (R, Ṙ) velocity grid with 40 × 40 resolution; Equations (11)-(12) on the instantaneous cavity spectrum. Time step Δt ≤ 10⁻¹¹ s during collapse; adaptive stepping otherwise. Convergence is demonstrated by halving the mesh and time step until the relative energy change is below 10⁻². Uncertainty is propagated via a Monte Carlo of 10⁴ samples.

The observed radiation decomposes as P_obs = P_SL + P_plasma + P_thermal + P_chemi + P_acou-leak + P_electrical + P_DCE. Excess optical power alone cannot establish a DCE mechanism, because SL-plasma emission [[5], [7]] is variable, gas- and pressure-sensitive. We require the combined signature: photon-number correlations g⁽²⁾ above the thermal baseline for paired modes summing to Ω_mod, plus two-mode squeezing below the vacuum limit for the same pair. This is the test class that established the superconducting-circuit DCE [[4]].

Measurement Chain per Quantum Signature (R2.6)

Two distinct chains are specified, one per signature. Chain A—Photon-number correlation ( g i , j ( 2 ) ( 0 ) ):

cavity output → cryogenic circulator → HEMT (T_N ≈ 2 K) → image-reject mixer → digitizer at 2 GS/s → FPGA time-tagging with 100 ps bins → HBT coincidence within τ_coinc = 1 ns on two orthogonal cavity ports. Null-model contributions enumerated: (a) HEMT amplifier noise; (b) detector afterpulsing; (c) detector cross-talk; (d) pump leakage at Ω_mod; (e) nonlinear mixing; (f) thermal radiation. Detection requires g⁽²⁾(0) − 1 > 5σ above the sum of (a)-(f). Chain B—Phase-referenced quadrature detection:cavity output → JPA in phase-sensitive mode, pumped at 2f_cav and phase-locked to the acoustic modulation carrier → homodyne detection with LO phase stepped in π/16 increments → digitized quadrature variances. Two-mode squeezing manifests as σ − 2 < 1/2 (vacuum units). Null-model contributions: (g) JPA added noise; (h) LO phase noise; (i) pump-tone leakage; (j) mixer image/imbalance; (k) thermal blackbody; (l) JPA nonlinearity. Only simultaneous success of Chain A and Chain B on the same paired modes constitutes a DCE detection.

The reference prediction is stated at a single, fully specified operating point. Target microwave readout band: f_MW ∈ [[4], [8]] GHz. Operating temperature: T = 20 mK at the microwave cavity, T_fluid = 280 K in the cavitation region. Thermal occupation of the microwave modes: n ¯ _th(6 GHz, 20 mK) ≈ 4 × 10⁻⁷. Material losses: internal Q_int ∈ [3 × 10³, 3 × 10⁴].

Reconciling MHz drive with GHz readout: the acoustic carrier at f₀ ≈ 2 MHz is not the DCE pump; the pump is the collapse waveform, whose spectral content peaks in the 0.1 - 10 GHz band because the Rayleigh collapse duration is τ_c ∼ 100 ps. Fourier-decomposing δd(t) gives a modulation spectral density (Equation (14)).

which at f = 2f_cav ≈ 12 GHz (parametric-resonance condition for a 6 GHz cavity) evaluates to S_δd(2f_cav)/d² ≈ 10⁻²⁷ Hz⁻¹ for d = 100 nm, δd = 0.4 nm, and Ṅ_b = 10⁵ cycle⁻¹. The gap-modulation spectral weight enters the predicted count rate through Equation (15).

in the weak-modulation limit, with the bracket capturing thermal enhancement.

Parameters: Ag/SiO₂ ALD stack with Drude ω_p = 1.4 × 10¹⁶ rad/s, γ = 3.2 × 10¹³ rad/s; SiO₂ ε_r = 3.9, tanδ = 3 × 10⁻⁴ at 20 mK; TE_011-like mode at f_cav = 6.0 GHz, V_mode = 1.2 × 10⁻⁸ m³, overlap 0.42, Q_int = 10⁴, Q_ext = 10⁴, Q_load = 5 × 10³; δd = 0.4 nm (Equation (6)); V_part = 5.0 × 10⁻⁹ m³; Ṅ_b/f₀ = 10⁵; losses TLS 10⁻⁴, radiative 10⁻⁴, acoustic ∼10⁻³. Substitution gives | h ˜ (2f_cav)|² ≈ 5.0 × 10⁻³⁸ m²s and S_δd/d² ≈ 2.5 × 10⁻²⁷ Hz⁻¹. Equation (15) then yields Equation (16).

translating to P_DCE ≃ 2.3 × 10⁻²¹ W, i.e., ∼10³ photons/s at ∼6 GHz. Over a 10⁵ s integration and 10⁻³ dark-count noise floor, this yields S/N ≈ 5 - 10, marginally detectable. The value Γ_DCE ≈ 5.7 × 10² photons/s is the reference prediction and supersedes the earlier 10⁴ - 10⁸ envelope.

A side-by-side comparison of how each key quantity was treated in the original manuscript versus the present revision is summarised in Table 3, immediately below.

Table 3.Original versus revised treatment of key quantities.