Optically rewritten fall of a levitated diamond

A commercial single-crystal diamond slab, 3 × 3 × 0.5 mm with 4.5 ppm of nitrogen-vacancy (NV) centres and every face cut along [100], hangs from a carbon-fibre rod 5 cm long and 0.5 mm across. The rod is glued to the centre of a slotted pyrolytic graphite plate, 10 × 10 × 1 mm, that levitates diamagnetically above four N52 cubes of side 12.7 mm magnetised in a checkerboard; a 4 mm iron plate under the array shields the diamond from it. The resonator weighs 128 mg in all, oscillates vertically at 17.6 Hz and has a quality factor of 55, at 300 K and atmospheric pressure. A cylindrical N52 magnet on a stage below the diamond supplies a field and a field gradient, and a fixed interferometer head reads a mirror on the graphite, sampled at 2440 Hz.

The source models each NV centre with the Hamiltonian ℋ = h𝔇𝒮_NV² + h𝔤_e𝓑·𝒮, with the zero-field splitting 𝔇 = 2.87 GHz and 𝔤_e = 28 GHz T⁻¹, and gives each spin the force 𝓕_z = 𝗆·∂_z𝓑 with the moment 𝗆 = −h𝔤_e⟨𝒮⟩. In the dark the ensemble sits in its thermal state at 300 K and carries a small moment. Green light at 532 nm pumps it. When an NV axis is aligned with the local field the pumping goes, in the source’s word, “predominantly” into m_s = 0, which carries no moment; at strong fields off the axis the eigenstates mix, and pumping leaves a net moment. A seven-level rate model, a ground triplet, an excited triplet and one intermediate state, gives the pumped populations. The four NV families, each at about 54.7° to the vertical, enter with equal weight, the moment is taken uniform over the ≈1 mm laser spot at its value at the spot’s centre, and the simulated force is divided by 1.2 for all the measurements. Switched at the resonance, the light swings the moment between its pumped and thermal values and drives the resonator.

At 50 mW and a 48.73% duty cycle the light drives a steady oscillation of about 100 nm, far above the nanometre-scale Brownian motion. Averaged period by period over 1200 s in a 56.8 ms window, one period at 17.6 Hz, the motion stays a clean sinusoid: it is phase coherent. With the light on the resonator moves toward the magnet, which the source reads as “indicative” of the attraction its model predicts. Under 10 mW of pulsed green light with the magnet in place, the resonance peak in the motional spectrum stands roughly four orders of magnitude above the Brownian noise. It disappears when the magnet is removed, and when a 980 nm laser at 50 mW, which cannot pump the NV centres, is chopped in its place; a small residual peak comes from the mechanical chopper, which ran during every measurement.

The source converts amplitude to force with its Eq. 2, ΔF = π𝓜(2πf₀)²𝒜/(2𝒬 sin πη). The inferred force rises with laser power toward saturation, as the model curves do. Against the magnet gap it passes through a maximum for each of two magnets, and the smaller magnet, with the steeper gradient, gives the larger force even where the larger magnet gives the larger field. The source reports forces over 5 nN in gradients of 10 to 100 T m⁻¹. Laboratory noise sets the position floor near 1 nm, and the source estimates that the light raises the diamond’s bulk temperature by about 3 K.

The resonator is one pattern of the omnium: plate, rod and diamond settle as one body, and the gradient moves them as one body. Each NV centre of that body is a nitrogen clamp beside a missing clamp, its levels the nodal modes that this clamp geometry sets (P-⁠3, P-⁠2). The 2.87 GHz splitting and the 637 nm zero-phonon line are differences of confinement eigenvalues, the operator that sets them to be derived (O-⁠9), and the optical pumping, the line and the return to the thermal state in the dark are same-n changes of confinement mode: excitation and eigenmode relaxation, each closure keeping its strand count, never a rebraiding. There are no intrinsic particle properties to carry. The light writes a configuration, the held moment, and the lattice erases it when the light goes off. The preparation runs up through the excited triplet and back through the intermediate state into the zero spin projection, and in the dark the lattice settles the ensemble by a third route. The forward and return routes are distinct events, and they do not commute (0.1a).

The laser pulses make the diamond act like an airplane wing: it is all about density differentials. A wing and this diamond are both bodies across a differential, read through one law of 0.1c, ∇_𝒰𝒰^μ = −𝔓^μν∂_ν ln 𝔪: a pattern free of every force in 𝔪²g_μν is read by the bookkeeper accelerating toward smaller 𝔪. Around a wing the air’s patterns follow the medium window, where for air of perfect form at fixed entropy per count the clock factor is the enthalpy per count, and the pressure differential left across the wing is its lift. The diamond follows a helical window that the analysis builds on its own energy,

with E_lev the diamagnetic energy of plate and slab. Positive E makes 𝔄 the geodesic action of 𝔪²g, and the Euler–Lagrange equation in a local one-dimensional inertial chart gives

Multiplied by Earth’s lapse, in the slow limit and with the moment held at its settled value, this is

beside the air’s drag. The last term is the source’s force: the window returns the source’s own dictionary on the same inputs. Wing and diamond are two applications of one law, with different constitutive energies.

The energy level carries no push. A uniform level, such as the light’s estimated 3 K of heating, about 2 × 10⁻² J or 3 × 10⁻¹⁹ kg with handbook values of diamond’s density and specific heat, counted once in 𝓜 (I-⁠7), changes the inertia and leaves −∂_zE unchanged. The first spatial variation gives the push, and the second gives the spring. A uniform field turns a held moment through its orientation energy and moves it nowhere. An aligned held moment lowers its energy where the field is stronger and falls toward that region, while the graphite’s induced diamagnetic energy rises there and pushes the plate up. The sign follows the energy, and no moment’s alignment is taken for a cone’s flow direction (P-⁠4); the push on the held moment is the settlement between diamond and magnet, system-wide and many-to-many (0.1a).

The fall is read between neighbours: the mirror against the fixed head, never off the resonator’s own clocks and rulers, which register no gradient they cross (0.1c, Use 7). Light on minus light off cancels the weight, the static levitation, the slab’s static diamagnetism and the static trap stiffness. Heat modulation, stiffness changes, light momentum and gas forces survive the difference. The light’s momentum flux 𝒫/c is 0.17 nN at 50 mW, under a twentieth of the force the light switches, and 0.033 nN at 10 mW; the weight of the heat is about 3 × 10⁻¹⁸ N.

On the A-⁠1 chart Earth at its surface reads α_ext = R_s/R_⊕ = 1.392 × 10⁻⁹, far below the exterior seam, in the general-relativistic regime. The resonator’s own window, 2G𝓜/(c²R_⊕) = 2.98 × 10⁻³⁸ at Earth’s centre, is a correct small reading (I-⁠1), and on the monopole pair domain F_maxα_ext,⊕α_ext,res = G𝓜_⊕𝓜/R_⊕² = 1.26 mN, the resonator’s weight. Patterns falling from rest far away pass the laboratory at the flow speed c√α_ext = 11.186 km s⁻¹ (0.1c), present at this size as at every other.

The cylinder’s field is the omnium’s helical density geometry (P-⁠5), not a separate entity. ∇·𝓑 = 0 reads as circulation without free ends, so a purely axial gradient over a finite spot requires transverse components; the source cancels the lateral forces by symmetry and leaves the finite-spot remainder unquantified. The source sets the moment uniform over the ≈1 mm spot at its centre value, while the force reads a profile-weighted mean over the illuminated region (I-⁠3), which a resolved moment map would test. Mixed eigenstates read as tilted windings between the NV axis, held by 𝔇, and the field axis, pulled by 𝔤_e𝓑; the crossover sits at 𝔇/𝔤_e = 0.10 T, and the source’s simulated 0.63 T, 1 mm above the magnet, is six times that.

By I-⁠2 the green pump, the 637 nm line and the 980 nm control are one carrier: 2.33, 1.95 and 1.27 eV, turning their grades at radii λ/2π of 84.7, 101 and 156 nm. The pump’s 0.38 eV excess over the line settles into the lattice through the same carrier. The infrared falls 0.68 eV short of the line and, as the source states, cannot polarize the spins.

The existing record measures no field from a static twist and no isotope dependence, so each test below adds one FDT input to the conventional account (Use 2) and compares neighbouring preparations. In the twist tests the conventional response cancels in the double-odd component and the comparator reads zero (Use 7); in the isotope tests the comparator’s density contribution is zero and its lattice and zero-point channels sit inside a stated envelope. The three twist tests read P-⁠5: imposing a physical twist, torsion or helical geometry, on a material is establishing a magnetic field. FDT reads that field along the twist axis, reversing with the hand. The two isotope tests read I-⁠7: a heavier carbon enters 𝓜 with more rest content and, at a fixed evaluation radius, raises the exterior compactness of each carbon site by Δ ln α_ext = Δ ln 𝓜_C, which a purely density-coupled equilibrium observable reads as a positive input.

For an observable 𝔬 read at hand 𝔶 = ±1 and bias 𝔲 = ±1, the double-odd component 𝒴 = ¼Σ𝔶𝔲𝔬(𝔶, 𝔲) and the zero-twist coil slope 𝔅, measured with a known uniform coil-field step, give the equivalent twist field 𝓑_tw = 𝒴/𝔅. Static torsion is even under time reversal, and both the conventional spin force and a strain shift of the resonance line are even in the bias, so the ideal conventional 𝒴 is zero. Residual uniform fields and gradients and the microwave ellipticity are nulled or stepped, and their measured slopes times the residuals correct the unit mean and enter its uncertainty 𝔡; the light enters by opposite faces in turn, so that its direction of travel cancels; and carried remanence is tested after demagnetisation. Detection is |𝒴̄| > 𝔷𝔡, with 𝔷 the Student quantile at the effective degrees of freedom: 2.093 for twenty complete units.

1. T-⁠1
   Static torsion of the NV slab. Grip a slab of the reported crystal, 3 × 3 × 0.5 mm with 4.5 ppm NV and [100] faces, by its two ends along [001] in a torsion stage, apply equal and opposite static elastic torsions, release them, and set a second magnitude. Read the lower optically detected magnetic-resonance line at the slab’s centre under 532 nm light, with a reversed bias of a few mT along [001]. FDT predicts a nonzero 𝒴 from a field along [001] that reverses with the hand; the conventional comparator is 𝒴 = 0. At the comparator a seeded simulation of twenty complete units, with an illustrative noise model, rejects 0.049 of the time against the nominal 0.05.
2. T-⁠2
   A helical support rod. Rebuild the resonator with carbon-fibre rods of opposite hand, each 5 cm long and 0.5 mm across, wound at equal angles and matched in mass, with a straight rod of the same dimensions as the released control and a second pair of equal angles as the second magnitude. At 50 mW, 48.73% duty and a 0.5 mm gap, infer ΔF for every run through Eq. 2 from that run’s measured mass, quality factor, frequency and amplitude, and turn the bias cylinder over between runs. FDT reads a hand-odd field reaching the defects below the rod; the conventional comparator is zero, and its false rejection is 0.050.
3. T-⁠3
   A bonded diamond blade. Cut two half-slabs at mid-height from one crystal of the reported kind, stack them along [001] and bond them at equal and opposite rotations about [001], with a blade bonded at zero rotation as the control and a second pair at a second rotation. Hang each blade from the reported rod and read ΔF as in T-⁠2. FDT reads the bonded relative rotation as helical geometry, with the same hand-odd field; the conventional comparator is zero, and its false rejection is 0.049.
4. T-⁠4
   Isotope dependence of the zero-field splitting. Grow three heavy (¹³C), three light (¹²C) and three mixed plates, about half heavy carbon, from enriched methane in one campaign, each with 4.5 ppm NV, and fit every plate’s bare 𝔇 at zero field and 300 K with its first-shell hyperfine terms. At the test point S_𝓜 = 1 the density contribution to Δ ln 𝔇 is ln(13.003354835/12) = 0.0803, against an illustrative conventional envelope of 0.01 for the lattice and zero-point channels. With an illustrative corrected plate scatter of 10⁻⁴ the heavy-minus-light difference resolves 2 × 10⁻⁴ (𝔡_HL = 8 × 10⁻⁵, 𝔷 = 2.776 at four Welch degrees of freedom), and the mixed plates must follow their assayed log-mass fraction of it within t_0.975,2𝔡_mix, with t_0.975,2 = 4.303.
5. T-⁠5
   Isotope dependence of static diamagnetism. Grow the same three groups NV-free, cut to 3 × 3 × 0.5 mm, and with the light off move the cylinder from far away to 0.5 mm; read the static diamagnetic force per volume, F_dia/V, from the displacement and the stiffness measured for every mount. At S_𝓜 = 1 the density contribution is again 0.0803, against the same 0.01 envelope, and an illustrative plate scatter of 2 × 10⁻³ resolves 5 × 10⁻³.

P-⁠5 fixes the existence of the twist field, and FDT reads it along the twist axis, reversing with the hand; its size is to be derived through I-⁠1 and I-⁠2 with a declared twist-to-field kernel. A finite null excludes a proposed field 𝓑_* of either sign when |𝓑_*𝔅| − |𝒴̄| > 𝔷𝔡_*, with 𝔡_* carrying the slope’s calibration error, and sets no lower bound on existence. Attribution requires the same sign at a second twist magnitude; for the slab, a bound on the torsion stage’s own hand-correlated field from the slab released at each hand; and, for rod and blade, disappearance with each straight or zero-rotation replacement and redetection at that sign after every piece is demagnetised in a decaying alternating field that peaks above the N52 coercivity. I-⁠7 fixes the sign of the mass input; each observable’s density sensitivity S_𝓜 is to be derived, and the observable’s sign follows from the product of the sensitivities. Isotope detection requires the heavy-minus-light shift to exceed the conventional envelope plus 𝔷𝔡_HL, and exclusion requires its confidence interval to miss the test-point value widened by the envelope on either side. The fractional mass change, 0.0836, is never substituted for the logarithm.

Several measurements test the conventional account with no FDT input added. A uniform coil field at fixed gradient changes the force only through the moment, read independently at the same spot. A [111] slab puts one NV family on the axis, where the source’s aligned pumping goes predominantly into m_s = 0, and three at 70.5°, and ΔF follows the source’s seven-level model at those angles. At equal force, power, duty and gap, evacuating the chamber changes the amplitude in proportion to 𝒬; the source’s “up to ∼1.5 μm” in vacuum against ∼100 nm in air is a ratio of 2 × 10¹, and the record prints neither the evacuated run’s drive nor its 𝒬. Reversing the beam at fixed spot, intensity and crystal-relative polarisation reverses the vertical part of the photon momentum’s contribution, at most 2𝒫/c = 0.33 nN at 50 mW, and leaves the spin force unchanged; the source’s green polarisation wanders by ±5°, so a beam tilt needs that control too. Microwave inversion of the held moment turns the gradient force repulsive. With ΔF counted positive toward a smaller gap 𝔥, the light also changes the stiffness by ∂_𝔥ΔF and the resonance by Δ_onf₀ = (f₀/2𝒦)∂_𝔥ΔF. An infrared control at matched absorbed heat, and an NV-free slab under green drive with the magnet present, close the heating channels that a change of wavelength alone leaves open.

X-⁠1, exploratory. The existing drive reads as three spatial readings of one clock factor, Earth’s lapse times the helical-window factor, relative to the fixed head: Earth’s slope G𝓜_⊕/(R_⊕c)² = 1.0927 × 10⁻¹⁶ m⁻¹, the trap’s curvature ω₀²/c² = 1.36 × 10⁻¹³ m⁻², and the light-switched slope ΔF/(𝓜c²) ≈ 4 × 10⁻²² m⁻¹; the level at a point is invisible (0.1c). The checkerboard holds the resonator in the laboratory flow, and the light rewrites its fall by rewriting its moment. The switched force is about 4 × 10⁻⁶ of the weight and moves the bottom of the well by ΔF/𝒦 ≈ 3 nm, and the resonant gain of 35 carries the response to the source’s 100 nm. An in-sample reconstruction of the existing drive, never counted with the tests.

On the source’s Hamiltonian, population rates, cylinder field and response dictionary the limiting map is the identity: the per-spin force, its volume integral and Eq. 2 return the source’s forces, amplitudes and spectra. The 128 mg, the 17.6 Hz and 𝒬 = 55 are apparatus inputs; the trace amplitude and the forces are estimates, the populations and the field simulations, and the common division by 1.2 makes the compared curves in-sample. The motion’s direction and the control spectra are accepted with their reported conditions. The moment’s sign, the adjoint where the source prints a transpose in its change of basis, and the source’s unstated peak-versus-rms amplitude convention are kept explicit. The helical window’s slow limit is the source’s own force equation, so its recovery adds no number. For the same stress tensor and data, I-⁠3 returns Earth’s curvature; the pair reduction that gives the weight is exact on its monopole domain (I-⁠1), and general relativity reduces to it at leading weak-field, slow-motion order (I-⁠4). Setting the twist source and the density sensitivity to their conventional values restores both comparators. On the source’s own inputs the map is the identity, and the five tests above are where FDT’s added inputs meet new data.