Superradiant MeV γ Scattered by a Room-Temperature Spinor Quantum Fluid

Recent reports have revealed the rich long-lived Mossbauer phenomenon of 93mNb, in which it has long been speculated that the delocalized 93mNb undergoes Bose-Einstein condensation following an increase in the 93mNb density beyond the threshold of 1012 cm−3 at room temperature. We now report on the superradiant Rayleigh of the M4 γ at 662 keV scattered into end-fire modes along the long axis of the sample, as evidence of Bose-Einstein condensation. We observed the Arago (Poisson’s) spot in order to demonstrate a near-field γ-ray diffraction from a mm-sized γ source, as well as a γ interference beyond the Huygens-Fresnel principle. During the 107-day monitoring period, seven Sisyphus cycles of mode hopping appeared in the superradiance, which demonstrates the optomechanic bistabilty provided by the collective interaction between the spinor quantum fluid and the impinging γs. Condensate-light interaction produces a pm matter-wave grating to become a Fabry-Pérot resonator with a Q-factor on the order of 1020, from which end-fired γs lase.


Introduction
Bose-Einstein condensation (BEC) [1,2] is among the most spectacular of the quantum phenomena to have been discovered over the last century.Micro-particles condense into one state generating a unique macroscopic feature, as defined by the off-diagonal long-range order (ODLRO) [3,4].The whole condensate gives a global response to an external drive, rather than being merely a superposition of random micro-activities.It is well known that eV-energy γ can undergo BEC at the relatively high temperature of 4 K [5,6], when the massless γ acquires a small mass in the quantum well to become an exciton-polariton.Room-temperature BEC occurs [7] when the Mossbauer keV γ of an eV mass exceeds a threshold density of 10 12 cm −3 .This work applies ORLRO scattering of impinging γ rays to prove the room-temperature BEC of delocalized 93m Nb γ excitations.
Long-lived Mossbauer nuclides emit biphoton γγ or a biphoton pair to preserve the energy from releasing a phonon [7][8][9][10][11].Once nuclear resonance occurs, a single nuclear γ excitation delocalizes to billions of identical nuclei in the crystal.Delocalized γ excitation acquires an eV mass and its spin orientation in the crystal, by absorbing the Nambu-Goldstone bosons of phonon and magnon [12].The nuclear exciton spreads in a µm region over the photonic crystal of identical nuclei, and it carries no Crystals 2017, 7, 196 2 of 13 charge, but does carry a spin.However, the term "nuclear exciton" [13] fails to capture the important character of an antiferromagnetic nesting spin chains [9].We, therefore, prefer the term "nuclear spin-density wave" (NSDW) [7].The NSDW BEC of the M4 93m Nb γγ in this report is, thus, a spinor quantum fluid carrying four spin quanta and odd parity.Coherent γs scattered by the spinor quantum fluid constitutes a γ laser, which is highly useful for biological nanoimaging.
Early NSDW investigations focused on the 103m Rh of one-hour half-life [7,[9][10][11] activated by a 6-MeV bremsstrahlung source in Beijing.The NSDW subsequently entered three regimes (I, II, and III) as its density increased, which were demonstrated by the behaviors of Rabi splitting [7,14].We summarized the phenomena observed in regimes I and II in the supplementary information of our previous report [9].In this report we concentrate on regime III, as well as an additional new regime beyond it.Either increasing the 103m Rh density above a threshold of 10 12 cm −3 or lowering the sample temperature using liquid nitrogen [8] caused a quantum phase transition into regime III, as characterized by the abrupt shrinkage of the Rabi splitting from 500 eV back to the 50-eV vacuum splitting.The Rabi splitting switched twice between two constant values every 20 minutes, revealing a "collapse and revival" [15] of the NSDW condensate in regime III.By cooling the sample already prepared in regime III to 78 K each time, the NSDW condensate of 103m Rh entered a new regime, in which the Rabi splitting of NSDW slowly varied between 50 and 500 eV with frequencies in the mHz range [11].
The support in Beijing for activating the 103m Rh was terminated in 2008, at which point we moved the NSDW experiments to Hsinchu and prepared the 93m Nb with a 16-year half-life using neutron activation [7].The activated Nb single crystal was an oval plate (13 mm × 12 mm × 1 mm), which was applied to measure the magnetoelectric effect (ME) induced by the BEC.The γ spectra of 93m Nb provided no biphoton information, until the γγ at the half-transition energy (15.4 keV) were recently released from another polycrystalline sample used in this report [9].We continued to verify the 93m Nb NSDW BEC for many years, until a transverse ME symmetry depending on the sample geometry was seen [7,11], recognized as the dynamics of low-lying bound states in the topological defects of NSDW quantum fluid.For example, Khaymovich and Silaev have discussed the nonlinear paramagnetic magnetization of vortices in the 3 He-B superfluid, which is induced by the Zeeman interaction and exhibits many low-frequency resonances [16].The smoking-gun resonances of Majorana fermions are caused by the interlevel spacing of n ω v ∼ nh10 5 Hz and the subharmonics of ω v /n, where n is an integer.A comprehensive review of recent progress on Majorana fermions in topological superfluids is given by Mizushima et al. [17], which introduces many possible magnetization sources, e.g., the spontaneous ferromagnetic vortex core and the exotic pairing of bound states.We take the pronounced magnetoinductance and the nonlinear ME resonances below kHz frequencies observed in a previous report [7] to be a fingerprint of topological defects in the spinor quantum fluid.A BEC threshold density of 10 12 cm −3 was then found, when the delocalization of 93m Nb and its corresponding ME signature spontaneously disappeared three years after the activation [7].However, BEC and the ME were immediately recovered by reactivating the sample again using neutron irradiation.Monitoring the transition from BEC to a normal state is still ongoing.Interestingly, 103m Rh and 93m Nb have similar BEC threshold densities at room temperature.
The 93m Nb ME showed a memory of the applied field.The 20-Hz transverse ME remained nearly symmetrical on the rotation of the sample at 1 T when the vertical field was increased from 0 to 1 T, and this rotating symmetry was broken at 1 T when the field was decreased from 9 T to 1 T [11].The fluxes trapped in the sample were demonstrated by distributing magnetic powders on the sample surface to reveal a spontaneous magnetization of the neutral vortices [11], which was similar to the magnetic vortex core in the 3 He-B superfluid [17].Furthermore, the ME response switched sign from an inductor to a negative inductor for an anticlockwise rotation of the sample through 360 degrees at 9 T, and vice versa.This geometric phase acquired by rotating sample is a hallmark of the non-Abelian statistics corresponding to Majorana zero modes [17], which probably survive at the high field and at room temperature.These three observations together reveal a topological nontrivial state of the spinor quantum fluid [17].A metal-insulator transition was revealed by the negative mV voltage crossing a residual resistance of 4 µΩ at 4 K [7].Magnetic fluxes in quantized vortices inhibit the electron transport inside the crystal; instead, electrons propagate at the edge states [17,18].The activated sample became an anisotropic insulator.Further investigation of the ME response beyond previous reports [7,11] reveals that the time reversal breaking quantum fluid is related to the symmetry-protected Weyl superfluid [17], which will be discussed in a future report.

Suppression of Atomic Channels
We recently reported superradiant Nb K-lines along a long sample axis for an active sample having a geometry of 1 mm × 1.5 cm × 3 cm [9].We draw an analogy with the superradiant Rayleigh scattered from cold and ultracold atoms [19,20] and, in particular, the atomic condensate in a high-finesse cavity [21].The impinging γ assumes the role of a laser, while the sample assumes the role of a γ cavity containing the NSDW condensate.There are three major mirrors in the cavity, namely the matter-wave grating induced by γs, the nuclei on the lattice sites, and the sample boundary.The boundary mirrors give rise to the end-fire modes, while the matter-wave mirror provides a high-finesse Fabry-Pérot resonator [22].The additional mirror of the lattice gives rise to some important physics of biphotons, which are absent in the cold-atom case.
We also previously reported the collective nuclear scattering of two E2 γs at 1173 keV and at 1333 keV emitted from a 60 Co source, as well as an M4 γ at 662 keV emitted from a 137 Cs source [9].Neither the 60 Co source nor a 152 Eu source provided the end-fire γs by lateral impinging (see Figure 1) due to the fact that their coherent lengths were shorter than the sample thickness of 1 mm.electron transport inside the crystal; instead, electrons propagate at the edge states [17,18].The activated sample became an anisotropic insulator.Further investigation of the ME response beyond previous reports [7,11] reveals that the time reversal breaking quantum fluid is related to the symmetry-protected Weyl superfluid [17], which will be discussed in a future report.

Suppression of Atomic Channels
We recently reported superradiant Nb K-lines along a long sample axis for an active sample having a geometry of 1 mm × 1.5 cm × 3 cm [9].We draw an analogy with the superradiant Rayleigh scattered from cold and ultracold atoms [19,20] and, in particular, the atomic condensate in a highfinesse cavity [21].The impinging γ assumes the role of a laser, while the sample assumes the role of a γ cavity containing the NSDW condensate.There are three major mirrors in the cavity, namely the matter-wave grating induced by γs, the nuclei on the lattice sites, and the sample boundary.The boundary mirrors give rise to the end-fire modes, while the matter-wave mirror provides a highfinesse Fabry-Pérot resonator [22].The additional mirror of the lattice gives rise to some important physics of biphotons, which are absent in the cold-atom case.
We also previously reported the collective nuclear scattering of two E2 γs at 1173 keV and at 1333 keV emitted from a 60 Co source, as well as an M4 γ at 662 keV emitted from a 137 Cs source [9].Neither the 60 Co source nor a 152 Eu source provided the end-fire γs by lateral impinging (see Figure 1) due to the fact that their coherent lengths were shorter than the sample thickness of 1 mm.The resonant scattering of a Mossbauer nuclide takes place over 10 −8 s, which is much longer than the thermal vibration.The coherent elastic scattering channels are enhanced at Bragg, also known as the suppression of inelastic channels [13].In our case, the condition of resonant Bragg scattering is no longer necessary because of the NSDW wave packet, in which the nuclei collectively enhance all kinds of elastic and inelastic nuclear scatterings.We, thus, use the term of SAC (suppression of atomic channels) rather than the "suppression of inelastic channels" in this report.
Nb K-lines have a coherent length of 0.3 μm [23] and a propagation depth of 50 μm in the Nb crystal.The nucleus-field coupling gK of K lines is no more than 10 -8 eV, while the allowed E1 transitions of 93 Nb lie above 1.3 MeV.Nearly 10 9 nuclei lie in the coherent length of 0.3 μm.The atom- The resonant scattering of a Mossbauer nuclide takes place over 10 −8 s, which is much longer than the thermal vibration.The coherent elastic scattering channels are enhanced at Bragg, also known as the suppression of inelastic channels [13].In our case, the condition of resonant Bragg scattering is no longer necessary because of the NSDW wave packet, in which the nuclei collectively enhance all kinds of elastic and inelastic nuclear scatterings.We, thus, use the term of SAC (suppression of atomic channels) rather than the "suppression of inelastic channels" in this report.
Nb K-lines have a coherent length of 0.3 µm [23] and a propagation depth of 50 µm in the Nb crystal.The nucleus-field coupling g K of K lines is no more than 10 -8 eV, while the allowed E1 transitions of 93 Nb lie above 1.3 MeV.Nearly 10 9 nuclei lie in the coherent length of 0.3 µm.The atom-field coupling is collectively enhanced by a factor of √ N for an ensemble of N ultracold atoms [21].In contrast, the 93m Nb NSDW entangles N 2 modes of N nuclei × N phonons, giving rise to an N-fold nucleus-field coupling [9].However, N × g K ~10 eV is still much weaker than large nuclear detuning.How can we determine the nature of the underlying physics to yield the superradiant Nb K-lines in a previous report [9], which shall be completely attenuated before reaching the sample boundaries?The penetration depth must be longer than 1 mm.We cannot understand the Nb K-line superradiance without considering the NSDW condensate, which enhances the Borrmann penetration [13] of γs emitted from an internal source.
A 152 Eu source provides E1 and E2 γs of coherent lengths longer than µm.It also provides atomic E1 γs of the Sm K-lines, which have very short coherent lengths of 60 nm [23].Table 1 shows a pronounced enhancement of the off-Bragg penetration at 122 keV by 170 σ.The E2 SAC showed up regardless of the large 622-keV detuning.In contrast, the E1 SAC of Sm K-lines at 40 keV and 45 keV were absent.The nucleus-field coupling of the E2 γ at 122 keV is of the µeV order and its coherent length lies between the intrinsic 0.6 m and the thermal broadening of 60 µm.Accordingly, we identify the amplification as N rather than √ N, otherwise the collective coupling is too weak to give any observable SAC effect from an external 152 Eu source.Note that the NSDW wave packet has a size of µm.There would be no SAC, unless ODLRO was present.
Table 1.Suppression of atomic channels (SAC) tested by a 30-µCi 152 Eu source.All measurements were undertaken with a 2.5 cm detector-source spacing, where the active sample always remained at the same position.Calibration: source atop the spacer; Pristine: source atop the pristine sample; Active: source atop the active sample; DAS: relative deviation between active and pristine data in %; DT: dead time of the HPGe detector.Assuming active and pristine count rates at 1408 keV are equal, we multiply the active data by a factor of 1.0036 to give the relative deviations of DAS.Three datasets in cps (counts per second) were undertaken by daily exchanging the 152 Eu source positions to maintain the equal 182 Ta activities.γ at 1121 keV is emitted from the 182 Ta decay.The half-lives at 368 keV and 779 keV denoted by * are not documented, however, they are estimated from the E1 transition at 1408 keV.The question remains of how to evaluate the N-fold coupling to provide the E1 SAC.The E1 N-fold coupling of an γ emitted from a two-level nuclide decreases very rapidly with its γ energy at zero temperature while it remains a constant value when the thermal broadening dominates.Furthermore, the photoelectric attenuation decreases with the γ energy.Regardless of which conditions apply, the E1 SAC is a monotonically decreasing function of the γ energy.We shall accordingly identify the insignificant E1 SAC from their trivial attenuations.Three E1 γs at 296 keV, at 368 keV, and at 444 keV provide pronounced deviations in their coherent lengths caused by the multi-level transitions.The thermal broadening dominates the coherent length of 16 µm at 368 keV, while the extreme short half-life of 27 fs reduces the coherent lengths of 10 µm at 296 keV and 9 µm at 444 keV, respectively.The N-fold coupling at 368 keV is stronger than that at 296 keV and at 444 keV.Table 1 shows Crystals 2017, 7, 196 5 of 13 a superior E1 SAC at 368 keV by 3 σ.The SAC coherence beyond the NSDW size reveals the forward ODLRO superradiance.

Superradiant Rayleigh Scattering and Its Near-Field Diffraction
As shown in Figure 1, the M4 γ at 662 keV from a 2.5-mCi 137 Cs source irradiated the active sample from its lateral side, while detecting γs along the 3-cm long sample axis.The superradiant M4 γ was verified with great care by two methods, in addition to the following Figures 2-5, namely by replacing the active sample by a pristine one and blocking the irradiation path with a 2-cm-thick Pb brick.Figure 2 and Table 2 show detailed measurements of one and a half circles around the active sample.One mirror symmetry along the 1-mm axis was found, while the mirror symmetry along the 1.5-cm axis was shown in rough terms.
Crystals 2017, 7,196 5 of 13 at 444 keV, respectively.The N-fold coupling at 368 keV is stronger than that at 296 keV and at 444 keV.Table 1 shows a superior E1 SAC at 368 keV by 3 σ.The SAC coherence beyond the NSDW size reveals the forward ODLRO superradiance.

Superradiant Rayleigh Scattering and Its Near-Field Diffraction
As shown in Figure 1, the M4 γ at 662 keV from a 2.5-mCi 137 Cs source irradiated the active sample from its lateral side, while detecting γs along the 3-cm long sample axis.The superradiant M4 γ was verified with great care by two methods, in addition to the following Figures 2-5, namely by replacing the active sample by a pristine one and blocking the irradiation path with a 2-cm-thick Pb brick.Figure 2 and Table 2 show detailed measurements of one and a half circles around the active sample.One mirror symmetry along the 1-mm axis was found, while the mirror symmetry along the 1.5-cm axis was shown in rough terms.Table 2. θ-dependence of superradiant Rayleigh in cps as a function of the source location of (r, θ), where r is the radius and θ is the impinging angle, as shown in Figure 1.2.5 mCi 137 Cs 3-mm point source was placed on the circle r = 5.4 cm radius; the second column of configuration was measured in the upwards sequence of −θ from the bottom number to the top number, while the other two columns were measured in a downward sequence.The data were obtained every five minutes.According to conventional wisdom, the pm-wavelength γ from a mm source provides no information on the cm geometry because its fringe visibility [22] vanishes.To demonstrate this highly-unusual near-field diffraction, we studied the superradiance in detail for a few specific angles: 0, 60, 80, and 90 degrees.The angular spread of impinging γ was reduced by moving the source away from the sample step-by-step, as shown in Figure 3 and Table 3.The superradiance decreased faster than 1/r 2 , as shown by the results for θ = 0 • .The superradiant efficiency at (1 cm, 0 • ) approached 40%, as estimated by the superradiance of 1.6 × 10 5 counts per second (cps) taking two end-fire modes, forward and backward superradiance in the 0-degree direction, a 50% pile-up loss and a 20% efficiency of the directional beam (0.5% efficiency of the coaxial HPGe detector at 662 keV using an isotopically emitting source, as provided by the vendor) into account, while the impinging γs are 4 × 10 5 s −1 in 0.5% of the 4π solid angle without considering polarizations of the M4 γ.Meanwhile, the 40% superradiant efficiency also manifested the M4 SAC, otherwise the photoelectric loss of 662-keV γ propagating 3 cm from the center to the end of sample would be more than 90%.According to conventional wisdom, the pm-wavelength γ from a mm source provides no information on the cm geometry because its fringe visibility [22] vanishes.To demonstrate this highly-unusual near-field diffraction, we studied the superradiance in detail for a few specific angles: 0, 60, 80, and 90 degrees.The angular spread of impinging γ was reduced by moving the source away from the sample step-by-step, as shown in Figure 3 and Table 3.The superradiance decreased faster than 1/r 2 , as shown by the results for θ = 0°.The superradiant efficiency at (1 cm, 0°) approached 40%, as estimated by the superradiance of 1.6 × 10 5 counts per second (cps) taking two end-fire modes, forward and backward superradiance in the 0-degree direction, a 50% pile-up loss and a 20% efficiency of the directional beam (0.5% efficiency of the coaxial HPGe detector at 662 keV using an isotopically emitting source, as provided by the vendor) into account, while the impinging γs are 4 × 10 5 s −1 in 0.5% of the 4π solid angle without considering polarizations of the M4 γ.Meanwhile, the 40% superradiant efficiency also manifested the M4 SAC, otherwise the photoelectric loss of 662-keV γ propagating 3 cm from the center to the end of sample would be more than 90%.  3 were taken every five minutes.Table 3. R-dependence of superradiant Rayleigh in cps as a function of source location of (r, θ).The source was 2.5 mCi 137 Cs.A nodal point appeared at (3.5 cm, 80°), while the nearby low superradiances were at (3.3 cm, 60°) and (3.7 cm, 90°).Note that the normal value of the internal γ at 1121 keV was 6.92 ± 0.01 cps.The readings at (1 cm, 0°) were lower than the real values, because of the pile-up loss of 48% dead time.The internal γ at 1121 keV was slightly increased by the 137    3 were taken every five minutes.
Table 3. R-dependence of superradiant Rayleigh in cps as a function of source location of (r, θ).
The source was 2.5 mCi 137 Cs.A nodal point appeared at (3.5 cm, 80 • ), while the nearby low superradiances were at (3.3 cm, 60 • ) and (3.7 cm, 90 • ).Note that the normal value of the internal γ at 1121 keV was 6.92 ± 0.01 cps.The readings at (1 cm, 0 Note that the superradiance in Table 3 was not a monotonic function at θ = 80 • .A nodal point was found at r = 3.5 cm.We also traced the nodal "low" that developed in the nearby positions.Vanishing superradiance was also found at the corresponding dark position (1 cm, 60 • ) using a large disk source of 10-µCi 137 Cs.Rotating this disk source changed the superradiance, i.e., a small angular distribution in the r-axis gave a large superradiance by an order of magnitude.Interference, thus, occurred 25,630 ± 90, which is significantly beyond the variance of the shot noise, i.e., 24,000, to reveal plenty of hidden signals in the fluctuation.Here, we concentrate on the quantized oscillations in this report, while the systematic analysis of signals of periods ranging from days to minutes will be discussed in greater detail elsewhere.

Mode Hopping of the Matter-Wave Grating
We found a systematic oscillation of the superradiance in time.Figure 4 shows the first longterm monitoring experiment of the superradiant M4 γ irradiated by the 2.5-mCi 137 Cs source located at (0.3 cm, 90°).Removing the smoothed results of Figure 4 from the raw data gives a variance of 25,630 ± 90, which is significantly beyond the variance of the shot noise, i.e., 24,000, to reveal plenty of hidden signals in the fluctuation.Here, we concentrate on the quantized oscillations in this report, while the systematic analysis of signals of periods ranging from days to minutes will be discussed in greater detail elsewhere.Assuming an unknown tick-tack clock Ξ in the NSDW condensate drove seven Sisyphus cycles by increasing the effective susceptibility χ, as the optomechanical bistability of the ultracold atoms driven by sweeping the laser frequency upwards [21,26].Taking the upside-down Sisyphus "fall" to be a step function, the relaxation time is about half an hour.However, the M4 γ at 662 keV immediately extinguished and reestablished within one second when we removed and returned the 137 Cs source, respectively.
Two air conditioners that maintained the temperature within 27 ± 0.5 °C were turned off at day 72.No extra fluctuation was observed by increasing the temperature fluctuation by an order of magnitude to ±6 °C.The superradiant Rayleigh indeed consisted of pairing γγ, e.g., γ11γ12 and γ21γ22 with energies E11 + E12 + E21 + E22 = E0 (662 keV) and the wavelengths in vacuum ~4λ0.A slight shift of the standing-wave energy of E11 − E12 = E21 − E22 = 4.34 keV + δ [9] absorbs the 0.01% temperature elongation of the lattice constant a0 = 2.86 Å between two nearest-neighbor Nb atoms.Free γ11γ12 + γ21γ22 combination to match different lattice constants together with entangled phonons and magnons provides the soft mechanism to launch the self-organized Dicke phase [21].
The mode hopping at ~1% per Sisyphus cycle indicates a grating number of the matter-wave, i.e., a0/λ0.Four possible quantizeFid levels are also highlighted in Figure 4.If the tick-tack Ξ is the collection of 0.01% impurities and the vacancy at lattice sties, missing or replacing 93 Nb all together gives a net effective χ on the new vacuum of NSDW condensate, i.e.: Assuming an unknown tick-tack clock Ξ in the NSDW condensate drove seven Sisyphus cycles by increasing the effective susceptibility χ, as the optomechanical bistability of the ultracold atoms driven by sweeping the laser frequency upwards [21,26].Taking the upside-down Sisyphus "fall" to be a step function, the relaxation time is about half an hour.However, the M4 γ at 662 keV immediately extinguished and reestablished within one second when we removed and returned the 137 Cs source, respectively.Two air conditioners that maintained the temperature within 27 ± 0.5 • C were turned off at day 72.No extra fluctuation was observed by increasing the temperature fluctuation by an order of magnitude to ±6 • C. The superradiant Rayleigh indeed consisted of pairing γγ, e.g., γ 11 γ 12 and γ 21 γ 22 with energies E 11 + E 12 + E 21 + E 22 = E 0 (662 keV) and the wavelengths in vacuum ~4λ 0 .A slight shift of the standing-wave energy of E 11 − E 12 = E 21 − E 22 = 4.34 keV + δ [9] absorbs the 0.01% temperature elongation of the lattice constant a 0 = 2.86 Å between two nearest-neighbor Nb atoms.Free γ 11 γ 12 + γ 21 γ 22 combination to match different lattice constants together with entangled phonons and magnons provides the soft mechanism to launch the self-organized Dicke phase [21].
The mode hopping at ~1% per Sisyphus cycle indicates a grating number of the matter-wave, i.e., a 0 /λ 0 .Four possible quantizeFid levels are also highlighted in Figure 4.If the tick-tack Ξ is the collection of 0.01% impurities and the vacancy at lattice sties, missing or replacing 93 Nb all together gives a net effective χ on the new vacuum of NSDW condensate, i.e.: where j runs for all kinds of impurities, N j is the number of impurities or vacancies, and χ i;j are their effective susceptibilities of the ith order as functions of the E and the B fields.Every β decay increases one nuclear charge at impurity sites and meanwhile increases their χ 1;j susceptibilities.Slight variation of χ 1;j repartitions the biphoton γγ energy like the temperature elongation, which hardly changes the γ intensity.In contrast, the χ 2 variation can be directly compensated by the blue-detuned γ intensity, as demonstrated by the downward region in Figure 4. Hyper-Raman processes accelerated the 94 Nb β decay (see Figure S3C in the Supplementary information), which revealed a significant χ 2 contribution.Note that the M4 field of 10 17 V/m and 10 5 T are available in cavities of impurity atoms, which may catalyze chemical reactions and nuclear reactions [8,9].Evaporation of impinging γs at impurities, impurity decays, activation of 93m Nb, and matter-wave back-action balanced each other during the downward region of the Sisyphus cycles until the mode hopping, which quickly advanced to the next resonance.Figure 5 shows the cross correlation among the 182 Ta γs, and three broad-band γs of BBL, BBM, and BBH, which represent three broad-band γs below 662 keV, between 662 and 1121 keV, and above 1121 keV.The hopping of broad-band γs provides further evidence of matter-wave grating.Figure 5 also reveals two hidden hopping in Figure 4. where j runs for all kinds of impurities, is the number of impurities or vacancies, and ; are their effective susceptibilities of the ith order as functions of the E and the B fields.Every β decay increases one nuclear charge at impurity sites and meanwhile increases their ; susceptibilities.Slight variation of ; repartitions the biphoton γγ energy like the temperature elongation, which hardly changes the γ intensity.In contrast, the variation can be directly compensated by the bluedetuned γ intensity, as demonstrated by the downward region in Figure 4. Hyper-Raman processes accelerated the 94 Nb β decay (see Figure S3C in the Supplementary information), which revealed a significant contribution.Note that the M4 field of 10 17 V/m and 10 5 T are available in cavities of impurity atoms, which may catalyze chemical reactions and nuclear reactions [8,9].Evaporation of impinging γs at impurities, impurity decays, activation of 93m Nb, and matter-wave back-action balanced each other during the downward region of the Sisyphus cycles until the mode hopping, which quickly advanced to the next resonance.Figure 5 shows the cross correlation among the 182 Ta γs, and three broad-band γs of BBL, BBM, and BBH, which represent three broad-band γs below 662 keV, between 662 and 1121 keV, and above 1121 keV.The hopping of broad-band γs provides further evidence of matterwave grating.Figure 5 also reveals two hidden hopping in Figure 4.Additional broad-band γs (BBL, BBM, and BBH in Figure 5) were created at the moment of applying the M4 γ, as shown in Figures S2 and S3 of the Supplementary Information.For example, the BBL broad-band γs of energy below the 662 keV in Figure 5 was mainly contributed by the scattering of the 662-keV biphoton fragments occurring at the sample site, rather than the Compton plateau occurring at the detector site.The BBL spectra in Figure S3A of the Supplementary Information show two bumps located near 630 keV and 470 keV, which revealed a loss of 30 keV by Raman pumping from 93 Nb to 93m Nb and broad distributions of the coincident arrival of four and three γs of the biphoton fragments, respectively.
The highest γ energy emitted from the 182 Ta decay is 1.453 MeV, which is limited by the Qβ of 1.814 MeV.However, we found decaying counts of a broad-band background which extended exponentially to an energy above 1.5 MeV, as shown in Figure S3C of the Supplementary Information, which strongly supports the reported 4-γ picture of standing-wave M4 γs trapped in the photonic Additional broad-band γs (BBL, BBM, and BBH in Figure 5) were created at the moment of applying the M4 γ, as shown in Figures S2 and S3 of the Supplementary Information.For example, the BBL broad-band γs of energy below the 662 keV in Figure 5 was mainly contributed by the scattering of the 662-keV biphoton fragments occurring at the sample site, rather than the Compton plateau occurring at the detector site.The BBL spectra in Figure S3A of the Supplementary Information show two bumps located near 630 keV and 470 keV, which revealed a loss of 30 keV by Raman pumping from 93 Nb to 93m Nb and broad distributions of the coincident arrival of four and three γs of the biphoton fragments, respectively.
The highest γ energy emitted from the 182 Ta decay is 1.453 MeV, which is limited by the Q β of 1.814 MeV.However, we found decaying counts of a broad-band background which extended exponentially to an energy above 1.5 MeV, as shown in Figure S3C of the Supplementary Information, which strongly supports the reported 4-γ picture of standing-wave M4 γs trapped in the photonic on the problem.The remaining questions are to construct a model of the β decay encountering multiple branching, splittings of which are opened by the highly-concentrated mixing of M4 and E2 fields.(4) The impinging M4 γ activated the 93m Nb excitation, as shown by Figure S3A in the Supplementary Information.Increasing NSDW density provides a stronger matter-wave grating to promote the collective forward scattering of 182 Ta γs.(5) Superradiance of both the impinging γs and the internal 182 Ta γs showed systematic oscillations, which were most likely driven by external sources.
The NSDW ME symmetry reveals a pseudo-scalar axion field of arbitrary winding numbers [7,27], the underlying physics of which may have been unveiled by the recent progress on the Weyl superfluid [17].The axion electrodynamics gives rise to the electric and the "magnetic" charges on the facets, the edges, and the vertices, where the axion gradient appears [27,28].It is worthwhile investigating the boojum texture with the axionic superradiance together in the near future.1-D superradiant tubes were concentrated at the eight long edges of the sample sheet, i.e., four 3-cm edges plus four 1.5-cm edges, which allowed the reconstruction of the mutual coherence of impinging γs to provide the near-field diffraction, as well as to recover the fringe visibility [22] from a mm-sized source.The Arago spot revealed a 0.3 µm aperture, which was probably none other than the compressed diameter of the superradiant tubes at the edges.

Materials and Methods
The preparation of active samples and the applied HPGe detector have been detailed in the references [9], while the setup of the superradiant measurements is shown in Figure 1.An Nb sample of 99.99% purity is described in this report, which has a dimension of 1 mm × 1.5 cm × 3 cm.We applied four kinds of radioactive sources in the measurements, i.e., one 2.5-mCi 137 Cs, two 10-µCi 137 Cs, two 1-µCi 60 Co, and one 2.3-µCi 152 Eu.The 2.5-mCi 137 Cs was a 3-mm point source encapsulated in a cylinder 8 mm long and 6 mm in diameter.Two 10-µCi 137 Cs were thin disks 10 mm in diameter and 1 mm thick.Two 1-µCi 60 Co and the 30-µCi 152 Eu were also thin disks 3 mm in diameter and 1 mm thick.The applied HPGe detector CANBERRA GC0518 (coaxial-Ge type) with ϕ 4.25 cm and a length of 4.1 cm was located in a stabilized box with 3-cm Pb shielding.A calibration using the 10-µCi 137 Cs source yielded 110 cps with 2.5 cm detector-source spacing.
Two long-term monitoring experiments of superradiance were undertaken, beginning on the 1 August 2016 (1st day 0) and on 16 November 2016 (2nd day 0).We moved the 137 Cs source from (0.3 cm, 90 • ) to (0.3 cm, 270 • ) in the second monitoring experiment.Two air conditioners maintained the environmental temperature at 27 ± 0.5 • C. The air conditioners were turned off at 3 pm on 11 October 2016.They were switched on and off twice during the following week and kept off after 18 October 2016.Afterwards, the temperature varied naturally between 18 • C and 30 • C. The long-term monitoring of the M4 superradiance was stopped and restarted several times.We calibrated the computer clock each day; it was faster than normal by ~1.5 s per day.

Conclusions
The NSDW of the delocalized 93m Nb state entered the BEC regime at room temperature.We observed pronounced condensate-field interactions, in particular the coherent superradiance and the suppression of atomic channels.Two M4 standing waves of 4-entangled γs at 30 and 662 keV coexist in the superradiant region, which construct a 1-D pm matter-wave grating.Seven Sisyphus falls occurred at the moments of forward hopping in 107 days, in addition to two other backward hopping episodes also shown.
We pay particular attention to the BEC matter-wave grating of length d in cm driven by the γs of λ 0 in pm.The number of matter-wave mirrors is nd/λ 0 , where n is the effective reflective index.Assuming n ∼ 1, the Q-factor of the Fabry-Pérot resonator is proportional to (d/λ 0 ) 2 > 10 20 [29],

Figure 1 .
Figure 1.Schematic representation of the experiment.The activated sample (1 mm × 1.5 cm × 3 cm, yellow color) with the normal vector in the E-W direction was surrounded by the Pb blocks (8 × 4 cm × 8 cm × 16 cm).A 2.5-mCi 137 Cs point source was placed at the location of (r, θ).The Pb blocks protected the HPGe detector from the leak exposure of 137 Cs source, except the end-fire mode (dark blue arrow) along the z axis.The lateral impinging multipolar MeV γs from 137 Cs gave rise to the superradiant Rayleigh scattering into the end-fire mode detected by a HPGe detector, which was located below the Pb shielding.

Figure 1 .
Figure 1.Schematic representation of the experiment.The activated sample (1 mm × 1.5 cm × 3 cm, yellow color) with the normal vector in the E-W direction was surrounded by the Pb blocks (8 × 4 cm × 8 cm × 16 cm).A 2.5-mCi 137 Cs point source was placed at the location of (r, θ).The Pb blocks protected the HPGe detector from the leak exposure of 137 Cs source, except the end-fire mode (dark blue arrow) along the z axis.The lateral impinging multipolar MeV γs from 137 Cs gave rise to the superradiant Rayleigh scattering into the end-fire mode detected by a HPGe detector, which was located below the Pb shielding.

Figure 2 .
Figure 2. The θ-dependence of the superradiant Rayleigh scatterings of M4 γ at 662 keV in counts per second (cps) as a function of the 137 Cs source location of (5.4 cm, θ).Data listed in Table2were taken every five minutes for a sequence from 0 to 540 degrees (−θ means 360 − θ to demonstrate the mirror symmetry).

Table 2 .
θ-dependence of superradiant Rayleigh in cps as a function of the source location of (r,θ), where r is the radius and θ is the impinging angle, as shown in Figure1.2.5 mCi 137 Cs 3-mm point source was placed on the circle r = 5.4 cm radius; the second column of configuration was measured in the upwards sequence of −θ from the bottom number to the top number, while the other two columns were measured in a downward sequence.The data were obtained every five minutes.

Figure 2 .
Figure 2. The θ-dependence of the superradiant Rayleigh scatterings of M4 γ at 662 keV in counts per second (cps) as a function of the 137 Cs source location of (5.4 cm, θ).Data listed in Table2were taken every five minutes for a sequence from 0 to 540 degrees (−θ means 360 − θ to demonstrate the mirror symmetry). 140

Figure 3 .
Figure 3.The r-dependence of superradiant Rayleigh scatterings of M4 γ at 662 keV in cps as a function of the 137 Cs source location of (r, θ).Data listed in Table3were taken every five minutes.

Figure 3 .
Figure 3.The r-dependence of superradiant Rayleigh scatterings of M4 γ at 662 keV in cps as a function of the 137 Cs source location of (r, θ).Data listed in Table3were taken every five minutes.

Figure 4 .
Figure 4. Seven Sisyphus cycles of superradiant Rayleigh scattering generated by a 2.5-mCi 137 Cs source at the position (0.3 cm, 90°) in the first long-term monitoring experiment begun on 1 August 2016 (day 0), the raw data of which are given in Figure S1 of the Supplementary information.The time sequence of data was γ counts per minute, while the presentation incorporates hourly smoothing (61 data points).Four horizontal lines indicate the potential quantization.Air conditioners were turned off after day 71, indicated by the gray area.Inset diagrams show the mode hopping periods (blue), the hopping amplitudes (red) and the raw + smoothed data (blue + red) of the Sisyphus falls.

Figure 4 .
Figure 4. Seven Sisyphus cycles of superradiant Rayleigh scattering generated by a 2.5-mCi 137 Cs source at the position (0.3 cm, 90 • ) in the first long-term monitoring experiment begun on 1 August 2016 (day 0), the raw data of which are given in Figure S1 of the Supplementary information.The time sequence of data was γ counts per minute, while the presentation incorporates hourly smoothing (61 data points).Four horizontal lines indicate the potential quantization.Air conditioners were turned off after day 71, indicated by the gray area.Inset diagrams show the mode hopping periods (blue), the hopping amplitudes (red) and the raw + smoothed data (blue + red) of the Sisyphus falls.

Figure 5 .
Figure 5. Fluctuations of 182 Ta γs and broad-band γs.The right-hand axis shows fluctuation of total counts of γs at 1121, 1189, 1221, and 1231 keV, which are collected to represent the 182 Ta decay.Note that counts of 182 Ta γs were less than the calibrated baseline.The left-hand axis shows counts of 0.015 × BBL, BBM, and 10 × BBH, which represent three broad-band γs below 662 keV, between 662 and 1121 keV, and above 1121 keV.Fluctuations in these are presented against calibrated baselines, as detailed in the Supplementary Information.BBL incorporates hourly smoothing, while BBM, BBH, and 182 Ta γs incorporate daily smoothing.An inset diagram gives raw and smoothed (white) data of the Sisyphus falls.Air conditioners were turned off after day 71, indicated by the gray area.

Figure 5 .
Figure 5. Fluctuations of 182 Ta γs and broad-band γs.The right-hand axis shows fluctuation of total counts of γs at 1121, 1189, 1221, and 1231 keV, which are collected to represent the 182 Ta decay.Note that counts of 182 Ta γs were less than the calibrated baseline.The left-hand axis shows counts of 0.015 × BBL, BBM, and 10 × BBH, which represent three broad-band γs below 662 keV, between 662 and 1121 keV, and above 1121 keV.Fluctuations in these are presented against calibrated baselines, as detailed in the Supplementary Information.BBL incorporates hourly smoothing, while BBM, BBH, and 182 Ta γs incorporate daily smoothing.An inset diagram gives raw and smoothed (white) data of the Sisyphus falls.Air conditioners were turned off after day 71, indicated by the gray area.