INTRODUCTION

PRD

PRVDAQ

Physical Review D

Phys. Rev. D

2470-00102470-0029

American Physical Society

10.1103/kgk5-bf5x

ARTICLES

Astrophysics and astroparticle physics

Prospects for relic neutrino detection using nuclear spin experiments

PROSPECTS FOR RELIC NEUTRINO DETECTION USING …

GARCIA DEL CASTILLO, PIEROBON, SENGUPTA, AND WONG

https://orcid.org/0009-0005-9159-205X

Garcia del Castillo

Yeray

https://orcid.org/0000-0001-6231-9461

Pierobon

Giovanni

^†

https://orcid.org/0000-0002-5107-0098

Sengupta

Dipan

^‡Wong

Yvonne Y. Y.

^§Sydney Consortium for Particle Physics and Cosmology School of Physics, The University of New South Wales

https://ror.org/03r8z3t63

, Sydney New South Wales 2052, Australia

Contact author: y.garcia_del_castillo@unsw.edu.au

^†

Contact author: g.pierobon@unsw.edu.au

^‡

Contact author: dipan.sengupta@unsw.edu.au

^§

Contact author: yvonne.y.wong@unsw.edu.au

27February2026

15February2026

1134

043061

9April2026

Correction: An incorrect version of Fig. 7 was used for publication and has now been replaced with the correct version.

26October202511February2026

Published by the American Physical Society

2026

authors

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Direct detection of the cosmic neutrino background (CνB) remains one of the most formidable experimental challenges in modern physics. In this work, we extend recent studies of CνB-induced coherent transitions in polarized nuclear spin ensembles. Adopting an open quantum system framework, we model coherent neutrino effects in large spin ensembles using a Lindblad master equation that also incorporates realistic experimental imperfections such as local dephasing and imperfect polarization. We solve the Lindblad equation numerically by way of a fast and computationally inexpensive method that can be extended to an arbitrarily large number of spins. Using our numerical solutions, we forecast the sensitivities of future experiments such as CASPEr to the local CνB overdensity parameter δν. Our findings indicate that a CASPEr-like experiment, though primarily aimed at axion dark matter search, could also constrain the CνB overdensity to δν∼1013 in configurations achievable by currently planned experimental efforts, and down to δν∼1011 in the most optimized scenario. While CνB detection remains out of reach in the foreseeable future, our results highlight the potential of using quantum sensing to probe fundamental physics.

Australian Research Council

10.13039/501100000923

DP240103130

I.INTRODUCTION

Standard hot big bang cosmology predicts a cosmic neutrino background (CνB), a relic from the early universe analogous to the cosmic microwave background (CMB), that decoupled from the primordial plasma when it cooled to temperatures of T∼1 MeV [1]. At decoupling these neutrinos had a nearly thermal spectrum of the relativistic Fermi-Dirac form nF=[exp(p/Tν)+1]-1, which is preserved by cosmological redshifting induced by the spatial expansion of the universe. Together with the minimum neutrino mass values implied by flavor oscillation experiments, we expect the CνB today to be largely nonrelativistic, with a temperature of Tν,0≃1.7×10-4 eV≃1.95 K, slightly cooler than the 2.73 K CMB photons. Its mean number density is predicted to be nν=nν¯≃56 cm-3 per flavor, but can be enhanced locally, i.e., in the solar neighborhood, by factors of O(1–10), depending on the neutrino mass, through gravitational clustering [2–14].

Notwithstanding its abundance, the CνB has so far evaded direct detection in the laboratory owing to its extremely low energies and weak interactions. To date, the only evidence we have of its existence is indirect and based entirely on its gravitational effects in the early Universe and subsequent impact on precision cosmological observables. Specifically, measurements of the primordial light element abundances (e.g., [15]) and the CMB anisotropies (e.g., [16]) consistently prefer an expansion rate—at minutes and 400,000 years postbig bang, respectively—compatible at the 10% level with the standard prediction of the CνB energy density.¹

In cosmology parlance, the effect of the CνB energy density on the early universal expansion rate is parametrized by the effective number of neutrinos Neff. The inferred value of Neff from primordial light element abundances and the CMB are Neff=2.88±0.27 (68% CL) [15] and Neff=2.99±0.17 (68% CL) [16], respectively, consistent with the energy density in three light neutrino flavors.

Anisotropic stress constraints from the CMB likewise support the weakly interacting nature of the CνB [17,18].

In the face of ever-improving cosmological/gravity-based constraints, it nonetheless remains of great interest to detect or at least constrain the CνB in a more direct way, using methods based on particle scattering. In this connection, a number of possible probes and constraints have been discussed. Below is a nonexhaustive list. (i)

At leading order in the Fermi constant GF, the so-called Stodolsky effect [19] refers to a spin-dependent shift in the energy of a standard-model (SM) fermion sitting in a CP asymmetric bath of neutrinos and antineutrinos. While cosmological neutrino-antineutrino asymmetries of ην≡(nν-nν¯)/nγ≲O(10-2) are not excluded by observations (e.g., [20]), it is also not well motivated within the SM, where O(10-10) particle-antiparticle asymmetries are generically expected in light of the observed matter-antimatter asymmetry in the Universe. We remark however on a recent claim of a large ην∼10-2-0.25 being preferred by extragalactic He4 observations [21]. Note also discussions of the survival of such asymmetries in the local Universe [22,23].

(ii)

At next-to-leading order O(GF2), coherent elastic scattering of a CνB wind with a macroscopic target can produce a mechanical force that causes the target to accelerate [24–29]. This force does not require the presence of a neutrino-antineutrino asymmetry, but works best for Dirac neutrinos. For Majorana neutrinos, the absence of vector currents causes the interaction cross section to be suppressed by a factor v2∼10-6, where v is the Earth-CνB relative speed. Current torsion balance setups cannot measure this acceleration unless the CνB number density is enhanced by a factor δν∼1012 or more [29]. Advancement on this front consists in using new techniques, e.g., interferometry [28], to enhance the sensitivity to the acceleration.

(iii)

Another O(GF2) effect, neutrino capture by β-decaying nuclei remains unique to CνB detection [30–33] and goes hand-in-hand with direct neutrino mass searches based upon β-decay end-point spectrum measurements [34–36]. The flagship tritium β-decay neutrino mass experiment KATRIN currently limits the local CνB overdensity to δν≲1011 [34]. The proposed PTOLEMY experiment aims to constrain this parameter to δν∼O(1) [37].

(iv)

An accelerator-based variant of the above has also been discussed in Ref. [38], which proposes to use accelerated beams of ions in ion storage rings to capture CνB neutrinos at nuclear β-resonances to enhance the capture cross section.

(v)

Inelastic O(GF2) scattering of the CνB on a target system can also induce transitions between different energy states. These include atomic transitions [39,40], as well as nuclear spin flips [41,42]. Of the latter, hydrogen spin flip induced by absorption of a relic neutrino-antineutrino pair may be sensitive to δν∼109 for neutrino masses mν∼10-3 eV [41].

(vi)

On astrophysical scales, probes of the CνB range from small-scale (i.e., sub-1 AU) perturbations to planetary orbits (current constraint δν≲5.34×1010 [43]) and the cooling of old neutron stars from CνB capture (potential sensitivity δν∼109 [44]), to large-scale (≳O(10) Mpc) up-scattering of the CνB by cosmic rays [45–47] (current constraint δν≲104 [47]) and features in the energy spectra of ultrahigh energy neutrinos due to CνB scattering (current constraint δν≲1014 [48]). Also proposed are resonant absorption dips in the cosmogenic neutrino energy spectrum when these neutrinos interact with the CνB on a meson resonance [49], a variant of the original Z-resonance absorption dips for ultrahigh energy neutrinos [50].

We are particularly interested in the proposal of Ref. [42], which exploits spin ensembles such as in nuclear magnetic resonance (NMR) experiments to detect coherent effects induced by the CνB through inelastic neutrino-spin interactions. Specifically, Ref. [42] argued that the momentum transfer q in such interactions can be macroscopic in wavelength terms, causing the collective CνB-induced spin flip rates across the spin ensemble to scale as N2, where N is the number of spins, if the spin sample’s physical size R is smaller than or comparable to q-1 (this situation is to be compared with an ∝N scaling in an incoherent, qR≫1 setting). Such macroscopic coherence of the target system can significantly enhance sensitivity to weak interactions and may conceivably yield a constraint on the CνB overdensity parameter comparable to the current KATRIN bound, δν≲1011 [34], for specific initial spin configurations [42]. Added to its appeal is that planned NMR experiments to search for dark matter axions and axionlike particles, e.g., CASPEr [51,52], are in principle sensitive to such coherent signals. Besides opening up a new avenue for ultralow-threshold searches for fundamental physics, the potential for these experiments to simultaneously search for and/or constrain both the axion and CνB parameter space is a most tantalizing prospect.

In this work, we expand on the study of Ref. [42] as follows: (i) We explore realistic experimental effects such as local dephasing and imperfect initial polarization of the spin ensemble, both of which can degrade the coherent CνB signature. (ii) Modeling these effects using an open quantum system framework, we solve the Lindblad master equation numerically, both for the full density matrix of N spins, and using a computationally efficient second-order approximation that tracks only a limited set of observables. Using these solutions, we are able to confirm for large N the validity of the perturbative solutions put forward in Ref. [42]. (iii) We forecast the sensitivity of realistic future experiments to the CνB overdensity parameter δν, especially the reach of the CASPEr experiment [51–53], by including a more realistic modeling of the instrumental noise, initial polarization of the spin ensemble, and sample size.

In addition to the sample size, which restricts the degree of CνB coherence across the spin ensemble, we find the initial spin polarization to be a particularly important limiting factor—and also the primary reason why the current generation of NMR axion searches cannot place competitive constraints on δν. In the most optimistic case (no instrumental noise, R∼10 cm, 100% polarization), a sensitivity of δν∼1011 may be achievable using Xe129 nuclei. In more realistic settings (instrumental noise, R∼1 cm, 25% polarization), however, the sensitivity to the CνB deteriorates to δν∼1013.

The paper is structured as follows. In Sec. II we introduce the scattering rates of the CνB with a single two-level system, such as a nuclear spin with I=1/2, and with an ensemble of N spins, highlighting a regime where a coherent ∼N2 enhancement can be achieved. In Sec. III, we review the Lindblad master equation commonly used to model open quantum systems, and describe the dynamics in the Dicke basis. Sections IV and V discuss, respectively, the emergence of N2 effects in collective observables and the limitations arising from incoherent local effects that eventually degrade the signal. We show in Sec. VI how the Lindblad master equation combining coherent and incoherent local effects can be efficiently solved using a numerical approximation. We use these numerical solutions to forecast the sensitivity of currently planned experiments to the CνB overdensity parameter δν, as detailed in Sec. VII. Section VIII contains our conclusions. The computational details of this work are reported in eight Appendices.

II.NEUTRINO-SPIN INTERACTION

Consider a spin-1/2 fermion—such as an electron or a nucleus—interacting with neutrinos via the weak interaction. We assume coherence over nuclear distances, allowing us to treat the fermion as a pointlike particle and neglect its internal structure. At low energies, this interaction can be described by an effective 4-fermi interaction specified by the Lagrangian L=Ψ¯(i∂-m)Ψ-qΨ¯γμΨAμ+Lint,Lint=-∑fGF2Ψ¯γμ(gVf-gAfγ5)Ψν¯fγμ(1-γ5)νf,(1)where Ψ denotes the fermion field of mass m and electric charge q, Aμ=(A0,A→) is the electromagnetic 4-potential, GF the Fermi constant, νf the neutrino field of flavor f=e, μ, τ, and gVf and gAf are the vector and axial-vector couplings of the fermion to νf.

The couplings gVf and gAf generally depend on the neutrino flavor as well as the nature of the fermion. Here, we focus exclusively on neutrino–nucleus interaction through the neutral-current channel. The corresponding couplings are therefore independent of flavor. The values of gV and gA are nucleus-dependent. In what follows, we consider the axial coupling gA, which is only sensitive to the spin of the unpaired nucleon (see, e.g., [54]).

It is convenient to express the neutrino fields in the mass basis. Performing the basis transformation νf=∑iUfiνi, where νi are the mass eigenstates (for i=1, 2, 3) and Ufi is the Pontecorvo-Maki-Nakagawa-Sakata matrix, the interaction Lagrangian becomes Lint=-∑ijGF2Ψ¯γμ(gVi,j-gAi,jγ5)Ψν¯iγμ(1-γ5)νj,(2)where the effective couplings in the mass basis are given by gV(A)i,j=∑fgV(A)fUfi*Ufj. In the flavor-independent case under consideration, the effective couplings are diagonal, i.e., nonzero only for i=j, and equal for all mass eigenstates.²

If instead of nuclear spins we consider electron spins, then the neutrino-spin interaction can also proceed via the charged-current channel. In that case, the couplings gV(A)i,j generally contain nonzero off-diagonal terms, allowing the neutrino to change mass eigenstates via the interaction. This case has been considered in Ref. [42].

In a typical NMR setup, a sample of (nonrelativistic) nuclear spins is placed in a constant magnetic bias field B→=∇×A→, which we choose to point in the negative-z direction, i.e., B→=-Bz^. We can therefore take the nonrelativistic limit and derive from the relativistic Lagrangian (1) the corresponding Hamiltonian for a single spin, H(1/2)=HS(1/2)+Hint(1/2)=ω0Jz+ΣzJz+Σ-J-+Σ+J+.(3)Here, HS(1/2)=ω0Jz describes the interaction of the spin with the magnetic field, where ω0=γB is the energy splitting between adjacent spin states—also called the Larmor frequency, and γ=gμN is the gyromagnetic ratio specific to the nucleus under consideration, defined via the nuclear magneton μN=e/2mp and the nucleus-specific g-factor. The Hint(1/2) terms describe interaction of the spin with the CνB, where the spin raising and lowering operators are defined as J±≡Jx±iJy. In our flavor-independent case the neutrino terms Σ±,z read Σz=∑i2GFgAν¯iγ3(1-γ5)νi,Σ±=∑iGF2gAν¯i(γ1∓iγ2)(1-γ5)νi,(4)and they arise exclusively from the axial part of the weak interaction. We refer to the reader to Appendix A for the derivation of the low-energy Hamiltonian, including the flavor-dependent case.

Physically, the term ΣzJz corresponds to elastic scattering of neutrinos off a spin, which alters the phase evolution of the spin but does not change its energy. In contrast, the Σ±J± terms describe inelastic processes, in which the neutrino exchanges energy with the spin, either exciting it (Σ+) or deexciting it (Σ-). The single-spin excitation and deexcitation rates γ±, defined as the transition rate of a spin from down to up (+) and vice versa (-), can be estimated using Fermi’s golden rule. Accounting only for inelastic neutrino-spin scattering, the rates read γ±=4GF2(2π)3gA2∑i∫pi±low∞d|p→||p→|2pi±mi2+|p→|i±2×[(1-Ni(pi±0))Ni(p0)+(1-N¯i(pi±0))N¯i(p0)],(5)where pi±=[(|p→|2+mi2∓ω0)2-mi2]1/2, pi±0=|p→|2+mi2∓ω0, mi is the mass of the ith mass eigenstate, and the lower integration limits are given by pi+low=(ω0+mi)2-mi2 and pi-low=0.³

All neutrino momenta throughout this work are 3-momenta. We do however use a superscript “0”, e.g., p0≡|p→|2+m2 to denote energy.

This expression applies to both Dirac and Majorana neutrinos, but with the understanding that the neutrino and antineutrino occupation numbers, Ni and N¯i, in the Dirac case are a factor two smaller than in the Majorana case.⁴

Although at decoupling relativistic Dirac neutrinos (antineutrinos) are exclusively in the left- (right-) helicity state, as they redshift to nonrelativistic velocities at late times, the generic expectation is they should relax into an equal mixture of left- and right-helicity states via gravitational interactions.

The detailed calculation of γ± from the low-energy Hamiltonian (3) can be found in Appendix B. Assuming a standard, CP-symmetric CνB with Ni=N¯i∼nF, and a spin sample consisting of Xe129 nuclei with gA=1/2, the rates (5) evaluate numerically to γ-SM≃1.1×10-48 Hz,γ+SMγ-SM≃0.994,(6)for a typical NMR energy splitting of ω0=1×10-8 eV and a normally ordered neutrino mass sum of ∑mν=0.15 eV.⁵

We adopt mass values consistent with global neutrino oscillation fits [55–57]. For normal ordering, the masses are fixed by the total mass sum and the measured mass-squared splittings, Δm212≃7.4×10-5 eV2 and Δm312≃2.5×10-3 eV2, giving m1≃0.0419 eV, m2≃0.0428 eV, and m3≃0.0652 eV for ∑mν=0.15 eV.

In the case of N spins, the low-energy Hamiltonian (3) generalizes to H=∑α(ω0Jzα+ΣzαJzα+Σ-αJ-α+Σ+αJ+α),(7)where J±,zα are understood to operate only on the spin at spatial position x→α, and the neutrino terms Σ±,zα are the same as those given in Eq. (4) but now with the neutrino fields specified by νi(t,x→α). The total excitation and deexcitation rates of the ensemble, defined here as the rate at which the total energy of the ensemble increases or decreases by one unit of ω0, can be written as⁶

Throughout this work, we always denote integrated (ensemble-level) rates by an upper-case Γ, while single-spin interaction rates—relevant for the Lindblad dynamics discussed in Sec. III—are represented by a lower-case γ.

Γ±=2GF2(2π)3gA2∑i∫pi±low∞d|p→||p→|2pi±mi2+pi±2×[(1-Ni(pi±0))Ni(p0)+(1-N¯i(pi±0))N¯i(p0)]×∫-11dμF±(p→i±-p→),(8)where μ≡p^i±·p^, is the polar angle between the momentum vectors p→ and p→′, and F±(q→)≡⟨ispin|∑α,βJ∓βJ±αe-iq→·(x→α-x→β)|ispin⟩(9)are the excitation and deexcitation form factors.

Depending on the initial spin configuration and the momentum transfer q→≡p→i±-p→ from the neutrinos to the ensemble, the form factors F±(q→) can evaluate to a range of values. For example, a system prepared in the ground state |G⟩≡|↓⟩⊗N in which all spins align initially with the magnetic field has F+G(q→)=N and F-G(q→)=0, and hence a nonvanishing total excitation rate Γ+=γ+N. In contrast, for an initial spin state aligned with the equatorial plane, |P⟩=∏α12(|↑⟩+|↓⟩), the form factors evaluate to F±P(q→)=N2+N249q2R2|j1(qR)|2,(10)with j1(x) the spherical Bessel function of order one. Clearly, F±P(q→) can range from O(N) in the fully incoherent regime, to O(N2) in the fully coherent regime, discussed in the following subsection.

A.Coherent regime

An inspection of the form factor (10) reveals that there are essentially two physical scales that determine the spatial coherence of the neutrino-spin interactions. The first is the characteristic size of the spin ensemble R, which sets the spatial extent over which the spins are distributed. The second is the typical momentum transfer induced by the relic neutrinos q.

Since the incoming neutrinos follow a momentum distribution Ni(p0), the momentum transfer q→=p→i±(p→)-p→ is also dictated by the distribution. The coherence properties must therefore be understood in terms of the typical momentum transfer q→typ extracted from this distribution. When the condition qtypR≪1 is satisfied, we see in Eq. (10) that j1(qtypR)≈qtypR/3, leading to F±∼O(N2). This condition can always be fulfilled if the typical neutrino momentum pν itself is small compared with the inverse system size, i.e., pνR≪1. For relic neutrinos with a typical incoming momentum of pν∼3.15Tν,0≃5.3×10-4 eV∼27 cm-1, the condition pνR≪1 therefore requires a very small R≪0.04 cm. However, even for the larger-size spin samples to be considered in this work, e.g., R∼O(1) cm, where formally pνR≫1, it is still possible to fulfill the broader condition qtypR≪1 and hence maintain coherence across the sample in some limits. Specifically, the typical CνB kinetic energy is much larger than the energy splittings ω0∼10-8 eV considered in this work. In this limit, the typical momentum transfer is qtyp≈2pνsin(θ/2), where θ is the angle between the incoming and outgoing neutrino. Thus, we see that coherence can be maintained for those scattering angles satisfying θ≪(pνR)-1.

In terms of the total excitation and deexcitation rates (8), the pνR≫1 limit is effectively equivalent to demanding that only outgoing momentum directions falling in the range cos((pνR)-1)<μ<1 contribute to the N2 enhancement. Thus, evaluating the μ-integral immediately yields Γ±≈N216pν2R2γ±,(11)which we note is suppressed by a factor ∼(2pνR)-2 relative to the naïve coherent rate. The typical momentum transfer in this regime is qtyp∼ω0mν/pν. This will be the case of interest in our study and was first introduced in Ref. [42] as an analogy to the Rayleigh-Gans regime [58]. See Appendix B for details.

As a concrete example, consider a spin ensemble composed of liquid Xe129 characterized with a spin density of ns∼N/R3∼1022 cm-3. In this case, assuming an energy splitting ω0=10-8 eV and a standard CνB with three normally ordered masses summing to ∑mν=0.15 eV, the total excitation rate of the spin ensemble evaluates to Γ±SM≃1.2×10-6 Hz(R1 cm)4,(12)where the R4 scaling represents the aforementioned forward limit of the N2 enhancement [42]. The dependence of the total interaction rate (Γtot=Γ++Γ-) on the system parameters, including ω0 and the sample size R, is illustrated in the left panel of Fig. 1, where regions exhibiting enhanced coherence can be identified. The right panel of the same figure shows the dissipation ratio Γ+/Γ-≡γ+/γ- across the relevant parameter space, and highlights a slight dominance of neutrino-induced deexcitation. This dominance arises because deexcitation is always kinematically allowed as it has no energy threshold: any neutrino regardless of its momentum can induce the transition; see Eq. (5). Excitation is on the other hand not always possible. The fractional difference is approximately γ+/γ--1∼-2mνω0/pν2 [42] for ω0≪pν2/(2mν)≪mν. See also Appendix B.

10.1103/kgk5-bf5x.f1

FIG. 1.

Typical neutrino-spin interaction rates for a single neutrino species at a standard number density nν∼110 cm-3 and with a typical momentum pν∼27 cm-1. Left: the total rate Γtot≡Γ++Γ- in the parameter space of system energy splitting ω0 and sample size R, where we have assumed a spin density of ns∼3×1022 cm-3 and a single neutrino mass of mν=0.1 eV. The regions that exhibit coherent effects are in the top-left corner. Right: dissipation ratio γ+/γ- in the parameter space of neutrino mass mν and system energy splitting ω0 in units of neV=10-9 eV. The ratio quantifies the imbalance between neutrino-induced excitation (γ+) and deexcitation (γ-) single-spin processes, setting the dynamics of the system in the absence of other interactions. We find γ+/γ-<1 across the entire parameter space shown, indicating a net deexcitation effect from relic neutrinos on the spin ensemble. The two horizontal red dashed lines denote the splitting energies for an ensemble of Xe129 spins achieved with two different sub-Tesla magnetic field values.

III.AN OPEN QUANTUM SYSTEM

Although our full system comprises a spin ensemble S and a relic neutrino bath B that interact with each other, in practice we are interested only in the dynamics of the former. To this end, we adopt the framework of open quantum systems, which assumes unitary evolution of the density matrix of the full system, ρ=ρS⊗ρB, under a total Hamiltonian H=HS+HB+Hint,(13)where HS and HB denote, respectively, the free Hamiltonian of the spin ensemble and of the bath, and Hint describes their interaction. Tracing out the bath degrees of freedom, the evolution of the reduced density matrix ρS is described by the Lindblad master equation [59–61], dρSdt=-i[HS+HLS,ρS]+∑kγkDOk[ρS(t)],(14)where the so-called Lamb-shift Hamiltonian HLS is a small Hermitian correction to HS arising from Hint, and DO[ρS(t)]=OρSO†-12{O†O,ρS}(15)is a superoperator that acts on the reduced density matrix and describes the dissipative effects of the system-bath interaction. Dissipation can occur through multiple channels, each specified by its own operator Ok and the associated interaction rate γk.

The Lindblad equation (14) serves well under the condition of weak coupling of the system to a large bath with no memory and when the Larmor frequency ω0 is much larger than the interaction rates. These conditions enter the derivation of Eq. (14) at various stages via the so-called Born, Markovian, and rotating-wave/secular approximation, which are discussed in more detail in Appendix C. See also, e.g., Ref. [61] for a general introduction to the Lindblad formalism.

A.The master equation

In the case of a single spin-1/2 interacting with a neutrino bath, the Hamiltonians of the spin system and its interaction with the neutrinos, HS(1/2) and HI(1/2), are given in Eq. (3). The corresponding Lindblad equation for the spin density matrix in the interaction picture takes the form dρS(1/2)dt=-iΔ[Jz,ρS(1/2)]+γ-(J-ρS(1/2)J+-12{J+J-,ρS(1/2)})+γ+(J+ρS(1/2)J--12{J-J+,ρS(1/2)})+γz(JzρS(1/2)Jz-12{Jz2,ρS(1/2)}),(16)where we have highlighted ρS(1/2) as the reduced density matrix for a single two-level system. We refer the reader to Appendix C for an ab initio derivation of this equation.

Inspecting Eq. (16), we see first an energy shift due to the Δ, which is a combination of the O(GF), Stodolsky effect that is present only when the neutrino-antineutrino asymmetry is nonzero, and O(GF2) terms from the Hermitian part of the Lindblad equation. The J± terms and the corresponding coefficients γ± represent the non-Hermitian interaction of the spin with the neutrinos, causing the spin to either excite or deexcite via inelastic scattering, analogously to absorption and emission in quantum optics.⁷

While we draw parallels between our system and quantum optics and introduced terminology commonly used in the latter subject, we always use “excitation” and “deexcitation” to refer to the effects of the CνB.

The Jz/γz term is, on the other hand, energy conserving and represents the effect of dephasing. The computations of the coefficients γ±,z can be found in Appendix D. Their general forms including both neutrino scattering and pair emission/absorption processes can be found in Eqs. (D10)–(D12), and, for γ±, match the heuristic single-spin excitation and deexcitation rates (5) under the same settings.

The single-spin equation (16) can be generalized to N spins, where the Hilbert space of the spin ensemble is now ρS=⊗Nρ(1/2) and Eq. (7) gives the corresponding Hamiltonian. As we show in Appendix E, omitting the Hermitian part, the Lindblad equation for the N two-level systems reads dρSdt=∑αβγ-αβ(J-αρSJ+β-12{J+αJ-β,ρS})+∑αβγ+αβ(J+αρSJ-β-12{J-αJ+β,ρS})+∑αβγzαβ(JzαρSJzβ-12{JzαJzβ,ρS}),(17)where α and β run over all the spins of the system, and the angular operators J±,zα act only on the spin α. The coefficients γ±,zαβ again encode information of the spin-bath interaction, but now correlate the interaction at spatial position x→α to that at x→β. For instance, assuming only inelastic neutrino-spin scattering, the γ±αβ coefficients read γ±αβ=2GF2πgA2∫d3p→(2π)3∫d3p→′(2π)3e-iΔx→αβ·(p→-p→′)×δ(1)(ω0±p′0∓p0)×[(1-Nk(p′0))Nj(p0)+(1-N¯k(p′0))N¯j(p0)].(18)Depending on the extent to which correlations can be maintained across spatial distances Δx→αβ≡x→α-x→β of the spin ensemble, qualitatively very different dynamics can arise. Two contrasting limits are the fully coherent and the fully incoherent regimes.

1.Fully coherent regime

Full coherence is achieved when e-iΔx→αβ.(p→-p→′)≈1 across the whole spin ensemble, or, equivalently, when the condition qR≪1 is satisfied for momentum transfer q and a sample size R. In this limit, the interaction coefficients are effectively the same for all spin pairs. The Lindblad equation then reduces to dρSdt=γ-(J-ρSJ+-12{J+J-,ρS})+γ+(J+ρSJ--12{J-J+,ρS})+γz(JzρSJz-12{Jz2,ρS}),(19)where we have introduced the collective spin operators J±,z=∑αJ±,zα. This collective Lindblad equation is structurally identical to that for a single spin and the coefficients γ±,z are the same single-spin interaction rates of Eqs. (D10)–(D12). In the language of quantum optics, these terms describe, in the order of appearance, collective emission, collective absorption, and collective dephasing (see, e.g., Ref. [62]). Since the system Hamiltonian is HS=ω0Jz, the collective dephasing term commutes with HS and has no dynamical effect on the observables of interest; we will therefore neglect it in the following.

2.Incoherent regime

In the opposite, qR≫1 limit, the phase differences across the system are large, leading to a complete loss of coherence. This defines the incoherent regime, in which neutrino-spin interactions are fully local. The master equation in this case factorizes into N independent Lindblad equations, i.e., dρSdt=∑αγ-(J-αρSJ+α-12{J+αJ-α,ρS})+∑αγ+(J+αρSJ-α-12{J-αJ+α,ρS})+∑αγz(JzαρSJzα-12{(Jzα)2,ρS}).(20)Here, each spin evolves independently under local interactions with the bath, and the rates γ±,z are again the single-spin rates. This local Lindblad form is general for any uncorrelated ensemble-bath interaction and applies to a wide class of physical systems for which the Lindblad approximation holds. In Sec. V, we will use Eq. (20) to model other (non-neutrino) environmental effects that eventually drive the spin ensemble toward thermal equilibrium in a realistic NMR experiment.

B.Dicke basis and initial states

Suppose all N spins are identically prepared and indistinguishable, and experience the same interactions with the environment. It is then convenient to use the framework of Dicke states [63], which are eigenstates of the collective operators J2 and Jz, and form a basis for the permutationally invariant subspace of the full Hilbert space. As permutational invariance is preserved by the dynamics of the scenarios of interest, adoption of the Dicke basis allows us to drastically reduce the computational complexity of the problem, both analytically and numerically: instead of working in the full 2N-dimensional Hilbert space, we can restrict the dynamics to a much smaller subspace, spanned by the Dicke states labeled by the total spin and its projection, |j,m⟩, where j=N/2,N/2-1,…,jmin, with jmin=0 if N even or jmin=1/2 if N odd, and |m|≤j. The full density matrix therefore contains nD=∑jminN/2(2j+1)2=O(N3)(21)elements. In the presence of purely coherent interactions, where the dynamics are confined to the fully symmetric subspace (j=N/2), this reduces further to nD=N+1.

The left panel of Fig. 2 illustrates the intuition provided by the Dicke representation—commonly visualized as the Dicke triangle—which organizes the Hilbert space according to total spin j, including both the symmetric subspace (j=N/2) and the nonsymmetric sectors (j<N/2) that do not exhibit cooperative or superradiant behaviors. When the coherence condition is satisfied, the neutrino-spin interaction is represented by collective, “vertical” transitions along the Dicke ladder in steps of |N/2,m⟩→|N/2,m±1⟩. This behavior contrasts with that of incoherent local processes such as dephasing, which induce only “horizontal” motion along the j-axis of the Dicke triangle. For more details on the Dicke basis and the effects of local interactions, we refer the reader to Ref. [62] and references therein. See also Refs. [64,65] for an exposition on Dicke superradiance.

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FIG. 2.

Visual representations of collective spin states in an ensemble of spin-1/2 particles. Left: the Dicke triangle shows the structure of permutational invariant Dicke states |j,m⟩, including the ground state |j,-j⟩, excited state |j,j⟩, and intermediate states such as the superradiant state |j=N/2,0⟩. The vertical coordinate m parametrizes the energy of the system, while the horizontal coordinate j represents the cooperativity of the system, i.e., the degree of coherence in the system. Blue shading indicates regions of enhanced collective effects, as determined by the expectation value of the emission rate operator ⟨Γ+Γ-⟩. Center: the density matrix for the coherent spins state (or product state) |P⟩ in the symmetric subspace basis |N/2,m⟩⟨N/2,m′|, with m∈(-N/2,N/2) and for N=30. We also depict the excited and the ground states, which lie in the opposite extremities of the matrix. Right: Bloch sphere depiction of collective spin states. The ground and the excited Dicke states point in the negative and positive z-directions, respectively. The product state |P⟩ also has a well-defined classical direction: in this figure, it aligns with +x and relates to the ground state by a π/2 pulse.

We are interested in three types of permutation-symmetric initial states, whose density matrices in the symmetric Dicke basis and Bloch sphere visualizations are shown in the center and right panels of Fig. 2. (i)

The ground state |G⟩≡|↓⟩⊗N=|N/2,-N/2⟩ represents the case in which all N spins are in the low-energy state of the splitting (pointing down in our notation), such that the polarization is unity in the Bloch sphere. From an experimental perspective, such a state is called hyperpolarized, as the polarization far exceeds the thermal equilibrium expectation (see Sec. V). Nonetheless, hyperpolarization can be achieved experimentally for, e.g., Xe129-enriched ensembles, via spin exchange optical pumping, as planned in the upcoming stages of the CASPEr experiment [53].

(ii)

The fully excited state |E⟩≡|↑⟩⊗N=|N/2,N/2⟩ has all N spins in the high-energy state of the splitting (pointing up). Starting from a hyperpolarized state like |G⟩, the |E⟩ state can be produced by applying a π pulse, i.e., applying a weak magnetic field oscillating at the Larmor frequency, until the polarization is inverted.

(iii)

The preparation of a hyperpolarized ground state |G⟩ is critical for all three initial configurations. However, achieving a pure state with near-unity polarization approximating |G⟩ may not always be possible under realistic experimental settings. A more faithful description of the spin ensemble’s initial configuration should therefore be given in terms of a density matrix, shown here in the Dicke basis in a generic form,⁸

The complete Dicke basis should in principle include a degeneracy label for each |j,m⟩ state, i.e., |j,m,α⟩, with 1≤α≤djN, where djN can be found in, e.g., Eq. (5) of Ref. [66], in order to recover the full Hilbert space. For simplicity, however, we have omitted this label, since it does not affect our arguments.

ρ=∑j,m∑j′,m′f(j,j′,m,m′)|j,m⟩⟨j′,m′|.(23)The initial polarization of the spin ensemble prior to the application of any pulse can then be trivially specified via the expectation value of the collective operator Jz, i.e., p≡-2N⟨Jz⟩=-2N∑j,mmf(j,m),(24)which always evaluates to |p|<1 if the configuration is not purely the ground (m=-N/2) or purely the excited (m=N/2) state.

We caution however that the definition (24) of p alone does not completely specify the effect of partial polarization on the spin ensemble. To see this, consider also the expectation values ⟨Jz2⟩=∑j,mm2f(j,m)≤N24,⟨J2⟩=∑j,mj(j+1)f(j,m)≤N2+N24,(25)where the upper bounds correspond to the expectations for the ground (j=N/2,m=-N/2) and the excited (j=N/2,m=N/2) states. Clearly, for ⟨Jz2⟩ to be suppressed relative to the optimal value we must have admixtures of |m|<N/2 states (i.e., states along the Dicke ladder in Fig. 2), while for ⟨J2⟩ to become suboptimal the ensemble must contain j<N/2 states (i.e., states along the j-axis in the Dicke triangle). Crucially, neither ⟨Jz2⟩ nor ⟨J2⟩ can be expressed in general in terms of p as defined in Eq. (24).

We shall return to the issue of polarization in Sec. VI A, where we numerically evaluate in the Lindblad equation and connect our solutions with the hyperpolarization protocols and level of polarization achievable in current and future experimental setups.

IV.COHERENT OBSERVABLES

Given an initial state, the Lindblad equation (17) can be solved to find the evolution of the spin ensemble. Several observables are of interest. The most common first-order observable is the collective magnetization, whose expectation value can be constructed from ⟨Jz⟩=tr(ρSJz), and is related to the ensemble’s energy. At second order, relevant observables include ⟨Jz2⟩ and ⟨Jx2⟩, where the latter represents the squared transverse spin component.

Exact analytical solutions for these observables generally do not exist. Numerical solutions, on the other hand, can become prohibitive for large values of N. We therefore consider first approximate analytical solutions in different limits and under different settings in order to map out the limiting behaviors of the system, before attempting a numerical treatment. We discuss first in this section approximate analytical solutions in the presence of coherent neutrino effects only. Section V considers the effects of non-neutrino “noise” from local interactions. We present numerical solutions including both neutrino-induced coherent and local noise effects in Sec. VI.

Before we delve into the solutions, we first define two timescales that characterize the evolution of the spin ensemble in the presence of coherent neutrino interactions. (i)

The superradiant timescale, tSR≡1N2γ±∼8.7×10-3 s(1025N)2,(26)characterizes the timescale for a single collective spin-flip, i.e., the transition |j,m⟩→|j,m±1⟩. The numerical estimate follows from setting γ±=γ±SM, where γ±SM is given in Eq. (6).

(ii)

The dissipative timescale, tdiss≡1Nγ±∼8.7×1022 s(1025N),(27)is the dynamical timescale of the system, defined as the duration over which an O(1) change in a collective observable such as ⟨Jz⟩ becomes apparent. This timescale also marks the end of the coherent superradiant regime, and limits the validity of the perturbative solution discussed below. The numerical estimate assumes again γ±=γ±SM from Eq. (6).

A.Perturbative solution

Considering the weakness of the neutrino-spin interactions, Ref. [42] proposed a perturbative/iterative solution to Eq. (19) around the initial state. Assuming only coherent neutrino interactions are present and to linear order in t, the solution reads ρS(t)≈ρS(0)+tγ-DJ-[ρS(0)]+tγ+DJ+[ρS(0)],(28)recalling the notation of DO[ρ] from Eq. (15), and we have again omitted the collective dephasing term.

In this perturbative limit, the evolution of ⟨Jz⟩ for an ensemble initially prepared in the |P⟩ state is given by ⟨Jz⟩|P⟩≈-(N2+1)(N2+12)γnett≈-N24γnett,(29)with γnet≡γ--γ+. As already noted in Ref. [42], ⟨Jz⟩ benefits from an N2 enhancement when the ensemble is initialized in the |P⟩ state. However, this enhancement is partially offset by a suppression arising from the near cancellation between the γ+ and γ- coefficients—typically γnet∼2(mνω0/pν2)γ±∼O(10-2-10-3)γ±, and can be even more strongly suppressed for smaller values of ω0 and mν, as illustrated in Fig. 1.

In a similar vein, the second-order observables evolve in the perturbative limit according to ⟨Jz2⟩|P⟩≈N4+γtotN(N+1)t,(30)⟨Jx2⟩|G⟩≈N4+2γ+N(N+1)t,(31)where γtot=γ-+γ+. Note that in the case of ⟨Jx2⟩|G⟩, we have assumed the ensemble to be prepared in the ground state: this observable can be advantageous, as it captures the same neutrino-induced signal as ⟨Jz2⟩|P⟩ (for the mν and ω0 values considered here, 2γ+≃γtot) but without the need to rotate the spins to the equatorial plane.⁹

As pointed out in Ref. [67], pulse imperfections can lead to errors in the manipulation of the collective spin.

It is also a quantity directly accessed by experiments like CASPEr that measure the transverse magnetization. The perturbative solutions (29)–(31) should remain valid as long as the state of the system does not deviate too much from the initial state. For the states considered in this work, we therefore expect the perturbative solutions to hold up to t≲tdiss, even in the case of the superradiant state.¹⁰

In the case of the superradiant state, although perturbation theory formally breaks down at times t≳tSR [42], the coherent effect persists up to the dissipation timescale tdiss, i.e., until the system has evolved significantly away from the initial state |N/2,0⟩. This suggests that, even though the perturbative solution no longer accurately describes the full density matrix, it can still capture the expectation values of the relevant observables within tdiss.

Inspecting the solutions (29)–(31), we note first of all that the well-known momentum-transfer condition qR≲1 is not, by itself, sufficient to guarantee a coherent effect. One must also select an observable suited to the given initial state. For a first-order observable, measuring ⟨Jz⟩ is advantageous when starting from the |P⟩ state, as the quantum states before and after scattering are not fully orthogonal. This requirement was already emphasized in Ref. [42]. Nevertheless, coherence can also build up dynamically through the action of collective neutrino-spin interactions, even if the initial state has no transverse coherence at t=0, such as in the ground state |G⟩. This is evident from Eqs. (30) and (31), where we have shown that measuring ⟨Jx2⟩|G⟩ produces the same coherent effect as measuring ⟨Jz2⟩|P⟩.

Second, while Eqs. (29)–(31) clearly demonstrate that both the first- and second-order observables benefit from an N2 enhancement, to assess the degree of observability we need also to consider the intrinsic quantum noise of the states as quantified by the variance of the observable. For the first-order observable ⟨Jz⟩|P⟩, the variance is σJz2≡⟨Jz2⟩|P⟩-⟨Jz⟩|P⟩2≈N4+γtotN(N+1)t+O(t2),(32)where we have supplied at the second equality the perturbative expression. The first, N/4 term in the second line is commonly known as the spin projection noise (SPN), and represents (uncorrelated) noise in the initial state. Figure 3 displays the perturbative solution for σJz alongside its numerical solution (see Sec. VI A) and illustrates that deviations from the SPN value of N/4 emerge around the dissipative timescale tdiss. Therefore, at t≲tdiss, the signal and noise scale as ⟨Jz⟩∼γnetN2 and σJz∼N, respectively, leading to a highly favorable signal-to-noise ratio scaling of ∼γnetN3/2.

10.1103/kgk5-bf5x.f3

FIG. 3.

Time evolution of the standard deviation of the first order observable Jz, as defined in Eq. (32), for N=108 and γ+/γ-=0.997. The perturbative solution is taken from the approximation in Eq. (32), σJz=N/4+γtotN2t, whereas the numerical solution is obtained with the second-order approximation method, introduced in Sec. VI A. We highlight the departure from the spin projection noise at t≳tdiss and the breakdown of the perturbative solution at t≳(Nγnet)-1.

On the other hand, at t≲tdiss, the second-order observables ⟨Jz2⟩|P⟩ and ⟨Jx2⟩|G⟩ have an intrinsic variance of σJ22∼N2, such that the signal-to-noise ratio scales as ∼γtotN. Thus, while higher-order observables may appear at first glance to contain more information, they do not necessarily imply an improvement in sensitivity to the CνB signal unless the intrinsic noise of the observable can be reduced, for instance, through squeezing [42]. Thus, we expect the ⟨Jz⟩|P⟩ observable to yield the best constraint on the CνB overdensity because of its favorable scaling with the number of spins.

B.Steady-state solution

At t≫tdiss, we expect the ensemble to reach a steady state characterized by saturation of the observable. This steady state is not captured by the perturbative solution (29)–(31), but can be found by solving the Lindblad equation (19) in the limit ρ˙S=0. Writing the density matrix in the Dicke basis as ρS=∑mm′ρSmm′|j,m⟩⟨j,m′| and assuming the stationary solution to be unique (i.e., ρSmm′=0 for m≠m′), a recursive relation for the diagonal elements ρSm,m≡ρSm can be obtained, γ-(N2+m+1)(N2-m)ρSm+1+γ+(N2-m+1)(N2+m)ρSm-1-γ-(N2+m)(N2-m+1)ρSm-γ+(N2-m)(N2+m+1)ρSm=0,(33)for m∈[-N/2-1,N/2+1], and ρS-N/2-1=ρSN/2+1=0. Equation (33) is effectively a linear equation for the entries of the diagonal with a tridiagonal matrix, which can be solved iteratively for ρt→∞m, from which we can compute the steady-state values of the observables via ⟨Jz⟩|P⟩t→∞=∑mmρt→∞m,⟨Jz2⟩|P⟩t→∞=∑mm2ρt→∞m,⟨Jx2⟩|G⟩t→∞=∑m⟨j,m|Jx2|j,m⟩ρt→∞m.(34)Note that, since we consider only coherent neutrino interactions, the final state of the spin ensemble implied by Eqs. (33) and (34) is not a thermal equilibrium state one would generically expect the system to tend to: for this to happen local interactions are required, to be discussed in Sec. V.

Figure 4 shows the late-time values of the normalized collective observable σx≡(⟨Jx2⟩|G⟩)1/2, as a function of the number of spins N. Although not shown here, we also find that the steady-state value of ⟨Jz⟩|P⟩ approaches the ground-state expectation -N/2 around t∼(Nγnet)-1 in the large N limit, following a similar scaling behavior in N as σx. For all observables convergence is reached for N≳O(103), but at larger values of N in those cases where a near cancellation between excitation and deexcitation, i.e., γ+≈γ-, exists.

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FIG. 4.

Steady-state scaling of the collective expectation value ⟨Jx2⟩|G⟩ as a function of the number of spins N, in the presence of neutrino collective interactions only. The nine curves shown here correspond to different values of the dissipation ratio γ+/γ- as indicated by the color scale. For a given γ+/γ-, the observable exhibits convergence at N≳O(103), with the convergence point shifting to larger values of N as γ+/γ- moves closer to unity.

V.LOCAL NOISE EFFECTS

Our analysis so far has focused exclusively on the coherent interaction between the spin ensemble and the CνB. Realistic experimental setups, however, are inevitably subject to other, non-neutrino environmental interactions that relax the system, i.e., drive it to a state of thermal equilibrium, by locally exciting or deexciting the spins, as well as locally dephasing the spin precessions. Among these, local dephasing plays a particularly important role.

This form of noise originates from random, uncorrelated fluctuations acting independently on each spin, due to, e.g., spin-spin interactions and magnetic field fluctuations, and, phenomenologically, can be described by the operator γϕlocJzα, where α labels the individual spin. As illustrated in Fig. 2, the key feature of local dephasing is that it acts incoherently and forces the system to move into lower-j sectors where collective effects such as superradiance are suppressed or entirely lost. The relevant timescale is tϕ∼1/γϕloc, which is easily much shorter than the dissipative timescale tdiss from Eq. (27). Consequently, local dephasing can wash out coherent signatures induced by the neutrino-spin interaction, effectively setting a limit on the observable impact of the CνB.

In a purely phenomenological approach, let us assume that relaxation is dominated by local (single-particle) effects, i.e., we do not model spin-spin interactions. This simplification allows us to “repurpose” the Lindblad equation (20) in the incoherent regime to model the relaxation characteristics of a realistic experimental sample. Identifying γz in Eq. (20) with the local dephasing rate γϕloc and the rates γ± with the local emission and absorption rates γ±loc, we can tune γ±,ϕloc to match the experimental longitudinal and transverse relaxation times, T1 and T2, and the final equilibrium state.

Explicitly, starting from Eq. (20), we take the trace tr[ρSJi], with i=x, y, z, on both sides of the equation, in order to obtain an evolution equation for the collective magnetization ⟨Ji⟩ in the ith direction, ddt⟨Ji⟩=Nγ-loctr[JiJ-ρS(1/2)J+-12Ji{J+J-,ρS(1/2)}]+Nγ+loctr[JiJ+ρS(1/2)J--12Ji{J-J+,ρS(1/2)}]+Nγϕloctr[JiJzρS(1/2)Jz-12Ji{JzJz,ρS(1/2)}],(35)where J±,z,i are single-spin operators, and ρS(1/2) is a single-spin density matrix. Solving this equation analytically, we find for the longitudinal magnetization ⟨Jz(t)⟩=⟨Jz⟩eq-[⟨Jz⟩eq-⟨Jz⟩(0)]e-t(γ+loc+γ-loc),(36)with ⟨Jz⟩(0)=0,-N/2 for the |P⟩ and the |G⟩ state, respectively. The longitudinal magnetization tends to an equilibrium value of ⟨Jz⟩eq=N2γ+loc-γ-locγ+loc+γ-loc,(37)at a rate that can be matched to the longitudinal relaxation time T1 via T1-1=γ+loc+γ-loc.(38)Similarly, starting from the |P⟩ state, we find the magnetization in the transverse direction to be ⟨Jx(t)⟩=N2e-t(γ+loc+γ-loc+γϕloc)/2,(39)yielding a match T2-1=12(γ+loc+γ-loc+γϕloc)(40)to the transverse relaxation time T2.

Observe in Eq. (37) that the equilibrium magnetization ⟨Jz⟩eq depends only on the ratio γ+loc/γ-loc. This is a consequence of the detailed balance condition γ+loc/γ-loc=e-βω, which ensures that the ensemble approaches the well-known thermal steady state ⟨Jz⟩eq=N2tanh(βω02),(41)as long as all interaction channels couple the system to a thermal reservoir at the same temperature T=1/β. Equation (41) also highlights the inherent difficulty in achieving a strong initial polarization: even under cryogenic temperatures and extremely large magnetic fields, the polarization remains small. Reinstating all physical constants, the thermal polarization takes the form p=tanh(ℏγB2kBT)≈ℏγB2kBT≃10-3(B10 T)(4 KT),(42)where the numerical estimate assumes the gyromagnetic ratio γ of Xe129 nuclei. At O(10-3), this estimate demonstrates that thermal polarization alone does not suffice to reach the strongly polarized regime, necessitating the use of advanced techniques to hyperpolarize the spin ensemble.

As established in Eqs. (38) and (40), the local relaxation times, T1 and T2, depend on the absolute values of the local rates γ±,ϕloc. In typical solid-state NMR systems, the longitudinal relaxation timescale is much larger than the transverse relaxation timescale, i.e., T1≫T2. This implies γ±loc≪γϕloc, and the transverse relaxation time T2 is well-approximated by the effective dephasing time set by 1/γϕloc.¹¹

In standard NMR terminology, the experimentally observed dephasing time is denoted T2* and includes contributions from magnetic field inhomogeneities and chemical shifts. In contrast, T2 refers to the intrinsic dephasing time due to dipolar spin-spin interactions, and typically satisfies T2≫T2*. For our purposes, we do not distinguish between T2 and T2*, but adopt the convention T2=T2*=γϕ-1, which we always use to refer to the actual (measured) dephasing time.

Note however that in a generic (i.e., not specific to solid state) NMR setup, T1 can be as small as T2. In what follows, we model the γ±loc and γϕloc rates according to the ratio T1/T2, with the additional stipulation that the ratio γ+loc/γ-loc should reproduce the equilibrium magnetization achievable at cryogenic temperatures.

VI.NUMERICAL SOLUTIONS

Having discussed both coherent neutrino and incoherent local effects, we are now in a position to explore a more complete, numerical solution of a master equation that accounts for both effects. The strong separation of timescales established above, T2≪T1, implies that coherent effects such as those induced by neutrino-spin interactions are primarily limited by local dephasing. However, because our observable of interest, ⟨Jz⟩|P⟩, experiences longitudinal relaxation, we must also account for the effects of T1, particularly on the correlations of the signal between different times. Therefore, a minimum master equation that should capture the essential behaviors of the system would read dρSdt=γ-DJ-[ρS]+γ+DJ+[ρS]+γ-loc∑αDJ-α[ρS]+γ+loc∑αDJ+α[ρS]+γϕloc∑αDJzα[ρS]≡L{ρS(t)},(43)where L is the Lindblad superoperator of this system, and we remind the reader that γ± are the CνB-induced excitation and deexcitation rates, while γ±loc and γϕloc are the local absorption/emission and dephasing rates.

Given Eq. (43) and assuming the CνB-induced excitation and deexcitation rates are tunable by an overdensity factor δν, i.e.,¹²

The overdensity parameter δν is strictly speaking not an independent parameter, in the sense that other characteristics of the CνB must also change when δν is varied. For example, enhancement of δν due to local gravitational clustering generally results in a larger typical momentum [3]. For excessively large values of δν (e.g., ∼1011, such as the current KATRIN δν limit [34]), physically consistent modeling can in fact lead to a signal that is completely different from the δν∼O(1) case [68]. We nonetheless adopt the definition (44), first in order to facilitate comparison with constraints and forecasted sensitivities in the literature that have been derived under similar assumptions. Second, for extremely large values of δν, there is also no model-independent way to consistently relate δν to other CνB properties. Consequently, the interpretation of δν must be limited to a mere characterization of how away an experimental technique is from achieving detection, rather than a necessarily physically meaningful measure of the local CνB overdensity.

γ±≈δνγ±SM,(44)we can now identify three distinct dynamical regimes defined by the competition between collective neutrino-induced interactions and local dephasing, in the limit where T1≫T2.

(i)

Coherence-dominated regime: When γ±≫γϕloc≈T2-1, coherent effects dominate even at the single-spin level, keeping the dynamics well confined to the symmetric subspace. The system exhibits collective behaviors with enhancement scaling as ∼N2, providing optimal conditions for signal-to-noise amplification. This regime is depicted in blue in Fig. 5 and is unattainable in realistic experiments under realistic dephasing, unless δν is very large.

(ii)

Intermediate regime: In the case γ±≪γϕloc, but Nγ±≫γϕloc is still maintained, individual spins are dominated by local dephasing. Yet collective effects are stronger in the global dynamics. Thus, collective effects remain observable, even while symmetry is partially broken by local noise. This regime is depicted in Fig. 5 in green, highlighting a complete suppression of the coherent signal only at t∼γ±-1. The intermediate regime is also the most relevant experimentally for detecting weak collective signatures under realistic dephasing, although an enhanced δν≫O(1) will still be necessary.

(iii)

Dephasing-dominated regime: When Nγ±≪γϕloc, incoherent processes dominate over the collective interaction. In this regime, the system rapidly departs from the symmetric subspace, the signal-to-noise ratio saturates well below unity, and sensitivity to the neutrino–spin interaction is effectively lost. This region, shown in orange in Fig. 5, corresponds to the case of a standard CνB, where δν∼O(1) is the generic expectation.

10.1103/kgk5-bf5x.f5

FIG. 5.

Time evolution of collective spin variance ⟨Jx2⟩. We highlight three distinct regimes: (i) coherence-dominated (γ±≫γϕloc), where collective dynamics are preserved; (ii) intermediate (γ±≪γϕloc, but Nγ±≫γϕloc), where partial coherence remains; and (iii) dephasing-dominated (Nγ±≪γϕloc), where local noise suppresses collectivity. Time is scaled by Nγ-, and the spin variance is shown in units of N/4 for N=100.

In the following, we first solve Eq. (43) in Sec. VI A numerically by way of an efficient method that tracks explicitly the observables of interest: we call it the “second-order approximation” because of a truncation that needs to be applied at second order. This is to be compared to a complete numerical solution for the full density matrix ρS, whose details can be found in Appendix F. In Sec. VI B we apply the same method to investigate the evolution of the two-time correlation functions under Eq. (43), which will be used later in Sec. VII to form the covariance matrix that quantifies the theoretical uncertainties in the observables.

A.Second-order approximation

While it is possible to numerically solve the Lindblad master equation (43) at the level of the density matrix for a system of N spins, it can quickly become impractical or even intractable for values of N larger than about 100 (see Appendix F). Rather than dealing with the full density matrix, then, we shift the focus to tracking just a handful physically meaningful quantities.

Specifically, as was done in, e.g., Refs. [66,69], for the case of Dicke superradiance and also touched upon in Sec. V, we construct differential equations from the master equation (43) directly for the expectation values of the relevant observables, namely, the collective magnetization ⟨Jz⟩, ⟨Jz2⟩, and ⟨Jx2⟩. This is akin to tracking the moments of a continuous system, which is widely used in many different fields. However, differential equations thus constructed generally do not form a closed system, in the sense that the equation of motion for, e.g., ⟨Jz⟩ depends on a higher moment ⟨Jz2⟩, whose own equation of motion depends in turn on ⟨Jz3⟩, and so on, giving rise an infinite hierarchy of coupled equations. To solve the system, therefore, one needs to truncate the hierarchy at a chosen order [66].

Here, we opt to truncate the hierarchy at second-order. For the observables ⟨Jz⟩ and ⟨Jz2⟩, this means we adopt the closure conditions ⟨Jz3⟩≈⟨Jz⟩⟨Jz2⟩ and ⟨JzJ2⟩≈⟨Jz⟩⟨J2⟩ to arrive at [66] ddt⟨Jz⟩=∑±±γ±[⟨J2⟩-⟨Jz2⟩∓⟨Jz⟩]+∑±±γ±loc[∓⟨Jz⟩+N2],ddt⟨J2⟩=-∑±γ±loc[⟨J2⟩+⟨Jz2⟩∓(N-1)⟨Jz⟩-N]-γϕloc[⟨J2⟩-⟨Jz2⟩-N2],ddt⟨Jz2⟩=∑±γ±[⟨J2⟩-3⟨Jz2⟩∓⟨Jz⟩∓2⟨Jz⟩⟨Jz2⟩±2⟨Jz⟩⟨J2⟩]-∑±γ±loc[2⟨Jz2⟩∓(N-1)⟨Jz⟩-N2].(45)Note that our closure conditions are by no means unique and other choices exist. For example, closure can be achieved alternatively by demanding that all cumulants (i.e., connected pieces of the moments) vanish at third order and above, e.g., ⟨Jz3⟩=3⟨Jz2⟩⟨Jz⟩+2⟨Jz⟩3+⟨Jz3⟩c, where ⟨Jz3⟩c is the connected piece to be set to zero. We observe however that all closure-dependent terms in Eq. (45) are proportional to γnet and vanish in the limit γnet→0. Indeed, given the small γnet≪γ± in our scenario, we have verified that adopting a different closure condition (e.g., the cumulant truncation scheme mentioned above) has no discernible impact on our numerical solutions within in the time range of interest (i.e., t≲T2) even for extremely large N values.

1.Initial conditions including partial polarization

As discussed in Sec. III B, in experimental settings spin ensembles are prepared in a hyperpolarized configuration that approximates the ground state and, if need be, rotated to the desired orientation by applying an appropriate pulse. Common hyperpolarization protocols such as spin-exchange optical pumping achieve close-to-unit polarization by way of spin excitation and deexcitation processes that can be modeled as local interactions. See Appendix G for details. Thus, in order to establish the initial conditions of a partially polarized spin ensemble prepared in the aforementioned manner, we can simply take Eq. (45) assuming only the local rates γ±loc are nonzero, reinterpret these rates as the excitation and deexcitation rates γ±hyper of the hyperpolarization technique in use, and compute the corresponding steady-state solutions for ⟨Jz⟩ and ⟨Jz2⟩.

These solutions then serve as initial conditions for the partially polarized “ground state,” which read ⟨Jz(0)⟩|G⟩=Np2,⟨Jz2(0)⟩|G⟩=N4+N(N-1)4p2,⟨J2(0)⟩=3N4+N(N-1)4p2,(46)where we have used ⟨J2⟩=⟨Jx2⟩+⟨Jy2⟩+⟨Jz2⟩=N/2+⟨Jz2⟩, and p=γ-hyper-γ+hyperγ-hyper+γ+hyper(47)is the initial polarization of the ensemble expressed in terms of the rates of the hyperpolarization method. See also Eq. (37). If on the other hand we wish to initialize in a product statelike configuration, a π/2 rotation of the coordinates should be applied to Eq. (46) to get ⟨Jz(0)⟩|P⟩=0,⟨Jz2(0)⟩|P⟩=N4,(48)while ⟨J2⟩ remains the same under rotation. Then, together, Eqs. (45) and (46) or (48) form a closed set of equations that completely determine ⟨Jz⟩,⟨Jz2⟩, and ⟨J2⟩.

To obtain approximate numerical solutions for the transverse observable ⟨Jx2⟩, we supplement the differential equations (45) with an additional equation of motion for the non-Hermitian variables ⟨J±⟩≡⟨Jx⟩±i⟨Jy⟩, i.e., ddt⟨J±⟩=γ-⟨Jz⟩⟨J±⟩-γ+⟨Jz⟩⟨J∓⟩-12(γ-loc+γ+loc+γϕloc)⟨J±⟩,(49)where we have applied the closure conditions ⟨J+Jz⟩≈⟨J+⟩⟨Jz⟩ and ⟨JzJ-⟩≈⟨Jz⟩⟨J-⟩. Here, we are interested in a spin ensemble initially prepared in the ground state |G⟩, such that the initial conditions are ⟨J±(0)⟩|G⟩=0. Then, a solution for ⟨Jx2⟩ can be constructed from ⟨Jx2⟩=2⟨J2⟩-2⟨Jz2⟩+⟨J+⟩⟨J-⟩+⟨J-⟩⟨J+⟩,(50)where ⟨Jz2⟩, ⟨J2⟩, and ⟨J±⟩ are solutions of Eqs. (45) and (49).

Figures 6–8 show the numerical solutions obtained from the above equations of motion under different settings, which we discuss in more detail below.

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FIG. 6.

Time evolution of ⟨Jz⟩ from the initial state |P⟩, in the presence of neutrino collective interaction only, with γ+/γ-=0.95. The solid blue, green, and gray lines denote our numerical solutions of the Lindblad equation (43) using the second-order approximation (45) for N=102,103, and 1022. In the case of N=102,103 (blue and green), the second-order solutions are compared with the full density matrix solutions displayed in long-dashed lines of the same colors, and show excellent agreement. The perturbative solution (29), represented here by the red short-dashed line, reproduces the large N numerical solution well at t≲(Nγnet)-1.

2.The ideal case: No local noise and perfect polarization

We consider first the case in which there is no local noise and the polarization of the initial state is perfect, i.e., p=1. Figure 6 shows the solutions for the collective observable ⟨Jz⟩ under these settings for several choices of N. These are contrasted with numerical solutions of the full density matrix for those choices of N values where a full numerical solution is tractable. Details of the full density matrix solution can be found in Appendix F.

Remarkably, where the full density matrix solution can be found, the second-order solution is in excellent agreement with it at all times. This agreement follows from the small γnet≪γ± of our system, where, as discussed above, the hierarchy of Eq. (45) becomes exact and independent of the truncation scheme in the limit γnet→0, and indicates that the second-order method is indeed a reliable way to obtain predictions for the collective observables. Importantly, the method remains valid for arbitrarily large N values—N=1022 in this figure—at no additional cost in terms of computational runtime or memory relative to the small N case. In contrast, the computational cost of numerically solving the full density matrix scales as ∝N3, and becomes prohibitively expensive even on modern computers when N≳104. For completeness, we also show in the same figure the perturbative solution (29). Evidently, our numerical solutions are well approximated by the perturbative solution at t≲(Nγnet)-1, especially for large N values.

3.With local noise and partial polarization

Having established that the second-order method is reliable, we now use it to investigate more realistic cases in which local noise is present and the polarization of the initial state may only be partial.

Figure 7 illustrates the impact of nonzero dephasing on the observable ⟨Jz⟩ for a large system size N (red curve). As shown, dephasing suppresses the growth of ⟨Jz⟩ at late times. In this example, the suppression begins around the dissipative timescale tdiss due to our specific choice of local dephasing rate, γϕloc=Nγ-. Different choices for this rate would lead to alternative behaviors, as previously demonstrated in Fig. 5. While dephasing does not directly alter the dynamics of ⟨Jz⟩, it inhibits the coherent dissipation mediated by neutrino interactions. This dissipation then drives the system toward its own equilibrium value, given by ⟨Jz⟩=-(N/2)γnet/γtot in the limit T1→∞.

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FIG. 7.

Time evolution of ⟨Jz⟩ from the initial state |P⟩ in the presence of neutrinos and local interactions. The black curve shows the evolution with neutrinos only, as in Fig. 6. The red curve includes local dephasing effects with γϕloc=Nγ-, such that tdiss coincides with T2. In this case, dephasing suppresses the superradiant neutrino interaction, resulting in a neutrino-induced relaxation characterized by γ+/γ-=0.99. The green curve includes both dephasing and relaxation effects specified by γ+loc/γ-loc=0.995 and T1=10T2 (i.e., γ+loc+γ-loc≃0.05γϕloc). The latter effect leads to the equilibrium magnetization ⟨Jz⟩eq defined in Eq. (37). Finally, the blue curve represents the evolution of ⟨Jz⟩ in the absence of neutrinos. All solutions have been computed for N=103 using the second-order approximation.

Finite T1 effects are introduced via the local emission and absorption rates γ±loc in the master equation. We select a ratio γ+loc/γ-loc=0.995, corresponding to an equilibrium polarization of approximately peq∼10-3—a value achievable at cryogenic temperatures. The strengths of the local relaxation processes are chosen such that T1 is ten times longer than T2, and we allow this ratio to vary in the forecasting analysis in Sec. VII. The blue and green curves in Fig. 7 show the effects of local relaxation and dephasing, with and without superradiant neutrino interactions, respectively.

Figure 8 shows the variance σJz2 of Jz and the transverse observable ⟨Jx2⟩|G⟩. Here, in addition to demonstrating the effects of local dephasing, we also show how partial polarization affects the time evolution of the observables. Clearly, in the presence of dephasing, the variance σJz2 shows minimal growth unless p>0.1. In the case of ⟨Jx2⟩|G⟩, the polarization requirement is even stronger—p>0.3—although, as discussed in Sec. IV, we do not expect this observable to be competitive against ⟨Jz⟩|P⟩. This result therefore highlights the importance of polarization for the purpose of CνB detection. For instance, a liquid methanol-based setup [53], with a polarization of order p∼10-5, is currently insufficient for NMR experiments to target CνB effects. More generally, thermal polarization levels achievable at typical experimental conditions—O(10-3) for Xe129 from Eq. (42)—remain too low to yield detectable signals. Advanced polarization techniques are thus essential to this venture.

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FIG. 8.

Left: time evolution of σJz2 starting from the product state |P⟩ (green lines), and from a state with an initial 10/50% polarization in the z-axis (blue/red lines), with γ+/γ-=0.95. In all cases, we illustrate in dashed lines the impact of local dephasing at a rate γϕloc=Nγ-. Right: similar to σJz2, but for ⟨Jx2⟩ starting from the fully polarized ground state |G⟩ (blue lines), and from a state with 30% polarization in the z-axis (orange lines). We have set γ+/γ-=0.997, and assumed local dephasing at a rate γϕloc=Nγ- for the dashed lines.

B.Two-time correlation functions

We envisage a future experiment that measures an observable, e.g., ⟨Jz⟩|P⟩, at a series of time points. To estimate the sensitivity to the CνB overdensity parameter of such an experiment, besides the observable’s variance σ2(t) at each time t, we will also need its covariance σ2(t,t+τ) at two different times. This covariance can be constructed from the two-time correlation function, generally written as ⟨O1(t)O2(t+τ)⟩, where O1 and O2 are two operators that act only on the spins.

Following the quantum regression theorem [70], the two-time correlation function can be computed as ⟨O1(t)O2(t+τ)⟩≡tr[ρS;O1(t+τ)O2(0)],(51)where the modified density matrix ρS;O1 is defined at t via ρS;O1(t)=ρS(t)O1(0). The evolution of ρS;O1(t+τ) from t to t+τ for τ≥0 is governed by the same Lindblad master equation (43) used to compute our observables but for a change of the time variable t→τ, i.e., ddτρS:O1(t+τ)=L{ρS;O1(t+τ)},(52)where L is the Lindblad superoperator of Eq. (43). A more detailed discussion about this equation of motion and the construction of two-time correlation functions can be found in Appendix H.

Since our main observable is ⟨Jz⟩|P⟩, we are particularly interested in the two-time correlation function ⟨Jz(t)Jz(t+τ)⟩. Following the second-order method of Sec. VI A—or, equivalently, using the correspondence between Eqs. (H8) and (H9) in Appendix H—a hierarchy of differential equations can be constructed from Eq. (52) to describe its time evolution, namely, ddτ⟨Jz(t)Jz(t+τ)⟩=∑±±γ±[⟨Jz(t)J2(t+τ)⟩-⟨Jz(t)Jz2(t+τ)⟩∓⟨Jz(t)Jz(t+τ)⟩]+∑±±γ±loc[∓⟨Jz(t)Jz(t+τ)⟩+N2⟨Jz(t)⟩],ddτ⟨Jz(t)J2(t+τ)⟩=-∑±γ±loc[⟨Jz(t)J2(t+τ)⟩+⟨Jz(t)Jz2(t+τ)⟩∓(N-1)⟨Jz(t)Jz(t+τ)⟩-N⟨Jz(t)⟩]-γϕloc[⟨Jz(t)J2(t+τ)⟩-⟨Jz(t)Jz2(t+τ)⟩-N2⟨Jz(t)⟩],ddτ⟨Jz(t)Jz2(t+τ)⟩=∑±γ±[⟨Jz(t)J2(t+τ)⟩-3⟨Jz(t)Jz2(t+τ)⟩∓⟨Jz(t)Jz(t+τ)⟩∓2⟨Jz(t)Jz2(t+τ)⟩⟨Jz(t+τ)⟩±2⟨Jz(t)J2(t+τ)⟩⟨Jz(t+τ)⟩]-∑±γ±loc[2⟨Jz(t)Jz2(t+τ)⟩∓(N-1)⟨Jz(t)Jz(t+τ)⟩-N2⟨Jz(t)⟩],(53)where we have truncated the hierarchy by approximating ⟨Jz(t)Jz3(t+τ)⟩≈⟨Jz(t)Jz2(t+τ)⟩⟨Jz(t+τ)⟩ and ⟨Jz(t)Jz(t+τ)J2(t+τ)⟩≈⟨Jz(t)J2(t+τ)⟩⟨Jz(t+τ)⟩, with ⟨Jz(t+τ)⟩≡tr[ρS(t+τ)Jz]. We also remind the reader that the time variable here is τ, and the function ⟨J(t)⟩ which appears in all three equations above does not evolve with τ.

Equation (53) can be solved from any time point t in the duration of an experiment, given the “initial conditions” at τ=0, ⟨Jz(t)⟩, ⟨Jz2(t)⟩, ⟨J2(t)Jz(t)⟩, and ⟨Jz3(t)⟩. For a spin ensemble originally prepared in the |P⟩ state, these initial conditions amount at t=0 to ⟨Jz2(0)⟩|P⟩=N4,⟨Jz(0)⟩|P⟩=⟨J2(0)Jz(0)⟩|P⟩=⟨Jz3(0)⟩|P⟩=0.(54)For t>0, on the other hand, ⟨Jz(t)⟩ and ⟨Jz2(t)⟩ can be obtained directly from the solutions to Eq. (45), while we use the same solutions to approximate ⟨J2(t)Jz(t)⟩≈⟨J2(t)⟩⟨Jz(t)⟩ and ⟨Jz3(t)⟩≈⟨Jz(t)⟩⟨Jz2(t)⟩.

We are particularly interested in the covariance of the observable ⟨Jz⟩|P⟩ at different times, i.e., the connected piece of the two-time correlation function, defined as σJz2(t,t+τ)≡⟨Jz(t)Jz(t+τ)⟩-⟨Jz(t)⟩⟨Jz(t+τ)⟩=σJz2(t+τ,t),(55)which we will use in Sec. VII to characterize the theoretical uncertainties in the observable. This connected piece can be obtained from the solutions of Eqs. (45) and (53) for every pair of (t,t+τ). It is however also possible—and instructive—to construct an equation of motion directly for σJz2(t,t+τ) from Eqs. (45) and (53), which reads ddτσJz2=-γnet[⟨Jz(t)J2(t+τ)⟩-⟨Jz(t)⟩⟨J2(t+τ)⟩-⟨Jz(t)Jz2(t+τ)⟩+⟨Jz(t)⟩⟨Jz2(t+τ)⟩]-(γtot+γ+loc+γ-loc)σJz2(t,t+τ).(56)Where the ∝γnet terms are suppressed relative to the ∝(γtot+γ-loc+γ+loc) term, this equation of motion is solved approximately by σJz2(t,t+τ)=σJz2(t+τ,t)≈σJz2(t)e-(γtot+γ-loc+γ+loc)τ≈σJz2(t)e-(γ-loc+γ+loc)τ,(57)where we have used γ±≪γ±loc at the second equality, and σJz2(t)≡⟨Jz2(t)⟩-⟨Jz(t)⟩2 is the usual variance at time t also defined earlier in Eq. (32).

Figure 9 shows the time evolution of the two-time correlation function σJz2(t,t+τ), obtained from solving Eq. (53), for N=103 and t=0. In the case γ-loc=Nγ- (dark blue solid line), we see an exponential decay at a characteristic time τ∼1/(2Nγ-)≈1/(γ-loc+γ+loc), while in the absence of local emission/absorption the correlation decays at τ∼1/(2γ-)≈1/γtot due to neutrino dissipation alone, confirming the analytical estimate (57). Observe also that local dephasing effects do not influence the decay in the longitudinal correlation. This is primarily a consequence of the small γnet in our scenario, which inhibits the effects of γϕloc on the evolution of σJz2. Lastly, while Fig. 9 presents the numerical solutions for t=0, the two-time correlation for other choices of t exhibits the same decay behavior, as suggested by Eq. (57).

10.1103/kgk5-bf5x.f9

FIG. 9.

Time evolution of the covariance of Jz, i.e., the connected piece of the two-time correlation function, σJz2(t,t+τ), defined in Eq. (55), for t=0. The correlation decays away on a timescale specified by T1=1/(γ-loc+γ+loc) as predicted in Eq. (57), while the dephasing timescale T2 does not play a role. All solutions have been obtained from solving Eqs. (53) and (56) for N=103. We observe however that they agree with the full density matrix solution at the 0.1% level.

VII.FUTURE CONSTRAINTS ON THE <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mi>ν</mml:mi><mml:mi mathvariant="normal">B</mml:mi></mml:mrow></mml:math></inline-formula>

To assess the sensitivity of future experiments to collective effects of the CνB, we consider an idealized NMR experiment using polarized Xe129 nuclei, as envisioned in future stages of the CASPEr experiment [53]. Xe129 is a noble gas isotope with a nuclear spin of I=1/2, making it naturally suited to modeling as a spin-1/2 system. In liquid form xenon has a density of ρ=2.94 g/cm3, which gives nXe≃1.35×1022 cm-3(58)as the number density of xenon nuclei given an atomic weight of Ar(Xe)=131.29 [71]. Assuming 100% isotopic enrichment of Xe129 can be achieved, we use the same number to approximate the Xe129 spin density ns.

The gyromagnetic ratio of Xe129 is approximately γXe≃11.8 MHz/T, which gives a maximum energy splitting of ω0≃5.8×10-7 eV for a maximum magnetic field strength of Bmax=12 T. For a fixed neutrino mass, the CνB-induced net interaction rate γnet∝1-γ+/γ-∼2mνω0/pν2 generally increases with ω0, as also shown in Fig. 1. However, ω0 cannot be made arbitrarily large: for the typical momentum transfer qtyp∼ω0mν/pν, the coherence condition must eventually break down when qtyp reaches ∼R-1. Then, setting the coherence limit at qtypR<1 immediately yields an upper bound on the energy splitting ω0<pν/mνR. Thus, in our modeling, the optimal energy splitting is determined by the smaller of the maximum value achievable in the laboratory and the coherence limit, namely, ω0⋆=min(γXeBmax,pνm1R),(59)where we have set mν=m1. Assuming the normal hierarchy, this choice of mν gives the largest possible value of ω0⋆, leading to the best sensitivity to δν.

As already mentioned in Sec. III B, the preparation of the ground state, with all spins aligned along the magnetic field direction (i.e., the -z-direction in this work), can be achieved via spin-exchange optical pumping. This technique is commonly used to hyperpolarize noble gases like xenon [72,73] and can reach polarizations up to 90% [74]. Notably, hyperpolarized Xe129 exhibits exceptionally long transverse relaxation times, with values estimated to be up to T2∼1000 s [53,72]. After the nuclear spin ensemble has attained the desired polarization, a coherent spin (product) state can be prepared by applying a resonant π/2 radio-frequency pulse at the Larmor frequency. This effectively rotates the collective spin vector into the equatorial plane (see Fig. 2), thereby allowing us to use the ⟨Jz⟩|P⟩ observable to constrain collective neutrino effects. Under certain conditions, the longitudinal relaxation time T1 can also be long, e.g., T1≳ hours for solid-state Xe129 [75]. In the following we take T1/T2=10, which corresponds to a value T1≃2.8 hours for T2=1000 s, modeled according to Eqs. (38) and (40).

A.<inline-formula><mml:math display="inline"><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula> analysis

To estimate the sensitivity to the CνB overdensity parameter δν, we simulate a mock experimental dataset J^z(ti) of a fiducial model that does not contain the coherent neutrino signal,¹³

A precessing transverse magnetization can in principle induce a current in the detection coil and generate a feedback magnetic field that accelerates the decay of the magnetization. This effect is commonly known as radiation damping and may exhibit a collective behavior scaling. In this work, we adopt a fiducial model that does not include such collective background effects, which can be justified under certain conditions [76]. A full modeling of the experimental environment lies beyond the scope of this study.

comprising Nshots measurements at nt discrete time points sampled from t1=1/fs to tnt=T2. The sampling frequency is fixed to fs=14.3 kHz, in line with the most recent CASPEr setup [53]. Then, assuming the likelihood function (i.e., probability of data given a theory) to be Gaussian, we can define a χ2-statistics via χ2(δν)=Nshots∑ijntdi[C-1]ijdj,(60)where C is the covariance matrix to be defined below, and di≡⟨Jz(ti;δν)⟩|P⟩-⟨J^z(ti)⟩|P⟩(61)is theoretical prediction minus the mock data at time ti.

The theoretical prediction ⟨Jz(ti;δν)⟩|P⟩ includes neutrino effects tuned by the overdensity parameter δν. To ensure consistency, we take ⟨Jz(ti;δν)⟩|P⟩ from the numerical solutions of Sec. VI, with the time variable rescaled by a factor (2pνR)2 to account for partial coherence in the 2pνR>1 regime (note that the local rates γ±,ϕloc must also be simultaneously rescaled by the same factor in order to preserve their effects). For the mock data J^z(ti), on the other hand, we assume for simplicity that they coincide with the expectation value ⟨J^z(ti)⟩|P⟩ under the fiducial model (i.e., without neutrino effects), as given by Eq. (36) with γ-loc≃0.026γϕloc and γ+loc/γ-loc=0.995 (i.e., blue line in Fig. 7).

The covariance matrix Cij=Cji is given by Cij=Cijth+σSQUID2δij,(62)and includes contributions from the total instrumental noise σSQUID2, as well as the theoretical covariance of the model including neutrinos, Cijth={σJz2(ti;δν),i=j,σJz2(ti;δν)e-(γ-loc+γ+loc)|(tj-ti),i<j,σJz2(tj;δν)e-(γ-loc+γ+loc)|(ti-tj),i>j,(63)where σJz2(ti;δν) is the theoretical variance of Jz, and we have used the approximation (57) for the two-time correlation functions σJz2(ti,tj;δν) in the off-diagonal (i≠j) entries. Again, we take σJz2(ti;δν) from the numerical solutions of Sec. VI, with a rescaling factor (2pνR)2 applied to the time scale and the local rates γ±,ϕloc.

The total instrumental noise, which we refer to as “SQUID noise,” includes intrinsic SQUID contributions, thermal (Johnson-Nyquist) noise from the input circuit, as well as other noise sources originating from, e.g., the readout strategy. These noises are assumed to be random, uncorrelated between independent measurements, and uncorrelated with the signal. We model their total variance as σSQUID2=N4FSQUID,(64)so that the ratio FSQUID≈PSQUIDPSPN(65)quantifies the ratio of instrumental noise relative to the spin projection noise. The SQUID and SPN power spectral densities, PSQUID(f) and PSPIN(f), are specific to the experimental setup. For the CASPEr experiment [53], we estimate that FSQUID∼85.¹⁴

Using the specifications given in Ref. [53], we estimate that at f=1.2 MHz (corresponding to B=0.1 T for Xe129), the SQUID spectral density is approximately independent of frequency and takes on the value of PSQUID∼10-12Φ02/Hz. The SPN power is on the other hand PSPN∼1.2×10-14Φ02/Hz. Combining these numbers per Eq. (65) yields FSQUID∼85.

In the following analysis, however, we shall retain FSQUID as a free parameter.

B.Forecasted constraints

By construction, the χ2 of Eq. (60) is automatically zero when δν=0. Then, to determine an upper limit on δν we simply need to find the smallest value of δν that satisfies χ2(δν)>Δχ2,(66)where Δχ2=2.71(3.84) corresponds to a 90% (95%) confidence level for one degree of freedom.

We consider first the best possible constraint on δν. This can be achieved by assuming perfect polarization p=1 and an optimal splitting energy ω0⋆ as defined in Eq. (59). In this idealized scenario, we find that a 10 cm-radius spherical sample of Xe129 nuclei would be able to constrain the CνB overdensity at δν⋆≲1.7×1012,90% CL,(67)for the normally ordered ∑mν=0.15 eV benchmark scenario. The limit assumes a coherence time of T2=103 s, T1=10T2, and Nshots=103 along the lines of Ref. [42], and is comparable to the estimated sensitivity quoted therein (which assumes hydrogen spins) within an order of magnitude. At face value, such an optimistic sensitivity appears uncompetitive against the current KATRIN limit on the neutrino overdensity of δν≲1011 for the benchmark scenario (but may be marginally better for a small range of neutrino masses, depending on the ratio T1/T2; see Fig. 10). Note however that this conclusion may change if the system achieves at least an O(10) value in squeezing [76].

10.1103/kgk5-bf5x.f10

FIG. 10.

Projected sensitivity to the CνB overdensity parameter δν as a function of the lightest neutrino mass m1, assuming normal ordering. The two blue solid lines correspond to realistic experimental setups planned by the CASPEr experiment [53], with partial polarizations p=0.3 and p=1, a sample radius R=1 cm, a SQUID noise-to-SPN ratio FSQUID=50, and the fiducial value T1/T2=10. The blue dashed lines represent the sensitivity to δν of an idealized spin-projection-noise-limited experiment, working at the standard quantum limit, with perfect initial polarization (p=1) using a 10 cm-radius spherical sample of Xe129, for several choices of the ratio T1/T2. In all cases we assume an effective coherence time of T2=103 s, Nshots=103 repeated measurements, and an optimal energy splitting ω0⋆ according to Eq. (59) for all values of m1. All exclusion projections have been derived at 90% confidence level. We additionally show the regions excluded by the 90% CL KATRIN bounds on the CνB overdensity [34] (horizontal gray region) and the neutrino mass [35] (vertical gray region).

Realistic NMR experiments face even more limitations: imperfect initial polarization, smaller sample volumes, and instrumental noise. Assuming an experimental setup likely to be realized by CASPEr—instrumental-noise-limited and using a smaller sample volume of V∼R3∼cm3—the sensitivity to δν reduces significantly to δνCASPEr≲5.3×1013(0.5p)2,(68)at 90% confidence level, again for the normally ordered ∑mν=0.15 eV benchmark.

Figure 10 shows the dependence of the projected δν sensitivities on the lightest neutrino mass m1, assuming normal ordering. From m1≃0.05 eV up to the KATRIN neutrino mass limit <0.45 eV [35], future setups following the specifications of CASPEr will be able to set limits of δν≲1012–1014, depending on the SQUID noise floor and provided that the initial sample polarization reaches O(1) values. Because the sensitivity to δν increases with neutrino mass for m1≳0.05 eV, our forecasted CASPEr constraints are generally better than the benchmark value (68) for larger neutrino masses.

For m1≲0.05 eV, the mass dependence in the rates is lost, and the sensitivities to δν asymptote to δν∼1013. On the other hand, while the m1 dependence also drops out in the optimistic (R=10 cm) constraint at very small masses, we still observe a somewhat favorable scaling in the range m1∈(2×10-3,3×10-2) eV. This arises because, for a larger sample size R=10 cm, the optimal energy splitting can be tuned to higher values and only reaches the ceiling imposed by the maximum magnetic field Bmax at around m1∼10-3 eV. The effect of this ceiling can be clearly discerned in Fig. 10, where the optimistic sensitivity plateaus at a value of δν∼5×1010 for m1≲10-3 eV.

We display in Fig. 11 the parametric dependence of the experimental sensitivity to δν on the system size R and the transverse coherence time T2 in the SQUID-noise-dominate regime (i.e., FSQUID≫1). As can be seen, when the condition pνR<1 is satisfied, the system is in the fully coherent regime, where the interaction rates Γ± scale as ∝R6. Taking into account the scaling of the total uncertainty, σ∝N1/2∝R3/2, this leads to a scaling of R-9/2 for the δν sensitivity. This behavior applies to both a fixed energy splitting and the optimal case, since the optimal splitting here is effectively constant and determined by the maximum magnetic field achievable in the experiment. As the radius increases, coherence is eventually lost around R∼pν-1 except in the forward direction, resulting in a reduced interaction rate that scales as Γ±∝R4 and hence a δν constraint scaling of R-5/2. This scaling remains the same even at pνR≫1 in the case of a fixed energy splitting.

10.1103/kgk5-bf5x.f11

FIG. 11.

Parametric dependence of the experimental sensitivity to the CνB overdensity parameter δν. Left: dependence on the spin ensemble radius R. Here, we illustrate that, despite a weaker scaling at large R, choosing the optimal energy splitting ω0⋆, as given by Eq. (59) for a fixed radius R, yields a better sensitivity to δν than choosing a fixed splitting with B<1 T. See the main text for a detailed explanation of the scaling behaviors with R. Right: dependence on the transverse coherence time T2, for both a quantum-limited experiment and a SQUID-noise-dominated experiment with FSQUID=50.

In the case of optimal splitting, however, a change in the scaling behavior occurs around R∼0.5 cm for Xe129 spins. This transition arises because, at these radii, values of the magnetic field required to tune γBmax to the optimal value pν/(m1R) are achievable experimentally. As such, the energy splitting ω0 becomes dependent on R, which leads the net rate γnet≈(2m1ω0/pν2)γ± itself to directly scale with R-1 at the optimal ω0. Combined with the loss of coherence at pνR≫1, the collective net interaction rate here scales as ∝R3. Consequently, the sensitivity to δν scales as R-3/2 in this regime. Note that, had we used a different target material with a different gyromagnetic ratio, the transition in the scaling behavior would occur at a different value of R.

In the right panel of Fig. 11, we present the T2 scalings. In the standard quantum-limited scenario, where SQUID noise can be suppressed, the sensitivity to δν scales as T2-1. For a fixed total integration time t, this is equivalent to the standard sensitivity scaling of tT2. In contrast, when uncorrelated SQUID noise dominates the uncertainties, we observe a more favorable scaling of δν∝T2-3/2. In the case of T1, we find a scaling of δν∝(T1/T2)-1/2. Scaling with other parameters can be straightforwardly deduced from Eq. (60). Combining all information, we find that the sensitivity to δν in the SQUID-noise-dominated regime scales as δν∝ns-3/2T2-3/2Nshots-1/2p-2FSQUID1/2r,(69)where r denotes the R dependence from Fig. 11, which can be parametrized as r={R-9/2,R≲pν-1,R-5/2,R≳pν-1,if ω0⋆=γXeBmax,R-3/2,R≳pν-1,if ω0⋆=pν/(mνR).(70)This shows, as expected, that the sensitivity to δν improves for larger sample sizes and higher spin densities, as well as longer coherence and measurement times. The sensitivity is, however, expected to degrade quadratically with imperfect polarization and is further limited by instrumental noise.

Finally, it is worth mentioning that one could also constrain δν using the second-order observable ⟨Jx2⟩|G⟩. While this observable has the advantage of probing a CνB-induced signal without requiring spin rotation into the equatorial plane, its signal-to-noise ratio scales as ∼γtotN, as pointed out in Sec. IV. As a result, it provides a weaker sensitivity to δν compared to ⟨Jz⟩|P⟩ for spin ensembles with N≳(γtot/γnet)2≃106. Indeed, even in the optimistic R=10 cm scenario, the sensitivity of ⟨Jx2⟩|G⟩ to the CMB overdensity is only δν≳1020, about ten orders of magnitude weaker than what can be achieved with ⟨Jz⟩|P⟩.

VIII.CONCLUSIONS

We have investigated in this work the dynamics of large spin ensembles interacting with the cosmic neutrino background, with a particular focus on identifying the conditions under which coherent collective effects—scaling as N2, where N is the number of spins—can amplify the signal from an otherwise extremely weak neutrino-spin coupling. Such collective neutrino-spin interactions are expected to contribute to dissipating a spin ensemble and can potentially be exploited as a competitive way to set limits on the local CνB overdensity, as first put forward in Ref. [42].

After briefly reviewing the interaction rates of the CνB with a system of N two-level systems (e.g., spin-1/2 particles) and the Lindblad master equation in the open quantum system framework, we reproduced the perturbative solutions presented in Ref. [42], valid when the spin ensemble has not deviated too much from its initial state. We then extend the analysis by considering steady-state analytical solutions as well as numerical simulations of the Lindblad master equation for different initial spin configurations. Notably, our numerical simulations account for realistic noise sources that can degrade the collective CνB signal, such as local dephasing as well as imperfect initial spin polarization. We make use of the Dicke basis—illustrated in Fig. 2—to facilitate this analysis, which provides an intuitive representation of how collective and local interactions influence the spin ensemble’s evolution under the Lindblad master equation.

We have furthermore presented a fast and computationally inexpensive method to derive approximate numerical solution from the Lindblad equation. Specifically, the method solves directly for the observables and correlation functions of interest while capturing both coherent neutrino effects and local interactions, and is able to reproduce the full density matrix solution to O(0.01)% accuracy. This fast method enables us to use statistical inference methods to forecast the sensitivity of future NMR experiments to the CνB overdensity, while incorporating the complete quantum evolution of the spin ensemble.

Considering the projected experimental specifications of the forthcoming CASPEr experiment [51–53] and nuclear-spin-based experiments more generally, we compute future experimental sensitivities to the CνB overdensity parameter δν using a χ2 analysis. We find that achievable near-future setups along the lines of CASPEr could constrain the CνB overdensity parameter at the level of δν∼1013 for neutrino masses currently within the KATRIN neutrino mass bound [35], with larger masses yielding a more favorable sensitivity. Notably, in addition to the scaling with the sample volume and the obvious dependence on instrumental noise, we find that the initial polarization p of the spin ensemble plays a prominent role in our projected sensitivities: scaling as δν∝p-2, a competitive constraint requires at least p∼0.25. This requirement also means that thermal polarization levels achievable at typical experimental conditions—p∼10-3 for Xe129 at T∼4 K and B∼10 T—are not suited to CνB detection. Advanced polarization techniques are needed to reach the desired p≳0.25.

In optimistic experimental scenarios—with a sample size up to R=10 cm, and 100% polarization—the constraint may improve to δν∼1011 for a small range of neutrino mass values. This number is largely consistent with the estimate of Ref. [42], and also at face value comparable to the current KATRIN limit on the CνB overdensity [34].

Our results indicate that CνB direct detection is likely beyond the reach of current NMR technology, even in the most optimistic scenario. Indeed, even meeting the “realistic” projected δν constraints represent significant experimental challenges. We emphasize however much effort is currently being devoted to overcoming these challenges—in the case of CASPEr, in the context of dark matter axion searches. Should the desired experimental configurations materialize, we have shown that, by means of a pulsed-NMR measurement such as performed recently in [53], the potential exists for such axion searches to simultaneously set potentially competitive constraints on the CνB for free. Constraining the CνB with nuclear spin experiments therefore remains an attractive possibility.

ACKNOWLEDGMENTS

We thank Caterina Braggio, Giovanni Carugno, Jan Hamann, Alex Sushkov, and Oleg Sushkov for enlightening discussions, and Marios Galanis for insightful comments on the manuscript. Y. G., G. P., and Y. Y. Y. W. are supported by the Australian Research Council’s Discovery Project (DP240103130) scheme.

DATA AVAILABILITY

The data that support the findings of this article are openly available at [77].

APPENDIX A:LOW ENERGY NONRELATIVISTIC SPIN HAMILTONIAN

We outline the derivation of the nonrelativistic Hamiltonian for the neutrino-fermion spin interaction. We begin with the Lagrangian for a fermion Ψ in the presence of an external electromagnetic 4-potential Aμ=(A0,A→) and a 4-fermi interaction with a neutrino νi in the mass basis, L=Ψ¯(i∂-m)Ψ-qΨ¯γμΨAμ-∑ijGF2Ψ¯γμ(gVi,j-gAi,jγ5)Ψν¯iγμ(1-γ5)νj,(A1)where i, j label the neutrino mass eigenstate, and q is the electric charge of the fermion. The corresponding equation of motion is (iγμ∂μ-m-qγμAμ-∑ijGF2γμ(gVi,j-gAi,jγ5)ν¯iγμ(1-γ5)νj)Ψ=0,(A2)or, equivalently, ((p0γ0-qA0γ0)-γ→.(p→-qA→)-∑ijGF2(gVi,j-gAi,j)γ0Σ0i,j+∑ijGF2(gVi,j-gAi,j)γ→·Σ→i,j)Ψ=0,(A3)where we have used p→=-i∇→, and p0=i∂t, and defined Σμi,j≡ν¯iγμ(1-γ5)νj.

We write the fermion spinor in 2-component notation, Ψ=[ϕ0χ0],(A4)such that in the nonrelativistic limit, we can make the ansatz Ψ(x→,t)=[ϕ0χ0]e-imt,(A5)which yields the solution χ0≈[σ→·(p→-qA→)/(2m)]ϕ0, where m is the mass of the fermion. Substituting this solution back into the equation of motion (A3) in 2-component form, we find a Schrödinger-Pauli-like equation with additional neutrino-fermion interaction terms, ((p→-qA→)22m-q2mB→·σ→+qA0+∑ijGF2gVi,jΣ0i,j+∑ijGF2gAi,jΣ→i,j·σ→)ϕ0=Eϕ0,(A6)where B→=∇→×A→ is the external magnetic field, and p0=E is the energy.

In the form (A6), the total Hamiltonian can be immediately identified with the terms in parentheses on the left-hand side of the equation. We concentrate exclusively on the spin interaction of the fermion with the CνB. Assuming the fermion (spin) is at rest, this interaction is described by the spin Hamiltonian, H=-μ→·B→+2GF∑ijgAi,jJ→·Σ→i,j,(A7)where we have used J→=σ→/2, and the magnetic moment is defined as μ→=(q/m)J→. Although this result has been obtained assuming a charged fermion, an analogous spin-magnetic field interaction is also present for neutral particles like the neutron or nuclei. Here, the interaction of a magnetic field with the magnetic moment due to the particle’s internal structure can be described by an effective Lagrangian of the form ψ¯σμνψFμν. In the nonrelativistic limit, this interaction term also reduces to a Hamiltonian of the form H=-μ→·B→. For nuclei, the convention is to define μ→=gμNJ→, where g is the g-factor of the nucleus under consideration, and μN=e/2mp is the nuclear magneton.

Supposing without any loss of generality that the magnetic field points along the negative z-axis, i.e., B→=-Bz^, the spin Hamiltonian can be recast into a more suggestive form, H=ω0Jz+ΣzJz+Σ-J-+Σ+J+,(A8)where the Larmor frequency ω0=γB is also the energy splitting between two adjacent spin states, with γ the gyromagnetic ratio, and we have defined J±≡J1±iJ2, and Σz=2GF∑ijgAi,jν¯iγ3(1-γ5)νj,Σ±=GF2∑ijgAi,jν¯i(γ1∓iγ2)(1-γ5)νj.(A9)Then, splitting the Hamiltonian into a spin ensemble and an interaction term as H=HS+HI,(A10)we identify HS≡ω0Jz, HI≡ΣzJz+Σ-J-+Σ+J+, and HII(t)=eiHstHIe-iHst=ΣzJz+e-iω0tΣ-J-+eiω0tΣ+J+(A11)gives the interaction Hamiltonian in the interaction picture.

In the case of a system with N spins, the Hamiltonians of the spin ensemble and the interaction can be straightforwardly generalized from Eqs. (A10) and (A11) to HS=ω0∑αJzα,HII(t)=∑αΣzαJzα+e-iω0t∑αΣ-αJ-α+eiω0t∑αΣ+αJ+α,(A12)where the operators J±,zα act only on the spin α at the spatial position x→α, and Σ±,zα are defined as per Eq. (A9) but with the understanding that the neutrino wave functions must now be specified as νi(t,x→α).

APPENDIX B:EXCITATION AND DEEXCITATION RATES1.Single spin

Consider first a single spin immersed in the cosmic neutrino bath. Following Fermi’s golden rule the excitation and deexcitation rates are given by γ±=|Tfi±|22πδ(1)(Ef-Ei)ρ±(Ef),(B1)where the superscripts “+” and “-” refer respectively to excitation and deexcitation, |Tfi+|2 is the transition probability from the initial state |i⟩=|↓⟩⊗|iν⟩ to the final state |f⟩=|↑⟩⊗|fν⟩, while |Tfi-|2 is its counterpart for the transition |i⟩=|↑⟩⊗|iν⟩ to |f⟩=|↓⟩⊗|fν⟩. The quantity ρ(Ef) is the density of states at the final energy Ef, which in our case can be written as, ρ±(Ef)=∑jk∑a,c∫d3p→(2π)3d3p→′(2π)3d3n→(2π)3(2π)3δ(3)⁢(p→-p→′∓n→)PiνPfν,(B2)where p→ and p→′ are the 3-momenta of the incoming and outgoing neutrinos respectively, n→ is the momentum of the final-state fermion (assuming it is initially stationary), Piν and Pfν represent respectively the thermal weight of the initial-state neutrino and any Pauli blocking factor associated with the final state, j and k are mass indices of the initial- and final-state neutrinos, while a and c index their helicities. Since the recoil of the fermion is expected to be tiny, the angle of the outgoing neutrino is essentially unrestricted. We therefore proceed to use the Dirac delta in Eq. (B2) to eliminate the integral over the nuclear momentum, to arrive at ρ±(Ef)=∑jk∑a,c∫d3p→(2π)3d3p→′(2π)3PiνPfν,(B3)where it is now understood that the relation between |p→| and |p→′| is determined by energy conservation alone.

At first order in perturbation theory (i.e., the Born approximation), the transition amplitude is given by Tfi±=⟨f|Σ±J±|i⟩.(B4)Focusing on excitation, consider an initial-state neutrino specified by a wave function νj(t,x→)=uj,a(p→)e-i(p0t-p→·x→),(B5)where a is a helicity index. Suppose it scatters off a spin inelastically, thereby transferring energy ω0 to it, and then exits the interaction in a final-state plane wave given by νk(t,x→)=uk,c(p→′)e-i(p′0t-p→′·x→). This process is represented by the leftmost diagram in Fig. 12, and its transition probability can be written as |Tfi+|2=|⟨↑|J+|↓⟩|2|⟨fν|Σ+|iν⟩|2=GF22gAj,kgAk,j12p02p′0⁢|u¯k,c(p→′)(γ1-iγ2)(1-γ5)uj,a(p→)|2,(B6)where we have used |⟨↑|J+|↓⟩|2=1 at the second equality, and the factors of 1/2p0 originates from the normalization of the u-spinors to 2p0. The thermal weight factors in Eq. (B2) are then Piν=Nj,a(p0) and Pfν=1-Nk,a(p′0), where Ni,a(p0) is the neutrino occupation number, while the Dirac delta δ(1)(Ef-Ei) becomes δ(1)(ω0+p′0-p0), enforcing the change in energy between the initial- and final-state neutrinos to be equal to ω0, the energy splitting due to the external magnetic field. A similar calculation can also be performed in the case of an antineutrino exciting a spin, as represented by the middle diagram in Fig. 12.

10.1103/kgk5-bf5x.f12

FIG. 12.

Feynman diagrams of the processes that contribute to the excitation of a spin in a 4-fermi effective interaction.

Putting it altogether, we then find the excitation rate of one spin to be γ+=GF2π∑jkgAj,kgAk,j∑ac∫dp˜∫dp˜′δ(1)(ω0+p′0-p0)×[(1-Nk,c(p′0))Nj,a(p0)u¯k,c(p→′)(γ1-iγ2)(1-γ5)uj,a(p→)u¯j,a(p→)(γ1+iγ2)(1-γ5)uk,c(p→′)+(1-N¯k,c(p′0))N¯j,a(p0)v¯j,a(p→)(γ1-iγ2)(1-γ5)vk,c(p→′)v¯k,c(p→′)(γ1+iγ2)(1-γ5)vj,a(p→)],(B7)where we have defined the phase space measure dp˜≡d3p→/((2π)32p0). Ignoring the relative motion between the spin and the CνB, and assuming the neutrino occupation numbers to be independent of helicity, the expression simplifies to γ+=8GF2π∑jkgAj,kgAk,j∫dp˜∫dp˜′δ(1)(ω0+p′0-p0)p0p′0[(1-Nk(p′0))Nj(p0)+(1-N¯k(p′0))N¯j(p0)],(B8)which, using the Dirac delta, can be further reduced to γ+=4GF2(2π)3∑jkgAj,kgAk,j∫(ω0+mk)2-mj2∞d|p→||p→|2pk+mk2+pk+2[(1-Nk(pk+0))Nj(p0)+(1-N¯k(pk+0))N¯j(p0)],(B9)with pk+=[(|p→|2+mj2-ω0)2-mk2]1/2, and pk+0=mk2+pk+2.

Computation of the deexcitation rate from inelastic neutrino-spin scattering proceeds in a similar fashion, and yields γ-=GF2π∑jkgAj,kgAk,j∑ac∫dp˜∫dp˜′δ(1)(ω0+p0-p′0)×[(1-Nk,c(p′0))Nj,a(p0)u¯k,c(p→′)(γ1+iγ2)(1-γ5)uj,a(p→)u¯j,a(p→)(γ1-iγ2)(1-γ5)uk,c(p→′)+(1-N¯k,c(p′0))N¯j,a(p0)v¯j,a(p→)(γ1+iγ2)(1-γ5)vk,c(p→′)v¯k,c(p→′)(γ1-iγ2)(1-γ5)vj,a(p→)].(B10)Again, ignoring any neutrino-spin relative motion and assuming helicity-independent neutrino occupation numbers, we find γ-=8GF2π∑jkgAj,kgAk,j∑ac∫dp˜∫dp˜′δ(1)(ω0+p0-p′0)p0p′0[(1-Nk(p′0))Nj(p0)+(1-N¯k(p′0))N¯j(p0)]=4GF2(2π)3∑jkgAj,kgAk,j∫(ω0-mk)2-mj2∞d|p→||p→|2pk-mk2+pk-2[(1-Nk(pk-0))Nj(p0)+(1-N¯k(pk-0))N¯j(p0)],(B11)with pk-=[(|p→|2+mj2+ω0)2-mk2]1/2, and pk-0=mk2+pk-2. The lower integration holds if it evaluates to a real number. Otherwise, it needs to be set to zero.

Suppose for simplicity mj=mk=m and ω0≪m. The lower integration limits of the γ+ and γ- integrals, Eqs. (B9) and (B11), are (ω0+m)2-m2≈2mω0 and zero, respectively. We can take the former to zero too if ω0≪|p→|2/(2m)≪m is satisfied by the bulk of the neutrino distribution. The same condition also allows us to approximate pk±≈|p→|2∓2mω0≈|p→|∓mω0/|p→|. Then, to leading order, we find γ+≈γ-∝∫0∞d|p→||p→|2pk+m2+pk+2[N(p0)+N¯(p0)]≈∫0∞d|p→||p→|2m|p→|[N(p0)+N¯(p0)],(B12)and γnet≡γ--γ+∝∫0∞d|p→||p→|2[pk-m2+pk-2-pk+m2+pk+2][N(p0)+N¯(p0)]≈∫0∞d|p→||p→|22m2ω0|p→|[N(p0)+N¯(p0)],(B13)where we have omitted for brevity the Pauli-blocking terms in both expressions. Thus, we see immediately that γ+<γ-, and γnet is suppressed by a factor ∼2mω0/|p→|2 relative to γ±.

2.<inline-formula><mml:math display="inline"><mml:mi>N</mml:mi></mml:math></inline-formula> spins

Consider now a system of N spins described by the interaction Hamiltonian (A12). We are interested in the excitation and deexcitation rates of the whole spin ensemble, which can be defined via Γ±=∑fspin|Tfi±|22πδ(1)(Ef-Ei)ρ±(Ef),(B14)where Tfi±=∑α⟨fspin|J±α|ispin⟩⟨fν|Σ±α|iν⟩(B15)are the excitation and deexcitation amplitudes of the N-spin ensemble from an initial state |ispin⟩ to an unspecified final state |fspin⟩. The summation over fspin encompasses a complete basis of final states; the ladder operators will project out the final states that have one unit of energy ω0 more (+) or less (-) than the initial state.

Again, focusing first on spin excitation due to inelastic scattering of a plane-wave neutrino from νj(t,x→α)=uj,a(p→)e-i(p0t-p→·x→α) to νk(t,x→β)=uk,c(p→′)e-i(p′0t-p→′·x→β), we find |Tfi+|2=∑αβ⟨fspin|J+α|ispin⟩⟨ispin|J-β|fspin⟩×GF22gAj,kgAk,j12p02p′0|u¯k,c(p→′)(γ1-iγ2)(1-γ5)uj,a(p→)|2e-i(p→′-p→)·(x→α-x→β).(B16)A similar expression can be found for |Tfi-|2. Then, following the same procedure as for a single spin, it is straightforward to arrive at the collective excitation and deexcitation rates Γ+=2GF2(2π)3∑jkgAj,kgAk,j∫(ω0+mk)2-mj2∞d|p→||p→|2pk+mk2+pk+2[(1-Nk(pk+0))Nj(p0)+(1-N¯k(pk+0))N¯j(p0)]×∫-11dμF+(p→k+-p→),Γ-=2GF2(2π)3∑jkgAj,kgAk,j∫(ω0-mk)2-mj2∞d|p→||p→|2pk-mk2+pk-2[(1-Nk(pk-0))Nj(p0)+(1-N¯k(pk-0))N¯j(p0)]×∫-11dμF-(p→k--p→),(B17)where μ≡p^′·p^=cosθp→p→′, with θp→p→′ the angle between the incoming and outgoing momenta, and F±(p→′-p→)≡∑fspin|⟨fspin|∑αJ±αe-i(p→′-p→)·x→α|ispin⟩|2(B18)are form factors that depend on the initial state of the whole N spin ensemble. Since the operator ∑αJ±α already projects the initial state into the subspace of final states that has the correct energy requirements, we can use the completeness relation of all possible final states ∑fspin′|f′spin⟩⟨f′spin|=1 to simplify the form factors to F±(p→′-p→)=⟨ispin|∑βJ∓βe+i(p→′-p→)·x→β⁢∑αJ±αe-i(p→′-p→)·x→α|ispin⟩,(B19)where it is understood that F±(p→′-p→) are to be evaluated for p→′=p→k+ or p→′=p→k-.

Suppose the spin ensemble is initially in the ground state, i.e., |ispin⟩=|↓1,↓2,…,↓γ,…,↓N⟩. We see immediately that the deexcitation form factor must always yield FG(p→′-p→)=0, while the excitation form factor evaluates to F+G(p→′-p→)=⟨↓1,↓2,…,↓N|∑βJ-βe+i(p→′-p→)·x→β⁢∑αJ+αe-i(p→′-p→)·x→α|↓1,↓2,…,↓N⟩=N,(B20)irrespective of the momentum transfer. Thus, in this case there is no coherent effect from the neutrino-spin interaction. If, on the other hand, the spin ensemble is initially in the product state |ispin⟩=|→1,→2,…,→γ,…,→N⟩, where |→⟩≡(|↑⟩+|↓⟩)/2, then the excitation and deexcitation form factors evaluate to F±P(p→′-p→)=⟨→1,→2,…,→N|∑βJ∓βe+i(p→′-p→)·x→β⁢∑αJ±αe-i(p→′-p→)·x→α|→1,→2,…,→N⟩=N2+14∑α,β≠αe-i(p→′-p→)·(x→α-x→β),(B21)which features possible coherent effects in the second term. When the spatial separation between spins |x→α-x→β| is much smaller than V1/3, where V is the volume of the sample, we can take the continuum limit and, for a constant spin density ns=N/V, approximate the last expression as F±P(p→′-p→)=N2+N24V2|∫d3xei(p→′-p→)·x→|2=N2+N249|p→′-p→|2R2|j1(|p→′-p→|R)|2,(B22)where j1(x) is the spherical Bessel function of order one, and we have assumed at the second equality the spin sample to be a sphere of radius R. Clearly, the second, N2 term dominates if |p→-p→′|R≪1, with |j1(x)|2/x2→1/9 as x→0. In this limit, the system exhibits coherent behaviors.

Given the typical momentum of the neutrino bath p→ν, in order for the coherence condition to hold over the whole length R of the spin sample, we must demand |p→ν-p→′(p→ν)|R≪1. Assuming for simplicity mi=mj=m, the typical momentum transfers during excitation and deexcitation are |p→ν-p→′(p→ν)|±2=pν2+(pν2+m2∓ω0)2-m2-2pν[(pν2+m2∓ω0)2-m2]1/2⁢cosθp→p→′,(B23)with pν≡|p→ν|. If we want coherence over the whole object, we must demand that |p→ν-p→′(p→ν)|2≪1/R2 even for cosθp→p→′=-1. For nonrelativistic neutrinos, where pν2+m2≈m+pν2/(2m), this yields the condition pν2+(m+ε±)2-m2+2pν(m+ε±)2-m2≪1/R2,(B24)or, equivalently, pν+(m+ε±)2-m2≪1/R,(B25)with ε±≡pν2/(2m)∓ω0. This in turn leads to pν[1+1∓ω02mpν2]≪1/R,(B26)when expanded to linear order in ε±.

From Eq. (B26), we see that excitation can only happen if ω0<pν2/(2m). But, if excitation happens at all, the coherence condition is also automatically fulfilled for any incoming momentum satisfying pνR≪1. On the other hand, while deexcitation has no kinematic threshold, coherence also does not come for free. Rather, in order to have coherent deexcitation, the condition ω0≪pν2/(2m) must be satisfied if pνR≪1. Physically, this means that the energy splitting must be much smaller than the incoming neutrino kinetic energy. If this is not the case, the momentum transfer of the neutrino is large and coherence is lost. For a typical CνB momentum of pν∼5.3×10-4 eV∼27 cm-1 and a neutrino mass of m=0.05 eV, this condition amounts to demanding that ω0≪3×10-6 eV.

For typical spin samples of size R∼1 cm, the condition pνR≪1 cannot be realized by the CνB across the whole sample volume. However, even in the case of pνR≫1, coherence is still possible under limited conditions. Specifically, assuming ω0≪pν2/(2m), Eq. (B23) evaluates to |p→ν-p→′(p→ν)|2≈4pν2sin2θp→p→′2.(B27)Demanding |p→ν-p→′(p→ν)|2R2≪1, we see that coherence can be achieved even in the case pνR≫1 for small angles θp→p→′≪1/(pνR) between the incoming and outgoing neutrino momenta. Under this condition, we can approximate the coherent part of the form factors as ∫-11dμF±P(p→′-p→)≈N24∫0(pνR)-1dθp→p→′sinθp→p→′≈N2412(pνR)2.(B28)In other words, the collective excitation and deexcitation rates Γ± in the pνR≫1 regime are suppressed by a factor (2pνR)-2 relative to the fully coherent rates. Evaluating Eq. (B23) for the momentum transfer squared q2≡|p→ν-p→′(p→ν)|2 in the forward direction (i.e., cosθp→p→′=1) and expanding in ε± as above, it is straightforward to show that the momentum transfer in this regime is q∼ω0m/pν.

APPENDIX C:DERIVATION OF THE LINDBLAD EQUATION

The Lindblad formalism assumes unitary dynamics of the total neutrino-spin system described by a Hamiltonian that can be decomposed into three parts, H=HS+HB+HI.(C1)Here, HS is the free Hamiltonian of the spin ensemble, HB the free Hamiltonian of the neutrino bath, and HI specifies the interaction between the spin ensemble and the neutrino bath. The density matrix ρ acts on the Hilbert space of the whole neutrino-spin system, and we assume it can be split into a direct product of the spin-system density matrix ρS and the bath density matrix ρB, i.e., ρ=ρS⊗ρB.(C2)Such a separation can be made in the limit of weak system-bath interactions and large baths (relative to the spin ensemble), where no significant correlations develop between the spin ensemble and the bath. This is the so-called Born approximation.

The dynamics of the total neutrino-spin system is governed by the Liouville-von-Neumann equation, dρdt=-i[H,ρ],(C3)where the density matrix ρ is understood to be in the Schrödinger picture. In the interaction picture this equation can be equivalently written as dρIdt=-i[HII,ρI],(C4)where HII is given by Eq. (A11), and has the formal solution ρI(t)=ρI(0)-i∫0t[HII(s),ρI(s)]ds.(C5)Plugging (C5) into (C4) and doing a partial trace over the bath, we arrive at an equation of motion for the spin ensemble alone that reads dρSdt=-itrB{[HI(t),ρ(0)]}-∫0ttrB{[HI(t),[HI(s),ρ(s)]]}ds,(C6)where we have omitted the interaction label (superscript “I”) for notational simplicity.

Observe that, implicit in the interaction Hamiltonian HI, the first term is formally O(GF), while the second term is O(GF2), and repeatedly plugging Eq. (C5) into (C6) would yield in principle a perturbative series in GF. Considering however that the interactions of interest are very weak, it suffices to iterate only once to capture the leading-order effects. Physically, the first, O(GF) term in Eq. (C6) corresponds to the so-called Stodolsky effect [19], which can be most easily seen in the Schrödinger picture where it manifests as a shift in the energy levels of the spin ensemble, provided that a nonzero neutrino-antineutrino asymmetry is present. We shall therefore not be concerned with this shift and will henceforth drop the O(GF) term.

To discern the role of the second, O(GF2) term in Eq. (C6), we apply the Markovian approximation, which is valid under two assumptions: (i) The bath’s correlation time is much shorter than the spin ensemble’s relaxation time. This ensures the bath quickly loses memory of the interaction, thereby preventing information from flowing back to the spin ensemble; and (ii) the bath does not change much over time. This is the case for the relatively weak interactions under consideration. Implementing the Markovian approximation consists formally in changing ρ(s)→ρ(t) on the right-hand side of Eq. (C6), i.e., dρSdt=-∫0ttrB{[HI(t),[HI(s),ρ(t)]]}ds,(C7)such that any step taken by the spin ensemble at any time t depends only on the state of the system at that time. Equation (C7) is the so-called Redfield equation. Next, we define a new integration variable s′=t-s and take the integration limit to ∞ (or, equivalently in terms of the original integration variable, we take the lower limit to s→-∞). This approximation leads us to dρSdt=-∫0∞trB{[HI(t),[HI(t-s),ρ(t)]]}ds,(C8)and corresponds to assuming that the correlations decay fast enough that the dynamics of the system remains the same irrespective of what happened at earlier times.

At this point we can simply evaluate Eq. (C8) for the Hamiltonian (A11). A brute-force evaluation will yield an intractably long expression. However, because the neutrino-spin interaction rate is expected to be much smaller than the Larmor frequency ω0=gμNB (i.e., many oscillations can occur over a timescale dictated by the interaction rate), we may adopt the so-called rotating-wave/secular approximation, which amounts to neglecting fast-oscillation terms of the form e±iω0t or e±2iω0t. Doing so yields dρSdt=(J-ρSJ+-J+J-ρS)Γ+-(2)(-ω0)+(J+ρSJ--ρSJ-J+)Γ-+(1)(-ω0)+(J+ρSJ--J-J+ρS)Γ-+(2)(ω0)+(J-ρSJ+-ρSJ+J-)Γ+-(1)(ω0)+(JzρSJz-JzJzρS)Γzz(2)(0)+(JzρSJz-ρSJzJz)Γzz(1)(0),(C9)where Γαβ(1)(ω0)≡∫0∞e-iω0strB{Σα(t-s)Σβ(t)ρB}ds,Γαβ(2)(ω0)≡∫0∞e-iω0strB{Σα(t)Σβ(t-s)ρB}ds.(C10)Note that Σα is an operator in the interaction picture and hence depends on time.

Rearranging Eq. (C9) to separate the Hermitian and non-Hermitian terms yields the Lindblad equation for our system dρSdt=-i[G-J+J-+G+J-J++GzJzJz,ρS]+γ-(J-ρSJ+-12{J+J-,ρS})+γ+(J+ρSJ--12{J-J+,ρS})+γz(JzρSJz-12{JzJz,ρS}),(C11)with γ+≡Γ+-(1)(ω0)+Γ+-2(-ω0)=∫-∞∞eiω0strB{Σ+(0)Σ-(s)ρB}ds,γ-≡Γ-+(2)(ω0)+Γ-+(1)(-ω0)=∫-∞∞eiω0strB{Σ-(s)Σ+(0)ρB}ds,γz≡Γzz(1)(0)+Γzz(2)(0)=∫-∞∞trB{Σz(0)Σz(s)ρB}ds,(C12)and G-≡Γ+-(2)(-ω0)-Γ+-(1)(ω0)2i,G+≡Γ-+(2)(ω0)-Γ-+(1)(-ω0)2i,Gz≡Γzz(2)(0)-Γzz(1)(0)2i,(C13)and we have assumed that the bath is constant in time. Observe that the first, Hermitian term in Eq. (C11) commutes with HS. Its form therefore remains unchanged in Schrödinger picture and represents an energy shift in the free Hamiltonian that is suppressed by GF2 through the bath-dependent coefficients G±,z. The remaining terms are non-Hermitian and characterize dissipative effects in the spin ensemble due to the bath via the γ±,z coefficients. Note that if the g-factor was negative, e.g., in the case of electrons, the Larmor frequency ω0 would be positive, which would effectively exchange γ+ and γ- in the Lindblad equation.

APPENDIX D:COMPUTATION OF <inline-formula><mml:math display="inline"><mml:msub><mml:mi>γ</mml:mi><mml:mo>±</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>γ</mml:mi><mml:mi>z</mml:mi></mml:msub></mml:math></inline-formula>

Our objective now is to compute γ+, γ- and γz, which requires that we calculate correlators of Σα that depend on the neutrino fields. To this end, we assume that neutrinos are Dirac fermions, whose expansion in modes is given by νi(t,x→)=∑s∫dp˜(ai,p→sui,s(p→)e-i(p0t-p→·x→)+bi,p→s†vi,s(p→)ei(p0t-p→·x→)),(D1)where dp˜≡d3p→/((2π)32p0), us(p→) and vs(p→) are the spinor solutions of the Dirac equation, and the creation/annihilation operators obey the anticommutation relations {ai,p→s,aj,q→r†}={bi,p→s,bj,q→r†}=(2π)32p0δ(3)(p→-q→)δrsδij.(D2)All other anticommutators of ai,p→s,aj,q→r†,bi,p→s,bj,q→r† are zero.

We start with γ+ given in Eq. (C12), which evaluates using (A9) to γ+=2GF∑ijkl∫-∞∞dseiω0strB{gAi,jν¯i(0,x→)ϒ-PLνj(0,x→)gAk,lν¯k(s,x→)ϒ+PLνl(s,x→)ρB},(D3)where have introduced the notations PL≡(1-γ5)/2 and ϒ±≡γ1±iγ2. Substituting Eq. (D1) into this expression, the nonvanishing terms read γ+=2GF∑ijklgAi,jgAk,l∑abcd∫-∞∞dseiω0s∫dp˜∫dq˜∫dp˜′∫dq˜′×[ei[(q0-q′0)s-(p→-p→′+q→-q→′)·x→]u¯i,a(p→)ϒ-PLuj,b(p→′)u¯k,c(q→)ϒ+PLul,d(q→′)trB{ai,p→a†aj,p→′bak,q→c†al,q→′dρB}+ei[(-q0+q′0)s-(p→-p→′-q→+q→′)·x→]u¯i,a(p→)ϒ-PLuj,b(p→′)v¯k,c(q→)ϒ+PLvl,d(q→′)trB{ai,p→a†aj,p→′bbk,q→cbl,q→′d†ρB}+ei[(-q0-q′0)s-(p→+p→′-q→-q→)·x→]u¯i,a(p→)ϒ-PLvj,b(p→′)v¯k,c(q→)ϒ+PLul,d(q→′)trB{ai,p→a†bj,p→′b†bk,q→cal,q→′dρB}+ei[(q0+q′0)s-(-p→-p→′+q→+q→′)·x→]v¯i,a(p→)ϒ-PLuj,b(p→′)u¯k,c(q→)ϒ+PLvl,d(q→′)trB{bi,p→aaj,p→′bak,q→c†bl,q→′d†ρB}+ei[(q0-q′0)s-(-p→+p→′+q→-q→)·x→]v¯i,a(p→)ϒ-PLvj,b(p→′)v¯k,c(q→)ϒ+PLvl,d(q→′)trB{bi,p→abj,p→′b†ak,q→c†al,q→′dρB}+ei[(-q0+q′0)s-(-p→+p→′-q→+q→)·x→)]v¯i,a(p→)ϒ-PLvj,b(p→′)v¯k,c(q→)ϒ+PLvl,d(q→′)trB{bi,p→abj,p→′b†bk,q→cbl,q→′d†ρB}].(D4)To compute the thermal traces of the bath, we assume that each neutrino species assumes a thermal distribution, such that the background density matrix is given by ρB=e-βHBtrB{e-βHB},(D5)where T=1/β is the neutrino temperature, and the bath Hamilton can be expanded as HB=∑i∑s∫dp˜p0(ai,p→s†ai,p→s+bi,p→s†bi,p→s).(D6)Using this information and the anticommutation relations (D2), the traces evaluate to trB{ai,p→a†aj,p→′bak,q→c†al,q→′dρB}=δpp′δqq′Ni,a(p0)Nk,c(q0)+δq′pδp′q(1-Nk,c(q0))Ni,a(p0),trB{ai,p→a†aj,p→′bbk,q→cbl,q→′d†ρB}=δpp′δqq′Ni,a(p0)(1-N¯k,c(q0)),trB{ai,p→a†bj,p→′b†bk,q→cal,q→′dρB}=δpq′δp′qNi,a(p0)N¯k,c(q0),trB{bi,p→aaj,p→′bak,q→c†bl,q→′d†ρB}=δpq′δp′q(1-Nk,c(q0))(1-N¯i,a(p0)),trB{bi,p→abj,p→′b†ak,q→c†al,q→′dρB}=δpp′δqq′Nk,c(q0)(1-N¯i,a(p0)),trB{bi,p→abj,p→′b†bk,q→cbl,q→′d†ρB}=δpp′δqq′(1-N¯i,a(p0))(1-N¯k,c(q0))+δqp′δpq′(1-N¯i,a(p0))N¯k,c(q0),(D7)where we have introduced the notation δpq≡(2π)32p0δ(3)(p→-q→)δrsδij, and Ni,a(p0) [N¯i,a(p0)] represents the occupation number of the ith neutrino (antineutrino) mass eigenstate with helicity a.

Substituting the thermal traces into Eq. (D4) and integrating over s, we find γ+=GF2π∑ikgAi,kgAk,i∑ac∫dp˜∫dq˜×[δ(1)(ω0+q0-p0)u¯i,a(p→)(γ1-iγ2)(1-γ5)uk,c(q→)u¯k,c(q→)(γ1+iγ2)(1-γ5)ui,a(p→)(1-Nk(q0))Ni(p0)+δ(1)(ω0-q0-p0)u¯i,a(p→)(γ1-iγ2)(1-γ5)vk,c(q→)v¯k,c(q→)(γ1+iγ2)(1-γ5)ui,a(p→)Ni(p0)N¯k(q0)+δ(1)(ω0-q0+p0)v¯i,a(p→)(γ1-iγ2)(1-γ5)vk,c(q→)v¯k,c(q→)(γ1+iγ2)(1-γ5)vi,a(p→)(1-N¯i(p0))N¯k(q0)],(D8)where we have assumed the neutrino occupation numbers to be independent of helicity. Compared with the single-spin excitation rate (B7) computed in Appendix B, we recognize immediately that the first and third terms correspond to inelastic scattering of a neutrino and an antineutrino, respectively, off a spin, thereby transferring energy to it. The second term describes the absorption of a neutrino-antineutrino pair, which we did not include in the heuristic calculation of Appendix B, but can also excite a spin if kinematically allowed; the rightmost diagram in Fig. 12 represents this process.

Neglecting the relative movement between the Earth and the CνB and using the properties of the spinors, Eq. (D8) can be further simplified to γ+=8πGF2∑ikgAi,kgAk,i∫dp˜∫dq˜p0q0[δ(1)(ω0+q0-p0)(1-Nk(q0))Ni(p0)+δ(1)(ω0-q0-p0)Ni(p0)N¯k(q0)+δ(1)(ω0-q0+p0)(1-N¯i(p0))N¯k(q0)].(D9)Using the Dirac deltas to eliminate one integral, the final expression reads γ+=4GF2(2π)3∑ikgAi,kgAk,i[∫(ω0+mk)2-mi2∞d|p→||p→|2qk+mk2+qk+2((1-Nk(qk+0))Ni(p0)+(1-N¯k(qk+0))N¯i(p0))+∫0(ω0-mk)2-mi2d|p→||p→|2qk+mk2+qk+2Ni(p0)N¯k(qk+0)],(D10)with qk+=[(|p→|2+mi2-ω0)2-mk2]1/2, and qk+0=mk2+qk+2. Except for the appearance of the second integral, this expression is the same as the heuristic rate (B9) estimated earlier. Observe also that the second integral contributes only if mk+mi<ω0. This is the kinematic requirement for neutrino-antineutrino pair absorption alluded to above. For typical values of the magnetic field considered in this work, ω0∼10-9 eV. Thus this process plays a role only for extremely small neutrino masses mi≲10-9 eV.

The computations of γ- and γz proceed in an analogous fashion. Contrary to γ+, however, γ- describes deexcitation of a spin via inelastic scattering with a neutrino or an antineutrino, i.e., the spin loses energy. In addition, a spin (up) can deexcite (to a spin-down) by emitting a neutrino-antineutrino pair if kinematically permitted. The final expression for γ- reads γ-=4GF2(2π)3∑ikgAi,kgAk,i[∫(ω0-mk)2-mi2∞d|p→||p→|2qk-mk2+qk-2((1-Nk(qk-0))Ni(p0)+(1-N¯k(qk-0))N¯i(p0))+∫0(ω0-mk)2-mi2d|p→||p→|2qk′mk2+qk′2(1-N¯i(p0))(1-N¯k(qk′0))],(D11)with qk-=[(|p→|2+mi2+ω0)2-mk2]1/2, qk-0=mk2+q-2, qk′=[(|p→|2+mi2-ω0)2-mk2]1/2, and qk′0=mk2+qk′2. Again, this expression can be contrasted with the heuristic rate (B11), which does not contain the second integral that contributes only if mk+mi<ω0.

For γz we find γz=16GF2(2π)3∑ikgAi,kgAk,i∫mk2-mi2∞d|p→||p→|2|p→|2+mi2-mk2⁢mi2+|p→|2(Ni(p0)(1-Nk(p0))+N¯i(p0)(1-N¯k(p0))).(D12)Contrary to γ±, this term does not describe energy exchange processes. Rather, it describes the collective dephasing of the spin ensemble due to elastic scattering.

In the case of Majorana neutrinos, the main difference in the computation of γ±,z is that the expansion in modes of the neutrino fields is given by νi(t,x→)=∑s∫dp˜(ai,p→sui,s(p→)e-i(p0t-p→·x→)+ai,p→s†vi,s(p→)ei(p0t-p→·x→)),(D13)and the corresponding bath Hamiltonian is HB=∑i∑s∫dp˜ p0ai,p→s†ai,p→s.(D14)If we neglect pair absorption and emission (which, as discussed above, are normally not kinematically allowed in NMR experiments except for close-to-zero neutrino masses), the expressions for γ±,z—Eqs. (D10)–(D12)—are exactly identical for Dirac and Majorana neutrinos. The only difference between the two cases occurs in the value of the occupation number Ni,a(p0). In the Dirac case, when ultrarelativistic neutrinos decouple from the early universe plasma, they are always in a left-helicity state. However, as the universe cools down and the neutrinos become nonrelativistic, we would generically expect them to relax via gravitational interactions into an approximately equal number of left- and right-helicity states with an average occupation number (assuming no neutrino-antineutrino asymmetry) given by Ni,a(p0)=N¯i,a(p0)≈nF(|p→|)/2, where nF(|p→|)=1e|p→|/T+1(D15)is the relativistic Fermi-Dirac distribution. On the other hand, Majorana neutrinos are their own antiparticles. Thus, relaxation into left- and right-helicity states merely turn an neutrino into an antineutrino and vice versa, leading to an occupation number of Ni,a(p0)=N¯i,a(p0)≈nF(|p→|).

APPENDIX E:LINDBLAD EQUATION FOR <inline-formula><mml:math display="inline"><mml:mi>N</mml:mi></mml:math></inline-formula> SPINS

Consider now a system comprising N 1/2-spins, where the Hilbert space is spanned by ρS=⊗Nρ1/2, and the Hamiltonian of neutrino-spin system is given in Eq. (A12). The corresponding Lindblad equation can be generalized from Eq. (C11). In the interaction picture (and omitting the Hermitian terms) it reads dρSdt=∑αβγ-αβ(J-αρSJ+β-12{J+αJ-β,ρS})+γ+αβ(J+αρSJ-β-12{J-αJ+β,ρS})+γzαβ(JzαρSJzβ-12{JzαJzβ,ρS}),(E1)with γ+αβ=∫-∞∞eiω0strB{Σ+α(0)Σ-β(s)ρB}ds,γ-αβ=∫-∞∞eiω0strB{Σ-α(s)Σ+β(0)ρB}ds,γzαβ=∫-∞∞trB{Σ3α(0)Σ3β(s)ρB}ds,(E2)which are generalizations of the γ±,z coefficients from Eq. (C12).

Consider first the coefficient γ+αβ. Following the procedure outlined in Appendix D, we find γ+αβ=2GF∑ijklgAi,jgAk,l∑abcd∫-∞∞dseiω0s∫dp˜∫dq˜∫dp˜′∫dq˜′×[ei[(q0-q′0)s-(p→·x→α-p→′·x→α+q→·x→β-q→′·x→β)]u¯i,a(p→)ϒ-PLuj,b(p→′)u¯k,c(q→)ϒ+PLul,d(q→′)trB{ai,p→a†aj,p→′bak,q→c†al,q→′dρB}+ei[(-q0+q′0)s-(p→·x→α-p→′·x→α-q→·x→β+q→′·x→β)]u¯i,a(p→)ϒ-PLuj,b(p→′)v¯k,c(q→)ϒ+PLvl,d(q→′)trB{ai,p→a†aj,p→′bbk,q→cbl,q→′d†ρB}+ei[(-q0-q′0)s-(p→·x→α+p→′·x→α-q→·x→β-q→′·x→β)]u¯i,a(p→)ϒ-PLvj,b(p→′)v¯k,c(q→)ϒ+PLul,d(q→′)trB{ai,p→a†bj,p→′b†bk,q→cal,q→′dρB}+ei[(q0+q′0)s-(-p→·x→α-p→′·x→α+q→·x→β+q→′·x→β)]v¯i,a(p→)ϒ-PLuj,b(p→′)u¯k,c(q→)ϒ+PLvl,d(q→′)trB{bi,p→aaj,p→′bak,q→c†bl,q→′d†ρB}+ei[(q0-q′0)s-(-p→·x→α+p→′·x→α+q→·x→β-q→′·x→β)]v¯i,a(p→)ϒ-PLvj,b(p→′)v¯k,c(q→)ϒ+PLvl,d(q→′)trB{bi,p→abj,p→′b†ak,q→c†al,q→′dρB}+ei[(-q0+q′0)s-(-p→·x→α+p→′·x→α-q→·x→β+q→′·x→β)]v¯i,a(p→)ϒ-PLvj,b(p→′)v¯k,c(q→)ϒ+PLvl,d(q→′)trB{bi,p→abj,p→′b†bk,q→cbl,q→′d†ρB}].(E3)After evaluating the thermal traces, the expression simplifies to γ+αβ=4GF2π∑ikgAi,kgAk,i∑ac∫dp˜∫dq˜×[e-iΔx→αβ·(p→-q→)δ(1)(ω0+q0-p0)u¯i,a(p→)ϒ-PLuk,c(q→)u¯k,c(q→)ϒ+PLui,a(p→)(1-Nk,c(q0))Ni,a(p0)+e-iΔx→αβ·(p→+q→)δ(1)(ω0-q0-p0)u¯i,a(p→)ϒ-PLvk,c(q→)v¯k,c(q→)ϒ+PLui,a(p)Ni,a(p0)N¯k,c(q0)+eiΔx→αβ·(p→-q→)δ(1)(ω0-q0+p0)v¯i,a(p→)ϒ-PLvk,c(q→)v¯k,c(q→)ϒ+PLvi,a(p→)(1-N¯i,a(p0))N¯k,c(q0)],(E4)where Δx→αβ≡x→α-x→β.

From Eq. (E4) we see immediately that in the limit |p→-q→|≪|1/Δx→αβ| and |p→+q→|≪|1/Δx→αβ|, the exponentials evaluate to unity such that γ+αβ loses its dependence on α and β, and approximates γ+ of Eq. (D8) computed previously for the single-spin system. The same limiting behavior is also seen in γ-αβ and γzαβ (not shown here). Then, in this coherent regime, the Lindblad equation reads dρSdt=γ-(J-ρSJ+-12{J+J-,ρS})+γ+(J+ρSJ--12{J-J+,ρS})+γz(JzρSJz-12{JzJz,ρS}),(E5)where we have defined J+≡∑αJ+α, J-≡∑αJ-α, and Jz≡∑αJzα. If on the other hand |p→-q→|≫|1/Δx→αβ|, the cross (i.e., α≠β) terms in Eq. (E4) average to zero and only the α=β terms survive. (The condition on |p→+q→| versus |1/Δx→αβ| is not important, as neutrino-antineutrino pair absorption and emission are not kinematically allowed for neutrino masses mi≳10-9 eV given typical values of the magnetic field B∼0.1 T.) In this limit the Lindblad equation reads dρSdt=∑αγ-(J-αρSJ+α-12{J+αJ-α,ρS})+γ+(J+αρSJ-α-12{J-αJ+α,ρS})+γz(JzαρSJzα-12{JzαJzα,ρS}),(E6)a form that is generally applicable for any “local” interaction that is not coherent over the length of the spin sample.

APPENDIX F:FULL NUMERICAL SOLUTIONS

We demonstrate here how we find a full numerical solution of the Lindblad master equation (43) for the spin ensemble’s density matrix ρS. We show first the case of collective neutrino interactions only. This effectively restricts the dynamics to the fully symmetric subspace of total angular momentum j=N/2, where numerical solutions can scale with N3 (see Ref. [62] for a detailed discussion) and are hence tractable for relatively large values of N. We then extend the numerical treatment to include local interactions, where the dynamics can no longer be restricted to the j=N/2 manifold.

In the presence of coherent interactions alone, we set γϕloc=0, and solve dρSdt=γ-DJ-[ρS]+γ+DJ+[ρS]=LD[ρS],(F1)where we have introduced the Liouvillian superoperator at the second equality, expressed in the Dicke basis. We work in Liouville space by vectorizing (i.e., column-stacking) the density matrix ρS→|ρS⟩ and using the Liouvillian superoperator LD=∑±γ±(J∓⊤⊗J±-12I⊗J∓J±-12(J∓J±)⊤⊗I),(F2)where the action of the operators on ρS from the left and the right becomes a Kronecker product, namely, XρSY→(Y⊤⊗X)|ρS⟩.

To numerically integrate the master equation (F1), we implement standard time-integration schemes such as the Euler method, as well as higher-order solvers such as the fourth-order Runge-Kutta (RK4) algorithm. To ensure numerical stability, especially for such explicit integration schemes, we compute the spectral radius of the Liouvillian, i.e., the largest eigenvalue of LD, λmax, and set the time step Δt such that Δtλmax≪1. This criterion ensures convergence and bounds the error at each step, particularly for stiff systems or rapidly decaying modes. In practice, we choose Δt=h/λmax with h∼0.1 depending on the desired accuracy. We have additionally tested implicit integration methods and found no substantial improvement in convergence compared to explicit methods, provided the timestep factor h is sufficiently small. Given their significantly lower computational cost, explicit schemes remain the optimal choice for efficiency in this context.

Our simulations are implemented in the publicly available code opennu,¹⁵

The source code is available at https://github.com/gpierobon/OpenNu.

written in c++ and based on the eigen¹⁶

Eigen is a high-performance c++ library for linear algebra: https://eigen.tuxfamily.org.

library for efficient sparse matrix operations. This implementation achieves high computational performance and a low memory footprint, which is essential for solving large-scale Lindblad dynamics. In particular, it allows us to simulate on a single modern CPU collective spin ensembles with Hilbert space dimensions up to the equivalent of N∼104. To avoid numerical instabilities due to very large or small values, we normalize time by the characteristic dissipation timescale tdiss, achieved by rescaling the Liouvillian with the product of the number of spins and the deexcitation rate γ-. As a result, the ratio γ+/γ- remains as the sole control parameter governing the dynamics in this regime. The left panel of Fig. 13 displays our numerical solutions obtained using opennu for N=100 and different initial conditions.

10.1103/kgk5-bf5x.f13

FIG. 13.

Left: time evolution of the emission rate observable ⟨Γ+Γ-⟩, showing different initial conditions under the influence of coherent neutrino effects only. Solid lines represent collective deexcitation (γ+=0), highlighting the delayed burst associated with the excited state, i.e., Dicke superradiance. These solutions can be reproduced using the public code opennu (see text). Right: time evolution of the ⟨Jx2⟩|G⟩ observable for N=30, 40, 50, including both neutrino collective effects (γ+/γ-=0.997) and local dephasing (γϕ=Nγ-), computed using the qutip code (see text). The bottom panel shows the residual between the full solution from qutip and the second-order approximate methods, demonstrating that the two methods match at the O(0.01)% level.

To include local interactions and fully solve Eq. (43), we use the permutational-invariant quantum solver (PIQS) introduced in Ref. [66] and implemented in the qutip library.¹⁷

The qutip source code is available at github.com/qutip/qutip.

PIQS also reduces the computational complexity of simulating open quantum dynamics from exponential to polynomial in N by exploiting the permutational symmetry of two-level systems, under the assumption that they are identical and identically prepared. However, when the dynamics involve population outside the fully symmetric subspace, i.e., when j<N/2 states are included, the symmetric Dicke basis must be extended to account for the action of local operators such as Jzα in Eq. (43). As a result, with respect to opennu the size of the Liouvillian superoperator increases by a factor of N2, significantly raising the computational cost and limiting tractable simulations to system sizes of N∼O(100). In the right panel of Fig. 13, we present numerical solutions computed with qutip for values up to N=50, demonstrating excellent agreement between the full density matrix solution and the fast second-order approximation.

APPENDIX G:HYPERPOLARIZATION

Hyperpolarized spin systems, which are out of thermal equilibrium, enhance signal intensity in NMR experiments [78]. A technique widely used in NMR applications is spin-exchange optical pumping (SEOP) [73], particularly for noble gases such as Xe129 or He3. SEOP can be broken down into two steps: (i)

Optical pumping of alkali metal atoms. A circularly polarized laser excites alkali atoms (typically Rb or Cs vapor) in a low magnetic field. A large electron spin polarization pA≈1 in the alkali ground state develops rapidly via repeated excitation-relaxation cycles and from angular momentum conservation in the atomic transition. The dynamics of this process is governed by a rate equation [79], dpAdt=Γpump(1-pA)-ΓA,relpA,(G1)where Γpump is the pumping rate, and ΓA,rel the total alkali spin relaxation rate. The steady-state pA is given by pAss=ΓpumpΓpump+ΓA,rel,(G2)which shows that to obtain a large pA requires Γpump≫ΓA,rel.

(ii)

Spin-exchange collisions. The polarized alkali atoms collide with the target noble gas, e.g., Xe, atoms in a gas cell, effectively transferring angular momentum to the noble gas nuclear spins (via a Fermi-contact interaction). The polarization of the noble gas spins pXe can be computed from dpXedt=ΓSE(pA-pXe)-ΓrelpXe,(G3)where γSE is the spin-exchange rate, and Γrel is the nuclear spin relaxation rate. Then, pXess=ΓSEpAssΓSE+Γrel(G4)gives the steady-state value of pXe.

SEOP typically operates in dilute gas phases, where nuclear spin-spin interactions are negligible and polarization accumulates through random spin-transfer collisions that do not generate coherences between spins. The overall mechanism is analogous to the local interaction channels described in Sec. V. Matching pXess of Eq. (G4) to the polarization parameter in Eq. (47) then yields a mapping γ-hyperγ+hyper=ΓSE(1+pAss)+ΓrelΓSE(1-pAss)+Γrel(G5)between the various rates to the hyperpolarization excitation and deexcitation rates used in Sec. VI A.

APPENDIX H:TWO-TIME CORRELATION FUNCTIONS

We outline here the computation of the two-time correlation functions ⟨C(t)A(t+τ)⟩ via the quantum regression theorem [70], where τ≥0, and the operators A and C act on the spin ensemble alone (i.e., not the bath). See also Ref. [69] for a more comprehensive introduction. This correlation function is defined in the Heisenberg picture via ⟨C(t)A(t+τ)⟩≡trT[ρ(0)C(t)A(t+τ)],(H1)where ρ(0) is the density matrix of total system of spin and bath at time t=0, and the trace trT pertains to the total system. Applying time evolution to the operators, i.e., A(t)=eiHtA(0)e-iHt, C(t)=eiHtC(0)e-iHt, and ρ(t)=e-iHtρ(0)eiHt assuming a constant H, allows us to rewrite the expression in the Schrödinger picture, ⟨C(t)A(t+τ)⟩=trT[e-iHτρ(t)C(0)eiHτA(0)],(H2)where the trace is now carried out over the state at time t+τ, with A(0) and C(0) as time-independent Schrödinger-picture operators.

Assuming the total density matrix can be split into a spin ensemble and a bath piece, i.e., ρ(t)=ρS(t)⊗ρB(t), at all times allows us to “swap the order” of ρB(t) and C(0) to get ⟨C(t)A(t+τ)⟩=trT[e-iHτ[ρS(t)C(0)⊗ρB(t)]eiHτA(0)]=trT[e-iHτ[ρS;C(t)⊗ρB(t)]eiHτA(0)],(H3)where we have defined in the last line a modified spin density matrix ρS;C(t)≡ρS(t)C(0). Writing out the total trace trT in terms of individual traces over the bath (subscript B) and spin ensemble (no subscript, for consistency with the notation of the main text), i.e., trT=trB⊗tr, we arrive at ⟨C(t)A(t+τ)⟩=tr[ρS;C(t+τ)A(0)],(H4)where ρS;C(t+τ)=trB[e-iHτ[ρS;C(t)⊗ρB(t)]eiHτ](H5)is simply the modified spin matrix ρS;C forwarded in time in time from t to t+τ under the total Hamiltonian H with the bath degrees of freedom traced out.

Using τ as the time variable, the formal equation of motion for ρS;C(τ) is ddτρS;C(t+τ)=L{ρS;C(t+τ)},(H6)where L is some superoperator, which in our case would be the Lindbladian. The formal solution reads ρS;C(t+τ)=eLτ{ρS;C(t)}, such that the two-time correlation function can also be written as ⟨C(t)A(t+τ)⟩=tr[eLτ{ρS(t)C(0)}A(0)].(H7)Note that while we have assumed a time-independent H in the above derivation, the equation of motion (H6) applies also to a time-dependent H. The qutip library contains functions that compute numerically two-time correlation functions for a given L.

We are interested to derive equations of motion directly for the two-time correlation functions ⟨C(t)Aμ(t+τ)⟩, Aμ is a set of operators indexed by ν=1,2,3,…. Suppose ⟨Aμ⟩ evolves under the Lindblad master equation as ddt⟨Aμ⟩=tr[dρSdtAμ]=∑αMμαtr[ρSAα]=∑αMμα⟨Aα⟩,(H8)where Mμα is square matrix of constant coefficients. It then following straightforwardly from Eq. (H6) that ddτ⟨C(t)Aμ(t+τ)⟩=∑αMμα⟨C(t)Aα(t+τ)⟩.(H9)In other words, the “same” set of equations of motion describe both the single-time expectation values and the two-time correlations.

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