arXiv:1101.2749v2 [hep-ph] 24 Jan 2011
Solitons and Precision Neutrino Mass Spectroscopy
Center of Quantum Universe, Faculty of Science, Okayama University
Tsushima-naka 3-1-1 Kita-ku Okayama 700-8530 Japan
We propose how to implement precision neutrino mass spectroscopy using radiative neutrino pair
emission (RNPE) from a macro-coherent decay of a new form of target state, a large number of
activated atoms interacting with static condensate field. This method makes it possible to measure
still undetermined parameters of the neutrino mass matrix, two CP violating Majorana phases, the
unknown mixing angle and the smallest neutrino mass which could be of order a few meV, determining
at the same time the Majorana or Dirac nature of masses. The twin process of paired superradiance
(PSR) is also discussed.
neutrino mass), the nature of masses (whether they have Majorana or Dirac type masses), and their
relation to the leptogenesis theory ,  are not clarified experimentally. Experimental efforts
to unravel these properties are mainly focused on nuclear targets. Nuclear targets, however, are
problematic at least in one important aspect, the mismatch of energy scale: the released energy of
nuclear transition is of order several MeV, and this is far separated from the expected neutrino mass
range of O[0.1]eV.
We proposed a few years ago the idea of using atomic targets to overcome this difficulty; RNPE
from a metastable state |e?, |e? → |g? + γ + νiνj. This is an elementary process predicted by
the ordinary electroweak interaction, and its detection opens a path towards the neutrino mass
spectroscopy , , by precisely measuring the photon energy spectrum, thereby resolving neutrino
mass eigenstates νi,i = 1,2,3.
With smaller released energies of atomic transitions, the atomic decay involving neutrino pair
emission has a demerit of tiny weak rates, unless a new idea of rate enhancement is taken into
account. Our enhancement mechanism uses a coherent cooperative effect of a large number of atoms
interacting with a common field , . A similar idea goes back to the superradiance (SR for short)
 of a single photon emission, where the decay rate from many atoms is in proportion to n2V ,
the target number density squared times a coherent volume V , unlike the target number nV in the
Atoms in a metastable state |e? may have a lifetime for a long time measurement. If these atoms
further have a developed coherence, macro-coherent two photon emission, called paired superradiance
(PSR for short), |e? → |g?+γ+γ, becomes easily detectable , its rate ∝ n2V , with V a macroscopic
target volume, unlike the case of usual SR limited by V ∝ the photon wavelength squared. PSR has
a distinct signature: two photons are back to back emitted and have exactly the same energy.
We propose in this work to use for the target of RNPE a coherent state of atoms interacting with
static field condensate (we call this as condensate for simplicity). The condensate is a limiting case
of multiple soliton solutions, as presented below. Both solitons and condensate are proved stable
against PSR, but unstable for RNPE.
PSR, emitting a highly correlated pair of two photons, is interesting from points of application
such as quantum entanglement. Artificial destruction of solitons and condensate, which can be easily
realized by a sudden application of electric pulse (thus abruptly changing the dielectric constant),
provides the most efficient mechanism of PSR emission yet to be discovered. If we successfully destroy
solitons for PSR under complete control, solitons may become qubits for quantum computing.
On the other hand, creation and subsequent long time control of the condensate removes the
most serious PSR background for RNPE. We compute macro-coherent RNPE rate ∝ n2V of conden-
sate decay and study sensitivity of spectral rates (spectral shape and event rate) to parameters of
the neutrino mass matrix, most importantly the fundamental parameter of CP violating Majorana
phases; the parameter of central importance in explaining the matter-antimatter imbalance of the
universe. RNPE spectrum shape from the condensate decay is time independent after condensate
formation and the most unambiguous tool for this process.
Our method uses laser to trigger RNPE at non-resonant frequencies, which should be a great
merit since the trigger is not destructive to target atoms.
The natural unit ? = c = 1 is used in formulas of this paper.
Effective atomic Hamiltonian and Maxwell-Bloch equation
consist of three levels of energies ǫg< ǫe< ǫp. The state |e?, for example1D2-state of Ba low lying
levels, is forbidden to decay to |g? by E1 transition, while E1 transitions from |p? to |e? and |g? may
both be allowed. The important part of Hamiltonian is derived ,  by eliminating time memory
Neutrinos are still mysterious particles: their absolute mass scale (or the smallest
We consider atoms that
effects of |p?,1P1in Ba, and by making a slowly varying envelope approximation of one field mode
propagating in a direction. The resulting effective Hamiltonian is restricted to two levels, |e? and
|g?, interacting with field E of frequency ω and a definite polarization. The 2 × 2 matrix elements
µab,a,b = e,g, are Stark energies; a product of two dipole (E1 or M1) transition elements to |p?
times the electric field squared. Dipole transition elements are related to measurable decay rates
γpa,a = e,g from |p? to |a?, thus
2(ǫpe− ω)(ǫpg+ ω)
We ignored the spin multiplicity factor 2Ja+1 in the relation d2
should be multiplied by (2Jp+ 1)/(2Je+ 1) if one includes this multiplicity.
The equation for the polarization vector?R (3 bilinears of amplitudes times the target number
density n), called the Bloch equation, is derived from the Schr¨ odinger equation, and may be written
as ∂t?R = |E|2M?R, where elements of 3×3 anti-symmetric matrix M are linear combinations of µab.
When this equation is combined with the Maxwell equation, written as (∂t+∂x)|E|2= ωµge|E|2R,
with R a component of?R, a closed set of equations follows, to describe spacetime evolution of
polarization and propagating field , .
When relaxation processes are ignored, one can introduce the tipping angle θ(x,t) by R(x,t) =
ncosθ(x,t). The Bloch equation is then reduced to a relation of θ(x,t) to the electric field strength;
|E(x,t)|2= ∂tθ/µ with µ =
field propagation in the two-level problem . The Maxwell equation in terms of θ(x,t) is
pa− ω2),(a = e,g),(1)
abto γab. The final PSR rate formula
(µee− µgg)2+ 4µ2
ge/4. The field θ(x,t) is an analogue of the area for
(∂t+ ∂x)θ = αm(−cosθ + A),
where ǫba= ǫb−ǫais the atomic energy difference. For the Ba D-state, αm∼ 2.4×10−6cm−1(n/1012cm−3)
at ω = ǫeg/2. Both αmand µ depend on ω. The non-linear equation (3) describes dynamics of a
fictitious pendulum under friction periodically varying ∝ αmsinθ at its location θ.
For |A| ≤ 1, the tipping angle is restricted to a finite θ−region of ≤ 2π. The propagation problem
in this case has been analytically solved in  in terms of arbitrary initial data. Hence the system
appears integrable in the mathematical sense. Typical solutions describe multiple splitting of pulses
and their compression when they propagate in a long coherent medium, as fully explained in .
The number of split pulses is given by the initial pulse area θ(−∞,∞) divided by 2π. This behavior
of pulse in medium is a symptom of instability, and pulses stabilize via PSR. It is thus anticipated
that stable objects against PSR exist; solitons.
There are two types of analytic solutions for solitons; |A| = 1 giving a single
soliton of quantized area 2π and |A| > 1 the multiple soliton. The case |A| < 1 is unphysical since
an excited state of population ncosθ ?= −n exists at ξ = ±∞. The case of |A| = 1 solution of area
2π has been obtained in  by using a different method.
We look for soliton solutions by assuming one variable dependence of x−vt for a soliton of velocity
v and by reducing the partial differential equation to an ordinary one. The solution for A = 1 thus
obtained has a Lorentzian shape of flux and the population given by
v(1 − v)
m(x − vt)2+ (1 − v)2
m(x − vt)2+ (1 − v)2, (5)
m(x − vt)2+ (1 − v)2.(6)
The soliton size is O[1/αm], and its field flux is of order, αm/µ ∼ 30Wmm−2(n/1018cm−3) for the Ba
soliton at ω = ǫeg/2.
This method applied to the |A| > 1 case, on the other hand, gives a new class of solutions given
1 − v
(x − vt).
A − cosX,
cosθ(x,t) =AcosX − 1
A − cosX
1 − v
Unlike the single peak for |A| = 1, the field flux given by (7) has infinitely many peaks equally
spaced, describing multiple soliton solutions in medium.
For a finite length of medium one may impose the boundary condition of no excited state at two
target ends of x = ±L/2. This gives a condition, αmL√A2− 1/(1−v) = 2π(2ns−1),ns= 1,2,···.
The quantity√A2− 1 is of order, and the soliton number density ∼ ns/(αmL).
Solitons may both emit and absorb photons within medium, their rate difference ∝ cosθ|E(x,t)|2.
This quantity, when integrated in the entire medium supporting a soliton, gives an integral of a total
derivative ∝ ∂xsinθ, hence vanishes for the quantized area of ∆θ = 2π. This proves the soliton
stability against PSR.
∞, simultaneous with the limit v → 0. Denoting A =?(η/v)2+ 1 with η kept constant, one has
One may consider the limit of large soliton density, ns/αmL ∼√A2− 1/4π →
µ(η − vcos(αmη(x − vt)/v))≈ηαm
thus an almost constant field flux is derived. The population ∝ cosθ oscillates, with the time period
τ = 2π/(αmη) and the space period τv. The parameter η is 4π× soliton density × soliton velocity.
Practically, the shortest spatial period is limited by the inter-atomic distance d. By identifying
the period τv with d, one finds η ∼ 2πv/(αmd), hence τ = d/v. As v → 0, τ → ∞, and the
target becomes fully excited with cosθ = 1. For the Ba1D2-state, the relevant numerical value is
αmd ∼ 4 × 10−7(n/1018cm−3)2/3.
The limit taken here gives a constant field ηαm/µ and the full excitation of target everywhere
(strictly, this is true for an infinitely long medium). This is the state of field condensate we use for
RNPE. Field condensate can be created by trigger laser irradiation from multiple directions, since it
has no memory of a particular direction.
The stability analysis around the condensate can be made, taking E = Ec+δE ,θ = θc+δθ with
Ec,θc= 0 the condensate solution. By keeping linear terms ∝ δE,δθ in the Maxwell-Bloch equation,
with δE,δθ ∝ e−iωtfor time dependence, the perturbation equation ∂xδE = i(ω + α2
a bounded and purely oscillatory solution, indicating the stability of field condensate.
PSR rate at soliton and condensate destruction
out soliton creation. The PSR rate without trigger is µ2
We first mention PSR rate with-
egn2V/(29π2), which is numerically ∼
0.5MHz(n/1012cm−3)2V/cm3for Ba. Under a strong trigger of flux |E|2, the rate for a target of
length L becomes 
Although the rate for |E|2≈ 106Wcm−2is large, time structure of PSR is complicated .
PSR after soliton formation occurs only at its destruction, without absorption from |g?. The
emission rate from |e? is ∝ (1+cosθ)|Es|2/2. One may compute rates based on perturbative methods,
in which one of the photons belongs to the soliton pulse. The other photon is emitted backward to
the soliton propagation direction, with exactly the same energy. The large rate enhancement ∝ n2V
is understood by the momentum conservation among emitted particles, implying ei(?k+?k′)·? x= 1.
The PSR rate at soliton destruction is (taking L = dx in eq. (10) )
v(1 − v)
m(x − vt)2+ (1 − v)2dx.(11)
The rate remains large during a time of
∆t =1 − v
∼ 14µsec1 − v
The space integrated rate per soliton is
∼ 5 × 1015Hz
(numbers for Ba) a formula valid for a target of length L ≫ 1/αm. For a short target of L ≤ 1/αm
the rate is reduced by αmL/(π(1−v)). The integrated rate for long target is by many orders (∼ 108)
larger than the trigger-less PSR rate. The prolonged time of Oµsec and its simple profile structure
has a number of merits of easier PSR identification such as the back to back two photon coincidence
measurement. PSR rate at condensate destruction is larger by ηαmL/(vπ) than at the single soliton
Effect of relaxation
There are a number of processes that might destroy coherence. One of
them is given by a field decay, introduced by a term κ|E|2in the Maxwell equation. This modifies the
basic equation (3) by an additional term −κθ. With the ansatz of variable dependence of ξ = x−vt,
this equation is
(1 − v)dθ
dξ= −αmcosθ − κθ.(14)
Direct numerical integration of eq.(14) gives distorted quasi-soliton solutions. Their profile, al-
though distorted, is unchanged as they propagate.
κc∼ 0.725αm, indicating a threshold of dissipation. Calculation gives the PSR rate = (PSR rate at
pure soliton destruction) ×∆(θ + sinθ)/2π (the difference ∆ to be taken at two target ends). This
rate is smaller than the one without dissipation, but not very much less, unless κ is very close to the
threshold κc. The condition of a sizable PSR rate, the relaxation constant κ < O[αm], implies that
κ < O[0.07]MHz(n/1012cm−3) for the Ba target.
The effective Hamiltonian for RNPE,
Quasi-solitons exist only for κ < κc where
?d ·?E ,
gives the amplitude for a single atom, where?Seand?d are electronic spin and dipole operators. To
give large matrix elements for these, we consider deexcitation of |e? of the angular momentum J = 2
or J = 0 to |g? of J = 0 via |p? of J = 1, realized by rare gas and alkhali earth atoms. Six measurable
constants, cij’s, given by cij= U∗
to the mass eigenstate, contain mixing angles and Majorana CP phases , .
The field operator νifor the Majorana neutrino is a superposition of annihilation (bi) and creation
operators; particle annihilation (bi) and anti-particle creation (d†
amplitude for i ?= j has the form, b†
for the Dirac case. Condensate RNPE decay rate of field |Ec|2∼ ηαm/µ is a sum of 6 νiνj pair
eiUej− δij/2 with U the unitary matrix relating the neutrino flavor
i) operator of the same Majorana particle, while for the Dirac neutrino it is a sum of two distinct
i). Thus, the νiνj pair emission
jfor the Majorana case, and cijb†
j(cij−cji)/√2 = i√2ℑcijb†
For i ?= j, Bij= (ℑcij)2for the Majorana case and Bij= |cij|2for the Dirac case, while Bii= |cii|2
for both cases. Factors, αmand µ, attributed to condensate parameters, involve intermediate |p?.
The state |p? that gives the largest condensate factor may be different from the intermediate state
that gives the largest RNPE rate. In the Yb case, |p? = 6s6p3P1for the largest condensate factor
and |p? = 6s6p1P1for the largest RNPE rate.
Yb RNPE rate: D vs M with CP phases
Figure 1: Yb RNPE photon energy spectrum in (11) ∼ (33) region from condensate decay. Dirac
case and 3 Majorana cases of different (α,β) are plotted; Dirac in blue, (π/2,0) Majorana in dotted
or short dashed red, (0,π/2) Majorana in broken black, and (π/4,−π/4) Majorana in dashed purple.
Neutrino masses of (m3,m2,m1) = (50,10,1)meV, and cosine angles 1/√2,√3/2,√0.97 are assumed
for all. The Majorana pair emission rate of (α,β) = (0,0) below (3,3) neutrino threshold is by
∼ 10−3smaller than the Dirac rate for these masses. Assumed target parameters are n = 1021cm−3,
V = 1cm3, and η = 103.
The function Iij(ω) in the formula (16) is given by an energy integral arising from the two neutrino
phase space. The integral in a symmetric form is given in terms of neutrino energies Ei,i = 1,2,
Iij(ω) = ω
dE1dE2δ(E1+ E2+ ω − ǫeg)θ(Cij(E1,E2))
ij = 2G(1)
ij = −G(1)
ij(E1,E2) + G(2)
ij(E1,E2) +E1E2− δMmimj
where the boundary region is given by Cij(E1,E2) ≥ 0 with
Cij(E1,E2) = (E2
j− ω2)2− 4(E2
and δM= 1 for the Majorana and δM= 0 for the Dirac case. In this calculation, averaged electron
spin matrix elements, ?(?k ·?Se/ω)2? = 1/12,??S2
(16) sharply rises at each (ij) threshold, a feature characteristic of 3 particle emission of massless γ
and nearly massless νi,νj, when both the momentum and the energy conservation hold.
The limiting case of 3 massless neutrinos gives RNPE rate of the condensate decay,
e? = 3/4, are used. The resulting spectrum given by
ǫeg),f(x) =9(324 − 540x + 245x2)
32(1 − ǫegx/(2ǫpg))2. (22)
The coefficient in front of the function f(2ω/ǫeg) is ∼ 8 × 10−5Hz for Yb of n = 1021cm−3,V =
1cm3,η = 103.
Experiments for the neutrino spectroscopy are to be performed keeping the macro-coherence of
the condensate. The initial trigger frequency ω ≤ ω11for RNPE of ω11= ǫeg/2−2m2
smallest neutrino mass, is reset each time for measurements of rate and parity violating quantities 
at different γ energies of the continuous spectrum. The energy resolution of RNPE spectrum is thus
determined by the precision of trigger frequency ω, and not by detected photon energy. This is a key
element for successful implementation of the precision neutrino mass spectroscopy, which must resolve
photon energies at the µeV level or less, since the (ij) threshold rise at ωij= ǫeg/2−(mi+mj)2/(2ǫeg)
is separated only a little from the half energy ǫeg/2 of dangerous PSR.
Calculated rates are sensitive to Majorana CP phases α,β defined by Ue2∝ eiα,Ue3∝ eiβ. Rate
rises at (12),(13),(23) thresholds are ∝ sin2α,sin2β ,sin2(α−β), respectively. For example, 4 cases of
(α,β) = (0,0),(π/2,0),(0,π/2),(π/4,−π/4) give 3 large threshold factors of (sin2α,sin2β,sin2(α −
β)) = (0,0,0),(1,0,1),(0,1,1),(1/2,1/2,1) at (12),(13),(23). We know of no other measurable
quantity of this high sensitivity to α and β. Within a given range of neutrino parameters, the easiest
observable might be the Majorana phase, as illustrated in our figures. Our proposed experiment
is not sensitive to the other CP phase δ, which however may be determined by future neutrino
oscillation experiments. Determination of all low energy phases, α,β,δ, is a requisite for a better
understanding of leptogenesis .
Distinction of Majorana and Dirac neutrinos is possible by the interference effect of identical Ma-
jorana fermions , giving different rates in the vicinity of thresholds. Rate difference of Majorana
and Dirac pair emission is larger for larger Majorana CP phases, as illustrated in Fig(1). Experi-
mentally, the spectral rate is fitted under an assumption of Majorana or Dirac neutrino and either
hypothesis is verified by a good quality of fitting.
1/ǫeg, with m1the
1.0680 1.06851.0690 1.0695 1.0700
Yb RNPE rate: 13 angle
with CP phase
Figure 2: Yb RNPE photon energy spectrum for different sin2θ13 values, 0.03 in blue, 0.02 in
dotted red, 0.01 in broken black, and 0 in dashed purple, all for (α,β) = (π/2,0). Neutrino masses
of (m3,m2,m1) = (50,10,5)meV, and cosine angles 1/√2,√3/2 for cosθ23,cosθ12 are assumed.
Assumed target parameters are n = 1021cm−3, V = 1cm3, and η = 103.
Yb RNPE rate: m1?1,2,4,6 meV
Figure 3: Yb RNPE photon energy spectrum for different m1values; 1 meV in blue, 2 meV in dotted
red, 4 meV in broken black, and 6 meV in dashed purple, all for (α,β) = (π/2,0). Other neutrino
masses are constrained by neutrino oscillation experiments, and cosine angles 1/√2,√3/2,√0.97 for
cosθ23,cosθ12,cosθ13are assumed. Assumed target parameters are n = 1021cm−3, V = 1cm3, and
η = 103.
Yb RNPE rate: Inverted vs normal hierarchy
Figure 4: Yb RNPE rate; case of normal and inverted mass hierarchies for different (α,β) values;
normal (0,0) (blue), inverted (0,0) (dotted red), inverted (π/2,0) (broken black), inverted (π/4,−π/4)
(dashed purple), all assuming (m3,m2,m1) = (50,10,5)meV and the same mixing as Fig(3). n =
1021cm−3,V = 1cm3,η = 103.
One possible serious background against RNPE might be the trigger-less SR due to the achieved
excellent coherence. This process has a monochromatic spectrum at ǫeg/2 different from RNPE,
nevertheless it might become dangerous, destroying the initial state. This can be avoided by choosing
J = 0 → 0 transition, which forbids single photon emission, complete to any order, hence SR
altogether. Alkhali earth atoms have level structure of this angular momentum configuration. Yb
and Hg atoms have levels of a similar nature, giving state candidates of |e? = (6s6p)3P0,|g? =
(6s2)1S0,|p? = (6s6p)1P1. Incidentally, two photons emitted by 0 → 0 RSR have perfectly correlated
polarizations, and may serve as an excellent device of quantum entanglement.
The calculated Yb 0 → 0 RNPE rate for (α,β) = (0,0) averaged over all photon energies is
∼ 3 × 10−4Hz for n = 1021cm−3,V = 1cm3,η = 103(a factor to be better understood) and is by
∼ 70 larger than the corresponding Xe 2 → 0 rate. When the Yb experiment at each photon energy ω
lasts for a day, its event number becomes O if ω is in the energy range of Fig(1). This event number
is further increased by f if one repeats condensate formation with a cycle time of 1/f sec. We show
in Fig(1) and Fig(2) the spectral rate for various combinations of CP phases and the mixing angle
θ13. Sensitivity to neutrino masses, in particular to m1values, is shown in Fig(3). Determination of
m1of a few meV range requires a high statistic data near (11) threshold. Distinction of normal vs
inverted hierarchies is most dramatic, as seen in Fig(4), hence its determination is easier.
RNPE rate of condensate decay increases like ∝ n3, effective with αm ∝ n, as the density n
increases. The event number from a single soliton decay is smaller than from the condensate decay,
typically by 1/(ηαmL) for a target length L ≫ 1/αmthat contains a soliton.
In an ideally coherent medium, field condensate never emits PSR. In practice, there may be a
variety of environmental effects that cause a leakage PSR, a potential background to RNPE. One of
these effects is a random fluctuation of dielectric constant, most simply due to a density fluctuation
√δn2. The resulting leakage PSR rate is estimated as
for a target length L. We used a Gaussian frequency distribution of width ∆mfor the trigger. We
have computed the Yb leakage PSR rate using
RNPE near (12) threshold of m1 = 2meV is found much larger than the background PSR. The
leakage PSR becomes larger than RNPE, only at photon energies ≤ 2µeV away from the first (11)
In summary, our proposed method of precision neutrino mass spectroscopy is most sensitive to
Majorana/Dirac distinction and to α,β measurements. It is worthwhile to experimentally investigate
both formation and long time control of solitons and condensate, which is of crucial importance for
controlled detection of PSR and RNPE. Some rudimentary method of efficient soliton formation has
been suggested in .
√δn2/n =5% and ∆m= 1GHz. The calculated Yb
for discussion on experimental aspects of this subject, and M. Tanaka for discussion on an aspect of
This research was partially supported by Grant-in-Aid for Scientific Research on Innovative Areas
”Extreme quantum world opened up by atoms” (21104002) from the Ministry of Education, Culture,
Sports, Science, and Technology.
I should like to thank N. Sasao and members of SPAN collaboration
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