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arXiv:1101.2749v2 [hep-ph] 24 Jan 2011

Solitons and Precision Neutrino Mass Spectroscopy

M. Yoshimura

Center of Quantum Universe, Faculty of Science, Okayama University

Tsushima-naka 3-1-1 Kita-ku Okayama 700-8530 Japan

ABSTRACT

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.

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Introduction

neutrino mass), the nature of masses (whether they have Majorana or Dirac type masses), and their

relation to the leptogenesis theory [1], [2] 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 [3], [4], 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 [5], [6]. A similar idea goes back to the superradiance (SR for short)

[7] 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

spontaneous decay.

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 [5], 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 [8], [6] by eliminating time memory

Neutrinos are still mysterious particles: their absolute mass scale (or the smallest

We consider atoms that

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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

6πγpa

ǫ2

3π(ǫpe+ ǫpg)

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 [8], [6].

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 [9]. The Maxwell equation in terms of θ(x,t) is

µaa=

pa(ǫ2

pa− ω2),(a = e,g),(1)

µeg= µge=

?γpeγpg

ǫ3

peǫ3

pg

.(2)

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),

αm= 6π

ǫ3

pg

(3)

?γpeγpg

peǫ3

ωn

ǫpe+ ǫpg,(4)

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 [6] 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 [6].

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.

Soliton solutions

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 [6] 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

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obtained has a Lorentzian shape of flux and the population given by

|Es(x,t)|2=2αm

cosθs(x,t) =−α2

µ

v(1 − v)

α2

m(x − vt)2+ (1 − v)2

m(x − vt)2+ (1 − v)2, (5)

α2

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

by

|E(x,t)|2=αm

X =αm

µ

v

1 − v

(x − vt).

A2− 1

A − cosX,

cosθ(x,t) =AcosX − 1

A − cosX

, (7)

√A2− 1

1 − v

(8)

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.

Field condensate

∞, simultaneous with the limit v → 0. Denoting A =?(η/v)2+ 1 with η kept constant, one has

|Ec(x.t)|2=

One may consider the limit of large soliton density, ns/αmL ∼√A2− 1/4π →

αmη2

µ(η − vcos(αmη(x − vt)/v))≈ηαm

µ

, (9)

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

mη/ω)δE gives

We first mention PSR rate with-

egn2V/(29π2), which is numerically ∼

geǫ4

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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 [6]

µ2

geǫegn2V L|E|2

32π

.(10)

Although the rate for |E|2≈ 106Wcm−2is large, time structure of PSR is complicated [6].

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) )

dΓ(x,t) =µ2

geαmǫegn2V

16πµ

v(1 − v)

α2

m(x − vt)2+ (1 − v)2dx.(11)

The rate remains large during a time of

∆t =1 − v

vαm

∼ 14µsec1 − v

v

1012cm−3

n

. (12)

The space integrated rate per soliton is

µ2

geǫegvn2V

16µ

∼ 5 × 1015Hz

vn2V

1024cm−3, (13)

(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 O[1]µsec 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

destruction.

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.

RNPE

The effective Hamiltonian for RNPE,

Quasi-solitons exist only for κ < κc where

GF?Se·?

ijcijν†

j? σνi

ǫpg− ω

?d ·?E ,

(15)

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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 [3], [4].

The field operator νifor the Majorana neutrino is a superposition of annihilation (bi) and creation

(b†

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

emission;

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†

ib†

j(cij−cji)/√2 = i√2ℑcijb†

ib†

id†

j

48n2V ηαmG2

ǫ3

Fγpg

pg(ǫpg− ω)2µ

?

ij

BijIij(ω). (16)

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.

1.0685 1.06901.06951.0700

eV

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

Hz

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

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phase space. The integral in a symmetric form is given in terms of neutrino energies Ei,i = 1,2,

Iij(ω) = ω

?∞

0

?∞

0

dE1dE2δ(E1+ E2+ ω − ǫeg)θ(Cij(E1,E2))

?

K(1)

ij

12

+3K(2)

ij

4

?

, (17)

K(1)

ij = 2G(1)

ij(E1,E2), K(2)

ij = −G(1)

i− m2

ij(E1,E2) + G(2)

ij(E1,E2) +E1E2− δMmimj

ω2

, (18)

G(1)

ij=1

8+E2

8−E2

1+ E2

2− m2

4ω2

2− m2

4ω2

j

− 3(E2

+(E2

1− E2

2− m2

8ω4

2− m2

8ω4

i+ m2

j)2

,(19)

G(2)

ij=1

1+ E2

i− m2

j1− E2

i+ m2

j)2

, (20)

where the boundary region is given by Cij(E1,E2) ≥ 0 with

Cij(E1,E2) = (E2

1+ E2

2− m2

i− m2

j− ω2)2− 4(E2

1− m2

i)(E2

2− m2

j), (21)

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

G2

Fγpgǫ2

egn2V ηαm

µǫ5

pg

f(2ω

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 [4]

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 [2].

Distinction of Majorana and Dirac neutrinos is possible by the interference effect of identical Ma-

jorana fermions [3], 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

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1.0680 1.06851.0690 1.0695 1.0700

eV

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

Hz

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.

1.0694 1.06961.0698

eV

1.?10?7

2.?10?7

3.?10?7

4.?10?7

5.?10?7

6.?10?7

7.?10?7

Hz

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.

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1.0671.0681.0691.070

eV

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

Hz

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[30] 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

3µ2

geǫegn√δn2V αmLe−(ω−ǫeg/2)2/∆2

32πµ

m

×

dω

√π∆m, (23)

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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)

threshold.

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 [6].

√δn2/n =5% and ∆m= 1GHz. The calculated Yb

Acknowledgements

for discussion on experimental aspects of this subject, and M. Tanaka for discussion on an aspect of

leptogenesis.

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

References

[1] M. Fukugita and T. Yanagida, Phys. Lett. B 174 45 (1986).

[2] S. Davidson and A. Ibarra, Nucl. Phys. B648, 345 (2003), and references therein.

[3] M. Yoshimura, Phys. Rev.D75, 113007(2007).

[4] M. Yoshimura, A. Fukumi, N. Sasao and T. Yamaguchi, Progr. Theor. Phys.123, 523(2010).

[5] M. Yoshimura, C. Ohae, A. Fukumi, K. Nakajima, I. Nakano, H. Nanjo, and N. Sasao, Macro-

coherent two photon and radiative neutrino pair emission, arXiv 805.1970[hep-ph](2008). M.

Yoshimura, Neutrino Spectroscopy using Atoms (SPAN), in Proceedings of 4th NO-VE Interna-

tional Workshop, edited by M. Baldo Ceolin(2008).

[6] M.Yoshimura,

arXiv:1012.1061 [hep-ph] (2010), and to appear in Progr. Theor. Phys.(2011).

Light Propagation andPairedSuperradiance inCoherent Medium,

[7] For an excellent review of both the theory and experiments of superradiance, M. Benedict,

A.M. Ermolaev, V.A. Malyshev, I.V. Sokolov, and E.D. Trifonov, Super-radiance: Multiatomic

coherent emission, Informa (1996). For a formal aspect of the theory, M. Gross and S. Haroche,

Phys.Rep.93, 301(1982). The original suggestion of superradiance is due to R.H. Dicke, Phys.

Rev.93, 99(1954).

[8] L.M. Narducci, W,W. Eidson, P. Furcinitti, and D.C. Eteson, Phys. Rev.A 16, 1665 (1977).

[9] S.L. McCall and E.L. Hahn, Phys. Rev.183, 457(1969). For a review, L. Allen and J.H. Eberly,

Optical Resonance and Two-level Atoms, Dover, New York, (1975). For comparison with exper-

imental results, R.E. Slusher and H.M. Gibbs, Phys. Rev.A4, 1634(1972).

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