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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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