1.1.7. Interference

Summary

Overall the system has at least two limits,

  1. strong interference regime (the wavelength of the perturbations are of the same orders);

  2. low-interference regime (one of the perturbations has a wavelength much larger than the others’).

For either case, the system is explained within the ralm of superpositions of multiple Rabi oscillations.

As for the first case, th criteria is related to both the resonance itself and the size of physical system we are interested in.

  1. Each mode has an associated resonance, and explained by a single Rabi oscillation. Higher orders have much smaller resonance width than low orders.

  2. Whether a mode is important to the system is determined by Q value, which is defined as the ratio of distance to resonance and resonance width.

  3. However a mode with a wavelength that is much larger than the system is not counted since the resonance has not accumulated any significant transition probabilities within the region of the physical system.

However, we do not have a cystal clear understanding of such a strong interference.

The second case is essentially some limit of the strong interference. With Komogorov spectra of matter, the resonance is simplified in some sense. One of the limits is that one of the matter perturbations has much larger wavelength than the others, which will behave as a shift of the background matter density, within one wavelength of the small wavelength perturbations. Here we use two frequencies as examples, c.f. Fig. 1.27.

  1. Even if the short wavelength one is exactly on resonance, adding the second perturbation as a background shift could move the system out of the resonance. The math behind it is related to the amplitude,

    |Bn|2|Bn|2+(nkωm)2,

    where nkωm=0 for the resonance situation. As we add in the second long wavelength perturbation, Bn, ωm will change. Nonetheless the relative change in Bn is small while the change in nkωm is huge compared to Bn value, if the amplitude of the new perturbation is large enough, i.e., A2|Bn|.

  2. The caveat is that the wavelength of the second perturbation k2 must be within a range much smaller than k1, and much larger than the resonance transition probability wavelength, roughly speaking, k1k2|Bn|.

  3. The newly added long wavelength merely has any effect when the system is in a region mod(k2x,π)0, since A2sin(k2x)0 for these values. Such regions probabily will appear multiple times within the size of physical system. Then we see accumulation effect, so that the transition amplitude will rise as we go furthure out.

From this point of view, the resonance is never destroyed since the accumulation will always be there.

In the calculation of multi-frequency matter background,

δλN=a=1NAasin(kax+ϕa),

we derived the equation of motion

ixΨ=HΨ,

where

Ψ=(ψb1ψb2),

is the wave function in the T basis (new basis unitary transform from background matter basis using T matrix) and

H=(0hNhN0),

where

hN=12n1=nN=BNΦNei(aNnakaωm)x,

and

BN=(i)ana(anaka)(aJna(Aakacos2θm)),ΦN=ei(anaϕa).

1.1.7.1. Apply Approximations to Two Frequencies

We could identify the important combinations of n’s {n1,,nN} so that we can approximate the actual result.

The important question is combining multiple lists of n’s is not intuitively easy to understand. Here we consider the effect of adding new list of n’s to the system.

For simplicity a simplified system will be used,

ix(ψb1ψb2)=(0A1eiϕ1x+A2eiϕ2xA1eiϕ1x+A2eiϕ2x0)(ψb1ψb2).

Notice that ϕn is always real but An are either real or pure imaginary.

The matrix equation can be rewritten as systems of differential equations,

ixψb1=(A1eiϕ1x+A2eiϕ2x)ψb2,ixψb2=(A1eiϕ1x+A2eiϕ2x)ψb1.

This set of equations is solved by rewrite them to a second order differential equation

(1.26)x2ψb2+A1eiϕ1x+A2eiϕ2x)x(A1eiϕ1x+A2eiϕ2x)xψb2+(A1eiϕ1x+A2eiϕ2x)(A1eiϕ1x+A2eiϕ2x)ψb2.

As a comparison, we also write down the equation for single n list,

x2ψb2+(iϕ1)xψb2+|A1|2ψb2=0.

Approximations?

For single n list equation, the second term dominates if ϕ1 is much smaller than 1. In the language of physics, the second term dominates if the system is very close to resonance.

ϕn and resonance

ϕn is in fact the deviation from exact resonance.

ϕn=aNnakaωm.

In the multi-n-list case, this domination is easily destroyed. As an example, suppose we have A1=A2=A and ϕ2=1010ϕ1, the second term in equation (1.26) becomes

eiϕ1x+ei1010ϕ1x)x(eiϕ1x+ei1010ϕ1x)xψb2=eiϕ1x+ei1010ϕ1x)(iϕ1eiϕ1xi1010ϕ1ei1010ϕ1x)xψb2,

which can be dropped on as long as ϕ1 is not as small as 1010.

We exaggerated the situation.

1.1.7.2. Resonance Destroyed

1.1.7.2.1. General Ideas

We first investigate two frequencies. The Hamiltonian for a single frequency matter perturbation is

H^=n=(012B^nei(nk^1)x^12B^nei(nk^1)x^0),

where

B^n=(i)ntan2θmnk^Jn(A^k^cos2θm).

In some conditions, even we have on of the matter frequancy at resonance, this resonance could be destroyed when a new matter frequency is destroyed. Numerical calculations show that this could happen if the new perturbation is off resonance and with larger Bn2.

Let’s first set this first perturbation at resonance. Suppose we add in another matter perturbation with a frequency which is higher, i.e., k2k. Since the wavelength of this new perturbation is much larger, we will assume it doesn’t change within one wavelength of the first perturbation. Thus it behaves as an additional background. Will this new background destroy the resonance? Illustration of this idea is shown in Fig. 1.27.

../../_images/second-freq-as-bg-illustration.png

Fig. 1.27 The two matter perturbations

To quantify this idea, we calculate the Q values for different modes with and without the new frequency. The procedure should be

  1. Prepare the parameters: ωv, λ0 (background matter profile), λ1 (perturbation amplitude for first perturbation), k1 (perturbation wavenumber for first perturbation)

  2. Calculate the Q values.

  3. Add in λ2 (perturbation amplitude for the second perturbation) and treat this new perturbation as a constant within one wavelength of the first perturbation. Just use some random phase for the new matter profile, i.e., set sin(k2x) to some resonanable value.

  4. Recalculate the Q values.

Without the new perturbation (Mathematica Code):

deltamsquare13 = 2.6*10^(-15);(*MeV^2*)
energy20 = 20;(*Energy in units of MeV*)
thetaV = ArcSin[Sqrt[0.093]]/2
omegaVKK = OmegaVacuum[energy20, deltamsquare13](*MeV*)(*deltamsquare13/(2 energy20)(*MeV*)*)

lambdaNKK = 0.5*Cos[2 thetaV] omegaVKK (*MeV*)

onekListKK = {1};
oneaListKK =(*{0.1};*){0.02 (MeVInverse2km[ 2 Pi/(omegaMKK onekListKK[[1]])]/1500)^(5/3)};
onephiListKK = {0};

Part[qValueOrderdList[listNGenerator[1, 10], onekListKK, oneaListKK, onephiListKK, thetamV], 1 ;; 10];
Grid@%

which returns the Q value results for each modes:

{1}  0.
{-1} 577810.
{2}  1.75394*10^10
{-2} 5.26182*10^10
{3}  4.25927*10^15
{-3} 8.51854*10^15
{4}  1.16361*10^21
{-4} 1.93935*10^21
{5}  3.76761*10^26
{-5} 5.65142*10^26

Adding in the new frequency:

phi2 = Pi/2/100;

twoaListKK =(*{0.1,0.1};*){0.02 (MeVInverse2km[2 Pi/(omegaMKK*twokListKK[[1]])]/1500)^(5/3),
0.02 (MeVInverse2km[2 Pi/(omegaMKK twokListKK[[2]])]/1500)^(5/3)};

lambdaNKK2 = 0.5*Cos[2 thetaV] omegaVKK + Sin[phi2]*twoaListKK[[2]]*omegaMKK(*MeV*)

omegaMKK2 = OmegaMatter2[lambdaNKK2, thetaV, omegaVKK](*MeV*)
thetamV2 = Mod[ArcTan[Sin[2 thetaV]/(Cos[2 thetaV] - (lambdaNKK2/omegaVKK)^2)]/2, Pi]

onekListKK2 = {1}*omegaMKK/omegaMKK2;
oneaListKK2 =(*{0.1};*){0.02 (MeVInverse2km[2 Pi/(omegaMKK2 onekListKK2[[1]])]/1500)^(5/3)};

Part[qValueOrderdList[listNGenerator[1, 10], onekListKK2, oneaListKK2, onephiListKK, thetamV2], 1 ;; 10]
Grid@%

and the results for Q values of different modes are:

{1}  246.833
{-1} 577687.
{2}  1.75752*10^10
{-2} 5.26655*10^10
{3}  4.27026*10^15
{-3} 8.53505*10^15
{4}  1.16758*10^21
{-4} 1.94507*10^21
{5}  3.78384*10^26
{-5} 5.67375*10^26

The corresponding plots are shown in Fig. 1.28 and Fig. 1.29

../../_images/second-freq-as-bg-example-1-first-freq-only.png

Fig. 1.28 Only the first frequency which is at resonance

../../_images/second-freq-as-bg-example-1-with-second-freq-as-bg.png

Fig. 1.29 With the second frequency added in but only as a shift in background density

What determines the amplitude is

|Bn|2|Bn|2+(nkωm)2.

In this example, the coefficient in unit of ωm for one perturbation only is

|B1|=3.46135×106,

while it shifts a little bit when the second frequency is added in as a background shift, in unit of ωm,

|B1|=3.46356×106.

We set the first frequency at resonance, which means

kωm=0.

With the apprearance of the second frequency, what we have now is, in unit of ωm,

kωm=0.000854192,

which is far beyond the width of the resonance.

1.1.7.2.2. Some Artifical Systems

../../_images/second-freq-as-bg-1-divided-10-10-20-30-100-1000-10000-matter-density.png

Fig. 1.30 Matter profiles with one wavelength of the first perturbation

The resonance from the first perturbation is destroyed as the second perturbation grows much larger than it.

../../_images/second-freq-as-bg-1-divided-10-10-20-30-100-1000-10000-full-numerical.png

Fig. 1.31 Full numerical solutions

The hint is, the shift of the background matter profile is related to the destruction effect, whether it’s true destruction or effective destruction (destruction within a region).

An investigation of the most important mode shows that it is destroyed due to the a shift of background matter density.

../../_images/second-freq-as-bg-1-divided-10-10-20-30-100-1000-10000-log.png

Fig. 1.32 Resonance destruction of the first mode

To verify how the interference actually works, we plot the full numerical calculation and the {1,0} mode, for comparision

../../_images/second-freq-as-bg-1-d-10-1-d-1000-1-d-10000-1-d-200000-1000-full-numerical-with-10-mode-with-gridlines.png

Fig. 1.33 The solid lines are the full numerical calculations of the system; Dashed lines are {1,0} mode of the corresponding parameters; Vertical grid lines are the positions of zero amplitudes of the second matter perturbation.

It is clearly shown in Fig. 1.33 that each vicinity of zero amplitude for the second perturbation, the transition amplitude will increase, due to the resonance of the first perturbation.

What’s more interesting is that as A2 becomes much larger than A1, the resonance seams to be destroyed. As an example, here we take A2=0.0346135,A1=0.0000357347 both in unit of ωm, which means A2 is almost three orders larger than A1. The results are shown in Fig. 1.34.

../../_images/second-freq-as-bg-1-d-10-1-d-1000-1-d-10000-1-d-200000-10000-full-numerical-with-10-mode-with-gridlines.png

Fig. 1.34 The solid lines are the full numerical calculations of the system; Dashed lines are {1,0} mode of the corresponding parameters; Vertical grid lines are the positions of zero amplitudes of the second matter perturbation.

1.1.7.3. Refs & Notes


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