6.8 Efimov States and Their Fano Resonances in a Neutron-Rich Nucleus
105
general resonance profile is asymmetric, as pointed out by Fano [103] over four
decades ago. While such asymmetric profiles have been widely observed in atoms
and molecules, resonances in nuclear and particle physics generally show symmetric
Lorentzian or Breit–Wigner shapes. It is the combination of the features of the Efimov
and Fano phenomena which, in fact, holds promise for the possible observation of
such resonances with the asymmetry being used as a diagnostic for the Efimov effect.
Our study of the Efimov bound state moving above threshold into the continuum is
very much analogous to the recent observation in ultracold cesium which tracked
a similar weakening of binding and disappearance as reflected in the loss rate of
cesium atoms from an optical trap [104].
In atoms, resonances occur widely due to doubly and multiply excited states,
such states often lying in the midst of a continuum built on ground or other low-lying
states [105]. Such resonances have also been increasingly common in recent studies
of exotic condensed matter systems, where they are sometimes associated with the
Kondo phenomenon [106].
Given its asymmetric profile, a resonance is described by three parameters, the
energy position E r , width and a so-called profile index q [104, 105]. Only when
this last parameter becomes large does the profile reduce to the symmetric Breit–
Wigner form, more familiar to physicists as a resonance, especially in nuclear and
particle physics. The reason for this reduction lies in q, which is the ratio of two
quantities, the amplitude through the discrete state and the direct amplitude to the
underlying continuum. In those instances, when the latter is small or negligible and
the embedded discrete state is dominant, the general profile reduces to Breit–Wigner,
characterized by just two parameters, E r and .
The study on n−
19 C elastic scattering predicting a resonance peak in the cross
section at energies of a few keV, discussed in the preceding section, showed a clearly
asymmetric resonance profile. By tuning parameters for the two-body energy of
n +
18 C, just above 220 keV when the first excited state of
20 C becomes unbound, the
elastic cross section exhibits a resonance as shown in Fig. 6.20. Upon fitting these
resonances to the Fano formula [103,105],
σ = σ 0
(q + ε)
2
/
1 + ε
2
,
where ε = (E − E r )/((/2) is a dimensionless, reduced energy measured from the
central position in units of the width, and σ 0 the background cross section far from
the resonance, we get the parameters shown in Fig. 6.20.
As pointed out in the earlier section, as we tune the two-body energy beyond
220 keV, the first Efimov state disappears to become quasi-bound state in n +
19 C
continuum and we observe the resonance in elastic scattering. A similar effect for
the second excited Efimov state was seen to disappear at above 140 keV binding
energy and we verified the appearance of a Fano resonance at energy above 140 keV,
as shown in Fig. 6.21, pointing to the generality of the phenomenon. Our obtaining
the same value of q for the two resonances is in conformity with their constituting a
sequence and lends further support to our Efimov interpretation.
105
general resonance profile is asymmetric, as pointed out by Fano [103] over four
decades ago. While such asymmetric profiles have been widely observed in atoms
and molecules, resonances in nuclear and particle physics generally show symmetric
Lorentzian or Breit–Wigner shapes. It is the combination of the features of the Efimov
and Fano phenomena which, in fact, holds promise for the possible observation of
such resonances with the asymmetry being used as a diagnostic for the Efimov effect.
Our study of the Efimov bound state moving above threshold into the continuum is
very much analogous to the recent observation in ultracold cesium which tracked
a similar weakening of binding and disappearance as reflected in the loss rate of
cesium atoms from an optical trap [104].
In atoms, resonances occur widely due to doubly and multiply excited states,
such states often lying in the midst of a continuum built on ground or other low-lying
states [105]. Such resonances have also been increasingly common in recent studies
of exotic condensed matter systems, where they are sometimes associated with the
Kondo phenomenon [106].
Given its asymmetric profile, a resonance is described by three parameters, the
energy position E r , width and a so-called profile index q [104, 105]. Only when
this last parameter becomes large does the profile reduce to the symmetric Breit–
Wigner form, more familiar to physicists as a resonance, especially in nuclear and
particle physics. The reason for this reduction lies in q, which is the ratio of two
quantities, the amplitude through the discrete state and the direct amplitude to the
underlying continuum. In those instances, when the latter is small or negligible and
the embedded discrete state is dominant, the general profile reduces to Breit–Wigner,
characterized by just two parameters, E r and .
The study on n−
19 C elastic scattering predicting a resonance peak in the cross
section at energies of a few keV, discussed in the preceding section, showed a clearly
asymmetric resonance profile. By tuning parameters for the two-body energy of
n +
18 C, just above 220 keV when the first excited state of
20 C becomes unbound, the
elastic cross section exhibits a resonance as shown in Fig. 6.20. Upon fitting these
resonances to the Fano formula [103,105],
σ = σ 0
(q + ε)
2
/
1 + ε
2
,
where ε = (E − E r )/((/2) is a dimensionless, reduced energy measured from the
central position in units of the width, and σ 0 the background cross section far from
the resonance, we get the parameters shown in Fig. 6.20.
As pointed out in the earlier section, as we tune the two-body energy beyond
220 keV, the first Efimov state disappears to become quasi-bound state in n +
19 C
continuum and we observe the resonance in elastic scattering. A similar effect for
the second excited Efimov state was seen to disappear at above 140 keV binding
energy and we verified the appearance of a Fano resonance at energy above 140 keV,
as shown in Fig. 6.21, pointing to the generality of the phenomenon. Our obtaining
the same value of q for the two resonances is in conformity with their constituting a
sequence and lends further support to our Efimov interpretation.