Superfluorescence in the presence of inhomogeneous
broadening and relaxation
B. Balko, I.W.Kay, J.D. Silk
Institute forDefenseAnalyses,Alexandria,VA22311,USA
R.Vuduc
CornellUniversity, Ithaca,NY14850,USA
and
J.W.Neuberger
University ofNorthTexas,Denton, TX76203,USA
In this paper we show how inhomogeneous broadening produces dephasing, inhibits
cooperative emission and thus reduces the intensity of the SF pulse. We also show how
electronic relaxation or time-dependent hyperfine interactions can mollify the effect of
inhomogeneous broadening so that SF can be recovered.
1. Introduction
One of the early concepts introduced for developing a c-ray laser [1] was based
on the idea of direct emission from isomeric nuclear levels. Such levels, because of
their long lifetime, permitted long pumping times and were considered to be good
storage levels. Unfortunately, because of the narrow natural linewidths associated
with the long lifetimes, and the relatively large environmental effects which cause
inhomogeneous broadening, resonance conditions for transitions involving such
levels are difficult to achieve. Several approaches to alleviate the problem have been
put forth. One of the approaches proposes to reduce the line broadening bymeans of
external fields [2,3].
Inhomogeneous broadening effects can also be overcome by internal fluctuating
fields. Such dynamic effects produce characteristic lineshape modifications in
nuclear resonance spectra. The process is commonly referred to as relaxation [4,5].
The purpose of this paper is to present a model of SF in the presence of inhomoge-
neous broadening and relaxation and to examine the proposition that time-depen-
dent hyperfine interactions can be used to recover SF in inhomogeneously
broadened systems. SF emission is interesting because when a c-ray laser is finally
Hyperfine Interactions 107 (1997) 369^379 369
Ä J.C. Baltzer AG, Science Publishers

developed it will probably operate in the SF mode. We show how the Maxwell^
Bloch equation used to model SF can be modified to account for inhomogeneous
broadening and determine to what extent relaxation in an inhomogeneously broad-
ened environment can modify the effect of broadening and how it can influence the
recovery of SF emission.
2. Superfluorescence
SF can be viewed as a spontaneous emission by N coupled, cooperating radiators
which produces the experimentally interesting effect depicted in fig. 1; a short pulse
with intensity proportional to N
2
, occurring at a fraction of the natural lifetime
… …1=N†
n
†. For a sample shaped like a needle the pulse is emitted along the axis.
The SF process is initiated by quantum fluctuations of the vacuum electromag-
netic field which gives rise to a phasing or buildup of correlations between the radia-
tors in the inverted region. SF is governed by the dynamics of the evolving system of
Maxwell^Bloch equationswith randomnoise sources [6]. No simple threshold condi-
tion connecting the pertinent parameters exists as in the case of lasing
#1
. Conditions
for the occurrence of SF are described in terms of characteristic times representing
the speed of competing processes: namely, the single nucleus lifetime 
n
, the radiative
lifetime 
r
, the cooperation time 
c
, the time of emission of the SF pulse 
SF
, the delay
time of the pulse 
D
, the dephasing time
#2

’
and the pumping time 

. The coopera-
tion time
#3

c
sets the absolute upper limit on the SF time 
SF
and consequently on
the delay time 
D
. Intuitively one expects that for SF to occur the following condi-
tions should hold:


< 
D
< 
’
; 
n
; 
c
:
More rigorous analysis [9] shows that


D

SF
p
< 
’
#1
For the steady-state nuclear laser the Schawlow^Townes gain condition given by Trammell and
Hannon [7] is k ˆ …
2
=2p†…f
n

=…1‡ †…1‡ a†† ÿ  > 0, where k is the gain coefficient,  is the
wavelength of the emitted radiation, f is the recoilless fraction of the transmission,  the internal
conversion coefficient, a the inhomogeneous broadening parameters and  the linear attenuation
coefficient. This condition is modified for the pulsed laser to account for the reduction of the inver-
sion during the emission process, but a sharp cutoff is still present. As we show in ref. [10] this is not
the case for SF emission.
#2
This commonly is due to collisions or inhomogeneous broadening.
#3
The cooperation time is obtained from the Dicke formula for the emission rate [8],

c
ˆ
1
4
N
c
…1=t
r
†
2
=A, whereN
c
is the number of emitters,  the linewidth of the radiation andA the
cross section of the rod forming the inverted region, and a self-consistent argument determining
how many atoms can be covered by the emission of a single atom during the Dicke emission time,
1=
c
.WithN
c
ˆ …c=
c
†Awe get 
c
ˆ 2=

cÿ
2
p
.
B.Balko et al. / Superfluorescence370

is actually the appropriate condition between the delay time, SF time, and the
dephasing time.
The basis for our theoretical model of nuclear SF is described in detail in ref. [10]
and is the set of Maxwell^Bloch equations in dimensionless units in the form intro-
duced byHaake andReibold [6]
@
@t
N
3
ˆ
N
3
; …1a†
@
@t
N
2
ˆ ÿ…E
‡
R
‡
‡ E
ÿ
R
ÿ
† ÿ ÿN
2
‡
N
3
; …1b†
@
@t
N
1
ˆ ÿ…E
‡
R
‡
‡ E
ÿ
R
ÿ
† ÿ ÿN
2
; …1c†
@
@t
R

ˆ …N
2
ÿN
1
†E

ÿ
1
2
‰ÿ ‡ ÿ

ŠR

‡ 

; …1d†
@
@x
E

ˆ g
0
R

ÿ
1
2
E

: …1e†
In the above equations distance is normalized to the length of the active region l
and retarded time is used normalized to the superfluorescence time given by

SF
ˆ
8p
0
3
2
l
Fig. 1. Superfluorescent pulse emission. The top figure shows the angular distribution of the radiation
and the bottom figure the temporal variation of the pulse. In both cases a comparison with normal
non-cooperative spontaneous emissionwith each nucleus emitting independently ismade.
B.Balko et al. / Superfluorescence 371

where 
0
is the natural lifetime,  is the inversion density of nuclei, and  is the photon
wavelength.
The quantities N
1
, N
2
, N
3
in eq. (1) are the population density operators of
the lowest, middle, and upper levels of a three-level atomic or nuclear structure.
The quantities R

are the positive and negative frequency components of the
polarization operator, and the quantities E

are the positive and negative fre-
quency components of the electric field operator. The quantities 

similarly
represent noise operators due to quantum electrodynamic fluctuations of the
vacuum state. These quantities and their effect on the SF emission are discussed
in detail in ref. [10].
Three decay constants that are associated with emission linewidths occur expli-
citly in eqs. (1a)^(1e). One of the constants is the pumping rate
which is identified
with the natural linewidth of the pumping level, another is ÿ , the inverse of the life-
time (
ÿ1
0
), which is identified with the natural linewidth of the SF level, and the third
isÿ

, which is a dephasing parameter associatedwith the homogeneously broadened
line.
In addition, we can associate a fourth decay constant, ÿ

, with the inhomogen-
eously broadened SF linewidth. An inspection reveals that it does not appear expli-
citly in any of the eqs. (1a)^(1e). Its physical effect must therefore be different from
that of ÿ

, which, as indicated on the right side of eq. (1d), is simply to increment the
natural linewidth.
3. Time-dependent coupling parameter
Eq. (1e) has to bemodified before themodel can be applied to SF emission in inho-
mogeneously broadened systems. The original equations apply to identical nuclei
which see precisely the same environment. Inhomogeneous broadening introduces
decorrelation between different nuclei because of their slightly different resonant
energies. It turns out that the required modification is simply to change the coupling
constant g to a time-dependent parameter.
A number of authors [11^13] have derived the effect of inhomogeneous broaden-
ing in earlier models of SF. Although the derivations are essentially equivalent, the
one given by Bonifacio and Lugiato in ref. [11] is particularly convenient for adapta-
tion to ourmodel.
Those authors observe that the rotating wave approximation of the interaction
Hamiltonian
#4
leading to theMaxwell^Bloch equations for their case has the form
#4
Ref. [11] uses a special representation that removes the Schr

odinger picture time dependence of
the free field photon and atomic operators. The resulting interactionHamiltonian is therefore time
dependent.
B.Balko et al. / Superfluorescence372

H1
…t† ˆ ÿ
ih

V
p
X
N
a
X
k
g
k
‰a
‡
k
R
ÿ
…a†f

…kÿ a; t† ÿH:C:Š ; …2†
where a represents the reciprocal lattice modes and the coefficients g
k
outside the
brackets in the double sum on the right side are coupling constants relating the
photon and polarization operators, and the c number function appearing inside the
brackets is given by
f …g; t† ˆ
1
N
X
N
jˆ1
e
ig
x
e
…!
j
ÿ!
0
†t
: …3†
The outer sum in eq. (2) is over the individual atomic positions at the lattice points
in a crystal lattice, and the polarization operatorsR

in each term of the sum charac-
terize the state of the corresponding atom.
The time-dependent factor in each term in the sumon the right side of eq. (3) intro-
duces a phase shift due to a shift !
j
about the resonant frequency !
0
. Arriving
photons emitted by atoms at other locations, because of phenomena such asDoppler
shifts caused by differing relative motion of the remote atoms or different hyperfine
fields at different sites, contribute to the different !
j
. The resulting phase shifts,
which are random, are responsible for the inhomogeneous line-broadening effect.
Eq. (3) is clearly an average of a large sample of random phase factors from some
statistical distribution and should, therefore, reduce to the ensemble average of the
randomvariable representing a single phase factor, i.e.,
f …g; t† ˆ he
ig
x
e
i…!ÿ!
0
†t
i ˆ he
ig
x
ihe
i…!ÿ!
0
†t
i ˆ F…g†g
1
…t† : …4†
Note that this equation includes the case in which the atomic locations may also be
random^ as long as they are statistically independent of the randomphase shifts.
As indicated by the notation in eq. (4), the function g
1
…t† is the average value of a
time-dependent random variable exp‰i…!ÿ !
0
†tŠ with some probability distribution
of frequencies. That is,
g
1
…t† ˆ
Z
1
ÿ1
^
g…†e
it
d ; …5†
where the function
^
g…!ÿ !
0
† is a normalized frequency distribution, which is usually
assumed to be eitherGaussian or Lorentzian.
The effect of inhomogeneous line broadening then is to replace the coupling con-
stant g
k
by a function of time obtained by multiplying g
k
by the function g
l
…t† that
the relations (4) and (5) define. The same argument applied to the corresponding
Hamiltonian in ref. [14], which, according to ref. [15], motivated the derivation of the
Maxwell^Bloch equations for the Haake^Reibold model, leads to a similar result.
An analysis similar to that of ref. [6] shows that the result carries over into eq. (1e).
If the normalized frequency distribution that determines g
1
…t† is Lorentzian the
B.Balko et al. / Superfluorescence 373

result is an exponentially decaying function of time. Thus, assuming a Lorentzian
frequency distribution, the introduction of inhomogeneous broadening into the
Haake^Reibold model is accomplished by replacing the coupling constant g
0
on the
right-hand side of eq. (1e) by the function
#5
g…t† ˆ g
0
e
ÿ…ÿ

=2†t
: …6†
4. Effect of relaxation on g
1
…t†
Using the time-dependent coupling function g
1
…t† instead of the constant g in eq.
(1e) we can model the effect of inhomogeneous broadening on the SF pulse shape.
Fig. 2a gives some calculated results showing the modification of the SF pulse shape
due to inhomogeneous and homogeneous broadening for comparison.
Homogeneous broadening is accounted for by the decay rate ÿ
’
in eq. (1c) and thus
plays a different role in the emission processes. For the same effective amount of
broadening (when ÿ
’
ˆ ÿ

†, inhomogeneous broadening has a more drastic effect
on the pulse shapes as shown in fig. 2b.
An inhomogeneously broadened line is formed by a collection of resonances each
displaced from the center frequency by an amount determined by some distribution
as depicted graphically in fig. 3a. In the figure only a small number of lines is shown,
each centered at a frequency !
i
, displaced from the center frequency !
0
, and each
representing a sum of resonances corresponding to the number of nuclei with that
particular resonance energy. The intensity of each line is determined by the pre-
scribed distribution. Shifting the resonances closer together provides an increase in
overlap of lines as shown in figs. 3b and c. Homogeneous broadening of the lines also
provides an increase in overlap as shown in figs. 3d and e but since the maximum is
also reduced, as discussed in ref. [17], the resonant effect between a source and absor-
ber remains small. In general both effects can contribute, as shown in figs. 3f and g.
Relaxation between the lines (!
0
‡ !
i
; !
0
ÿ !
i
) on opposite sides of the average
resonance, !
0
has the combined effect of broadening the lines and shifting them
toward the center until a total collapse at the outer frequency occurs at high relaxa-
tion rates, i.e., in the ``motionally narrowed region''. This effect has been observed in
M

ossbauer experiments in many compounds. In fig. 4 we show some typical spectra
of a classical paramagnet in a temperature range where at the low relaxation rate
(low temperature) a typical Zeeman splitting occurs. As the relaxation rate increases
the lines broaden until at very high rates there is a collapse to a single central line
#6
.
#5
This is equivalent to a special case of the result that was obtained elsewhere [16]:
…@=@x‡ …1=v†@=@t†E

…x; t† ˆ
R
p…v†R

…x; t; v†dv, taking into account the fact that the Haake^
Reibold independent variables involve the retarded time while Haake et al. [16] use the true time
and that p…v† replaces g…!ÿ !
0
† as the frequency distribution associated with detuning contribu-
tions from all atoms.
B.Balko et al. / Superfluorescence374

There are some special compounds which exhibit this simple classical paramagnetic
behavior, but, in general, more complex and not easily predictable behavior can
occur as described in ref. [18]. Also, transmission spectra can hide some complex
dynamic behavior, because completely different phenomena can give deceptively
similar transmission spectra. To uncover the true relaxation mechanisms more dis-
L
Fig. 2. Effect of inhomogeneous and homogeneous broadening on SF emission. In (a) the solid line
shows the pulse without broadening and the dashed and dotted lines show the pulses emitted from
homogeneously broadened and inhomogeneously broadened systems each with broadening of 200
times the natural linewidth. In (b) we show the effect of increasing broadening on the pulse intensity
with curve L obtained using a Lorentzian distribution and curveG using aGaussian distribution.
#6
This behavior can be easily understood if one assumes a classical field flippingmodel for the relaxa-
tion [4]. The parameter that determines the lineshape is the ratio of the relaxation rate to the
Larmor precession frequency of the levels. The lifetime of the excited state only determines the
minimum linewidth [5]. At low relaxation rates the full Zeeman-split spectrum is observed because
the nucleus responds to a particular field at its site. At very high relaxation rates with respect to the
Larmor precession frequency the nuclear spins cannot respond to the changing field and so the
average effect of zero field is observed. At intermediate relaxation rates the lines broaden and col-
lapse in a complicated fashion and a complete theory has to be used to describe the detailed line-
shapes.
B.Balko et al. / Superfluorescence 375

criminating experiments have to be performed [19]. Nevertheless one thing is clear.
However complex the relaxation mechanisms, the effect is to produce a simpler,
higher-intensity peak in the transmission spectrum in the high relaxation or
``motionally narrowed'' limit. It seems reasonable to expect that in this limit one
would also obtain an increase in the peak intensity in SF emission.
As can be seen from the previous section, the inhomogeneous broadening pro-
duces a temporal variation of the coupling factor g
1
…t† (eqs. (3) and (5)), even though
the resonance frequencies at different sites !
j
may be time independent. If the hyper-
fine fields cause the energies !
j
in eq. (3) to jump between different values, then the
correlation function in eq. (5) is modified and can be calculated using the approach
introduced in the literature [5] for dealing with time-varying fields. For example,
according to Blume, if the hyperfine energy jumps between two values !
j
and ÿ!
j
,
then the correlation function required in eq. (4),
g
1
…t† ˆ he
i…!ÿ!
j
†t
i ;
becomes
Fig. 3. A graphic representation of inhomogeneous broadening (a) and the increase in overlap pro-
duced by shifting lines (b,c), homogeneously broadening the lines (d,e), and the combined effect of
both (f,g).
B.Balko et al. / Superfluorescence376

g1
…t† ÿ he
i
R
t
0
f …t
0
† dt
0
i
av
ˆ cos x
t‡
1
x
sin x
t
 
e
ÿ
t
ˆ G…
;; t† ; …7†
where ˆ 2!
j
,
is the relaxation rate and x ˆ ‰…
2
=

2
† ÿ 1Š
1=2
.
To model inhomogeneous broadening  should take on values from ÿ1 to ‡1
subject to some probability distribution
…; † that describes the broadening. The
effective inhomogeneously broadened coupling factor then is given by
G…
; ;T† ˆ
R
1
ÿ1
…; †G…
;; t† d and replaces g
0
in eq. (1e). These approaches,
when applied to SF pulse emission, provide a reduction of the effect of inhomoge-
neous broadening.
Consider fig. 5a where we show two SF pulses calculated assuming the same
amount of inhomogeneous broadening in the system but different relaxation rates.
The increased relaxation rate produces a sharper, more intense, main peak and also a
smaller secondary peakwhich is characteristic of a strong SF emission. Fig. 5b shows
the effect of relaxation on SF intensity for inhomogeneously broadened systems
Fig. 4. Effect of relaxation on lineshape in a classic paramagnet. The numbers in the figure give the
relaxation rate or the flipping rate of the magnetic field (550 kG) producing the Zeeman splitting in
s
ÿ1
.
B.Balko et al. / Superfluorescence 377

when the broadening is modeled by Gaussian and Lorentzian distributions. The
Lorentzian distribution ismore effective in depressing the SF peak but at high relaxa-
tion rates both broadening effects can be overcome.
5. Conclusion
Our results show that while SF emission is suppressed due to inhomogeneous
broadening it can also be recovered through fast relaxation between levels producing
the broadening. For full recovery the relaxation has to be fast enough to produce a
L
Fig. 5. Effect of relaxation on SF emission in an inhomogeneously broadened system. In (a) the solid
line shows the pulse suppressed due to inhomogeneous broadening modeled by a Gaussian distribu-
tion with  ˆ 10
6
times the natural linewidth and the dashed line shows the pulse recovered with
relaxation (
ˆ 10
3
ÿ ). In (b) we show the effect of relaxation on SF in inhomogeneously broadened
systems with a Gaussian distribution (solid line) and Lorentzian distribution (dashed line). The
relaxation rate is given in units of ÿ.
B.Balko et al. / Superfluorescence378

collapse in an absorption spectrum to a single narrow line.However, partial recovery
of the SFmay also provide a large enough signal to be observed.
References
[1] V. Vali andW.Vali, Proc. IEEE 51 (1963) 1822.
[2] A.V.Andreev, Y.A. Il'inskii andR.V.Khokhlov, Sov. Phys. JETP 40 (1975) 816.
[3] Y.A. Il'inskii andR.V.Khokhov, Sov. Phys. JETP 38 (1974) 809.
[4] H.H. Wickman, in: Hyperfine Structure and Nuclear Radiations, eds. E. Mathias and
D.A. Shirley (NorthHolland, Amsterdam, 1968) p. 928^947.
[5] M. Blume, in: Hyperfine Structure and Nuclear Radiations, eds. E. Mathias and D.A. Shirley
(NorthHolland, Amsterdam, 1968) pp. 911^927.
[6] F.Haake andR.Reibold, Phys. Rev. A29 (1984) 3208.
[7] Trammell andHannon, Opt. Commun. 15 (1975) 325.
[8] F.T.Arrechi and E. Courtens, Phys. Rev. A2 (1970) 1730;
R.M.Dicke, Phys. Rev. 93 (1954) 68.
[9] M.F.H. Schuurmans andD. Polder, Phys. Lett. 72A (1979) 306.
[10] B. Balko, I.W.Kay and J.W.Neuberger, Phys. Rev. B 52 (1995) 858.
[11] R. Bonifacio and L.A. Lugiato, Phys. Rev. A11 (1975) 1507.
[12] F.Haake et al., Phys. Rev.A23 (1981) 1322.
[13] R.H. Eberly, Acta Phys. Pol. A39 (1971) 633.
[14] D. Polder, Phys. Rev.A19 (1979) 1192.
[15] F.Haake andR.Reibold, Phys. Rev. A29 (1984) 3208.
[16] F.Haake, J.W.House,M.King,G. Schroder andR.Glauber, Phys. Rev. A23 (1981) 1322.
[17] B. Balko, I.W. Kay, J. Nicoll, J.D. Silk and G. Herling, these proceedings (1st Int. Gamma-
Ray LaserWorksh., 1995),Hyp. Int. 107 (1997) 283.
[18] G.R.Hoy,M.R.Corson andB. Balko, Phys. Rev. B27 (1983) 2652.
[19] B. Balko, Phys. Rev. B33 (1986) 7421.
B.Balko et al. / Superfluorescence 379