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\markboth{\small \quad \em H.~J.~Lipkin \hfill}{\hfill\small \em 
Coherent effects \quad}

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\begin{document}
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\title{COHERENT EFFECTS IN TRANSITIONS BETWEEN STATES CONTAINING SEVERAL NUCLEAR EXCITONS}

\author{HARRY J. LIPKIN}

\address{Department of Particle Physics
  Weizmann Institute of Science, \\ Rehovot 76100, Israel}

\address{School of Physics and Astronomy,
Raymond and Beverly Sackler Faculty of Exact Sciences,
Tel Aviv University, Tel Aviv, Israel}  
 
\address{Advanced Photon Source, Argonne National Laboratory,\\
Argonne, IL 60439-4815, USA}


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\maketitle\abstracts{  When ten separated identical nuclei are all excited to
the same first excited state they decay independently with the natural line
width and lifetime. But a standard Wigner-Weisskopf treatment of the initial
state as a pure quantum state says that it decays exponentially to a state with
nine excited nuclei with ten times the natural width and ten times the
exponent for a single decay. The resolution of this paradox would have amused
and interested Misha Marinov. We explore the possibility that a synchrotron
radiation beam may be sufficiently intense to excite more than a single
M\"ossbauer nuclear resonance in the same sample in the same pulse; i.e. to
excite a coherent state of several nuclear excitons.   We describe
multi-exciton states using a quasispin description of the states of the
nucleus. We then consider the decay and production of a two-exciton  state and
find that the two-exciton decay is twice as fast as a single exciton decay and
therefore that the width for exciting a second exciton when one exciton is
already present is twice the width for exciting a single  exciton. The
implications of this factor of two are examined and the  treatment is then
generalized to the three-exciton and multi-exciton cases. The conclusion that
the width for the excitation of an  additional nuclear exciton is enhanced by a
factor $n+1$ when $n$ excitons are already present is interesting and deserves
further investigation.}

%\vspace{0.4cm}
%\newpage

\tableofcontents

\newpage

\section{Introduction. A paradox in the quantum mechanics of multiply-excited
systems}

\subsection {Simple coherence effects in excitation transition probabilities}

The possibility that synchrotron radiation sources\,\cite{Gerdau} may be available with
sufficient intensity to allow the excitation of several nuclei in a 
sample\,\cite{HASYLAB}
by a single pulse leads to investigation of multiple excitations in solids.
New coherence phenomena arise which first lead to an apparent paradox. 
The essentials can be seen in the simple example of a system with two  nuclei,
denoted by $A$ and $B$ which can be excited from the ground state by a
synchrotron radiation pulse. Let $\ket{0}$, $\ket{A}$, $\ket{B}$ and $\ket{AB}$
denote respectively the states where neither nucleus  is excited, nucleus $A$
is excited, nucleus $B$ is excited and where both nuclei are excited. Let the
transition operator denoted by $T$ describing the excitation of a nucleus 
by a synchrotron radiation pulse be equal for the two nuclei and independent
of whether the other nucleus is excited. Then the matrix elements of the
transition operator satisfy the relation
\beq{twonuc} 
\bra{A}T\ket{0} =  \bra{B}T\ket{0} =  \bra{AB}T\ket{A} = \bra{AB}T\ket{B} 
\equiv T \,.
 \eeq

A state with a single excitation produced when a pulse of synchrotron radiation
strikes both nuclei equally is described as
\beq{twoex} 
\ket{1} \equiv {1\over \sqrt 2}\cdot \{\ket{A} + e^{i\phi} \cdot \ket{B}\};
 ~ ~ ~ ~ |\bra{AB}T\ket{1}|^2 = |T|^2 \cdot \{1 + \cos \phi\}\,,
 \eeq
where $\phi$ denotes the relative phase of the two excitations which depends
upon the production mechanism.

The transition probability $|\bra{AB}T\ket{1}|^2$ for producing a
double excitation from the singly excited state $\ket{1} $ depends
upon the relative phase $\phi$ between the excitations of nucleus $A$
and nucleus $B$. For the case where $\phi=0$ we see that the
transition probability is $2|T|^2$ or double the transition
probability for a single excitation. This enhancement factor of two
arises when the relative phase $\phi$ has just the value to produce
constructive interference between the two single excitations.

We shall see in detail below that the condition for this constructive
interference is exactly produced in realistic systems with many nuclei and  a
single excitation in the case where the first excitation  is a nuclear 
exciton\,\cite{Gerdau,Trammell}$^{\mbox -}$\cite{HanTram}
%,HTREVIEW,HanTram}  
produced by the same synchrotron
radiation pulse that produces the second excitation. However, in all other
cases the relative phase $\phi$ is random and the there is no enhancement. 

We now extend this to the case of three nuclei with the third nucleon denoted
by $C$. We define the coherent double excitation state where any pair of the
three  nuclei is excited, and the phase produces constructive interference for
the third excitation,   
\beq{threeex}
\ket{2_{coh}} \equiv {1\over \sqrt 3}\cdot \{\ket{AB} + \ket{BC} +  \ket{CA}\};
 ~ ~ ~ ~ |\bra{ABC}T\ket{2_{coh}}|^2 = 3 \cdot |T|^2\,. 
 \eeq
 
Extending this treatment to the case of an initial state where (n-1) coherent
excitations are already present;  e.g. a state containing (n-1) nuclear
excitons, leads to the result that the transition  probability  for producing a
state with n-excitations from  this state is enhanced by a factor n. This 
suggests a picture with a collective excitation which behaves like a boson,
where the creation of such a boson is enhanced by the same factor as the
stimulated emission of any boson when other bosons are  already present.

In the remainder of this paper we consider the production and decay of nuclear
excitations in detail. The result will show that this ``boson-enhancement"
picture has a general validity for the case of multiple production of nuclear
excitons, and that it is possible to define operators which create and
annihilate nuclear excitons which obey boson commutation rules to a very good
approximation.

\subsection {Simple coherence effects in decays of multiply excited states}

The decays of the states $\ket{AB}$ and $\ket{ABC}$ with double and triple 
excitations respectively  are simply described by the Fermi Golden Rule of
time-dependent perturbation theory as  proportional to the square of the
transition matrix element (\ref{twonuc}) of the operator $T$. This gives
the well-known result that each excited nucleus decays independently 
with the standard exponential time dependence $e^{-\lambda t}$ where 
$\lambda$ is the decay constant proportional to $|T|^2$ given by the Fermi 
Golden rule for a single decay.

However, we can also describe the decay as a cascade of several decays using
a basis of the coherent states,
\beq{caSC3}
\ket{ABC} \rightarrow \ket{2^\gamma_{coh}} \rightarrow \ket{1} \rightarrow
\ket{0}\,,
 \eeq
where the coherent doubly-excited state of three nuclei
is now redefined to explicitly include the photon emitted by  the third nucleus, 
\beq{threerec} \ket{2^\gamma_{coh}} \equiv {1\over \sqrt 3}\cdot
\{\ket{AB\gamma_C} + \ket{BC\gamma_A} +   \ket{CA\gamma_B}\}; ~ ~ ~ ~
|\bra{ABC}T\ket{2^\gamma_{coh}}|^2 = 3 \cdot |T|^2 \,, \eeq
where $\gamma_A$, $\gamma_B$ and $\gamma_C$ denote that nuclei $A$, $B$ and
$C$ respectively  are now in their ground states but a photon has been emitted
in their transition from the excited state.

We immediately find that the first decay of the cascade, $\ket{ABC} \rightarrow
\ket{2^\gamma_{coh}}$ is described by the squared transition  matrix element
(\ref{threeex}), $|\bra{ABC}T\ket{2^\gamma_{coh}}|^2 = 3 \cdot |T|^2$, and that
the time  dependence of this decay is $e^{-3\lambda t}$.  If the photons
emitted in the three terms in the  state $\ket{2^\gamma_{coh}}$  are identical,
then these three terms can contribute coherently to the subsequent decay as in
the decay of a nuclear exciton and its decay can be speeded up in the standard 
nuclear exciton manner.\cite{HanTram,HJLIP} This does not occur in a normal 
decay without exciton
correlations and  the decays of the three terms are incoherent. We therefore 
assume here that the subsequent decays are incoherent in order to examine the
additional speedup resulting from the simultaneous presence of several
excitations.  The singly-excited state  $\ket{1}$ in the cascade
(\ref{threerec}) is therefore a complicated state  with terms containing
different photons. Each term decays individually with  the normal time 
dependence $e^{-\lambda t}$.

The second decay of the cascade is described by the incoherent decays of each
doubly-excited term into a coherent singly-excited state , with a squared
transition matrix element given by Eq.\,(\ref{twoex}) with $\phi = 0$.
$|\bra{AB}T\ket{1_{coh}}|^2 = 2 \cdot |T|^2$. The time  dependence of this
decay is $e^{-2\lambda t}$.

At first this appears inconsistent with the well-known result that all decays
have the time dependence $e^{-\lambda t}$.  However, we shall now see that the
two descriptions are completely equivalent.

\subsection{The cascade decay of a multi-excited state}

The time behavior of the radiation from the decay of a multi-excited state 
where each transition has a different lifetime would be expected to be a linear 
combination of several exponentials, one corresponding to each lifetime. However,
for the case where the decay width of the $n \rightarrow n-1$  transition
is proportional to $n$ the expression simplifies in a surprising manner and the
observed radiation is just a single exponential, exactly as if there were no
cascade and there were $n$ independent nuclei decaying. We now examine 
in detail how this surprising result arises.

Consider the cascade decay  (\ref{caSC3}) of a state  $\ket{ABC}$ containing
three nuclear excitations.    Let $P_{3X}(t)$, $P_{2X}(t)$ and
$P_{1X}(t)$ denote the probabilities respectively for the system to be in the
triply-excited, doubly-excited and singly-excited states. The decay rates
from these states are respectively $3\lambda$, $2\lambda$ and $\lambda$
and the probabilities $P_{3X}(t)$, $P_{2X}(t)$ and $P_{1X}(t)$  
satisfy the differential equations and initial conditions: 
\beq{wYY1} 
{{dP_{3X}}\over{dt}} = - 3\lambda P_{3X}\, ; ~ ~ ~ P_{3X}(0) = 1\, , ~ ~ ~ P_{3X}(t)
= e^{-3\lambda t}\, ,
\eeq 
\beq{wYY2a}
{{dP_{2X}}\over{dt}} = - 2\lambda P_{2X}  - {{dP_{3X}}\over{dt}}\, ; ~ ~ ~
P_{2X}(0)=0\, , 
\eeq 
\beq{wYY2b} 
{{dP_{1X}}\over{dt}} =  - \lambda P_{1X}  -
{{dP_{2X}}\over{dt}}- {{dP_{3X}}\over{dt}}\, ; ~ ~ ~ P_{1X}(0)=0 \, ,
\eeq 
where we have also given the trivial solution of the equation for $P_{3X}(t)$.
These can be
rewritten in a compact general form in terms of the number $N$ of  initial
nuclear excitations, where $N=3$ in this case, but the results are  easily
extended to hold for all values of $N$. 
\beq{zYY2a} 
{{dP_{kX}}\over{dt}} = -k
\lambda P_{kX}  - \sum_{j=k+1}^{N}  {{dP_{jX}}\over{dt}}; ~ ~ ~ P_{kX}(0) =
\delta_{kN} \,. 
\eeq

The total radiation emitted from all nuclei is given by 
\beq{wYY4} 
 R(t) = 3\lambda P_{3X}(t) +2\lambda P_{2X}(t) + \lambda
P_{1X}(t) =   \sum_{k=1}^{N} k \lambda P_{kX}(t) ;~ ~ ~ ~  R(0) =  N\lambda\,. 
\eeq
Substituting Eqs.\,(\ref{wYY1}-\ref{wYY2b}) gives the differential equation for 
$R(t)$ which is easily solved,
\beq{wYY3} 
{{dR}\over{dt}} = - \lambda R ;~ ~ ~ ~ R(t) = Ne^{-\lambda t} \,.
\eeq
Thus, although the individual probabilities $P_{kX}(t)$ contain several exponentials, 
the total radiation decays exponentially with the single exponent of the decay
rate for a single nucleus. All other exponential terms drop out of the total 
radiation.

To show that this treatment applies to the cascade decay of a general N-fold
excited state we note that Eqs.\,(\ref{zYY2a}) and (\ref{wYY4}) will hold for
all values of N if we generalize the assumption that the decay rate  for the 
transition $kX \rightarrow (k-1)X$ is $k\lambda$  to hold for all values of 
$k$. Eq.\,(\ref{zYY2a}) can be rearranged to the form 
\beq{zYY2b}
\sum_{j\geq k}^{N} {{dP_{jX}}\over{dt}} 
=  -k \lambda P_{kX}; ~ ~ ~ P_{kX}(0) = \delta_{kN}\,.  
\eeq
This simply states that the change in the total probability for the system to 
be in a state of $k$ or more excitations is given by the decay of the 
$k$-fold excited
state with the initial condition that all $N$ nuclei are excited at $t=0$.
The general result that Eq.\,(\ref{wYY3}) holds for all values of N follows from
summing this over the index $k$, rearranging and evaluating the sums,  
\beq{zYY2c}
- R = - \sum_{k=1}^N k \lambda P_{kX} =  
\sum_{k=1}^N \sum_{j\geq k}^{N} {{dP_{jX}}\over{dt}} 
=  \sum_{j=1}^N \sum_{k=1}^{j} {{dP_{jX}}\over{dt}} 
=  \sum_{j=1}^N j {{dP_{jX}}\over{dt}} 
=  {{1}\over{\lambda}} \cdot{{dR}\over{dt}}\,. 
\eeq  

We have therefore resolved the apparent contradiction about different exponentials.
The radiation for the cascade decay of a state with $k$ excitations via coherent 
states is therefore the same as for $k$ independent decays with the normal
lifetime. Thus the radiation does not show the presence of a cascade decay with
a speeded-up first decay. 

However, one apparently new phenomenon remains. The natural line width for the 
excitation of the state with $k$ excitations from a state with  $k-1$ $coherent$ 
excitations is $k$ times the normal line width. 
For ordinary decays of systems with $n$ excited nuclei this analysis merely
confirms what we already know. That coherent states like $\ket{2_{coh}}$ and 
$\ket{1_{coh}}$ are produced in the decays of states with multiple excitations
is only of academic interest since such coherence is never observed in
real experiments. The one case where the coherence is of interest is in the
production and decay of nuclear excitons where the coherent state is actually
created experimentally. In this case the enhancement of the width for
excitation from a state which contains coherent excitations can be very
important for excitation  from a broad-band  synchrotron radiation source with
an energy band width much larger than the natural line width of the nuclear
resonance.

We now show in detail how these same results hold for systems of coherently
excited  nuclear excitons and their cascade decays. 

\section {Simple properties of multiphoton excitation}

\subsection {What exactly is a nuclear exciton?}
 
Coherent scattering and Bragg scattering of M\"ossbauer resonance
radiation are now well understood and well established experimentally.
\cite{Gerdau,Trammell}$^{\mbox - }$\cite{HanTram,Ruffer}$^{\mbox -}$\cite{Sumrule}
%\cite{Gerdau,Trammell,HTREVIEW,HanTram,Ruffer,vanBurck,LipQM,Sumrule}
But this is all at the one-photon level, where the coherent excitation
of many nuclei has been described as a nuclear 
exciton.\cite{Trammell}$^{\mbox -}$\cite{HanTram}
%\cite{Trammell,HTREVIEW,HanTram} 
Only one nucleus
is excited, but it can be one of many, and the amplitudes for the
excitations of different nuclei add coherently to produce a forward peak
in the outgoing radiation, a speedup of the nuclear lifetime, and peaks
at Bragg angles.\cite{Ruffer,vanBurck}
 
Synchrotron radiation beams of sufficient intensity 
may be available to enable the excitation of several nuclei
simultaneously in the same synchrotron pulse.   
The question arises what happens when several nuclear excitons are
present in a lattice as a result of excitation by several photons. How does the
presence of several excitons affect the cross section for additional excitation
by another photon? How does  the presence of several excitons affect the decay
lifetime? Do nuclear excitons behave like bosons? One can immediately see a
number of interesting analogies which are not quite right: Dicke
superradiance,\cite{Super} atomic excitons, spin waves, and gauge fields.

We  are still a long way from coherent $\gamma$-ray light and $\gamma$-ray
lasers. The energy difference between a state with one nuclear  excitation and
a state with two nuclear excitations is much too large to maintain any phase
coherence between these two states. There is a factor of $10^{12}$ between the
resonance energy of the order of tens  of kilovolts and the natural line widths
which are of order of tens of  nanovolts. Thus the relative phase between a
one-exciton and a two-exciton state reverses  $10^{12}$ times during the
lifetime of the nuclear excitation. There is little hope of maintaining any
phase coherence over such a long period.     

The presence of several excited nuclei which can decay coherently
suggests Dicke superradiance.\cite{Super} But the distances between nearest
neighbors are comparable to the wave length of the radiated photons.
This makes the physics very different and more complicated than simple
superradiance.\cite{Gammaray}
 
One can follow the Dicke example and define a quasispin for each nucleus which
is ``up" when the nucleus is excited and ``down" when the nucleus is in its
ground state. The nuclear exciton then appears as a kind of ``quasispin wave".
The wave function differs from the Dicke case because the quasispins are
located at definite lattice points and there is a definite phase between the
amplitudes at different lattice points. But the wave does not move; there are
no quasispin-quasispin interactions that can flip quasispins. The same is true
for the atomic exciton analogy. The nuclear exciton is created and decays, but
does not move in space. 
 
One can also describe the relative phases of the quasispins at different points
on the lattice as a kind of gauge field. Transformations of these phases
resemble gauge transformations since they do not change the lattice energy,
which depends only upon the number of excited nuclei; i.e. upon the z-component
of the total quasispin. Quasispin rotations about the z-axis can be described
as SU(1) gauge transformations under which the Hamiltonian of the lattice is
``gauge invariant". But the relevant physics of emission and absorption of
photons is not invariant under these gauge transformations. The coherence
properties of excitation and decay are very sensitive to these phases. 
 
So far no consistent framework has been developed to enable the inclusion of
these various coherence effects in the calculation of the production and decay
of these multi-exciton states. 

\subsection {Problems with a classical approach}

We now try to pinpoint the differences between conventional treatments of
the M\"ossbauer effect and what is needed to treat multiphoton processes and 
multiple excitations in the crystal. 

The first treatment by Hamermesh {\em et al}~\cite{Hamermesh} using a
classical radiation field obtains all the correct results for one photon processes.
The extensive development of this approach is  now summarized in an  excellent
review by Hannon and Trammell.\cite{HTREVIEW} One might expect that a
classical description of the electromagnetic field which works for even one
photon should work even better for many photons. However, the problem here
is not in the classical description of the photon; it is the quantum effects not
considered in the classical description of the lattice.

This problem is simply seen in the example of a photon incident on a  thick
crystal where so may photons have already passed through the first part of the
crystal that all the M\"ossbauer nuclei in this part of the crystal have 
already been excited. The photon will pass through this part of the crystal
without seeing any M\"ossbauer nuclei that can be excited and therefore without
seeing anything like the dielectric constant and index of refraction  that was
seen by the first photon. Clearly the classical formulation does not work here.
The nuclear excitation changes the properties of the crystal.

    The classical picture describes the photon propagation through a medium in
terms of parameters like a refractive index and a dielectric constant. To
determine these parameters from theory requires a
microscopic picture in which the medium consists of nuclei having
very sharp resonances. The presence of  different nuclei with different
hyperfine splittings produces a complicated structure in the frequency spectrum
of the dielectric constant and index of refraction. 

     All this is still well defined when there is only one photon and the
treatments of Hamermesh {\em et al}~\cite{Hamermesh} and all those following are
valid. But nuclear excitations in the medium can produce a variation in the
dielectric constant and index of refraction of a crystal as a function of the
number of excited nuclei present which changes with time during the nuclear
lifetime when photons are emitted and the number of nuclear excitations is
changing.  Perhaps there is a classical way to describe this with the
properties of the medium being affected by the strength of the electromagnetic
field in the sample. 

    This now suggests an anology with lasers and nonlinear optics. But here the
M\"ossbauer situation  introduces two important differences: (1) a very long
lifetime; (2) a very short wave length of the same order as the distance between
neighboring atoms. The parameters describing these lifetimes and distances
differ by many orders of magnitude from the values relevant to conventional
lasers and nonlinear optics. Thus extrapolations from this other physics are
highly questionable. This is already seen in the lifetime speedup at the one
photon level. There the speedup does not go as the number of excited atoms,
$N$, but as $N^{(1/3)}$ because the wave length is not very much longer than
the distances between atoms.\cite{HanTram,HJLIP}

We attempt here to avoid the complications of nonlinear optics by using the
microscopic quantum picture of the lattice with nuclear excitons. This solves 
one problem, and may offer some insight. But it raises another.

    The simple classical method with a medium described by bulk properties like
a dielectric constant and index of refraction describes the passage of a
radiation field through a medium and includes all the effects of multiple
scattering which must be taken into account in the microscopic photon picture. 

    Multiple scattering of nuclear excitons is not simple. In a thick
sample a nuclear exciton exists with amplitudes for nuclear excitation
over a finite range in the sample. When it decays and progresses through
the sample, it produces a new nuclear exciton further forward in the
sample with a different spatial distribution. Meanwhile, if there were
several nuclear excitations initially present there are still other
nuclear excitations present with the initial spatial distribution.
    
    The question of how to take all these complications into account is not
clear. Perhaps it can be handled with classical nonlinear optics with the
properties of the medium simply being functions of the electromagnetic field.
Perhaps it is necessary to describe the medium itself by a quantum-mechanical
wave function in which there are different probability amplitudes for each
state of the medium with each having different dielectric constants and
different indices of refraction. 

    Before attacking these problems we try to start in a simple way, with
simple toy models of states having a small number of nuclear excitons in a thin
sample with no multiple scattering.  We consider the case where the number $N$
of nuclei which can be excited to make a single nuclear exciton is large, while
the number $N_e$ of simultaneously excited nuclei; i.e. the number of nuclear
excitons is small but finite, $N \gg N_e > 1$. 

\section {A quasispin description of nuclear excitons }

\subsection{The quasispin description of two-level systems} 

It is convenient to introduce a quasispin formalism using the $SU(2)$ Lie
algebra approach to collective excitations in 2-level systems first developed
for the nuclear shell model.\cite{LMG} A  similar formalism was also used by
Dicke to treat superradiance.\cite{Super}

Consider a crystal containing $N$ atoms, in which each nucleus can be either in
an excited state or the ground state.  We describe the state of each nucleus by
a two-component spinor, and define Pauli spin matrices denoted by
$\sigma^i_\mu$ where $i=1,2,3$ and $\mu = 1 ......N$ and $\sigma^3_\mu$ has the
eigenvalue $+1$ if nucleus number $\mu$ is excited and has the eigenvalue $-1$
if nucleus number $\mu$ is in its ground state.  The interaction Hamiltonian
describing transitions between the excited and ground state of a nucleus  by
the emission or absorption of a photon with wave number $\vec k$  is 
 \beq{QQ2.9} 
 H_{int}(\vec k)=g\cdot \left(
\sum^N_{\mu=1} a^\dagger_{\vec k}e^{-i\vec k\cdot (\vec r_\mu + \vec \xi_\mu)} 
 \sigma^-_\mu + 
\sum^N_{\mu=1} a_{\vec k}e^{i\vec k\cdot (\vec r_\mu + \vec \xi_\mu)}
 \sigma^+_\mu\right) ,
\eeq
where $\vec r_\mu$ denotes the equilibrium position of nucleus $\mu$, 
$\vec \xi_\mu$ the displacement of the nucleus from its equilibrium position, 
and $\vec p_\mu$ its momentum, $a^\dagger_{\vec k}$ is a creation operator for a photon with momentum
$\vec k$, we neglect spin, $\sigma^\pm_\mu = ((\sigma^1_\mu +i
\sigma^2_\mu)/2$ is a quasispin raising  or lowering operator and g is a constant
specifying the strength of the interaction which multiplies all transition 
matrix elements and determines absolute decay rates.  The unperturbed
Hamiltonian for the system including the lattice dynamics can be written: 
\beq{QQ2.10} 
H = {\epsilon\over 2}
\sum^N_{\mu=1} \sigma^3_\mu + {{\vec p_\mu^2}\over {2M}} + \sum^N_{\mu,\nu=1}
V_{\mu \nu} \,, 
\eeq 
where $\epsilon$ is the nuclear excitation energy, $M$ is the mass of the atom 
or ion and $V_{\mu \nu}$ is some two-body interaction potential depending only
upon the co-ordinates of the atoms and not on their momenta.

The transition rate for elastic (M\"osssbauer) photon emission between some
initial state denoted by $\ket{\psi_i}$ and a final state denoted by
$\ket{\psi_f}$  in which the state of the lattice is unchanged is given by the
Fermi Golden Rule of time-dependent perturbation theory as  proportional to the
square of the transition matrix element of the operator Eq.\,(\ref{QQ2.9}),
\beq{QQ2.11} 
 W^{el}_{i \rightarrow f}(\vec k) = g^2 f_{LM}\cdot \mid \bra {\psi_f}
\sum^N_{\mu=1}
e^{-i\vec
k\cdot\vec r_\mu} \sigma^-_\mu
\ket{\psi_i}\mid^2 \,,
\eeq
where $f_{LM}$ denotes the Lamb-M\"ossbauer factor
\beq{QQLM} 
f_{LM} = \mid \bra {\psi_i}
e^{-i\vec
k\cdot\vec \xi\mu}
\ket{\psi_i}\mid^2\,, 
\eeq
which gives the probability that the transition is elastic and we assume that 
$f_{LM}$ is the same for all nuclei $\mu$. This must be corrected in the case
where the M\"ossbauer nuclei occupy different types of lattice sites and have
different values for $f_{LM}$, but does not affect our conclusions regarding 
the effects of multiple excitation coherence. 

Thus the sum of the elastic transition rates over all final states is
proportional to the sum of Eq.\,(\ref{QQ2.11}) and a common phase space factor ,
which we disregard. Evaluating this sum by closure gives 
\beq{QQ2.12a} 
W_i (\vec k) =
\sum_f W^{el}_{i \rightarrow f}(\vec k) = g^2 f_{LM}\cdot \bra {\psi_i}
\sum^N_{\mu,\nu=1}
e^{i\vec k\cdot\vec r_\nu} \sigma^+_\nu
e^{-i\vec k\cdot\vec r_\mu} \sigma^-_\mu
\ket{\psi_i} \, .
\eeq
The direct term in the sum with $\mu=\nu$ is can be trivially evaluated 
to give
\beq{QQ2.12b} 
W_i (\vec k) = g^2 f_{LM} \cdot N_e 
+ g^2 f_{LM} \delta W_i (\vec k)
\eeq
where $N_e$ denotes the number of excited nuclei in the initial state
and  the interference term $\delta W_i (\vec k)$
is given by
\beq{QQ2.13b}
\delta W_i (\vec k) =
\bra {\psi_i}
\sum^N_{\nu=1}
\sum^N_{\scriptstyle\mu=1\atop\scriptstyle\mu\neq\nu}
e^{i\vec k\cdot\vec r_\nu} \sigma^+_\nu
e^{-i\vec k\cdot\vec r_\mu} \sigma^-_\mu
\ket{\psi_i} \,.
\eeq

The direct term
arises from the independent emission from each excited atom and is proportional
to the number $N_e$ of excited nuclei. The interference term
expresses the quantum mechanics of the coherence between the excitations of a
nucleus at the point $\vec r_\nu$ and a nucleus at the point $\vec r_\mu$. This
is the direct analog of the famous two slit experiment in which the amplitudes
of the waves passing through the two slits must be added if the slit through
which the particle passed is not known. Here the amplitudes from the
excitations of two nuclei must be added if it is not known which nucleus was
excited. The relative phase of the two excitations is determined by the phase
factor $ e^{i\vec k\cdot\vec r_\nu} \cdot e^{-i\vec k\cdot\vec r_\mu} $
appearing in the interaction operator and by the relative phase in the initial
state $\ket{\psi_i} $; e.g.  by the phase in a nuclear exciton determined by
the way it was produced. When this produces constructive interference the
resulting  enhancement has sometimes been called superradiance, but this has
been confused in the literature. 

There are two different sources for enhancement. One source is the interference
between amplitudes for the emission of a photon by different nuclei from a
state when only one nucleus is excited; i.e. $N_e =1$, when the state is described
as the state of a single nuclear exciton. The enhancement factor
here is a function of the total number of nuclei $N$ which can be excited. 
This coherent emission is sometimes also called superradiance, but this
designation is controversial. The other source for enhancement arises from the 
presence of several excitations; i.e. $N_e > 1$. This
introduces a combinatorial factor proportional to $N_e$ which is generally  
called superradiance. This is just the enhancement factor that arose in the
simple examples (\ref{twoex}) and (\ref{threeex}) and can be considered as 
analogous to stimulated emission of excitons considered as bosons.


\subsection {The wave functions of states with one or two nuclear excitons}

We now construct explicitly the wave functions for states with one or two
nuclear  excitons and examine the coherence effects discussed in the simple
model with several independent excitations.

Let $\ket{\psi_{0i}}$ denote the initial state with no nuclei excited and the 
lattice in a state denoted by $i$. The elastic transition in the lattice
induced by the absorption of a photon with wave vector $\vec k$ is described by
the interaction Hamiltonian (\ref{QQ2.9}) and leads to a final state of the
lattice with one nuclear excitation
\beq{WW4.1}
\ket{\psi_{f1}} = H_{int}(\vec k)
\ket{\psi_{0i}}
= g \sqrt{f_{LM}}\cdot \sum^N_{\mu=1}
e^{i\vec k \cdot\vec r_\mu} \sigma^+_\mu 
\ket{\psi_{0i}}\,.
\eeq

This is just the nuclear exciton state. The norm of this state is 
\beq{QQ4.2}
\langle {\psi_{f1}} \ket{\psi_{f1}} = g^2 f_{LM}N \,.
\eeq
Thus the normalized state with one excitation (the one-exciton wave function) 
is  
\beq{QQ4.2a}
\ket{\psi_1} = {{1}\over {\sqrt {N}}} \cdot \sum^N_{\mu=1}
e^{i\vec k\cdot\vec r_\mu} \sigma^+_\mu
\ket{\psi_{0i}} = {{1}\over {g \cdot \sqrt {f_{LM}N}}}\cdot H_{int}(\vec k)
\ket{\psi_{0i}}\,.
\eeq

The transition rate given by the Fermi Golden Rule is proportional to 
the square of the transition matrix element 
\beq{FGR4.1}
|\bra{\psi_1} H_{int}(\vec k)
\ket{\psi_{0i}}|^2
= g^2 N f_{LM}\,.
\eeq
This proportionality to $g^2 Nf_{LM}$ is exactly the same as 
for the transition rate in the incoherent
excitation of $N$ nuclei. 

We now consider the transition in the lattice induced by the absorption of two
photons with wave vector $\vec k$ and described by the interaction Hamiltonian
(\ref{QQ2.9}) as a double excitation. This leads to a two-exciton final state
of the lattice:
\beq{QQ4.1}
\ket{\psi_{f2}} = [H_{int}(\vec k)]^2
\ket  {\psi_{0i}}
= g^2 f_{LM}\cdot \sum^N_{\mu, \nu =1}
e^{i\vec k \cdot\vec r_\mu} \sigma^+_\mu e^{i\vec k\cdot\vec r_\nu} \sigma^+_\nu
\ket{\psi_{0i}}\,, 
\eeq
\beq{QQ4.1a}
\ket{\psi_{f2}} = g\cdot\sqrt {f_{LM}N}\cdot  [H_{int}(\vec k)]\ket{\psi_1}\,. 
\eeq

The normalized two-exciton wave function is easily obtained by using some 
quasispin algebra 
\beq{QQ4.5}  
\sigma^-_\tau \sigma^-_\rho \sigma^+_\mu \sigma^+_\nu \ket{\psi_{0i}}=
\delta_{\tau \mu} \delta_{\rho \nu} + \delta_{\tau \nu} \delta_{\rho \mu} 
- 2 \delta_{\tau \mu} \delta_{\rho \nu} \delta_{\mu \nu} 
\ket{\psi_{0i}},
\eeq 
\beq{WW4.2}
\langle {\psi_{f2}} \ket{\psi_{f2}} = g^4 \cdot {2N(N-1)}f_{LM}\,,
\eeq  
\beq{WW4.22}
 \ket{\psi_2} = {{1}\over {\sqrt {2N(N-1)}}}\cdot \sum^N_{\mu, \nu =1}
e^{i\vec k \cdot\vec r_\mu} \sigma^+_\mu e^{i\vec k\cdot\vec r_\nu} \sigma^+_\nu
\ket{\psi_{0i}} \,,
\eeq 
\beq{WW4.22a}
 \ket{\psi_2} = {{1}\over {g \cdot \sqrt {2(N-1)f_{LM}}}}\cdot 
H_{int}(\vec k)\ket{\psi_1}\,. 
\eeq 

 The transition rate for the excitation of the second exciton from a one-exciton
state is given by the Fermi Golden Rule as proportional to 
the square of the transition matrix element 
\beq{FGR4.22}
|\bra{\psi_2} H_{int}(\vec k) \ket{\psi_1}|^2 =2g^2(N-1) f_{LM}\,.
\eeq
This is double that for the transition rate in incoherent double excitation of
$N$ nuclei. It appears that there is an enhancement of the cross section by a
factor of two because of the coherence between the two nuclear excitons.  This
is the same factor of two that appeared in the simple example
(\ref{caSC3}-\ref{zYY2c}). The Lamb-M\"ossbauer factor appears here just as in
the normal case of double excitation. 

\subsection {Decays of states with one or two nuclear excitons}

The cross section calculated above by first-order perturbation theory may not
be valid for the case of a resonance. For a more reliable estimate, we follow
the procedure used in the simple example described in 
Eqs.\,(\ref{caSC3}-\ref{zYY2c}) and examine the decays of states with 
one or two
nuclear excitons. We note  that the wave vector $\vec k'$ of the emitted photon
can be  different from wave vector $\vec k$ of the photon that created the
exciton. The transition matrix element for such a decay from a state with a
single nuclear exciton (\ref{QQ4.2a})  is 
\beq{WW4.22x}
\bra {\psi_{0i}} T(\vec k') \ket{\psi_1} =
\bra {\psi_{0i}} H_{int}(\vec k')\ket{\psi_1} =
g \cdot \bra {\psi_{0i}} \sum^N_{\rho=1} e^{-i\vec k'\cdot\vec r_\rho} 
\sigma^-_\rho
\ket{\psi_1} \,,
\eeq  
\beq{WW4.22y}
\bra {\psi_{0i}} T(\vec k') \ket{\psi_1} 
={{g}\over {\sqrt {N}}} \cdot\bra {\psi_{0i}} \sum^N_{\mu, \rho=1} 
e^{-i\vec k'\cdot\vec r_\rho} 
\sigma^-_\rho
e^{i\vec k\cdot\vec r_\mu} \sigma^+_\mu
\ket{\psi_{0i}} \,,  
\eeq  
\beq{WW4.22z}
\bra {\psi_{0i}} T(\vec k') \ket{\psi_1} 
= {{g}\over {\sqrt {N}}} \cdot\bra {\psi_{0i}} \sum^N_{\rho=1} 
e^{i(\vec k -\vec k')\cdot\vec r_\rho} 
\ket{\psi_{0i}} \,.  
\eeq  

The transition for photon emission from a two-exciton state (\ref{WW4.22})
is described as a 
decay in two steps : first by emission 
of a photon with wave vector $\vec k'$ to a singly excited state and 
subsequently to a state with no excitation. 

The transition matrix element between this two-exciton state 
$\ket{\psi_2}$ and a final one-exciton state by emission of a photon with
wave vector $\vec k'$ is 
\beq{QQ4.4}  
\bra {\psi_1} T(\vec k') \ket{\psi_2} =
g \cdot \bra {\psi_1} \sum^N_{\rho=1} e^{-i\vec k'\cdot\vec r_\rho} 
\sigma^-_\rho
\ket{\psi_2}  
\eeq 
\bea  
\bra {\psi_1} T(\vec k') \ket{\psi_2} =\hspace{6.1cm} \nonumber\\ 
={{g}\over {N \sqrt {2(N-1)}}}\cdot \sum^N_{\mu, \nu, \rho, \tau =1} 
\bra {\psi_{0i}}
e^{-i\vec k\cdot\vec r_\tau} \sigma^-_\tau
e^{-i\vec k'\cdot\vec r_\rho} \sigma^-_\rho
e^{i\vec k \cdot\vec r_\mu} \sigma^+_\mu e^{i\vec k\cdot\vec r_\nu} \sigma^+_\nu
\ket{\psi_{0i}}\,,
\eea 
\beq{QQ4.6}  
\bra {\psi_1} T(\vec k') \ket{\psi_2} =
g \cdot {{\sqrt {2(N-1)}}\over {N  }}\cdot \sum^N_{\rho =1} 
\bra {\psi_{0i}}
e^{i(\vec k-\vec k') \cdot\vec r_\rho} 
\ket{\psi_{0i}} \,,
\eeq 
\beq{QQ4.6a}  
\bra {\psi_1} T(\vec k') \ket{\psi_2} =
{{\sqrt {2(N-1)}}\over {\sqrt N  }}\cdot 
\bra {\psi_{0i}} T(\vec k') \ket{\psi_1} \,, 
\eeq 
where we have used Eq.\,(\ref{QQ4.5}). 

Comparing this result with the transition matrix element for a one-exciton
decay, we see that the double excitation enhances the decay 
rate by a factor $2(N-1)/N \approx 2$. This is exactly the same
enhancement that appeared in the simple example (\ref{caSC3}-\ref{zYY2c}) and 
(\ref{FGR4.22}) observed for the excitation of a two-exciton state 
from a one-exciton state where we have used the Fermi Golden Rule to describe
the excitation. While the Golden Rule may not be adequate to describe a 
resonance excitation, it is definitely adequate to describe the decay of an
excited state into the continuum by the emission of a photon, and the same
combinatorial factor describing the effects of exciton coherence appears in
both excitation and decay.  

The same factor of two applies both to inelastic photon emission and internal
conversion. For these cases the transition matrix element for inelastic
emission or internal conversion has the same form as in 
Eqs.\,(\ref{QQ4.4}-\ref{QQ4.6}) except that there is no sum over the index $\rho$.
Since it is known by the nuclear recoil and/or the vacancy produced by the
conversion electron which nucleus emitted the photon, the transitions in
different nuclei are incoherent, and the total transition rate is obtained by
squaring the transition matrix element for each nucleus and then summing over
all nuclei. The result gives the same factor of two. 

Regardless of whether the decay of the first excitation is coherent or 
incoherent, the system remains in a nuclear exciton state with a single excited 
nucleus in the lattice. However there is also a single photon present. The  
lattice transition matrix element is now exactly the same as for the single 
nuclear exciton. But there is now an additional possible enhancement
due to stimulated emission. 

This suggests that if a synchrotron radiation pulse is incident on a
crystal which contains one nuclear exciton, the width of the resonance for
excitation producing a second nuclear exciton is double the width for the
case where no nuclear exciton is initially present. This can be a very important
effect if the number of excitons present is appreciable. 


\subsection{Decays via internal conversion and inelastic scattering }

To consider the case of a decay via internal conversion, we write the normalized
wave functions for the final states after singly and doubly excited states
have decayed via internal conversion in an atom at $\vec r_\mu$ leaving a
hole in an electron shell,
\beq{QQ15.1} 
\ket{\psi_{0h\mu}} = h^\dagger (\vec r_\mu) \ket{\psi_{0i}}
\eeq 
\beq{QQ15.2} 
\ket{\psi_{1h\mu}} = {{1}\over {\sqrt {N-1}}}\cdot 
\sum^N_{\nu =1; \nu \not= \mu} 
h^\dagger (\vec r_\mu) e^{i\vec k\cdot\vec r_\nu} \sigma^+_\nu
\ket{\psi_{0i}}\,,
\eeq 
where  $h^\dagger(\vec r_\mu)$ denotes an operator that 
creates a hole in an electron shell of the atom at $\vec r_\mu$. 

The matrix element for an internal conversion transition is
\begin{eqnarray}
\label{QQ15.3a} 
\bra {\psi_{1h\mu}} T^\dagger (h) \ket{\psi_2} &=&
g' \cdot \bra {\psi_{1h\mu}} h^\dagger(\vec r_\mu) 
\sigma^-_\mu
\ket{\psi_2} \,, \\[2mm]
\label{QQ15.3b}
\bra {\psi_{1h\mu}} T^\dagger (h) \ket{\psi_2}
 &=& 
{{g'}\over {(N-1) \sqrt {2N}}}\qquad\qquad\qquad\quad\nonumber\\
\times \sum^N_{ \nu \not= \mu, \rho, \tau, =1} 
\bra {\psi_{0i}}
e^{-i\vec k\cdot\vec r_\nu} \sigma^-_\nu &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
h(\vec r_\mu)  h^\dagger(\vec r_\mu)\sigma^-_\mu 
e^{i\vec k \cdot\vec r_\rho } \sigma^+_\rho e^{i\vec k\cdot\vec r_\tau} 
\sigma^+_\tau 
\ket{\psi_{0i}}\,.
\end{eqnarray}
Thus,
\beq{QQ15.4}
\bra {\psi_{1h\mu}} T^\dagger (h) \ket{\psi_2} =
\sqrt {2} \cdot 
\bra {\psi_{0h\mu}} T^\dagger (h) \ket{\psi_1}\,.
\eeq 

%The double excitation enhances the decay 
%rate by a factor ${{2(N-1)}\over {N }}\approx 2$. 


Inelastic emission is treated in the same way as internal conversion. Since
it is known by the nuclear recoil as in the vacancy produced by the conversion
electron which nucleus emitted the photon, the transitions in different nuclei 
are incoherent, and the total transition rate is obtained by squaring the 
transition matrix element for each nucleus  and then summing over all nuclei.
The result gives the same factor of two.  



\subsection {The difference between coherent and incoherent cascade decays}

We now show by the use of explicit nuclear exciton wave functions  how the
conclusions from the simple example (\ref{caSC3}-\ref{zYY2c}) apply to the
exciton case, and the excitation widths are different for coherent and
incoherent excitations. The two-exciton cascade decay can be written: 
\beq{QQ94.6}  
\ket{\psi_2} \rightarrow \sqrt 2 \cdot \bra {\psi_{0i}} T(\vec k') \ket{\psi_1} 
\cdot \ket{\psi_1,\gamma} 
\rightarrow \sqrt 2 \cdot[ \bra {\psi_{0i}} T(\vec k') \ket{\psi_1}]^2 
\cdot \ket{\psi_0, 2\gamma} \,.
\eeq 

The analogous incoherent cascade beginning with two well-defined
excited nuclei at lattice points $\mu$ and $\nu$ is described by the simple
model in Eqs.\,(\ref{caSC3}) and (\ref{threerec}) for the more general case of
a triple cascade. 

Both the coherent and incoherent  transitions  have the same doubling of the
decay rate for the first transition from the doubly excited state to the singly
excited state in comparison with the decay rate for the singly excited state.
In the time reversed transitions the excitation of a second nucleus from the
intermediate state with one excitation has double the width of the resonance
for the excitation of a single nucleus.  But the state  describing the 
coherent single independent excitation at two well defined lattice sites 
cannot be produced in any realistic excitation experiment. However, the
single nuclear exciton excitation which occurs in the decay (\ref{QQ94.6}) is
easily achieved in an excitation experiment by exciting the crystal with a
synchrotron radiation photon. 

In the coherent case, the transition occurs between a coherent state with two
excitations and a coherent state with a single excitation, accompanied by a
single photon state. The factor of two implies a speedup of the decay of a
single transition and a doubling of the natural line width. There then follows
the decay of the single coherent state, now with the normal speeded up lifetime
for the single transition. 

The coherent two-step transition is therefore a cascade, whose  time behavior 
is exactly the same as in the simple example (\ref{caSC3}-\ref{zYY2c}). 

Regardless of whether the decay of the first excitation is coherent or
incoherent, the system remains in a coherent state with a single excited
nucleus in the lattice. The lattice transition matrix element is now exactly
the same as for the singly excited coherent case. However in the coherent case
there is also a single photon present and therefore an additional possible
enhancement due to stimulated emission. However, probability of such stimulated
emission is probably small. 

\section{The quasispin description of states with many nuclear excitons }

\subsection{Explicit description of multiple exciton states}

We construct the whole set of nuclear exciton states explicitly  using our quasispin algebra.
We immediately encounter a crucial difference  between the nuclear exciton and
Dicke superradiance.\cite{Super}  Because each excited nucleus is at a
different well-defined point on a lattice or in an  amorphous solid, an
additional degree of freedom arises here which does not exist in Dicke
superradiance. 

The physics of the difference is seen by noting that in any system of
$N$ nuclei, located at different points in a solid, there are $N$
independent states of the system in which one nucleus is excited and
all the others are in their ground state. The nuclear exciton created
by the absorption of a single photon from a synchrotron radiation
source is a well-defined linear combination of these $N$ states with
well defined phases between them determined by the production
mechanism.  With these phases, the amplitudes for the decay by
emission of a photon in the forward direction all interfere
constructively, while the amplitude for emission in the forward
direction vanishes for all the other $N-1$ states of the $N$ nucleon
system which are orthogonal to the nuclear exciton state.
  
This physics is conveniently described formally in our quasispin
description by noting that an arbitrary phase arises in defining the
total quasispin of the system, and that this phase can be chosen to
pick out the particular states for which interference between
different forward emission amplitudes is constructive.  We begin by
defining the quasispin operators for each nucleus as objects with
quasispin 1/2 defined in terms of the Pauli spin operators 
$\vec\sigma_\mu$ with the conventional factor (1/2),
\beq{QQ5.1a}     
\vec s_\mu = (1/2) \vec \sigma_\mu; ~ ~ ~ ~     
s^{\pm}_\mu = \sigma^{\pm}_\mu \,.            
\eeq
We now define the total quasispin of the system by choosing appropriate 
phase factors as follows: 
\beq{QQ5.2a}
S_{\vec k}^{\pm}= \sum^N_{\mu=1}
e^{\pm i\vec k\cdot\vec r_\mu} s^{\pm}_\mu = 
\sum^N_{\mu=1}e^{\pm i\vec k\cdot\vec r_\mu} \sigma^{\pm}_\mu; ~ ~ ~ ~
S^3= \sum^N_{\mu=1}s^3 _\mu =  (1/2) \sum^N_{\mu=1}
\sigma^3 _\mu \,,
\eeq
\beq{YY5.4a}
(S_{\vec k})^2 \equiv (1/2)(S_{\vec k}^+  S_{\vec k}^- + 
S_{\vec k}^-  S_{\vec k}^+)  + (S^3)^2\,. 
\eeq 

The total quasispin operators satisfy angular momentum commutation rules by 
construction,
\beq{YY5.3a}
[S^3, S_{\vec k}^{\pm}] = \pm S_{\vec k}^{\pm}; ~ 
[S_{\vec k}^+, S_{\vec k}^-] = 2S^3 ; ~
[(S_{\vec k})^2, (S_{\vec k}^+] =[(S_{\vec k})^2, S_{\vec k}^-] =
[(S_{\vec k})^2, S^3 ] = 0\,.
\eeq

Any arbitrary choice of phases with opposite signs for the raising  and
lowering quasispin operators will give total quasispin operators that satisfy
the angular momentum  commutation rules. Changing these phases by adding an
additional space-dependent phase gives a unitary transformation which 
resembles a gauge transformation. However, the  interaction  Hamiltonian
(\ref{QQ2.9}) is not invariant under such an apparent  gauge transformation.
The choice of phases (\ref{QQ5.2a}) chooses a ``gauge" in which the  interaction 
Hamiltonian (\ref{QQ2.9}) assumes the very simple form:
\beq{YY5.5a}
H_{int}(\vec k)=g\cdot (a^\dagger_{\vec k} S_{\vec k}^- + 
a_{\vec k}S_{\vec k}^+ )\,,
\eeq
and the total Hamiltonian (\ref{QQ2.9}-\ref{QQ2.10}) which includes the part of the 
interaction that describes emission and absorption of photons of wave vector
$\vec k$ can be written
\beq{YY5.5b}
H_{tot}(\vec k) = H + H_{int}(\vec k)={\epsilon\over 2}S^3
+ {{\vec p_\mu^2}\over {2M}} +
\sum^N_{\mu,\nu=1}
V_{\mu \nu} + g\cdot (a^\dagger_{\vec k} S_{\vec k}^- + 
a_{\vec k}S_{\vec k}^+ ) \,.
\eeq
We now note that this piece of the total Hamiltonian commutes with the total 
quasispin 
\beq{YY5.6} 
[(S_{\vec k})^2, H_{tot}(\vec k) ] = 0 \,.
\eeq
The operators $S_{\vec k}^{\pm}$ are seen to be operators which excite and de-excite
nuclear exciton states. The operator $S^3$ simply counts the number $N_e$
of excited nuclei. Its eigenvalues are $N_e- (N/2)$ and varies from $-N/2$
when no nuclei are excited to $+N/2$ when all nuclei are excited. 

The states of the system can be classified into quasispin multiplets 
labeled by the eigenvalue of the operator $(S_{\vec k})^2$ denoted in the 
common notation by $S_{\vec k}(S_{\vec k}+1)$. The individual states in the 
multiplet can be labeled by the eigenvalues of $S^3$ denoted by $m$.
We immediately note that the lowest state in any given multiplet with 
$m=-S_{\vec k}$ is annihilated by the interaction operator (\ref{YY5.5a}) if there 
are no photons initially present, 
\beq{YY5.7}
H_{int}(\vec k)\ket{S_{\vec k}, -S_{\vec k}} = 
g\cdot a^\dagger_{\vec k} S_{\vec k}^- \ket{S_{\vec k},-S_{\vec k}}  = 0 \,.
\eeq

Thus these lowest states in the quasispin multiplet cannot radiate photons in the
forward direction; i.e. in the direction of $\vec k$. This is trivially obvious for
the case $S_{\vec k} = N/2$ where the lowest state in the  multiplet contains no
excited nuclei. However, for other values of  $S_{\vec k}$ the lowest state
contains $N_e = (N/2) - S_{\vec k}$ excited nuclei which from Eq.\,(\ref{YY5.7})
are not allowed to decay by emitting forward photons. These excited states of
the system are thus called ``subradiant"\,\cite{HanTram} in the  forward
direction, in contrast to the coherent nuclear exciton states which  are
sometimes called ``superradiant" in the forward direction. 

The states in a given multiplet can be characterized by 
a ``seniority number" $v \equiv (N/2) - S_{\vec k} $ which is just the number
of subradiant nuclear excitations in the system which cannot radiate in the
forward direction. The states in the multiplet are created by adding nuclear
excitons. 

For the case where the number of excitations $N_e$ is small it is  convenient to
define ``normalized exciton creation and destruction operators", and verify
that in the approximation where the number of excited nuclei is small compared
to the total number of nuclei, $2 N_e/N \ll 1$ they indeed 
satisfy  boson  commutation rules. 
\beq{YY5.8a} 
X^\dagger = {{1}\over {\sqrt {N}}}\cdot S_{\vec k}^+; ~ ~ ~ ~   
X = {{1}\over {\sqrt {N}}}\cdot S_{\vec k}^- \,,           
\eeq 
\beq{YY5.9a} 
[X^\dagger,X]  = {{1}\over {{N}}}\cdot [S_{\vec k}^+,S_{\vec k}^-]= 
{{2}\over {{N}}}\cdot S^3 = -1 +{{2 N_e}\over {{N}}}= -1 
+O\left({{N_e}\over {{N}}}\right) \approx -1 \,,                
\eeq  
 \beq{QQ2.92z} 
[\sigma^-_\mu, X^\dagger] = 
{{-2}\over {\sqrt {N}}}\cdot e^{i\vec k\cdot\vec r_\mu} s^3_\mu 
={{1}\over {\sqrt {N}}}\cdot e^{i\vec k\cdot\vec r_\mu}\left [1
+O\left({N_e}\over {{N}}\right)\right] \approx 
{{1}\over {\sqrt {N}}}\cdot e^{i\vec k\cdot\vec r_\mu}\,,
\eeq
where we have noted that the operator $s^3_\mu$ has the eigenvalue $-1/2$ for
all terms in the multiexciton wave function in which the nucleus at the point
$\vec r_\mu$ is not excited.

 The Hamiltonian (\ref{YY5.5b}) is simply expressed in terms of these exciton
operators,
\beq{YY5.9b} H_{tot}(\vec k) = H + H_{int}(\vec k)= 
{\epsilon\over 2}S^3
+ {{\vec p_\mu^2}\over {2M}} +
\sum^N_{\mu,\nu=1}
V_{\mu \nu} + 
g{\sqrt {N}}\cdot (a^\dagger_{\vec k} X + a_{\vec k}X^\dagger )\,.
\eeq
 
The transition matrix elements for the absorption or emission 
of a photon from a state where there are already $n$ excitons present 
and no subradiant excitations; i.e. $S_{\vec k} = N/2$, is seen to be 
\beq{YY5.10a} 
\bra{n+1} H_{int}(\vec k) \ket{n,\vec k} = g 
\bra{m+1} S_{\vec k}^+\ket{m} = g \sqrt{(S-m)(S+m+1)} \,,
\eeq 
\beq{YZ5.10a}
\bra{n+1} H_{int}(\vec k) \ket{n,\vec k} = g 
\sqrt{(n+1)(N-n)} \approx g \sqrt{nN} \,,
\eeq
\beq{YY5.10b} 
\bra{n-1,\vec k} H_{int}(\vec k) \ket{n} = g 
\bra{m-1} S_{\vec k}^-\ket{m} = g \sqrt{(S-m+1)(S+m)} \,,
\eeq 
\beq{YZ5.10b} 
\bra{n-1,\vec k} H_{int}(\vec k) \ket{n}= 
g \sqrt{n(N-n+1)} \approx g \sqrt{nN} \,.
\eeq


\subsection{Explicit description of the two enhancements arising in a multiple
exciton state}

We now generalize the treatment of two-exciton decays in Eqs.\,(\ref{QQ4.4}-
\ref{QQ4.6}) to the general multi-exciton case using the formalism of 
Eqs.\,(\ref{QQ2.12a} - \ref{QQ2.13b}) and show the interplay of the two
enhancements.

We write the initial state with $N_e$ excitons 
\beq{QQ2.91} 
\ket{\psi(N_e)} \equiv {{(X^\dagger)^{N_e} \ket{\psi_{0i}}}\over {\sqrt
{N_e!}}}\,.
\eeq
 Then 
\beq{QQ2.93b} 
\sigma^-_\mu \ket{\psi(N_e)} \approx  \sigma^-_\mu\cdot 
{{(X^\dagger)^{N_e} \ket{\psi_{0i}}\over {\sqrt {N_e!}}}= 
\sqrt {{N_e} \over { N}}\cdot e^{i\vec k\cdot\vec r_\mu}
\cdot {{(X^\dagger)^{(N_e -1)}}\over
{ {(N_e-1)!}}}
\ket{\psi_{0i}}}\,.
\eeq 
Substituting Eqs.\,(\ref{QQ2.91}) and (\ref{QQ2.93b}) into Eq.\,(\ref{QQ2.13b}) 
gives
\beq{QQ2.94}
\delta W_{N_e} (\vec k') \approx 
\sum^N_{\nu=1}
\sum^N_{\scriptstyle\mu=1\atop\scriptstyle\mu\neq\nu}{{N_e} \over { N}}\cdot
{{\bra {\psi_{0i}}
 e^{i(\vec k'-\vec k) \cdot(\vec r_\nu - \vec r_\mu)} 
(X)^{(N_e -1)} (X^\dagger)^{(N_e -1)} 
\ket{\psi_{0i}}
\over
{\sqrt {(N_e-1)!}}}}\,,
\eeq
\beq{QQ2.94x}
\delta W_{N_e} (\vec k') 
\approx \sum^N_{\nu=1}
\sum^N_{\scriptstyle\mu=1\atop\scriptstyle\mu\neq\nu}{{N_e} \over { N}}\cdot
e^{i(\vec k'-\vec k) \cdot(\vec r_\nu - \vec r_\mu)} 
\approx N_e \cdot \delta W_{N_e =1} (\vec k')\,.
\eeq
The factor $N_e$ is the enhancement due to superradiance. The lattice sum over
the indices $\mu$ and $\nu$ determine the enhancement due to the coherence 
from the nuclear exciton. This enhancement factor depends upon the geometry
and is not proportional to the number of nuclei $N$ that can be excited. For
a three dimensional system this enhancement or speedup is proportional to 
$N^{1/3}$.  

The enhancement factor $N_e$ can also be seen as a ``boson enhancement" factor
as in Eqs.\,(\ref{twoex}) and (\ref{threeex}) or as a simple combinatorial
factor. A one-exciton state in which any one of $N$ nuclei can be excited with
equal  amplitude is the sum of $N$ equal terms each normalized by a factor  
$1/\sqrt {N}$. A two-exciton state in which any pair of the 
$N(N-1)/2$ possible pairs of nuclei can be excited with equal  amplitude  is
the sum of $N(N-1)/2$ equal terms each normalized by a factor   $\sqrt
2/[N(N-1)$. Each term in the two-exciton state can decay into
either of two terms in the one-exciton state. Thus there are a total of 
$N(N-1)$ equal terms contributing to the transition matrix element between the
two-exciton and the one-exciton state, and the product of the normalization 
factors for the two states produce an overall normalization factor of
 $\sqrt {2/[N^2(N-1)]}$. 
This gives an enhancement factor for the
transition rate of $2\times (N-1)$. The factor $N-1$ arises from the coherence
of the exciton decay in the exact forward direction. The factor 2 arises
because there are two excitons in the initial state, $N_e = 2$. A similar
combinatorial factor of $N_e $ arises for all numbers of excitations. 


\section{Conclusions}

We find that nuclear excitons behave like a collective boson excitation,
where production of a nuclear exciton can be enhanced by a ``stimulated 
emission factor" if there are already nuclear excitons present. Thus 
the excitation of the n-th nuclear exciton in a system which already contains
$n-1$ nuclear excitons is enhanced by a factor $n$ over the excitation of a
single exciton when no others are present.  This implies that the partial width
for the excitation of the n-th exciton is enhanced by a factor $n$ over the 
normal width for the decay and excitation of a single exciton.  

Since the excitation probability from a broad-band synchrotron beam with a width
much larger than the natural line width of the nuclear resonance depends upon
the integral of the cross section over the energy width of the beam, the
increased partial width of the resonance produces an increased excitation 
probability from the broad beam.

The decay of an n-exciton state is the sum of exponentials corresponding to the
enhanced decay widths of each state containing more than a single nuclear
exciton. However, the time dependence of the radiation emitted is exactly the
same as for n independent excitons, and contains only a single exponential
which is the same as that for a single exciton.


\section*{Acknowledgments}
\addcontentsline{toc}{section}{\numberline{}Acknowledgments}

This work originated as an invited contribution to the April, 2000 workshop at
HASYLAB\cite{HASYLAB}. It is a pleasure to thank E. E. Alp, A.Q.R.Baron, U.van
Buerck, R. Coussement H.Franz, E. Gerdau, W.Potzel, R.Roehlsberger, W.Sturhahn
and T.Toellner for helpful discussions and comments.

This work was supported
in part by grant from US-Israel Bi-National Science Foundation
and by the U.S. Department
of Energy, Basic Energy Sciences, Office of Science, under Contract
No.W-31-109-Eng-38.

\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}

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\end{thebibliography}


\end{document}

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