Symmetry, Integrability and Geometry: Methods and Applications
Vol. 2 (2006), Paper 069, 9 pages
Integrable Models of Interaction of Matter
with Radiation?
Vladimir I. INOZEMTSEV
†
and Natalia G. INOZEMTSEVA
†
Laboratory of Theoretical Physics, JINR, Dubna, Russia
E-mail: inozv@theor.jinr.ru
‡
Moscow Technical University, Dubna Branch, Russia
‡
Received July 18, 2006, in final form September 19, 2006; Published online October 13, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper069/
Abstract. The simplified models of interaction of charged matter with resonance modes of
radiation generalizing the well-known Jaynes–Cummings and Dicke models are considered.
It is found that these new models are integrable for arbitrary numbers of dipole sources
and resonance modes of the radiation field. The problem of explicit diagonalisation of
corresponding Hamiltonians is discussed.
Key words: integrability; radiation; Gaudin models
2000 Mathematics Subject Classification: 81R05; 81S99
1
Introduction
In the last four decades, much attention has been paid to the problem of describing processes
of interaction of charged matter with electromagnetic radiation at resonance [1, 2, 3, 4]. This
problem, being overcomplicated from the mathematical viewpoint in the general case, admits
some very attractive simplifications which allow one to construct some rather simple models
having even exact solutions. These models can be used for quantum statistical description of
real interaction processes.
The starting point for constructing such models is the quantum Hamiltonian [3]
X 1
e ~ 2
1 +
p~ − A + u(x) + HF ,
HF =
ω k bk bk +
,
H=
2m
c
2
k
where p~ and m are the momentum and mass of the electron in some atom, u(x) is the Coulomb
~ is the vector potential of the secondly
potential of interaction of the electron with nucleus, and A
quantized electromagnetic field which is given by the formula
X 2πc2 1/2 −i~k~
x
i~k~
x
~
0
A=
~ek b+
e
+
b
e
,
[bk , b+
k
k
k0 ] = δkk ,
ωk
k
where ~ek are polarization vectors and bk , b+
k are the usual operators of annihilation and creation
~
of the mode with wave vector k and the energy ωk . There are several steps of simplification of
this general Hamiltonian. First, one neglects the term which is nonlinear in the vector potential,
e2 ~ 2
A . Second, one considers only transitions between higher levels of an atom,
2mc2
1 2
p~ + u(x) → ω0 σ z + const,
2m
?
This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”.
The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html
2
V.I. Inozemtsev and N.G. Inozemtseva
where σ z is the Pauli matrix. Third, one considers only small ~k, ka 1, where a is the
~
typical size of an atom, e±ik~x ∼ 1. Fourth, one uses the dipole and so-called rotating wave
approximations [4], which result in
X
X
e ~
−
+
−
p~A →
gk σ x (b+
+ bk ) →
gk (b+
k
k σ + bk σ ),
mc
k
k
σ±
where
are the usual Pauli matrices. And finally, one considers only the case of resonance:
P
k → kfixed . By all these steps one can write down very simple Jaynes–Cummings Hamiltonian [2] which is linear in the operators of creation and annihilation of one resonant mode,
1
HJC = ω0 b+ b + ω0 σ z + g(σ + b + σ − b+ ).
(1)
2
The solution of eigenequation for the operator (1) is almost trivial, but it allows one to describetime evolution of the corresponding wave function in great detail, with transparent applications to physics of one-mode laser. This extremely simple model admits almost evident
generalisations in two directions.
2
The simplest generalisations of the JC model:
the n-level atom and Dicke model
The first of these generalisations is the model of one n-level atom with (n − 1) modes of resonant
radiation,
(n)
HJC =
n−1
X
ωα N̂α + ∆R̂nn +
α=1
n−1
X
gα bα R̂nα + b+
α R̂αn ,
(2)
α=1
where
N̂α = b+
α bα − R̂αα ,
R̂αβ
γδ
= δαγ δβδ ,
the sets {ωα }, {gα } are energies of the modes and their coupling constants, and ∆ is so-called
detuning parameter [5, 6]. The extended physical motivation for this kind of the generalization
of the Jaynes–Cummings model can be found in [6]. There are evident commutation relations
(n)
N̂α , N̂β = 0,
HJC , N̂α = 0,
which make the problem of diagonalisation of (2) at n ≥ 3 quite easy. The second way of
generalisation of the Jaynes–Cummings model consists in considering an arbitrary number N of
two-level sources of radiation,
HD = ωb+ b +
N
X
(j)
(j)
ωj σz(j) + λ σ+ b + σ− b+ .
(3)
j=1
The solution of this so-called Dicke model which has been proposed many years ago [1] is
far from being trivial. It turned out that the problem with the Hamiltonian (3) has some
hidden dynamical symmetry which allows one to construct some nontrivial operators commuting
with (3).
What happens if one will try to use both the ways of generalisation of the Jaynes–Cummings
model, i.e. consider the Hamiltonian of the problem of an arbitrary number N of multilevel
sources interacting with an arbitrary number n − 1 of resonant modes of radiation,
H
(N,n)
=
n−1
X
α=1
ωα N̂α +
N
X
k=1
(k)
∆k R̂nn
+
n−1
N
XX
α=1 k=1
(k)
(k)
gα(k) bα R̂nα
+ b+
α R̂αn ,
(4)
Integrable Models of Interaction of Matter with Radiation
3
(k)
where N ≥ 2, n ≥ 3 (the detuning parameters {∆} and coupling constants {gα } are assumed
to be arbitrary)? The answer is not clear till now. In the following two sections we will argue
that there is some simplification of the spectral problem defined by (4) in the case of equality of
all the modules of the coupling constants {g}: these constants should not depend on k and their
(k)
dependence on α is given by the formula gα ∝ εα , εα = ±1. Namely, we will construct in this
case explicitly some set of N − 1 nontrivial operators commuting with (4); their existence allows
one to call the model (4) quantum integrable under the above choice of the coupling constants.
3
The models of Gaudin type
In this section, let us consider some more general operators introduced first by Gaudin [7],
(N )
Hj
=
n N
+1
X
X
αβ j
s
fjs
T̂αβ T̂βα
,
j = 1, . . . , N + 1,
(5)
α,β=1 s6=j
o
n
j
are operators of absolutely arbitrary representations of SL(n), obeying the comwhere T̂αβ
mutation relations
j
j
j
s
δαµ ,
(6)
= δjs T̂αµ
δβλ − T̂λβ
T̂αβ , T̂λµ
αβ αβ
βα
and fjs
are arbitrary sets of numbers, obeying the evident restriction fjs
= fjs
. It is easy
to see that operators
N̂α =
N
+1
X
j
T̂αα
j=1
trivially commute with (5),
(N )
Hj , N̂α = 0.
Now let us pose the problem: how to find such the sets of numbers {f } that
(N ) (N ) Hj , Hk
=0
(7)
for all indices {j, k}. It is easy to see that in this case the “Hamiltonian”
Hn(N ) =
N
+1
X
(N )
ηj Hj
+
n
X
ωα N̂α ,
(8)
α=1
j=1
(N )
where ηj and ωalpha are arbitrary constants, commutes with all Hj
,
(N ) (N ) Hn , Hj
= 0.
αβ
βα
In the simplest nontrivial case of n = 2, one can always choose {f } such that fjk
= −fkj
, and
the equations (7) can be written in the form
N
+1 X
n
X
(N ) (N ) αβ αλ
αβ βλ αλ βλ
fjs + fjk
fks
Hj , Hk
=−
fjs fsk + fkj
s6=j,k α,β,λ=1
j
j
k
s
k
s
× T̂αβ
T̂λα
T̂βλ
− T̂βα
T̂αλ
T̂λβ
= 0,
(9)
4
V.I. Inozemtsev and N.G. Inozemtseva
which immediately gives a overdetermined set of bilinear equations for {f },
αβ αλ
αβ βλ
αλ βλ
fjs
fsk + fkj
fjs + fjk
fks = 0.
(10)
It has been shown by Gaudin [7, 8] that for n = 2 there is some so-called anisotropic solution
to (10),
αβ
fjs
=
αα
fjs
= cot(uj − us ),
1
,
sin(uj − us )
(11)
where {uj } are arbitrary numbers. Now, following Gaudin, let us choose them as
uk = −ελk +
π
+ uN +1 ,
2
k = 1, . . . , N.
N +1 αβ
αα
As ε → 0, then fj,N
as Jordan–Schwinger realization
+1 ∼ ελj , fj,N +1 ∼ 1. Let us choose T̂αβ
N +1
of SL(2), T̂αβ
= −bα b+
β and consider the limit N2 1, where N2 is an eigenvalue of the
operator N̂2 . It is easy to see that in this limit b-operators can be treated as c-numbers;
−1
√
b2 , b +
N 2 ; let us also take ε ∼ N2 2 and send N2 → ∞. In this limiting process, one easily
2 ∼
finds
(N )
lim H2
N2 →∞
= ω 1 b+
1 b1 +
N
X
(j)
(j) (j)
(j)
λj T̂11 − T̂22 + T̂12 b1 + T̂21 b+
.
1
(12)
j=1
(j) And finally, if we will choose T̂αβ as σ-representation of SL(2), we will obtain just the Dicke
Hamiltonian!
This derivation of the Dicke model proposed first by Gaudin [7] is not original of course. There
are some recent papers in which the connection between it and the Gaudin models was exploited
for construction the algebraic Bethe ansatz method for eigenvectors of the Dicke Hamiltonian
and some generalisations which might be considered as the interaction term without the rotating
wave approximation [9, 10, 11].
The above limiting procedure, however, does not work in the general case n ≥ 3. One also
has in this case the overdetermined system of bilinear equations
αβ βλ
αβ αλ
αλ βλ
fks = 0,
fjs + fjk
fjs
fsk + fkj
but, if n ≥ 3, there is a possibility of the choice of indices α 6= β 6= λ which excludes all
αα =
“anisotropic” solutions of the type (11). There are only “isotropic” solutions of the type fjs
αβ
1
fjs
= νj −ν
, and hence there are no parameters for Nn → ∞. Hence it seems that the Gaudin
s
approach does not lead to any integrable system of the type (4).
Let us prove absence of “anisotropic” solutions to (10) as N = 2, n ≥ 3. In this simplest
nontrivial case the set {f } can be parametrized as
αβ
fjs
=
3
X
εjsp Aαβ
p ,
αλ
fsk
=
p=1
3
X
εskp Aαλ
p ,
p=1
βλ
fjs
=
3
X
εjsp Aβλ
p .
p=1
As j 6= k 6= s, (10) reads:
αλ βλ
αβ βλ
αλ
Aαβ
k Aj + As Ak + As Aj = 0,
with general solution
Aαλ
1 =
A
,
sin ν1
Aαλ
2 =
A
,
sin ν2
Aαλ
3 =−
A
,
sin(ν1 + ν2 )
(13)
Integrable Models of Interaction of Matter with Radiation
Aαα
1 = A cot ν1 ,
Aαα
2 = A cot ν2 ,
5
Aαα
3 = −A cot(ν1 + ν2 ),
where A, ν1 , ν2 are some parameters. Now, if one takes α 6= β 6= λ, one gets from (10)
−
2 cos nu2 1
A2
=0
2
cos ν22 cos ν1 +ν
2
→
A = 0.
Surprisingly enough, one can construct integrable models of Gaudin type at n ≥ 3 by using only
“isotropic” solution of (10). It will be done in the next section.
4
New integrable model with N sources and n − 1 modes
The receipt of constructing the integrable model of the type (4) is quite unusual. Let us choose
the above H-operators as obeying the equality
(N ) (N ) s , {x} ,
Hj , Hk
= Ŝjk T̂αβ
(14)
where x is some parameter and Ŝjk have the following property: as x → ∞, there are only zero
s act nontrivially. Then we can introduce new
eigenvalues of them in a subspace in which T̂αβ
N
P
(N )
(N )
operators lim Hj
in this subspace which will commute, and HnN +1 = −
Hj is some
x→∞
j=1
nontrivial Hamiltonian. By direct calculation of the commutator of the operators (5), one finds
Ŝjk
s
T̂αβ
N
X
, {x} = −
n
X
αβ βλ
αβ αλ
αλ βλ
fks
fjs + fjk
fjs
fsk + fkj
s6=j,k α,β,λ=1
n
X
j
j
k
s
k
s
× T̂αβ
T̂λα
T̂βλ
− T̂βλ
T̂αλ
T̂λβ
−
×
j
N +1
k
T̂αβ
T̂λα
T̂βλ
−
αβ
αβ βλ
αλ
αλ βλ
fj,N
+1 fN +1,k + fkj fj,N +1 + fjk fk,N +1
α,β,λ=1
j
N +1
k
T̂βλ T̂αλ
T̂λβ
.
(15)
αβ
= A(νj − νk )−1 , 1 ≤ j, k ≤ N , i.e. use the isotropic Gaudin solution at
Let us choose fjk
j, s, k ≤ N . Then the double sum vanishes and we are left with
Ŝjk
s
T̂αβ
n
X
αβ
αλ
, {x} = −
fj,N +1 fN
+1,k +
A
βλ
βλ
fk,N
− fj,N
+1
+1
νj − νk
α,β,λ=1
j
j
N +1
N +1
k
k
× T̂αβ
T̂λα
T̂βλ
− T̂βλ
T̂αλ
T̂λβ
.
There are no other restrictions to the numbers
αβ
fj,N
+1
(16)
except symmetry. Hence we have
N n(n+1)
free parameters at given {A, {νj }} in the relation (16). Of course,
2
αβ
if all fj,N +1 = 0. Fortunately, one can construct the nontrivial ansatz:
there is trivial case
αβ
βn
αn
nn
fj,N
+1 = δβn fj,N +1 + δαn fj,N +1 (1 − δαn δβn ) fj,N +1 ,
αβ
i.e. fj,N
+1 = 0 unless α 6= n or β 6= n.
The commutator HjN , HkN (16) is still too complicated. Let us make the next assumption:
√
λn
fj,N
+1 = ελ AF ,
nn
fj,N
+1 = −F νj ,
|ελ | = 1,
(17)
6
V.I. Inozemtsev and N.G. Inozemtseva
where {νj } are arbitrary numbers. Then (16) can be recast in the form
n−1
X
(N ) (N ) j
j
N +1
N +1
k
k
T̂λβ
T̂βλ
− T̂βn
T̂nλ
Hj , Hk
=
T̂λn
AF εβ ελ T̂nβ
β,λ=1
−
n−1
X√
N +1
N +1
j
k
j
k
AF F νj ελ T̂nn
T̂λn
T̂nλ
− T̂nn
T̂nλ
T̂λn
λ=1
j
j
N +1
N +1
k
k
− F νk ελ T̂nλ
T̂nn
T̂λn
− T̂λn
T̂nn
T̂nλ
.
(18)
Let now choose
N +1
T̂βλ
= −bβ b+
λ
as Jordan–Schwinger representation of SL(n)). Hence the basis of the whole Hilbert space of the
j
problem consists of the direct products of the basis vectors of the spaces in which T̂βλ
,1≤j≤N
Q
+ lβ
|0i ,
act, and various basic vectors of above Jordan–Schwinger representation,
β,{l} bβ
{l} ∈ N. Consider the action of the right-hand side of (17) on the subspace spanned by the
(L)
vectors ϕnn with ln = L, L 1 such that


N
N
X
X
j
j 
+
2 (L)
(L)
N +1

T̂nn
.
T̂
=
b
b
−
N̂n ϕ(L)
∼
L
ϕ
,
N̂
ϕ
=
−
T̂
+
n
n
nn
n
nn
nn
nn
nn
j=1
j=1
Then
N +1
T̂nn
∼ −L2 ,
N +1
T̂nα
∼ −Lb+
α,
N +1
T̂αn
∼ −Lbα ,
1 ≤ α ≤ n − 1.
If
A∆
F = 2,
L
nn
fk,N
+1
A∆νk
=−
,
L2
λn
fk,N
+1
√
ελ A ∆
=
,
L
then
h
i
(N )
(N )
ϕ(L)
lim Hj , Hk
nn = 0
L→∞
and
(N )
H̃j
(N )
= lim Hj
L→∞
commute on this subspace! The explicit form of H-operators reads
(N )
H̃j
=A
N X
n
X
j
s
T̂αβ
T̂βα
(νj − νs )−1 + ŵj ,
s6=j α,β=1
ŵj = A
"n−1
X
√
ελ ∆
j
bλ T̂nλ
+
j
b+
λ T̂λn
#
+
j
∆νj T̂nn
,
(19)
λ=1
(N )
HN +1
=−
(N )
X
j=1
(N )
H̃j
=−
X
j=1
ŵj = −A
"n−1
N
X
X
k=1
√
ελ
#
k
∆(bλ T̂nλ
λ=1
N
(N ) P
j
They commute with H̃j
and N̂α = −
T̂αα
+ b+
α bα .
j=1
+
k
b+
λ T̂λn )
+
k
∆νk T̂nn
.
Integrable Models of Interaction of Matter with Radiation
7
j
And finally, let us choose as T̂αβ
the matrix representation of SL(n),
j
j
T̂αβ
→ R̂αβ
,
n−1
P
and add linear combination
α=1
(N )
ωα N̂α to HN +1 .
It is easy to see that we obtain just the operator of generalized Dicke model (4) for n ≥ 3,
H (N,n) =
n−1
X
ωα N̂α +
α=1
N
X
(k)
∆k R̂nn
+
(k)
(k)
gα(k) bα R̂nα
+ b+
α R̂αn
α=1 k=1
k=1
with coupling constants
√
gα(k) = −Aεα ∆,
n−1
N
XX
εα = ±1.
(20)
Hence, under the above choice of the coupling constants, the most general model (4) becomes
quantum integrable and the spectral problem might be simplified. However, we did not find
the way for doing it except the simplest nontrivial case of N = 2 which is described in the next
section.
5
Explicit solution for N = 2
In this case the Hamiltonian (4) under the conditions (20) can be written as
Hn(2) =
n−1
X
1
2
ωα N̂α + ∆1 R̂nn
+ ∆2 R̂nn
+
α=1
n−1
2
XX
k
k
bα R̂nα
+ b+
α R̂αn ,
(21)
α=1 k=1
N̂ α = b+
α bα −
2
X
j
R̂αα
.
j=1
Let us rewrite (21) as
Hn(2) =
n−1
X
ωα N̂α + k̂1 + k̂2 ,
(22)
α=1
−1
k̂1,2 = ± (∆1 − ∆2 )
P̂1,2 +
n−1
X
1,2
1,2
1,2
+ b+
bα R̂nα
α R̂αn + ∆1,2 R̂nn ,
P̂1,2 =
α=1
n
X
1
2
R̂αβ
R̂βα
.
α,β=1
Eigenfunctions of (22) are of course common eigenfunctions of (n + 1) operators {Nα }, k̂1 , k̂2 ,


n−1
n−1
X
X (1)
(2)
+
1 2
ψ=
Aλµ ϕ1λ ϕ2µ b+
(Bλ ϕ1λ ϕ2n + Bλ ϕ2λ ϕ1n )b+
λ bµ +
λ + Cϕn ϕn Zn |0i,
λ,µ=1
λ=1
1 }, {R̂2 } and Z =
where ϕ1λ , ϕ2µ are eigenvectors of {R̂λλ
n
µµ
n−1
Q
α=1
Nα
(b+
α)
.
One gets from the relations k̂1 ψ = k1 ψ, k̂2 ψ = k2 ψ the following set of algebraic equations
for the coefficients {A, B, C},
Aµλ (∆1 − ∆2 )−1 + Bµ(2) = k1 Aλµ ,
(2)
(1)
Bλ (∆1 − ∆2 )−1 + C = k1 Bλ ,
8
V.I. Inozemtsev and N.G. Inozemtseva
(1)
(2)
Bλ (∆1 − ∆2 )−1 + ∆1 Bλ +
X
(2)
Aµλ (Nµ + 1 + δλµ ) = k1 Bλ ,
µ
−1
C(∆1 − ∆2 )
+ C∆1 +
X
Bµ(1) (Nµ
+ 1) = k1 C,
µ
(1)
−Aµλ (∆1 − ∆2 )−1 + Bλ = k2 Aλµ ,
(1)
(2)
−Bλ (∆1 − ∆2 )−1 + C = k2 Bλ ,
X
(2)
(1)
(1)
−Bλ (∆1 − ∆2 )−1 + ∆2 Bλ +
Aλµ (Nµ + 1 + δλµ ) = k2 Bλ ,
µ
−C(∆1 − ∆2 )−1 + ∆2 C +
X
Bµ(2) (Nµ + 1) = k2 C.
µ
They can be easily reduced to two equations for k1 , k2 :
(k1 + k2 − ∆1 )(∆1 − ∆2 )−1 + (∆1 − ∆2 )−2 − k1 (k1 − ∆1 ) + 1 +
n−1
X
(Nµ + 1) = 0,
µ=1
(∆2 − k1 − k2 )(∆1 − ∆2 )−1 + (∆1 − ∆2 )−2 − k2 (k2 − ∆2 ) + 1 +
n−1
X
(Nµ + 1) = 0,
µ=1
and the eigenvalues of the Hamiltonian (21) are given by
h
2,n
=
n−1
X
ωα Nα + k1 + k2 .
α=1
6
Conclusions
In this paper, we found the most general quantum integrable model of interaction of (n − 1)
modes of radiation with N dipole sources which comprises all known models of that type: Jaynes–
Cummings model (N = 1, n = 2), its generalisation for arbitrary number of modes (N = 1),
(N )
Dicke model (arbitrary N , n = 2). The commutative ring of operators which includes Hn
is found in explicit form. We did not use any “anisotropic” form of the Gaudin solution; the
Hamiltonian of the model was obtained via some limiting procedure. We almost immediately
got solution for N = 2, arbitrary n with the use of additional integrals of motion, but the
most interesting case of solution for arbitrary N and n is unreachable at the present stage of
investigation. We hope to come back to this problem in the future.
This paper has been presented for memorial volume of Vadim Kuznetsov. We both knew him
personally — he was our guest in Dubna almost 17 years ago when he was a PhD student of
Professor I.V. Komarov in Leningrad. We remember him as bright young man, very active in
the field of quantum and classical integrable systems. Later he solved very complicated problem
of integration of equations of motion for classical Toda chains with non-exponential ends [12]
proposed by one of us (V.I.).
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