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Journal of Physics A: Mathematical and Theoretical
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PAPER
Time evolution of the atomic inversion for the
generalized Tavis–Cummings model—QIM
approach
- The su(1,1) Tavis-Cummings model
Andrei Rybin, Georg Kastelewicz, Jussi
Timonen et al.
To cite this article: N M Bogoliubov et al 2017 J. Phys. A: Math. Theor. 50 464003
- The Quench Action
Jean-Sébastien Caux
- Exact solution of generalized Tavis Cummings models in quantum optics
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View the article online for updates and enhancements.
This content was downloaded from IP address 129.8.242.67 on 28/10/2017 at 00:16
Journal of Physics A: Mathematical and Theoretical
J. Phys. A: Math. Theor. 50 (2017) 464003 (20pp)
https://doi.org/10.1088/1751-8121/aa8c6a
Time evolution of the atomic inversion
for the generalized Tavis–Cummings
model—QIM approach
N M Bogoliubov1,2 , I Ermakov2,3
and A Rybin2
1
St.-Petersburg Department of V. A. Steklov Mathematical Institute RAS Fontanka
27, St.-Petersburg, 191023, Russia
2
ITMO University, Kronverkskiy 49, 197101, St.Petersburg, Russia
3
Saint Petersburg State University, University Embankment, 7-9, St Petersburg,
Russia
Received 1 March 2017, revised 9 September 2017
Accepted for publication 13 September 2017
Published 23 October 2017
Abstract
A model describing the interaction between two-level atoms and a single
mode field in an optical cavity enclosed by a medium with Kerr nonlinearity
is considered in our paper. We study the model within the framework of the
analytical solution obtained by application of the quantum inverse method
(QIM). Time evolution of the atomic inversion is calculated and transition
elements of photons are represented in the determinant form. The obtained
answers depend on the solutions of the Bethe equations. We provide the
numerical solutions of these equations for the different parameters of the
model and study the behaviour of certain dynamical correlation functions.
Keywords: Tavis–Cummings model, Kerr-nonlinearity, quantum optics,
quantum inverse method, exactly solvable models, Bethe equations,
dynamical correlation functions
(Some figures may appear in colour only in the online journal)
1. Introduction
The exactly solvable models of quantum nonlinear optics [1–6] allow one to study the behaviour of strongly correlated systems in ways that otherwise would be impossible.
One of the most fundamental models in cavity quantum electrodynamics—exactly solvable
for its eigenstates and eigenvalues—is the Jaynes–Cummings (JC) model [7]. It describes the
interaction of the single mode of a cavity field with the two-level atom. In spite of its simplicity the model exhibits many nonclassical features caused by its intrinsic nonlinearity, such as,
for example, collapses and revivals of the atomic inversion [8–11]. The JC model has been
1751-8121/17/464003+20$33.00 © 2017 IOP Publishing Ltd Printed in the UK
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J. Phys. A: Math. Theor. 50 (2017) 464003
thoroughly studied both theoretically [12] and experimentally [13]. The fiftieth anniversary of
this model was recently noted [14]. Due to the progress of circuit quantum electrodynamics
[15] the JC model has again attracted the attention of physicists [16]. Many interesting applications of the JC model concerned with its generalizations go far beyond its original concept.
Natural generalization of the JC model is to consider an ensemble of N noninteracting twolevel atoms coupled to a single mode of a cavity field. The model describing such an ensemble
is known as the Tavis–Cummings (TC) model. It was solved by Tavis and Cummings [17] at
exact resonance, and by Hepp and Lieb for finite detuning [18]. The TC model may also be
expanded in many other ways. For example, one can consider an array of coupled cavities,
the so-called Jaynes–Cummings–Hubbard model [19], or abandon the rotating wave approx­
imation [20].
Recently, models that include either a nonideal cavity or a dynamical Stark shift have again
attracted attention [21–24]. Nonlinear effects in cavities can provide new tools in quantum
state engineering[25–28]. If the cavity is not ideal an effective Hamiltonian can be derived
[29–31], which adds a fourth-order term in the boson operators to the simple TC Hamiltonian.
We consider the integrable case of such a model—the integrable generalized Tavis–Cummings
(IGTC) model.
The presence of a Kerr-like medium changes the behaviour of the system significantly. For
example, it crucially changes the Rabi oscillations in the JC model [30, 31]. Kerr-type systems
can also be applied to the description of the nonlinear oscillators [32, 33] and Bose–Hubbard
dimers [34–36].
The TC model belongs to a class of integrable models known as Gaudin–Richardson systems [37, 38]. The IGTC model belongs to a different set of integrable models connected with
the so called XXX rational R-matrix. It was solved exactly in [39] by the Quantum Inverse
Method (QIM) [40–42]. The QIM allows us to obtain the exact expressions for the energy
spectrum of the model and its dynamical correlation functions. In this work we show that by
using the so-called determinant representation [43] it becomes possible to obtain some manageable expressions for correlation functions.
In the QIM approach it is considered that the model is solved if the analytical expression
for the eigenenergies, eigenvectors and correlation functions are expressed through the solutions of Bethe equations. The Bethe equations being a set of coupled nonlinear algebraic
equations dependent on the model under consideration. Because of their nonlinearity, Bethe
equations are not easy to solve analytically or even numerically. Several approaches to this
problem have been developed, for example [44], which have been successfully applied to the
investigation of Bethe equations of the Gaudin and Richardson models [45]. For the XXX
Heisenberg chain with spin-12 another technique was developed—the theory of deformed
strings (for its extensive review see [46]). In this paper we present some exact numerical solutions of Bethe equations for the IGTC model in the TC limit.
The paper is organized in the following way. In the second section we provide a description of the IGTC model as a generalization of the TC model. In the third section we discuss in
detail the application of the QIM to the model. We present the Bethe equations, the eigenvectors and find the spectrum of the model. In the fourth section the analytical expression for the
dynamical correlation functions are obtained and presented in determinant form. In the fifth,
a numerical analysis of the Bethe equations and correlation functions is presented in detail.
In the appendix we provide a number of explicit examples of the solutions of Bethe equations and present the numerical values of all the quantities we need in the calculation.
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2. Integrable generalized Tavis–Cummings model
As the starting point we consider an ensemble of N two-level non interacting atoms of one sort
in the cavity. We assume that only one mode of the cavity field has to be taken into account,
whereas the other ones are suppressed. Each atom has ground |ψi− and exited |ψi+ states. The
TC model model is defined by the Hamiltonian (in the units = c = 1):
HTC = ωa† a + ω0 Sz + g(a† S− + aS+ ),
(1)
where ω and ω0 are frequencies of the cavity mode and the atomic system, g is the cavity-atom
coupling constant. a and a† are the usual annihilation and creation operators for the cavity
field, which satisfy usual commutation relations [a, a† ] = 1. We also introduced the collective
N-atom Dicke operators S± , Sz (spin operators for which total spin S N/2 ) as:
N
N
N
N
1 i
Sz ≡
Siz =
S± ≡
Si± =
σi± ,
σz ,
(2)
2
i=1
where
Si+ = |ψi+ ψi− |,
i=1
i=1
Si− = |ψi− ψi+ |,
Siz =
i=1
1 −
(|ψ ψi− | − |ψi+ ψi+ |),
2 i
(3)
where σi± = 12 (σ x ± iσ y ), and σix , σiy , σiz are the Pauli matrices acting each on ith site. Operators
Sz and S± satisfy the commutation relations:
+ −
[S
, S ] = 2Sz , [Sz , S± ] = ±S± ,
(4)
of the su(2) algebra.
In the Hamiltonian (1) the first term describes single quantized mode of the cavity field, the
second one describes the atomic inversion of the whole system, and the first term of the interaction part describes an atomic transition from the excited state to the ground state accompanied by the emission of a photon, whereas the second one describes the reverse process. The
number of excitations M and the Casimir operator S2:
(5)
M ≡ Sz + a† a,
2
S
≡ S+ S− + Sz (Sz − 1)
(6)
are two nontrivial constants of motion [HTC , M] = [HTC , S2 ] = 0.
In this paper we consider the exactly solvable extension of the TC model that describes
the system in a complex environment and takes into account the spin–spin interaction. The
Hamiltonian for the IGTC model can be written in the form [39]:
HK = ωa† a + ω0 Sz + g(a† S− + aS+ ) + γ(a† a† aa + (Sz )2 ).
(7)
Here, the fourth-order term in the boson operators describes a Kerr-like medium, whereas
(Sz )2 describes the spin–spin coupling. When γ = 0, HK reduces to the TC model (1) and this
becomes the JC model [7] for N = 1 (S = 12 ). When S = 12 , (Sz )2 = 1 and HK with γ = 0 is
the JC model with Kerr nonlinearity [30].
It is easy to verify that operator M commutes with the Hamiltonian HK : [HK , M] = 0 , so
we can consider another Hamiltonian H = g−1 (HK + (γ − ω)M − γM2 ) and [HK , H] = 0.
We can then write
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J. Phys. A: Math. Theor. 50 (2017) 464003
H = ∆Sz + (a† S− + aS+ ) + ca† aSz ,
(8)
where c = −2g−1 γ is a new effective coupling constant and ∆ = g−1 (γ + ω0 − ω) is the frequency shifted detuning. One should note that the last term on the right-hand side of (8) causes
photon number dependent changes in the atomic transitions and therefore describes a Stark
shift. Henceforth, we shall consider H, but the same results can immediately be extended to
the model with Kerr nonlinearity, Hamiltonian (7), through the mapping given above.
3. The determinant representation of correlation functions
To apply QIM to the solution of the model we consider the two 2 × 2 matrix operators
LB (λ), LS (λ) for bosons and spins respectively [39]:
λ − ∆ − c−1 − ca† a
a†
LB (λ) =
,
(9)
a
−c−1
λ − cSz
cS+
L
(λ)
=
,
(10)
S
cS−
λ + cSz
in which λ is a complex number. Note that the elements of LB (λ) and LS (λ) mutually commute. For the monodromy matrix of the QIM T(λ) we can now set
A(λ) B(λ)
T(λ) = LB (λ)LS (λ) =
,
(11)
C(λ) D(λ)
so that
B(λ) = λX − Y;
X = a† + cS+ , Y = (1 + c∆)S+ − ca† Sz + c2 a† aS+ ;
(12)
[X, Y] = 0,
and
We also have
C(λ)
= λa − (S− + caSz ) ≡ λa − Z.
(13)
A(λ) = (λ − ∆ − c−1 − ca† a)(λ − cSz ) + ca† S− ,
(14)
D(λ) = caS+ − c−1 (λ + cSz ).
Multiplying L-operators in an order different than in (11) we obtain the following expression for the monodromy matrix
A(λ)
B(λ)
T(λ) = LS (λ)LB (λ) =
.
(15)
C(λ)
D(λ)
The entries of this matrix satisfy the involution relations [36]
B(λ)
= C+ (λ∗ ), C(λ)
= B+ (λ∗ ).
(16)
The introduced monodromy matrices satisfy the intertwining relation
R(λ, µ)T(λ)T(µ) = T(µ)T(λ)R(λ, µ)
(17)
R(λ, µ)T̃(λ)T̃(µ) = T̃(µ)T̃(λ)R(λ, µ)
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with the rational R-matrix


f (µ, λ)
0
0
0
 0
g(µ, λ)
1
0 
,
R(λ, µ) = 
(18)
 0
1
g(µ, λ)
0 
0
0
0
f (µ, λ)
with the enries
f (µ, λ) = 1 −
c
,
µ−λ
g(µ, λ) = −
c
.
µ−λ
The traces of the monodromy matrices (11)and (15) coincide:
TrT(λ) = TrT(λ)
= τ (λ),
and commute for arbitary complex numbers λ, µ:
[τ
(λ), τ (µ)] = 0.
(19)
In the explicit form:
τ (λ) = λ2 − λ(cM+∆ + 2c−1 ) + cH,
(20)
where M is the number operator (5) and H is the Hamiltonian (8). It can be checked that
H = c−1 τ (0),
(21)
∂τ (λ)
|λ=0 −c−1 ∆ − 2c−2 .
M = −c−1
∂λ
It can be shown that MB(λ) = B(λ)(M + 1), and likewise S2 B(λ) = B(λ)S2 . So B(λ)
acts as a creation operator for the quasi-particles, while C(λ) is an annihilation operator.
The M-particle state vectors are constructed in the usual fashion for the QIM method
M
M
|(22)
ΨS,M ({λ}) =
B(λj ) | ΩS =
(λj X − Y) | ΩS ,
j=1
j=1
where
the
vacuum
state |ΩS = |0|S, −S
(a|0 = 0; S− |S, −S = 0,
with
2
z
S |S, −S = S(S + 1)|S, −S, and S |S, −S = −S|S, −S) is annihilated by operator C
C(λ) | ΩS = 0,
and is the eigenstate of the operators A, D :
where
A(λ) | ΩS = a(λ) | ΩS , D(λ) | ΩS = d(λ) | ΩS ,
(23)
a(λ) = (λ − ∆ − c−1 )(λ + cS), d(λ) = −c−1 (λ − cS).
(24)
In formula (22) a short-hand notation {x} ≡ {λ1 , λ2 , . . . , λM } is used.
In the explicit form the state vector (22) has the form
M
|(25)
ΨS,M ({λ}) =
(−1)M−m em Xm YM−m | ΩS ,
m=0
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where
em =
λi1 λi2 . . . λim
i1 <i2 <...<im
is the elementary symmetric function [47].
The conjugated M-particle state vectors are equal to
ΨS,M ({λ}) | = ΩS |
M
j=1
C(λj ) = ΩS |
M
j=1
[λj a − (S− + caSz )]
(26)
M
= ΩS |
(−1)M−m em am ZM−m
m=0
and
ΩS | B(λ) = 0.
By the construction the state vectors (22) and (26) are symmetric functions of their arguments
{λ} ≡ {λ1 , λ2 , ..., λn }.
The state vectors (22), (26) are eigenvectors of the transfer matrix, and thus of the
Hamiltonian if {λ} are the roots of the Bethe equations, which here take the form, for
n = 1, 2, . . . , M ,
M
λn − λj + c
λn + cS
=
(1
+
∆c
−
cλ
)
,
n
(27)
λn − cS
λn − λj − c
j=1,j=n
where 0 S N2 . Evidently, there are K = min(2S, M) + 1 (modulo the permutation group)
sets of solutions of these M Bethe equations. The complex valued roots are pairwise conjugated [48].
The eigenvalues of the transfer matrix (20):
M
M
c
c
θ(28)
(1 −
) − d(µ) (1 +
).
S,M (µ) = a(µ)
µ − λj
µ − λj
j=1
j=1
From the equation (21) we see that the M-particle eigenenergies of the Hamiltonian (8) are
equal to:
H | ΨS,M ({λ}) = ES,M | ΨS,M ({λ}),
M M S
S c
c
(29)
.
ES,M =
− S∆ +
1−
1+
c
λj
c
λj
j=1
j=1
The ground state of the Hamiltonian (8) corresponds to the minimal value of eigenenergy. The
set of solutions of Bethe equations that defines this state will be labeled σg : {λσg }.
The form of the Bethe equations drastically depends on the parameter c. In the case of vanishing nonlinearity c → 0 the Bethe equations (27) transform to another set of equations (30),
which refers to the well-studied TC-model [37, 38].
M
2S
2
− λ̃n + ∆ =
.
(30)
λ̃n
λ̃ − λ̃j
j=1,j=n n
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N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
The energy spectrum of the TC-model can be expressed in the following form:


M
1
σ
.
ES,M
= −S ∆ + 2
(31)
σ
λ̃
j=1 j
Consider the state vectors constructed by operators (16)
M
M
j ) | ΩS , Ψ
j ).
S,M ({λ}) =
S,M ({λ}) |= ΩS |
|(32)
Ψ
B(λ
C(λ
j=1
j=1
It was proved in [41] that on the solutions of Bethe equations
S,M ({λ}) = νM | ΨS,M ({λ}), Ψ
S,M ({λ}) |= ν −1 ΨS,M ({λ}) |,
|(33)
Ψ
M
where for the model under consideration
M
M
λj − cS
−1
=
(1
+
∆c
−
cλ
)
=
ν
.
j
(34)
M
λj + cS
j=1
j=1
Note, that from Bethe equations (27) it follows that
M
λn + cS
= 1.
(1 + ∆c − cλn )
(35)
λn − cS
n=1
In [39] it was proved that Bethe state vectors form a complete orthogonal set
ΨS,M ({λσ1 }) | ΨS,M ({λσ2 }) ∼ δσ1 ,σ2
| ΨS,M ({λσ })ΨS,M ({λσ }) |
(36)
= I.
Nσ2
σ
Index σ denotes the independent sets of solutions of Bethe equations (27), and the summation
is over all K sets of solutions.
The scalar product of the Bethe state vectors for M = 1, 2 can be calculated explicitly:
and
S1 (µ, λ) ≡ ΨS,1 (µ)|ΨS,1 (λ) = −c2 s2 + s(2 + c(2∆ − λ − µ)) + λµ,
(37)
S2 (µ1 , µ2 , λ1 , λ2 ) ≡ ΨS,2 (µ1 , µ2 )|ΨS,2 (λ1 , λ2 ) =
2c4 s4 − 4c4 s3 + 2c4 s2 − (λ1 + λ2 ) −2c3 s3 + 2c3 s2 + c 4s2 − 2s (c∆ + 1)
− (µ1 + µ2 ) −2c3 s3 + 2c3 s2 + 4cs2 (c∆ + 1) − 2cs(c∆ + 1) + 4c2 s2 (c∆ + 1) + 2c2 λ1 λ2 s2
+ 2c2 µ1 µ2 s2 + (c∆ + 1) −8c2 s3 + 4c2 s2 − 2c2 s + 8s2 − 4s (c∆ + 1)
+ 2 (λ1 + λ2 ) (µ1 + µ2 ) s(c∆ + 1) − 2c (λ1 + λ2 ) µ1 µ2 s
(38)
The further straightforward calculation of scalar products for M > 2 is rather difficult because
the number of commutation relations (one needs to evaluate in order to get the answer) grows
exponentially.
To obtain the analytical expression for the correlators for the arbitrary M we shall use
Slavnov’s formula for the scalar products [49–51]. This formula, adopted for the model under
consideration, has the form:
− 2cλ1 λ2 (µ1 + µ2 ) s + 2λ1 λ2 µ1 µ2 .
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N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
SM ({µ}, {λ}) = ΨS,M ({µ}) | ΨS,M ({λ})
M
cM j=1 d(λj )
(39)
det T({µ}, {λ}),
=
j>k (µk − µj )
α<β (λβ − λα )
where the entries of the M × M matrix T({µ}, {λ}) are
∂
Tab ≡ T(µa , λb ) = −c−1
θ(µa , {λ})
∂λb



M
M

c
c
1
a(µa )
1−
− d(µa )
1+
.
=
(λb − µa )2 
µ a − λj
µ a − λj 
j=1,j=b
(40)
j=1,j=b
It is supposed here and below that {λ} are the solutions of Bethe equations, while {µ} is the
set of arbitrary parameters. Functions a(µ), d(µ) are defined in (23).
The square of the norm of the eigenvectors is calculated by the Gaudin formula [37, 43]
which is (39) in the limit {µ} → {λ}:
M
λα − λβ + c
N = SM ({λ}, {λ}) = c
d2 (λj )
det Φ({λ}),
(41)
λ α − λβ
2
M
j=1
α=β
where the entries of the M × M matrix Φ are


M

∂
a(λa )
λa − λk − c 
.
Φ
ln
(42)
ab = −
∂λb  d(λa )
λk − λa − c 
k=1,k=a
The determinant representation (39) may be used in calculation of the transition element of
the photon annihilation operator
n
Ψ
(43)
S,M−n ({λ }) | a | ΨS,M ({λ}),
where {λ} and {λ } are the solutions of Bethe equations (27) for M and M − n particles
respectively. We note that from the definition (13) it follows that limλ→∞ λ−1 C(λ) = a and
hence
SM ({µ}, {λ})
n
Ψ
lim
.
(44)
S,M−n ({µ}) | a | ΨS,M ({λ}) =
µ1 ,µ2 ,...,µn →∞ µ1 µ2 . . . µn
Replacing the arbitrary parameters {µ} by the solutions of Bethe equations {λ } we obtain
the answer for (43). Denoting
µ−M+a−1
∂ a−1
a
T(µa , λb )
µa →∞ (a − 1)! ∂µa−1
a
Vab = lim
we obtain the following answer
AM,n ({µ}, {λ}) ≡ ΨS,M−n ({µ}) | an | ΨS,M ({λ})
M−n M
M
(45)
j=1
α=1 (µj − λα )
M
det T(n) ({µ}, {λ}).
=c
d(λj ) j>k>n (µk − µj )
α<β (λβ − λα )
j=1
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N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
Here set {µ} ≡ {µ1 , µ2 , ..., µM−n } has the length M − n, whereas the set {λ} ≡ {λ1 , λ2 , ..., λM }
has the length M. The entries of the M × M matrix T(n) are equal to Vab for a n and are Tab
(40) for a > n, 1 b M . To obtain the answer for the transition element (43) we have to
change parameters {µ} on the solutions of Bethe equations for M − n particles.
For instance, for n = 1 the answer for the transition element is
ΨS,M−1 ({λ }) | a | ΨS,M ({λ})
M−1 M
M
j=1
α=1 (λj − λα )
M
det T(1) ({λ }, {λ}),
d(λj ) =c
− λ )
(λ
−
λ
)
(λ
β
α
j
α<β
j>k>1 k
(46)
j=1
where the entries of M × M matrix

−1
T
 21
T(1) ({λ }, {λ}) = 
 ..
 .
TM1
−1
T22
..
.
TM2
−1
T23
..
.
TM3
...
...
..
.
...

−1
T2M 

.. 
.
. 
TMM
To find the transition element of the creation operator one can take the complex conjugation of the transition element of the annihilation operator and obtain:
ΨS,M−n ({λ }) | an | ΨS,M ({λ})∗ = ΩS |
= ΩS |
= ΩS |
M
B+ (λj )(a† )n
j=1
M
j=1
M−n
j=1
j )(a† )n
C(λ
M−n
j=1
M−n
j=1
C(λj )an
M
j=1
B(λj ) | ΩS ∗
C+ (λj ) | ΩS ) | ΩS B(λ
j
ν
(47)
= M−n ΨS,M ({λ}) | (a† )n | ΨS,M−n ({λ }),
νM
where the definition (32) and the properties (16), (33) have been used, and λ are the solutions
of the Bethe equations (27) with M − n particles. The obtained formula allows us to express
the transition element of the creation operator in the determinantal form:
A†M,n ({λ}, {λ }) ≡ΨS,M ({λ}) | (a† )n | ΨS,M−n ({λ })
νM
(48)
= A∗M,n ({µ}, {λ}),
νM−n
where A∗M,n ({µ}, {λ}) is the complex conjugated coefficient (45) on the solutions of the Bethe
equations. A similar determinant representation for the transition elements (45) and (48) for
the bimodal Bose–Hubbard model has been studied in [36].
The obtained representations for the transition elements (45) and (48) allow us to calculate
different n-photon time-dependent correlation functions.
We define the n-photon Green function an (t)(a† )n M as the average taken over the
M-particle ground state of the model | ΨS,M ({λσg }):
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N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
an (t)(a† )n M =
σ
=
g
σ
−ES,M
eit(ES,M
)
σ
1
ΨS,M ({λσg }) | e−iHt an eiHt (a† )n | ΨS,M ({λσg })
Nσ2g
ΨS,M ({λσg }) | an | ΨS,M+n ({λσ })
2
Nσ2g Nσ(49)
× ΨS,M+n ({λσ }) | (a† )n | ΨS,M ({λσg })
σg
σ
2 ν σ
−ES,M
eit(ES,M
) M+n
σg
σ
n
=
({λ
})
|
a
|
Ψ
({λ
})
Ψ
σg ,
S,M
S,M+n
2 N2
N
νM
σ
σ
g
σ
where the sum is taken over K = min(2S, M + n) + 1 sets of solutions of Bethe equations (27)
for M + n particles λσ labeled by σ , and the complete orthogonal set (36) of eigenstates of
the Hamiltonian (8) was used. Substituting formulas (29), (34), (41) and (45) into (49) we
obtain the final answer.
The technique developed allows us to calculate different dynamical correlation functions of
the considered model and, respectively, of the TC model in the c → 0 limit.
4. Time evolution of the atomic inversion
Knowing the projection of the initial state on the state vectors of the model one can obtain
the answers for the correspondent dynamical correlation functions. Let us consider the time
evolution of the coherent states. Let us assume that in the initial state | Φ0 the spin system is
in the ground state | S, −S (S− | S, −S = 0 ) and the field is in the coherent state | α α ∈ C
(a | α = α | α)
|α|2
†
|(50)
Φ0 =| α | S, −S = e− 2 eαa | 0 | S, −S.
The time evolution of the initial state can be obtained by application of the evolution operator
U(t) = exp(iHt), so we have | Φ(t) = U(t) | Φ0 . We may use representations (26) and (25)
to find the projections of the initial states on the Bethe state vectors:
ΨS,M ({λ}) | Φ0 = (−1)M+1 (cαS)M e−
|α|2
2
M
m=0
em (cS)−m ≡ fM ({λ}),
(51)
M
|α|2 M − 2
Φ0 | ΨS,M ({λ}) = (cᾱS) e
(−1)M−m em (cS)−m ≡ gM ({λ}).
m=0
The evolution of field intensity can be constructed through the complete set of eigenfunctions | ΨS,M ({λσ }):
σ
σ
eit(ES,M1 −ES,M3 )
Φ(t) | (a ) a | Φ(t) =
Nσ21 Nσ22 Nσ23
(52)
M σ ,σ ,σ
† n n
1
σ1
× fM ({λ })gM ({λ
σ3
2
3
})A†M,n ({λσ3 }, {λσ2 })AM,n ({λσ2 }, {λσ1 }).
The summation here is over all sets of solutions of correspondent Bethe equations.
Knowing the answer for the field intensity we can find the evolution of atomic inversion
applying equality
Φ(t) | Sz | Φ(t) = Φ(t) | M | Φ(t) − Φ(t) | a† a | Φ(t)
= n − S − Φ(t) | a† a | Φ(t),
10
(53)
N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
Figure 1. Temporal evolution of the atomic inversion Sz . The atoms are initially in
the ground state with the total spin S = 32 , and the field is in the coherent state with
n = 20. The parameters are equal to c = 0.01/0.5/2.0 and ∆ = 0.0/1.0/2.0/3.0 .
where M is the number operator (5), and n = |α|2 is the average number of photons in a
cavity.
The examples of the numerical results for Sz (t) for different model parameters are given
in the figure 1. For the sake of simplicity we consider the case when the cavity contains only 3
atoms, so the total spin S changes in the range [− 32 , 32 ]. In the initial state | Φ0 =| α | S, −S
all the atoms are in ground state. It is evident that the state | Φ0 is not an eigenstate, so it
rapidly decays in time. We see that the collapses and revivals of Rabi oscillations for small
coupling constants (c = 0.01) are similar to those of the Jaynes-Cumming model [9] and the
time of the first revival is proportional to the average number of photons in a cavity t ∼ n.
As the effective coupling constant c increases the revivals become more localized in time,
their amplitudes and the time of the first revival decrease because of the dominating role of a
Kerr medium. The presence of detuning affects the form of the revivals for small values of the
effective coupling c.
Examples of neat evaluation of the Sz (t) are discussed in the next section.
11
N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
Figure 2. The roots distribution λσ
i of the Bethe equations (27) with the parameters
c = 0.5, M = 35, ∆ = 1.0 and S = 32 . Note that not all the roots are presented in this
figure.
Figure 3. The roots distribution λσ
i of the Bethe equations (27) with the parameters
c = 0.0, M = 35, ∆ = 1.0 and S = 32 .
12
N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
1
Figure 4. Behaviour of the single solution {λ } of the Bethe equations (27) for
different c. The represented solutions refer to the ground state of the system with
∆ = 1.0, S = 32 , M = 35.
5. Numerical analysis of the Bethe equations and evaluation of the correlation
functions
The solutions of the Bethe equations (27) give us complete information about the system
under consideration. The roots distribution of the Bethe equations (27) and the spectrum for
IGTC for the fixed values of parameters c, ∆, S, M are presented in figure 2, and in tables A5
and A6 of the appendix. The roots distribution of the Bethe equations (30) and the spectrum
for the TC model for the fixed values of parameters ∆, S, M are presented in figure 3.
Solutions of the equations (27) become close to the solutions of the equations (30), when
c → 0. The example of such transformation is presented in figure 4, for the solution {λ1 }
which refers to the ground state of the system.
We use an iterative approach to the solution of the Bethe equations (27). Within this framework it is crucial to choose the correct initial assumption for the roots distribution in the first
iteration. The algorithm for finding the correct initial assumption is the following: (I) Solve
the equations (30) for ∆ = 0.0, taking the initial assumption in a string form with all roots
having the same real part {a ± ib · 1, a ± ib · 2, a ± ib · 3, ...}. By varying the parameters a
and b we find the correct initial assumption for equations (30) which do not lead to nonphysical solutions containing coinciding roots. (II) Use the solution of the equations (30) as
an initial assumption for the equations (27) with small c and Δ (the optimal step can be found
experimentally). (III) Vary c and Δ slowly until they take the desired values.
In table A1 we present the sets of solutions of the Bethe equations (27) for this system,
σ
eigenenergies E3/2,M
, and norms of the Bethe wavefunctions Nσ2. Using the data from table A1
we evaluate the numerical values of the scalar products f15 ({λσ }), g15 ({λσ }), presented in
13
N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
table A2, and we also evaluate the numerical values for the photonic transition elements:
A15,1 ({λσ }, {λσ }) and A†15,1 ({λσ }, {λσ }), presented in tables A3 and A4.
6. Conclusions
In this paper the QIM approach to the IGTC model was considered. We have obtained the
determinant representation for the norm of the Bethe wavefunctions Nσ2 and for the trans­ition
elements of the photons AM,n ({λσ }, {λσ }), A†M,n ({λσ }, {λσ }). The knowledge of these elements allows us to investigate any dynamical correlation function of the model provided the
roots of the Bethe equations are known. The solutions of the Bethe equations for different
values of parameters of the model were calculated numerically. The obtained results were
applied for the evaluation of the atomic inversion Φ(t)|Sz |Φ(t).
Acknowledgments
This work was supported by the Russian Science Foundation (grant № 16-11-10218).
Appendix
Here we provide the solutions of the Bethe equations and the numerical values of the photonic
transition elements and norms of wavefunctions.
Table A1. Solutions of the Bethe equations for c = 0.5, ∆ = 1.0, S = 32 , M = 14, 15.
M
14
15
λ11
λ12
λ13
λ14
λ15
λ16
λ17
λ18
λ19
−0.239 824
−0.750 193
−0.241 867
0.176 368
1
E3/2,M
N12
λ21
λ22
λ23
λ24
λ25
λ26
λ27
λ28
λ29
−0.750 262
0.167 747
1.773 790
3.798 26 ± 4.182 16 I
2.422 63 ± 2.016 87 I
1.977 72 ± 1.038 23 I
3.022 76 ± 3.038 48 I
4.856 00 ± 5.581 94 I
2.812 93 ± 2.333 51 I
3.444 29 ± 3.334 88 I
4.247 73 ± 4.467 06 I
2.325 25 ± 1.401 43 I
5.333 30 ± 5.858 46 I
2.014 00 ± 0.477 29 I
14.3938
15.2107
22
1.328 07 × 10
−0.738 31
−0.365 95
3.128 92 + 1.922 48 I
2.736 71 + 1.071 69 I
2.512 42 + 0.333 28 I
3.678 73 + 2.886 82 I
4.411 73 + 4.000 60 I
5.430 90 + 5.382 55 I
1.111 14 × 1025
−0.740 738
−0.353 435
2.598 74
2.727 31 + 0.628 46 I
3.487 14 + 2.220 83 I
3.037 37 + 1.372 10 I
4.848 37 + 4.287 62 I
5.900 50 + 5.663 55 I
4.079 92 + 3.179 73 I
(Continued )
14
N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
Table A1. (Continued )
M
14
15
4.805 44
2
E3/2,M
N22
5.0842
25
1.053 91 × 10
9.918 35 × 1027
−0.830 452
5.016 24 + 3.900 16 I
3.332 98 + 1.1325 I
6.012 43 + 5.252 18 I
3.021 86 + 0.475 188 I
4.298 96 + 2.825 74 I
3.751 56 + 1.914 65 I
2.988 33 + 0.158 96 I
4.102 89 + 2.188 17 I
3.639 48 + 1.393 31 I
6.474 53 + 5.530 95 I
4.690 08 + 3.104 76 I
3.277 31 + 0.710 51 I
5.443 43 + 4.1804 I
−5.387 34
−5.659 03
3.872 31 + 1.188 78 I
5.601 29 + 3.852 93 I
4.3222 + 1.937 11 I
3.506 08 + 0.553 672 I
4.883 24 + 2.812 31 I
6.588 51 + 5.173 88 I
4.676 24 + 2.191 07 I
3.447 65 + 0.216 031 I
4.190 98 + 1.426 75 I
3.789 25 + 0.767 612 I
5.271 76 + 3.076 48 I
6.023 49 + 4.122 97 I
7.045 06 + 5.446 49 I
−16.3119
−17.1358
−0.823 073
2.849 73
λ31
λ32
λ33
λ34
λ35
λ36
λ37
λ38
λ39
3
E3/2,M
N32
4.496 29 × 1027
3.014 53
3.250 13
λ41
λ42
λ43
λ44
λ45
λ46
λ47
λ48
λ49
4
E3/2,M
N42
2.0875 × 10
4.9616 × 1030
2.996 98
30
2.718 93 × 1033
Table A2. Numerical values for the functions
c = 0.5, ∆ = 1.0, S =
3
2, M
= 15.
f15 ({λσ })
σ
g15 ({λσ })
−3.542 25 × 109
1.880 59 × 1012
−4.846 51 × 1014
8.586 41 × 1016
1
2
3
4
f15 ({λσ }) and g15 ({λσ }) for
−5.716 61 × 1011
1.314 35 × 1013
−1.474 09 × 1014
1.094 85 × 1015
Table A3. Numerical values for the transition element A15,1 ({λσ }, {λσ }) for
c = 0.5, ∆ = 1.0, S = 32 , M = 15.
σ, σ 1
2
3
4
1
2
3
4
−1.459 83 × 1024
1.657 28 × 1024
−2.810 49 × 1023
3.759 25 × 1022
3.739 37 × 1023
−1.194 78 × 1027
1.282 74 × 1027
−1.716 35 × 1026
−6.080 01 × 1022
2.306 75 × 1026
−5.405 08 × 1029
5.624 94 × 1029
7.897 56 × 1021
−2.996 71 × 1025
8.056 98 × 1028
−2.6862 × 1032
15
N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
†
σ
σ
Table A4. Numerical values for the transition element A15,1 ({λ }, {λ }) for
c = 0.5, ∆ = 1.0, S = 32 , M = 15.
σ, σ 1
1
2
3
4
−1.260 67 × 1024
6.8671 × 1022
−5.578 47 × 1020
3.413 26 × 1018
2
3
4
7.456 56 × 1024
−1.143 15 × 1027
5.879 13 × 1025
−3.598 44 × 1023
−2.785 92 × 1025
5.071 59 × 1027
−5.692 46 × 1029
2.709 89 × 1028
8.631 94 × 1025
−1.571 59 × 1028
2.024 05 × 1030
−3.0869 × 1032
Table A5. Example of the numerical solutions of the Bethe equations for
c = 0.5, ∆ = 0.1, M = 4 , and different values of S. Note: this example is easy to
reproduce.
S
1
2
3
2
1
λ11
1.048 08 ± 0.508 37 i 1.349 52 ± 0.426 617 i
λ12
1.291 19 ± 1.890 55 i 1.532 19 ± 1.617 69 i
2
1.577 16 ± 0.365 265 i 1.758 06 ± 0.310 326 i
1.758 01 i ± 1.408 80 i
1.954 84 ± 1.240 21 i
−6.420 34
−8.225 79
λ13
1
−2.328 55
ES,4
λ21
λ22
−0.299 68 ± 0.401 56 i 0.692 35 ± 1.446 50 i
0.435 412 ± 2.039 59 i 0.878 76
λ23
2
ES,4
λ31
λ32
λ33
3
ES,4
−4.463 41
2.078 55
−0.733 00
−0.230 48
1.167 94 ± 1.042 22 i
1.534 92 ± 0.872 592 i
1.244 67
1.528 05
−1.114 21
−1.550 59
−2.216 33
−0.248 57 ± 1.948 60 i −0.449 67 ± 1.030 01 i
−3.847 29
1.312 63 ± 0.402 60 i
−0.948 36 ± 0.529 91 i −1.074 33
−1.345 80 ± 0.584 813 i
3.693 89
−0.733 658
0.987 374
1.236 30
−0.863 032 ± 1.974 70 i −1.244 56 ± 1.277 96 i
λ41
−1.462 16 ± 0.578 535 i 1.340 77
λ42
λ43
4.900 38
4
ES,4
−1.667 71
2.016 06
−1.917 37 ± 0.611 298 i
λ51
−1.377 97 ± 2.0319 i
λ52
λ53
5.790 68
5
ES,4
16
N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
Table A6. Example of the numerical solutions of the Bethe equations for
c = 0.5, ∆ = 0.1, M = 15, and different values of S.
S
1
2
1
3
2
2
λ11
λ12
λ13
λ14
λ15
λ16
λ17
λ18
λ19
2.327 41 ± 0.353 73 i
2.651 60 ± 0.953 04 i
3.058 14 ± 1.657 19 i
3.562 51 ± 2.463 49 i
4.187 08 ± 3.385 09 i
4.974 93 ± 4.461 72 i
6.039 78 ± 5.810 56 i
2.061 13
2.475 83 ± 0.259 97 i
2.800 88 ± 0.826 68 i
3.196 18 ± 1.503 89 i
3.682 14 ± 2.288 67 i
4.284 84 ± 3.194 15 i
5.049 01 ± 4.258 94 i
6.087 96 ± 5.599 00 i
2.092 62
2.602 27 ± 0.168 87 i
2.942 25 i ± 0.709 56 i
3.330 06 ± 1.362 10 i
3.801 12 ± 2.125 04 i
4.384 31 ± 3.012 80 i
5.126 06 ± 4.063 80 i
6.139 58 ± 5.393 12 i
2.098 94
3.076 94 ± 0.599 44 i
3.459 53 ± 1.229 65 i
3.918 47 ± 1.971 24 i
4.484 44 ± 2.840 44 i
5.205 25 ± 3.876 15 i
6.194 09 ± 5.193 00 i
2.654 82
2.099 86
2.747 37
1 −5.464 07
ES,15
−10.846 27
−16.150 21
−21.379 55
λ21
λ22
λ23
λ24
λ25
λ26
λ27
λ28
λ19
1.536 27 ± 0.810 20 i
1.915 51 ± 1.628 51 i
2.426 62 ± 2.502 33 i
3.068 13 ± 3.467 91 i
3.876 83 ± 4.575 37 i
4.965 49 ± 5.947 66 i
−0.498 71
2.121 58 ± 0.128 31 i
2.410 32 ± 0.672 32 i
2.758 30 ± 1.351 72 i
3.210 46 ± 2.149 73 i
3.790 58 ± 3.071 52 i
4.539 77 ± 4.152 31 i
5.568 37 ± 5.507 53 i
−0.810 31
2.577 21 ± 0.550 29 i
2.917 33 ± 1.194 37 i
3.348 63 ± 1.963 57 i
3.902 05 ± 2.864 94 i
4.622 48 ± 3.932 12 i
5.620 50 ± 5.278 39 i
2.342 32
2.113 76
−5.388 47
−10.545 03
1.773 69 ± 0.297 77 i
2.029 09 ± 0.967 17 i
2.426 13 ± 1.727 41 i
2.940 90 ± 2.573 21 i
3.583 30 ± 3.522 57 i
4.392 18 ± 4.619 27 i
5.480 81 ± 5.983 46 i
−0.266 25
2
ES,15
5.214 07
λ31
λ32
λ33
λ34
λ35
λ36
λ37
λ38
λ19
3
ES,15
4
λ1
λ42
λ43
λ44
λ45
λ46
λ47
λ48
λ19
−0.032 60
1.370 64
−0.136 68
1.973 76 ± 0.219 89 i
2.228 42 ± 0.807 99 i
2.593 30 ± 1.527 68 i
3.073 33 ± 2.352 55 i
3.683 57 ± 3.290 84 i
4.462 59 ± 4.381 58 i
5.521 61 ± 5.742 75 i
−0.536 45
9.982 96
1.817 76 ± 0.620 933 i
2.119 19 ± 1.376 46 i
2.572 49 ± 2.234 50 i
3.171 01 ± 3.196 82 i
3.944 28 ± 4.305 03 i
5.000 11 ± 5.679 89 i
−0.327 67
1.702 80
−0.744 50
4.719 66
0.929 48 ± 0.622 77 i
1.368 96 ± 1.540 94 i
1.897 87 ± 2.446 14 i
2.546 98 ± 3.426 90 i
3.359 65 ± 4.543 50 i
4.450 75 ± 5.922 23 i
−0.750 16
0.165 66
−0.241 92
14.319 02
4
ES,15
17
−1.087 52
2.052 22 ± 0.487 06 i
2.326 58 ± 1.169 72 i
2.732 67 ± 1.992 29 i
3.287 59 ± 2.940 09 i
4.022 90 ± 4.042 95 i
5.043 27 ± 5.416 79 i
1.916 06
−0.984 28
−0.637 40
−0.424 88
1.440 14 ± 0.334 64 i
1.617 67 ± 1.185 14 i
2.049 60 ± 2.109 40 i
2.645 23 ± 3.106 05 i
3.418 65 ± 4.234 86 i
4.475 19 ± 5.624 12 i
−0.123 58
−1.000 49
−0.480 820
9.111 88
(Continued )
N M Bogoliubov et al
J. Phys. A: Math. Theor. 50 (2017) 464003
Table A6. (Continued )
S
1
2
3
2
1
2
λ51
λ52
λ53
λ54
λ55
λ56
λ57
λ58
λ19
0.207 63 ± 0.682 92 i
0.784 26 ± 1.509 81 i
1.357 24 ± 2.418 91 i
2.022 65 ± 3.405 99 i
2.843 01 ± 4.527 17 i
3.938 44 ± 5.909 29 i
−0.309 78
5
ES,15
18.237 59
−1.000 65
−0.433 68
ORCID iDs
N M Bogoliubov https://orcid.org/0000-0002-5834-5066
I Ermakov http://orcid.org/0000-0002-2535-5540
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