Journal of Quantum Information Science, 2013, 3, 10-15
http://dx.doi.org/10.4236/jqis.2013.31003 Published Online March 2013 (http://www.scirp.org/journal/jqis)
Thermodynamic Properties and Decoherence of a
Central Electron Spin of Atom Coupled
to an Anti-Ferromagnetic Spin Bath
Martin Tchoffo, Georges Collince Fouokeng*, Lukong Cornelius Fai, Mathurin Esouague Ateuafack
Mesoscopic and Multilayer Structures Laboratory, Department of Physics, Faculty of Science,
University of Dschang, Dschang, Cameroon
Received December 30, 2012; revised January 31, 2013; accepted February 9, 2013
ABSTRACT
The decoherence of a central electron spin of an atom coupled to an anti-ferromagnetic spin bath in the presence of a
time varying B-Field (VBF) is investigated applying the Holstein-Primak off and Bloch transformations approaches.
The Boltzmann entropy and the specific heat capacity at a given temperature are obtained and show the correlation of
the coupling of the spin bath and the electron spin of the central atom. At low frequencies, the coherence of the coupled
system is dominated by the magnetic field intensity. At low VBF intensity, there is decrease in entropy and heat capac-
ity at increase external magnetic field that show the decoherence suppression of the central electron spin atom. The
crossing observed in the specific heat capacity corresponds to the critical field point of the system which repre-
sents the point of transition from the anti-ferromagnetic system to the ferromagnetic one.
C
B
Keywords: Entropy; Specific Heat Capacity; Varying B-Field; Decoherence
1. Introduction
Thermodynamic properties of Quantum systems have
become very attractive due to their potential applications
in thermoelectric devices [1], tunneling and decoherence
[2,3]. Thermodynamic properties of quantum systems
now aid in the investigation of the dynamical entropy
[4-8]. Recently, different definitions of specific heat are
discussed [9] and the entropy for a quantum oscillator in
an arbitrary heat bath at finite temperature is examined
[10-12]. Experiments [13,14] show the feasibility of pro-
cessing Quantum Information (QI) via the manipulation
of optically excited electron spins [15] in Diamond. De-
coherence of electron spins coupled to nuclear spin baths
in quantum dots or solid-state impurity centers [16] is
crucial in spin-based Quantum Information (QI) process-
ing [17], magnetic resonance spectroscopy [18,19], and
magnetometry [20]. Recent quantum technologies show
that the relevant environments are of nanometer size [16-
19] and therefore their quantum nature is enhanced. Quan-
tum nuclear spin bath, in contrast to classical noises,
possessed to a great extent controllability and surpris-
ingly coherence recovery of an electron spin [21-23]. The
central-spin model, or Gaudin model, describes one spin
coupled to N − 1 bath spins via both isotropic and ani-
sotropic Heisenberg interactions, including a constant
magnetic field [24-27]. Decoherence leads to suppression
of spin tunneling in magnetic molecules and nanoparti-
cles [28,29] and also destroys the Kondo effect in a dis-
sipationless manner [30]. Decoherence may be investi-
gated with the help of the spin-echo-like techniques [31]
or spin wave approximation. Extension of the spin-echo-
like approach to quantum computations is known as the
“bang-bang control” [32].
In this paper, our objective is to evaluate the influence
of the external parallel VBF on the thermodynamic pro-
perties and Decoherence tailoring of a Central electron
spin of atom coupled to an anti-ferromagnetic spin bath
by the SWA method and compare our results with those
obtained via the partition function [4].
The organization of the paper is as follows: in Section
2, we present a brief description of the theoretical ap-
proach used and the model for simulation of open many-
spin systems. In Section 3, we evaluate the thermody-
namic properties of the Central electron spin of atom coupl-
ed to an Anti-ferromagnetic spin bath and then conclude
our findings in Section 4.
2. Theoretical Approach and the Model
Hamiltonian
The theoretical description of decoherence have been
*Corresponding author.
C
opyright © 2013 SciRes. JQIS

M. TCHOFFO ET AL. 11
studied numerically [33] and show that the evolution of
the central spin system from its initial pure state 0 to
the final mixed state, along with the corresponding trans-
formation of the environment, is a very difficult problem
of quantum theory. For some more complex models, dif-
ferent approximations can be employed, such as the Mar-
kov approximation [34]. A special case of environment
consisting of uncoupled oscillators, so-called “boson bath”,
is also rather well understood theoretically [35]. The bo-
son bath description though applicable for many types of
environment [36], is not universal. The boson bath model
is not applicable for the decoherence caused by an envi-
ronment made of spins [33]. The most direct approach to
study the spin-bath decoherence is to simulate numeri-
cally the evolution of the whole compound system by di-
rectly solving the time-dependent Schrödinger equation
[33]. This approach allows us to avoid any kind of ap-
proximation, except for the obvious limitation on the to-
tal number of spins models [33]. To make such simula-
tions feasible, high-performance computational schemes
are needed. The simulation method based on the Cheby-
shev’s polynomial expansion [35,36] was used in [33].
The spin wave approximation method was also used in
[26,27]. Both approaches are applicable when the Ham-
iltonian is not explicitly dependent on time. In this work,
we use a spin wave approximation by the Holstein-Pri-
makoff and the Fourier transformation method.
ψ
Let us consider a system of quantum spins describ-
ed by
N
H
and initially described by0
H
:
SSBB
H
HH H  (2.1)
where
0
z
SB
H
gBS   (2.2)
the Hamiltonian of the central spin atom,

0
0,
zzz
SBa ib i
i
J

,
H
SSS
N
 
 (2.3)
the Hamiltonian of the interaction of the central spin with
the spin bath



,,, ,,
,,
,
z
B
aibib ja jBAai
ij
z
BAbj
j
HJSSSSgBB S
gBBS





 

δδ
δδ i
(2.4)
The spin-bath Hamiltonian;
g
is the gyromagnetic fac-
tor,
B
μ
the Bohr magneton, J0 the coupling constant,
J
the exchange interaction and


0cosBB


t
is the
VBF applied in the direction and -z

 
b
a
the pulsa-
tion of the VBF. The effects of the next nearest neighbor
interactions are neglected. We assumed that the spin
structure of the environment may be divided into two in-
terpenetrating sub-lattices and with the property
that all nearest neighbors of an atom on lie on
and vice versa. and ,bi represents the spin op-
erators of and atom on sub-lattice and b.
The field A is anisotropic and assumed to be positive
which approximates the effect of the crystal anisotropy
of the energy with the property of tending for positive
magnetic moment
a
b
,ai
S
j
S
th
i
B
th a
B

to align the spins on sub-lattice
in the positive az

direction and spins on sub-lattice
in the negative bz

direction. Using the reduce Hol-
stein-Primakoff and Bogoliubov transformations [17], we
have the reduced Hamiltonian
0
z
SB
H
gBS   (2.5)

0
z
k
0
J
SB k
k
H
Sm
n
N
 (2.6)


0
B
kkk k
kk
H
En 
m




 (2.7)
where k is the frequency of the magnon in the system
and the energy of a central spin atom. The factors
kkk
ω
0
β
E
m
β

 and kkk are respectively the total
number of magnons on the branch b and . From
Equation (2.6), we see that if the two branches of the
Network have the same number of magnons, the Hamil-
tonian SB
nαα


a
H
vanish then there will be no coupling be-
tween the Network and the Networkb. In this case
the behavior of the system depends totally on the driven
external field.
a
3. Thermodynamic Properties of a Central
Electron Spin of Atom Coupled to an
Anti-Ferromagnetic Spin Bath
In this section, we find the Boltzmann entropy and the
specific heat capacity to show the influence of the ani-
sotropy of the field that characterize the anti-ferromag-
netic environment and the VBF on the dynamic of the
central electron spin system. We suppose that our system
is in a canonical ensemble [37] .To attempt to evaluate
the dynamical properties of the considered system, the
statistical sum is needed. Let 0 be the zero-point en-
ergy of the system obtained considering the harmonic
approximation given by,
E
1
21
KK
K
0kk
Em
ii








 (3.8)
with the mode of vibrations and k the atom mass
of the crystal, the statistical sum of the system has the
form:
km

0B
exp
Z
EKT

 (3.9)
where
B
K is the Boltzmann constant and the abso-
lute temperature. With the help of the Helmholtz free
energy
T
F
,
ln
B
F
KT Z (3.10)

M. TCHOFFO ET AL.
12
We find the Boltzmann entropy as S


V
SFT  (3.11)
And the specific heat capacity at a constant volume

vv
CTST

(3.12)
The indicated sum in (3.9) is easily done if ;
then the modes become closer together. In the canonical
distribution, according to the thermodynamic principle
(from which the entropy of a system depends on disorder
such as temperature...), of course using the Spin wave
approximation, the Boltzmann entropy and the specific
heat Capacity is evaluated respectively as:
N

0
12342 4
2ln
B
kkk
EgB
SDDDDD
TT


 

D
(3.13)
and
 
15 36
2
24
11
v
kk
TDD TDD
CDD
 


2
(3.14)
where



1exp k
DωTωT

 k

(3.15.a)


21exp k
DωT

  (3.15.b)

3exp k
DωTωT



k

(3.15.c)
 
41exp k
DωT

 
 
(3.15.d)

512expk
D 

 

T (3.15.e)



612exp 1k
DT 

 

(3.15.f)
From the analytical results obtained in Equation (3.13)
we have the plots in Figures 1 and 2 describing the behavior
of the Boltzmann entropy under the influence of the external
VBF. From the expression in Equation (3.14) we have the
plots in Figures 3 and 4 describing the behavior of the spe-
cific heat capacity under the influence of the external VBF.
It is shown that the Boltzmann entropy (Figure 1) in-
creases respectively with increase temperature and de-
creases as a function of the increase of the paralleled time
dependent external VBF intensity (see Figure 2). The
specific heat capacity in Figures 3 and 4 show the cross-
ing point for different temperatures and for different exter-
nal VBF intensity. This crossing point corresponds to the
critical magnetic field whose expression is given
Equation (17), [26]. This represents transition point of
the state of the crystal. Tending the entropy to zero
show that the external VBF brings the central spin sys-
tem to its coherent state. From the external VBF, the
plot of the Boltzmann entropy in (Figure 5) and of the
specific heat capacity in (Figure 6) show that the

c
B
050100150 200250 300 350400 450500
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
T=t em perature
S=entropy
B
0
=1.5
B
0
=2.5
Figure 1. Boltzmann entropy versus temperature for dif-
ferent external VBF intensity with
.

0 14,2
A
Bg
and

M6.
0 1 23 45 6 78 910
-20
0
20
40
60
80
100
B=magnetic field intensity
S= ent ropy
T=200
T=100
Figure 2. Boltzmann entropy versus the external VBF in-
tensity for different values of temperature with
.0 14,2
A
Bg

 and

M6.
Figure 3. Specific heat capacity versus the temperature for
different values of external VBF intensity with the constants
.0 14,2


A
Bg
and

M6.
presence of the anisotropic field creates a dephasing: this
describes decoherence phase observation in a coupled
central spin system. Thus the action of the temperature


T and of the external VBF are both opposite effects.
The decoherence of the spin of the central atom due to

M. TCHOFFO ET AL. 13
Figure 4. Specific heat capacity versus the external VBF
intensity for different values of temperature with
and . .0 14,2
A
BgM6
Figure 5. Boltzmann entropy versus the external VBF for
different anisotropy field intensity with the given
and . 150, 2Tg M6
Figure 6. Specific heat capacity versus the external VBF
intensity for different anisotropy field intensity with the
given and 150, 2Tg

M6.
the anisotropy of the field of the anti-ferromagnetic en-
vironment or by the temperature is thus reduced by the
increased of the external VBF intensity.
4. Conclusions
In this paper, we have investigated the thermodynamic
properties of quantum multistate-coupled systems. We
present the influence of the external VBF on the deco-
herence of the spin of the central atom coupled to an
anti-ferromagnetic spin bath where the number of envi-
ronmental atom vanishes. A Formalism is developed,
using Holstein-Primakoff and Bloch transformations ap-
proaches from where the density of state of the system,
the statistical sum and the free energy of a central elec-
tron spin are found. It is shown that the effect of the ex-
ternal systems in such formalism can always be included
in some general classes of Functions (statistical sum and
free energy). The resulting specific heat capacity ap-
proaches the classical (Pettit-Dulong law) result for high
temperatures and goes to zero for vanishing temperature.
The entropies of both systems also obey the second law
of thermodynamics as well as the third law of thermody-
namics.
We observed that both specific heat capacity and en-
tropy decreased with increased intensity of the external
VBF and decrease temperature. This shows that the de-
coherence aspect of the central electron spin atom ob-
served at the coupling with the anti-ferromagnetic spin
bath is reduced. We observe the numerical results for the
central spin of atom coupled to an anti-ferromagnetic
spin bath and confirm the results using the partition func-
tion methods [4]. Due to the frequency of the VBF, the
evolution of the system is controllable in time. This pro-
cess is very important in the efforts of suppressing deco-
herence in spin bath coupled system [21].
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