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Phys. Scr. 98 (2023)045113 https://doi.org/10.1088/14024896/acc3cc
PAPER
Coherence controlled generation of Gaussian quantum discord in a
quantum beat laser
Haleema Sadia Qureshi
1
, Shakir Ullah
2
and Fazal Ghafoor
3
1
Department of Physics, Fatima Jinnah Women University, The Mall Rawalpindi 46000, Pakistan
2
Institute of Nuclear Sciences, Hacettepe University, 06800 Ankara, Turkey
3
Department of Physics, COMSATS University Islamabad, Islamabad Campus 45550, Pakistan
Email: haleema.sadia@fjwu.edu.pk
Keywords: quantum discord, Gaussian states, quantum beat laser, quantum correlations
Abstract
Quantum discord, an appropriate measure of quantum correlation, is analyzed in a twomode
Gaussian state of the cavity ﬁeld evolved by a system of quantum beat laser. In the laser system, two
arbitrary singlemode Gaussian states of the cavity ﬁelds and an external classical ﬁeld couple to the
atomic lasing medium. We calculate the timedependent expression for the quantum discord both
analytically and numerically, by following the reduced density matrix equation of the resulting cavity
ﬁeld. In this framework, we investigate the generation and control of Gaussian quantum discord in the
twomode Gaussian state of the cavity ﬁeld at the output in terms of the purity and nonclassicality of
the two input cavity modes, the Rabi frequency of the classical driving ﬁeld, the relative phase of the
coupling parameters, and the damping rates of the cavity modes. The behaviour of quantum discord
appears oscillatory due to the quantum beats in the output cavity ﬁeld developed by the classical
driving ﬁeld in the medium as time passes. Moreover, we ﬁnd that quantum discord switches on and
off while adjusting the relative phase of the coupling parameters. Analysis of the analytical results
agrees well with our numerical simulations.
1. Introduction
Today, seeking for the most powerful measure and quantiﬁer of quantum correlations is one of the core subject
in quantum information Science. This quantiﬁcation, in case of pure quantum states, is entailed by quantum
entanglement [1]which is the basic ingredient for the successful implementation of various protocols of
quantum computation and communication [2,3]. However, it has been realized that quantiﬁcation of non
classical correlations is highly elusive in case of mixed quantum states [4,5]. In fact, numerous recent research
ﬁndings have shown that there are mixed quantum states which, although to be separable, might exhibit some
strong correlations with potential practical applications in quantum information science [6–9]. These
correlations cannot be reproducible by any classical probability distribution.
One such kind of correlation is now characterized by quantum discord [10]which provides us with the
amount of mutual information and accounts essentially all the quantum correlations within the system. The in
depth analysis of quantum discord has been signiﬁcantly explored in different contexts in open quantum
systems of qubits [11–14], both under the Markovian and nonMarkovian environments. In the case of
continuous variables, computing quantum discord is relatively tough because it contains solving of an
optimization problem which is extremely nontrivial even for the basic two qubits system [15]. However, for two
mode systems which are restricted to Gaussian states, an approximation to the quantum discord has been
computed and characterized, the socalled Gaussian quantum discord [16,17].
In this framework, quantum discord has been generated in superconducting waveguides by utilizing the
dynamical Casimir effect [18]. A realisable continuous variable quantum network code on the basis of quantum
discord has been proposed which is comparatively better than discrete variable ones [19]. Quantum discord is
found to be the basic ingredient for the quantum advantage in Gaussian quantum illumination [20]. In addition,
RECEIVED
25 October 2022
REVISED
8 March 2023
ACCEPTED FOR PUBLICATION
13 March 2023
PUBLISHED
23 March 2023
© 2023 IOP Publishing Ltd
Gaussian quantum discord has been investigated in a unique setup composed of two nonresonant bosonic
modes which are coupled to each other and also interacting with an external joint thermal reservoir [21].
Recently, a strict hierarchy relation of various kinds of quantum correlations in different passive and active
quantum systems is shown, for instance, in a linear beam splitter [22], correlated spontaneous emission laser
[23], and nonlinear parametric converter [24]. In these contributions, the authors analysed that quantum
discord [25,26]is the super set of quantum entanglement [27–31], steering [32,33], and nonlocality [34].Itis
worth mentioning that, as compared to entanglement, quantum discord is highly robust toward decoherences
such as depolarizing, dephasing, and in general to amplitude damping [35,36]. Furthermore, various other
works have also been reported on quantum discord with several operational interpretations and practical
applications, for example, remote state preparation [37], work extraction [38], entanglement distribution [39],
quantum state broadcasting [40], quantum state merging [41], and quantum metrology [42]. All these positive
attributes are the basic motives for further investigation of the nonclassical aspect of cavity ﬁeld radiation
evolved by a quantum beat laser employing Gaussian quantum discord in the present contribution.
In this paper, the existing mathematical procedure under Gaussian measurements [16,17]is readily
followed, for convenience, and applied to an experimentally viable system of quantum beat laser (QBL)[43]to
analyse the Gaussian quantum discord. In this system, a gain medium composed of threelevel atoms in a V
conﬁguration inside a doubly resonant cavity is assumed to interact with a strong external driving ﬁeld along
with the two quantized cavity modes being initially considered in the form of arbitrary singlemode Gaussian
states (SMGSs). At the output, the evolved cavity modes take the form of twomode Gaussian state (TMGS)are
expected to be strongly correlated that give rise to twomode Gaussian discord. In this connection, we
thoroughly investigate quantum discord with respect to various parameters of the QBL system. It is worth noting
that experimentally accessible parameter values are chosen for our proposed system such that they correspond
to the micromaser experiments [44,45]. Recently, collective quantum beats in a spontaneous emission process
without an initial superposition of the excited levels in a threelevel Vtype atomic system has been
experimentally demonstrated by exciting a dilute atomic gas of magnetooptically trapped
85
Rb atoms with a
weak drive resonant on one of the transitions [46]. In addition, various other experiments have been
demonstrated, although in different contexts, on the threelevel atomcavity coupled systems [47–49].
This work takes the following structure. In section 2, we describe the model formalism of our proposed QBL
system with Gaussian input and output modes of the cavity ﬁeld. In section 3, we present the mathematical
framework of Gaussian quantum discord and obtained an expression as a function of the various system
parameters. In section 4, we discuss the graphical results of the generated Gaussian quantum discord in the
TMGS of the evolved cavity ﬁeld. Finally, main conclusions of the present contribution is summarized in
section 5.
2. Model
In this section, we describe our proposed model of QBL wherein a gain medium consisting of threelevel atoms
in a Vconﬁguration is considered inside a doubly resonant cavity (see ﬁgure 1). A ground level
1ñ
∣
, excited level
3ñ
∣
, and middle level
2ñ
∣
constitute the model atom, as illustrated in ﬁgure 1(b). The atomic transitions from level
1ñ
∣
to
2ñ
∣
and
3ñ
∣
, being electrical dipole allowed, are respectively coupled to two quantized input cavity modes of
frequencies ν
1
and ν
2
. The atomic transition between levels
2ñ
∣
and
3ñ
∣
, being electrical dipole forbidden, is
continuously driven by an external classical ﬁeld of carrier frequency ν
3
and Rabi frequency Ωe
−if
, with phase f.
The total Hamiltonian of the proposed system, in the framework of rotatingwave and electricdipole approach,
takes the form
HH H,1
oint
=+ ()
with
H
aa aa
22 33
11 , 2
o12 2 11 1
11
112
22
wn wn
wnn
= + +D ñá + + +D ñá
+ñá+ +
[( )∣ ∣ ( )∣ ∣
∣∣ ] ()
††
Hga ga31 21 23 2 e H.c ., 3
t
int 1122i3
=ñá+ñá
Wñá +
fn+
∣∣ ∣∣ ∣∣ ()
()
⎡
⎣⎤
⎦
where a
i
ai
(
)
†and g
i
(with i=1, 2)stand for, respectively, the lowering (raising)operator of photons in the two
cavity modes, and the atomic coupling strengths with the two quantized cavity modes. The strong classical ﬁeld
is supposed to be in resonance with the atomic transition
3ñ
∣

2ñ
∣
, that is ν
3
=ω
3
−ω
2
, however, the two
quantized input cavity modes are tuned from the atomic transitions
3ñ
∣

1ñ
∣
and
2ñ
∣

1ñ
∣
by Δ
1
=ω
3
−ω
1
−ν
1
and Δ
2
=ω
2
−ω
1
−ν
2
, respectively. The two transitions
3ñ
∣

1ñ
∣
and
2ñ
∣

1ñ
∣
are considered fully quantum
mechanical in nature up to the second order in their respective coupling strengths g
1
and g
2
. The transition
2
Phys. Scr. 98 (2023)045113 H S Qureshi et al
3ñ
∣

2ñ
∣
is considered semiclassical in nature up to all orders in the Rabi frequency Ω. Note that, this assumption
allows us to operate the proposed system linearly and it is justiﬁable when Ωis greater than the coupling
strengths g
1
and g
2
.
2.1. Dynamical equations
In the interaction picture, the Hamiltonian given in equation (1)is transformed into the following form
Hgaga33 22 31 21 232e H.c. 4
I12
1122i
= D ñá + D ñá + ñá + ñá  Wñá +
f
∣∣ ∣∣ ∣∣ ∣∣ ∣∣ ()
⎡
⎣⎤
⎦
The equation of motion for the density matrix ρof the atomcavity ﬁeld coupled system is given by
H
i,. 5
I
rr= [] ()
Now the equation of motion for the reduced density matrix of the output cavity ﬁeld is obtained by tracing over
the atomic variables
H
ga ga
itr ,
i, i, H.c. 6
fIatom
1131 2221
rr
rr
=
=  +
[]
[][] ()
††
In the strongly driven limit, i.e. when Ω?g
1
,g
2
, and by keeping the coupling strengths g
1
and g
2
of the cavity
modes up to the ﬁrst order [50], the equation of motions for the two density matrix element operators ρ
31
and
ρ
21
can be approximately evaluated as
Figure 1. Schematic of QBL. (a)A gain medium, which consists of threelevel atoms, is placed inside a cavity that supports two modes.
(b)A threelevel model atom of the gain medium is coupled to two modes of the quantized cavity ﬁeld of frequencies ν
1
and ν
2
, and a
classical ﬁeld of carrier frequency ν
3
and Rabi frequency Ωe
−if
, with phase f.
3
Phys. Scr. 98 (2023)045113 H S Qureshi et al
gaa gaii
2ei i, 7
31 31 i21 133 11
11 232 2
rgr r r r r= + D + W++
f
() ( ) ()
gaa gaii
2ei i. 8
21 21 i31 222 22
11 123 1
rgr r r r r= + D + W++
f
() ( ) ()
In the above, we phenomenologically included the atomic decay at a rate γfrom the upper levels to the lower
levels. consequently, the zerothorder equations of motion for the corresponding relevant matrix element
operators ρ
33
,ρ
22
,ρ
32
, and ρ
11
are obtained as
ri2ee , 9
f33 33 i23 i32 3
rgr r r r= + W+
ff
() ()
i2ee, 10
22 22 i32 i23
rgr r r= + W
ff
() ()
i2e, 11
32 32 i22 33
rgr rr= + W
f() ()
0, 12
11
r=()
where we have assumed that the atoms which are prepared in level
3ñ
∣
are considered to be inserted into the gain
medium at a rate r
3
via a pumping process. The steadystate solutions for ρ
31
and ρ
21
can be obtained by setting
all the derivatives with respect to time on the lefthand side of equations (7)–(12)to be zero. On substituting the
steadystate solutions into equation (6)and by considering the damping of the cavity ﬁeld to the vacuum, we
obtain the equation of motion for the reduced density matrix of the output cavity ﬁeld as [51]
a a aa aa
a a aa aa
a a aa aa
a a aa aa
aa a a aa
aa a a aa
2
e
e
22
22 , 13
ffff
fff
fff
fff
fff
fff
11 11 1 111 1111 1 1
22 22 2 222 2222 2 2
12 21 1 221 1 212
12i
21 12 2 112 1221 1 2i
111
11
11
222
22
22
rVVr V rVr
VVr V rVr
VVr V rVr
VVr V rVr
rrrk
rrrk
=+  
++  
++  
++  
+
+
f
f
[( ) ]
[( ) ]
[( ) ]
[( ) ]
()
() ()
†† †
†† †
†† †
†† †
†††
†††
**
**
**
**
where κ
1
and κ
2
indicate the rates of damping of the two quantized cavity modes. The two diagonal terms ς
11
and
ς
22
represent the gain mechanism in the laser system, however, the other two offdiagonal terms ς
12
and ς
21
represent the coherence effect in the system generated due to the coupling of external classical ﬁeld with the
atomic transition
3ñ
∣

2ñ
∣
. The expressions of the diagonal and offdiagonal terms ς
11
,ς
22
,ς
12
, and ς
21
are as
follows
XY Z2i2i, 14
11 22 22
Vg gg g=+W+W++WW[( ) ( )] ( )
XY Z2i2i, 15
21 22 22
Vg gg g=+W+W+WW[( ) ( )] ( )
XY Zii, 16
12
Vgg=W W  W+[( ) ( )] ( )
XY Zii, 17
22
Vgg=W W + W+[( ) ( )] ( )
where
Xgr Y
2,i2
2,18
23
22 2 2
gg
g
g
=+W =DW
+DW()
()
() ()
and
Zi2
2.19
22
g
g
=D+W
+D+W
()
() ()
Here for convenience, we considered g
1
=g
2
=gand Δ
1
=Δ
2
=Δ.
Following a straightforward algebra, the various equations of motion for the variances and covariances
terms are readily obtained by employing the master equation (13)
d
dt aa aa aa
aa
1
24e
e, 20
1111
11 11112
i12
12 i1211 11
VV k V
VVV
áñ= + áñ+ á ñ
+áñ++
f
f

[( )
]()
†††
†
*
**
d
dt aa aa aa
aa
1
24e
e, 21
2222
22 22221
i12
21 i1222
22
VV k V
VVV
áñ= + áñ+ áñ
+áñ++
f
f
[( )
]()
†††
†
*
**
4
Phys. Scr. 98 (2023)045113 H S Qureshi et al
d
dt aa aa a a a a
1
22
e, 22
1211 22 121
212 2221 1 1
12 21 i
VV kk V V
VV
áñ= + + áñ+á ñ+áñ
++ f
[( ( )) (
())] ()
††††
**
*
d
dt aa aa aa aa
1
22
e, 23
1211 22 1 2 1212 2 221
11
12 21 i
VV kk V V
VV
áñ= + + áñ+áñ+áñ
++ f
[( ( )) (
())] ()
††††
**
*
d
dt aa aa aa aa
1
22ee,24
1 2 11 22 1 2 1 2 21 i11 12 i22
VV kk V Váñ= + + áñ+ áñ+ á ñ
ff
[( ( )) ] ( )
d
dt aa aa aa2e, 25
11 11 1 11 12 i12
Vk Váñ=  áñ+ á ñ
f
() ()
d
dt aa aa aa2e. 26
22 22 2 22 21i12
Vk Váñ=  áñ+ áñ
f
() ()
One can solve the above coupled rate equations numerically and analyzed the Gaussian quantum discord in the
TMGS of the resulting laser ﬁeld. Moreover, in the strong coupling limit, that is when Ω?γ, the master
equation (13)can be approximated to the following
aa aa aa aa aa a a aa a a
aa a a aa aa a a aa
2i i e
22 22 , 27
ff f f f
ffff ff
12
211221121
21
21
21
2i
111
11
11 222
22
22
rVr r Vr r
r r rk r r rk
=+++++
+  +
f
[( ) ( )] [( ) ( )]
()( )()
†† †† †† ††
††† † ††
where
gr LL
2,28
12
23
Vg
=
+
() ()
()
with L
±
=1/(Δ±Ω/2).
2.2. Cavity modes at the input
We choose two separable quantized modes, each one in the form of arbitrary SMGS, as the input cavity modes.
The corresponding characteristic function is
hh h h,exp
1
2,29
12 in
cs()≔ ()
†
⎛
⎝⎞
⎠
where
hhhhh,, ,
1122
=(
)
†
**
is a row matrix with complex amplitudes h
i
(with i=1, 2)and
0
0,30
in 1
2
ss
s
=()
⎛
⎝⎞
⎠
is the total covariance matrix of the two input quantized modes of the cavity ﬁeld. In the above matrix, the ﬁrst
and second input cavity modes are, respectively, represented by the subcovariance matrices 1
11
11
s
ab
ba
=*
⎜
⎟
⎛
⎝
⎞
⎠
and
2
22
22
s
ab
ba
=*
⎜
⎟
⎛
⎝
⎞
⎠
, where α
i
are real, however, e
ii
ii
bb=j
∣∣ are complex elements. Furthermore, these matrix
elements of the subcovariance matrices can be related to the purity μ
i
[52]and the nonclassicality τ
i
[53]of the
input quantized modes of the cavity ﬁeld [54–56]by the following expressions
11
41 2 ,14
12 .31
ii
i
i
iiii
i
2222
am
tbtt m m
t
=+
=+
()
∣∣ () ()
Here, the purity and the nonclassicality are, respectively, deﬁned as 12det
ii
m
s=()
and
max 0,
ii
1
2
tz=
{
}
, where ζ
i
denotes the minimum eigenvalue of subcovariance matrix σ
i
. For any physical
state, μlies in the range 0 μ1, however, nonclassicality falls in the range 0 τ1/2. It is to be noted that,
a state is pure if μ=1 and mixed if μ<1. Likewise, a state is fully nonclassical if τ=1/2.
2.3. Cavity modes at the output
For the evolved cavity modes, which are in the form of TMGS, the characteristic function in terms of the
complex probability amplitudes h
1
and h
2
is written as
hh h h,exp
1
2,32
12 out
cs=() ()
†
⎛
⎝⎞
⎠
5
Phys. Scr. 98 (2023)045113 H S Qureshi et al
where
,33
T
out
13
32
sss
ss
=˜˜
˜˜ ()
⎜⎟
⎛
⎝⎞
⎠
represents the total covariance matrix of the output twomode cavity ﬁeld, with
aa aa
aa a a
12
12,34
11111
11 11
s=áñ+ áñ
á ñ á ñ +
˜()
†
†
*
⎛
⎝
⎜⎞
⎠
⎟
aa aa
aa a a
12
12,35
22222
22 22
s=áñ+ áñ
á ñ á ñ +
˜()
†
†
*
⎛
⎝
⎜⎞
⎠
⎟
and
aa aa
aa aa .36
3
1212
12 12
s=áñáñ
á ñ á ñ
˜()
†
†
**
⎛
⎝
⎜⎞
⎠
⎟
In the above, the subcovariance matrices
1
s
˜
and 2
s
˜, respectively, represent the ﬁrst and second resulting cavity
modes, while the subcovariance matrix 3
s
˜denotes the correlations between the two resulting cavity modes of
the QBL system. By solving numerically the coupled rate equations (20)–(26), we obtain the matrix elements of
σ
out
, wherein, the initial conditions for the two cavity modes are as follows: aa 12
110 1
a
á
ñ= 
†,
aa 12
220 2
a
á
ñ= 
†,
aa
110 1
b
á
ñ=
,aa
220
2
b
á
ñ= , and aa aa a a 0
120 120120
á
ñ=á ñ=á ñ=
††
*. Moreover, in
the limit of strong classical driving ﬁeld, that is when Ω?γ, we can easily solve the coupled rate equations (20)–
(26)and the solutions are given by
aa 1
2cos 2 1 e , 37
t
1112 12 2
aa Jaaáñ= ++  
k
[()()] ()
†
aa 1
2cos 2 1 e , 38
t
2212 122
aa Jaaáñ= + 
k
[()()] ()
†
aa i
2sin 2 e , 39
t
1212
2i
Ja aáñ=  kf+
()( ) ()
†()
aa i
2sin 2 e , 40
t
1221
2i
Ja aáñ=  kf
()( ) ()
†()
aa 1
4ee1ee, 41
t
12 1i2 2ii42ii
12
bbáñ= +  ´
jf j J k f++Q
()() ()
() ()
aa 1
4e1e e1e e , 42
t
11 1 i2 2 i 2 2i2 2 i 2 i
12
bbáñ=
++´
Jjf Jj kf++Q+
[( ) ( ) )] ()
() (())
aa 1
4e1e e1e e , 43
t
22 1 i2 2 i 2 2i2 2 i 2 i
12
bbáñ=
++´
Jjf Jj k++Q
[( ) ( ) )] ()
() ()
where ϑ=g
2
r
3
Ωt/(Ω
2
−4Δ
2
)γand Θ=g
2
r
3
t/(2Δ+Ω)γ, and for convenience we consider equal damping
rates for both the cavity modes, i.e. κ
1
=κ
2
=κ.
3. Gaussian quantum discord
Here, we give the mathematical framework of Gaussian quantum discord which provides us with the
discrepancy between two classically equivalent deﬁnitions of mutual information [16,17]. It is investigated that
quantum discord accounts almost all kinds of nonclassical correlations [57,58], including quantum non
locality, steering, and entanglement [22–24]. Here, our main focus is to explore the quantum discord in a
bipartite continuous variable QBL system prepared in Gaussian states, therefore, we follow the already existing
approach which was carried out for TMGS [16,17]. This approach is restricted to an optimization process under
Gaussian positive operatorvalued measures (POVMs). In general, the elements of a Gaussian POVM can be
written as
GG
1
0
ddrd
P
=
p
() () ()
†, where ρ
0
is a density matrix that represents a physical arbitrary SMGS and
G
aaexp ii
ddd=() [
]
†
*with (i=1,2)is the displacement operator which is a function of a complex number δ.
Here, δis representingeach of the possible outcomes.
In connection to this, the underlying physical phenomenon is subjected to generalized Gaussian
measurements. The total covariance matrix σ
out
, given in equation (33), of the evolved two modes (say mode 1
and mode 2)of the output cavity ﬁeld provided that ζ
±
1/2, where ζ
+
and ζ
−
represent the symplectic
eigenvalues which are invariants under symplectic transformations and can easily be obtained as
6
Phys. Scr. 98 (2023)045113 H S Qureshi et al
I
1
24. 44
224
z=LL
() ()
Here,
II I2
12 3
L= + +
with
I
det
11
s=˜,
I
det
22
s=˜,
I
det
33
s=˜
, and
I
det
4out
s=are the local symplectic
invariants of the output covariance matrix σ
out
.
Hence, using the symplectic eigenvalues and local symplectic invariants, the general formula for the
Gaussian quantum discord can be expressed as [16,17]
Iffff,45
2
zz x=
¬+
() () () () ()
where
fs ssss1
2log 1
2
1
2log 1
2,46=++

() ( )
⎛
⎝⎞
⎠⎛
⎝⎞
⎠⎛
⎝⎞
⎠⎛
⎝⎞
⎠
and
II II III II
I
II I I I I II I I II
I
if 0
if 0
,47
21 2 1
1
2
2
3
2241 3
3
2241
22
12 3
243
4412
23
2412
2
x
h
h
=
>
+ + + 

+ +   +
()
()( )∣∣ ()( )
()
()()
⎧
⎨
⎪
⎩
⎪
with
III
412
2
h
=()
−
III1
214
++
(
)( )
. The leftarrow ‘←’in the superscript of Gaussian discord shows
that the measurement has been carried out on the mode 1. It is obvious that
shows Gaussian discord for the
measurement on the mode 2. Note that the case η>0(η0)represents those states for which the Gaussian
quantum discord is optimized by a set of homodyne (heterodyne)measurements [16].
4. Results
In the present section, we thoroughly analyse the nature of Gaussian quantum discord by plotting
¬
for
numerous parameter values of the proposed setup of QBL. From the Hamiltonian given in equation (1)of the
coupled atomcavity system, we calculate the reduced density matrix
f
r
for the cavity ﬁeld at the output of the
system. Further, we use the initial conditions from the input subcovariance matrices of the two quantized cavity
modes along with
f
r
of the resulting TMGS of the ﬁeld and obtained the dynamical rate equations, given in
equations (20)–(26), for the various elements of the resulting covariance matrix of the cavity ﬁeld. We solve the
dynamical rate equations of the system by two ways, that is, numerically in the arbitrary coupling limit of Ωand
analytically in the strong coupling limit when Ω?γfor which the solutions are given in equations (37)–(43)and
obtain a timedependent expression for the Gaussian quantum discord. It is worth mentioning that all the
graphical results of quantum discord via a QBL are traced using experimentally accessible parameter values such
that they correspond to the micromaser experiments [44,45].
In ﬁgure 2, Gaussian quantum discord
¬
is plotted versus dimensionless time gt for ﬁxed parameter values
of the system. In the ﬁgure, the solidblue and dottedred lines, respectively, correspond to the numerical
solutions in the arbitrary coupling limit of Ωand analytical solutions in the strong coupling limit when Ω?γ.A
small difference appears between the two cases, i.e., the arbitrary limit and the strong limit, wherein the amount
Figure 2. Gaussian quantum discord
¬is shown with respect to dimensionless time gt using different system parameter values, for
example, μ=1, τ=0.4, j
1
=j
2
=0, f=0, Δ=0, κ=0.012g,γ=0.279g,r
3
=0.511g, and Ω=13.953g. The results of solidblue
and dottedred lines correspond to the numerical solutions in the arbitrary coupling limit of Ωand analytical solutions in the strong
coupling limit when Ω?γ, respectively.
7
Phys. Scr. 98 (2023)045113 H S Qureshi et al
of quantum discord in the strong driven limit is observed greater than the one of the arbitrary limit. It is vividly
noticed that the proﬁle of quantum discord is underdamped and oscillatory versus the time of its development.
Essentially, the underdamped behaviour is due to the interaction of the quantized cavity modes with the
environment which appears in terms of the cavity damping rate κin the exponent of e
−2κt
, however, the
oscillatory behaviour is due to the
sin 2J()
and
cos 2J(
)
terms in the solutions of the dynamical equations.
Furthermore, it is not hard to realize from ﬁgure 3that the survival time of quantum discord increases as the
Rabi frequency Ωof the external classical ﬁeld enhances. However, it is also apparent that the number of
oscillations in a given time decreases with the increase of Ω. This, in other words, reveals that the external
classical ﬁeld generates a coherence effect between the energy levels
2ñ
∣
and
3ñ
∣
which plays an essential role for
the generation of quantum discord in the cavity ﬁeld via the proposed system of QBL.
The effect of relative phase fof the classical ﬁeld on the Gaussian quantum discord in the evolved modes of
the cavity ﬁeld is further analyzed, as shown in ﬁgure 4. An oscillatory response of quantum discord with respect
to fis clearly seen in the ﬁgure. It is evident that the quantum discord is maximum when f=0, π, and 2π,
however, it is zero when 2
f
=pand
3
2
p. This, more generally, infers that quantum discord is maximum at
Figure 3. Gaussian quantum discord
¬is shown versus dimensionless time gt and Rabi frequency Ω/gusing different system
parameter values, for example, μ=1, τ=0.4, j
1
=j
2
=0, f=0, Δ=0, κ=0.012g,γ=0.279g, and r
3
=0.511g.
Figure 4. Gaussian quantum discord
¬is shown versus dimensionless time gt and relative phase fusing different system parameter
values, for example, μ=1, τ=0.4, j
1
=j
2
=0, Δ=0, κ=0.009g,γ=0.465g,r
3
=0.511g, and Ω=13.953g.
8
Phys. Scr. 98 (2023)045113 H S Qureshi et al
f=nπand zero at n21
2
f
=+
p
()
, where nis a positive integer. Hence, this analysis entails that quantum
discord can be switched on and off by manipulating appropriately the relative phase fof the external classical
coupling ﬁeld. Luckily, this switching behaviour of quantum discord with respect to relative phase fis consistent
with the switching of quantum steering [33]and entanglement [27,51].
In ﬁgure 5, we analyse the inﬂuences of the controlling parameters of the input quantized modes of the cavity
ﬁeld on the generation of quantum discord. In this connection, we consider three different values for the non
classicality τand purity μof the input arbitrary SMGSs and plotted
¬
versus dimensionless time gt,as
respectively shown in ﬁgures 5(a)and (b). From ﬁgure 5(a), it is not hard to see a prominent effect of τon the
development of quantum discord. The amount along with the time development of quantum discord increases
as the nonclassicality enhances. This result infers that one can generate maximally discordant TMGS at the
output if the input two SMGSs are more nonclassical. On the contrary, quite remarkably, an opposite effect of μ
on the quantum discord is apparently observed in ﬁgure 5(b). From the quantitative comparison between
ﬁgures 5(a)and (b), we notice that both the amount and interval of time of quantum discord increase with the
decreases of μ. This clearly shows that maximally discordant states can be generated at the output if one chooses
mixed states for the input cavity modes. Note that our this result with respect to μis an agreement with the work
presented in [59].
Furthermore, in order to analyse the inﬂuence of cavity damping rates κ, the Gaussian quantum discord
¬
is plotted with respect to dimensionless time gt, as illustrated in ﬁgure 6. For this case, we consider three different
values of κ, while values of the other system parameters are kept ﬁxed. A considerable decrease in both the
amount and interval of time of quantum discord generation is noticed with the increase of cavity damping rate κ.
This analysis entails that the damping rate κadds a decoherence effect in our proposed laser system and
Figure 5. Gaussian quantum discord
¬is shown with respect to dimensionless time gt using different system parameter values, for
example, j
1
=j
2
=0, f=0, Δ=0, κ=0.009g,γ=0.465g,r
3
=0.511g, and Ω=13.953g.(a)The dottedred, dashedblack, and
solidblue lines show, respectively, the results for τ=0.25, 0.35, and 0.42, with μ=1. (b)The dottedred, dashedblack, and solid
blue lines show, respectively, the results for μ=0.6, 0.8, and 1, with τ=0.35.
Figure 6. Gaussian quantum discord
¬is shown with respect to dimensionless time gt using different system parameter values, for
example, μ=1, τ=0.4, j
1
=j
2
=0, f=0, Δ=0, γ=0.465g,r
3
=0.511g, and Ω=13.953g. The dottedred, dashedblack, and
solidblue lines show, respectively, the results for κ=0.0093g, 0.0186g, and 0.0302g.
9
Phys. Scr. 98 (2023)045113 H S Qureshi et al
consequently in the generation of Gaussian quantum discord in the TMGS of the output cavity ﬁeld. Hence, one
must reduce the cavity damping [60]in order to obtain maximal quantum discord at the output of the system
under consideration.
5. Summary
In this contribution, a thorough investigation of Gaussian quantum discord in an experimentally realizable QBL
system is presented. We considered a lasing medium, inside a doubly resonant cavity, which is coupled to two
quantized modes of the cavity ﬁeld, being initially in the form of arbitrary SMGSs, and also with an external
classical driving ﬁeld. Following the total energy of the coupled atomcavity system, we obtained the reduced
density matrix for the evolved cavity ﬁeld. In connection to this, the initial conditions from the input covariance
matrices of the two quantized cavity modes along with the reduced density matrix of the resulting cavity ﬁeld are
used to calculate the dynamical equations for the various elements of the evolved twomode covariance matrix.
The timedependent expression for the Gaussian quantum discord is obtained, from the output covariance
matrix, both analytically and numerically as a function of different parameters characterizing the QBL system.
To this end, we explore the generation and control of Gaussian quantum discord with respect to various
controlling parameters, for example, the major controlling parameters are the initial purity and nonclassicality
of the quantized cavity modes, the Rabi frequency of the classical coupling ﬁeld, the relative phase of the
coupling parameters, and the damping rates of the cavity ﬁeld.
From our results of ﬁgures 2–6, we found an underdamped and oscillatory proﬁle of quantum discord
against the time development. The reasons, of underdamped and oscillatory behaviours, are respectively the
appearance of cavity damping rate κin the exponent of e
−2κt
and
sin 2J()
and
cos 2J(
)
in the solutions of the
dynamical equations. We also observed that the amount, number of oscillations, and the time interval of
Gaussian quantum discord are directly proportional to the magnitude of the Rabi frequency Ωof the classical
coupling ﬁeld. This clearly reveals that a coherence is generated due to the coupling of the classical ﬁeld with the
energy levels
2ñ
∣
and
3ñ
∣
which in turn is very crucial in the generation and control of quantum discord.
Furthermore, the proﬁle of quantum discord shows oscillatory behaviour versus the relative phase fand,
interestingly, it can be switched on and off by adjusting fappropriately. From the indepth quantitative analysis,
a direct (inverse)relation of quantum discord with the nonclassicality τ(purity μ)of the initial cavity modes is
observed. In addition to this, we found a signiﬁcant decrease in the amount and also in the time of survival of
Gaussian quantum discord with the increase of cavity damping rate κ. This clearly shows that κdegrades the
quantumness in the resulting modes of the cavity ﬁeld.
The generation of Gaussian quantum discord is believed to be the most proper way of observing the non
classical nature of light modes via the system under consideration. Moreover, most importantly, we have chosen
values of the controlling parameters from micromaser experiments [44,45]for our proposed setup. Hence, the
manipulation of these practically accessible parameters would make QBL a practical means for the experimental
demonstration of Gaussian quantum discord.
Acknowledgments
SU is grateful for the support from TÜBİTAK1001 Grant No. 121F141.
Data availability statement
All data that support the ﬁndings of this study are included within the article (and any supplementary ﬁles).
ORCID iDs
Haleema Sadia Qureshi https://orcid.org/0000000278278403
Shakir Ullah https://orcid.org/0000000273717884
Fazal Ghafoor https://orcid.org/0000000224709183
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