Cent. Eur. J. Phys. • 12(12) • 2014 • 813-821
DOI: 10.2478/s11534-014-0520-5
Central European Journal of Physics
The influence of spontaneously generated coherence
on atom-photon entanglement in a Λ-type system with
an incoherent pump
Research Article
Xin Yang1∗ , Dong Yan2† , Qianqian Bao3 , Yan Zhang 4 , Cuili Cui 1
1 College of Physics, Jilin University, Qianjin Street 2699, 130012, Changchun, China
2 School of Science, Changchun University, Weixing Road 6543, 130022, Changchun, China
3 School of Physics, Liaoning University, Chongshan Middle Road 66, 110036 Shenyang, China
4 College of Physics, Northeast Normal University, People Street 5268, 130024, Changchun, China
Received 16 March 2014; accepted 15 July 2014
Abstract:
Owing to interference induced by spontaneous emission, the density-matrix equations in a three-level Λtype system have an additional coherence term, which plays a critical role in modulating the inversionless
gain and electromagnetically induced transparency effect. In addition, it is shown that spontaneously generated coherence (SGC) has an effect on the entanglement between an atom and a photon of the coupling
laser field by calculating the degree of entanglement (DEM) of the atomic system. In this paper, we investigate the influence of the SGC effect on atom-photon entanglement in a Λ-type system, which generally
remains a high entangled state. When an incoherent pump source is introduced, we find that the SGC
effect could exert considerable influence on the atom reduced entropy under certain conditions for both
transient and steady states. More interestingly, such an incoherent pump field could actively affect the
short-time dynamic behaviors of the transient quantum entangled state at a certain range of pump rate as
a typical coherent case.
PACS (2008): 03.65.Ud, 89.70.Cf,42.50.-p
Keywords:
spontaneously generated coherence • entanglement • quantum entropy • electromagnetically induced
transparency
© Versita sp. z o.o.
1.
Introduction
For two dacades, the research on quantum entanglement
has attracted a great deal of interests [1–7], which refers to
the interactive phenomenon in the quantum system com∗
†
E-mail: [email protected] (Corresponding author)
E-mail: [email protected]
posed of at least two particles. In particular, preparation
[1, 8], manipulation [9] and purification [10] of the quantum
entanglement are always important concerns in the field
of quantum information and quantum optics. As a kind of
physical resources, the quantum entanglement can play
a key role in all aspects of quantum information, such as
quantum teleprotation [11], quantum superdense coding
[12], quantum secret sharing [13], quantum algorithm [14],
etc.
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The influence of spontaneously generated coherence on atom-photon entanglement in a Λ-type system with an incoherent pump
As a nonlocality resource for quantum information protocols [15], we prefer getting high degree of entanglement
due to the adverse environment-induced decoherence effects. In accordance with the concept of von Neumann
entropy [16], it has been put forward that for a bicomponent system in a quantum state, the degree of the entanglement between the two components can be described
accurately through the utilization of the reduced quantum
entropy by Bennett, Phoenix and Knight et al. [17–21],
as the degree of the entanglement is proportional to their
reduced quantum entropy. In order to build a quantificationally comparable relationship among different entangled states, we quantify the DEM adopting the definition
of partial entropy in our system.
Since coherent states are regarded as semiclassical, the
superpositions of coherent states rapidly aroused great interests, such as consequent manifestation [22, 23], process
to generate such states [24], their properties and extensions to generalized coherent states [25], etc. Entangled
coherent state is a special and interesting case of multimode coherent superposition state. [15] The earliest entangled coherent state appeared in the research on charge
superposition [26], and then it was applied to the studies
of generating superpositions of coherent states [22], cavity
quantum electrodynamics realization [27], entanglement of
a coherent state with a vacuum state, etc. In particular,
generation of entangled coherent states [28, 29] could be
achieved with double electromagnetically induced transparency (EIT) [30–32] effect. Furthermore, nonlocal preparation of distant entangled coherent [33] states could be
possible utilizing the EIT effect.
On the other hand, several studies found that the relative
phase could modulate the EIT effect and the inversionless
gain. Here the relative phase stems from the SGC effect
[34, 35]. In particular, the SGC effect is one of the most
representative dissipative processes [36–38]. Generally,
narrowing down the decay rate is expected. However, a
previous work demonstrated that the SGC effect between
two spontaneous emissions in the V-type system has a
major role in establishing the atom-photon entanglement
[39]. It is found that in the presence of SGC effect, the
reduced entropy and the atom-photon entanglement are
phase-dependent.
In this paper, we consider a naturally well-entangled
three-level Λ-type system driven by a strong coherent
field and an incoherent pump field [32]. While considering
the SGC effect, we apply the EIT effect to the research on
the entorpy of entanglement. Due to the presence of the
incoherent pump, even though the probe is very weak, we
are surprised to find that the SGC effect can influence the
DEM [39–44] conditionally. Moreover, by comparing with
the gain or absorption of the atom system in both steady
Figure 1.
Sketch of the proposed three-level Λ-type model driven by
an incoherent pump field with rate 2Λ and a strong incoherent field with frequency ωc , which is probed by a weak
coherent field with frequency ωp . 2γ1 , 2γ2 and ∆p , ∆c are
decay rates and detunings of the two coherent fields, respectively. The polarizations are arranged so that one field
drive only one transition.
and transient states, we revealed the intrinsic physical
mechanism for the influence of SGC effect on the DEM
from the aspect of interaction.
2. Theoretical models and basic
equations
We consider a three-level Λ-type system (see Fig. 1(a))
with two near-degenerate low levels |2i and |3i, and an
exited level |1i. A strong coherent field with amplitude
Ec and carrier frequency ωc is applied to drive the transition |1i ↔ |2i. And a weak probe field with amplitude Ep
and carrier frequency ωp probes the transition |1i ↔ |3i.
Meanwhile, this transition is pumped with a rate 2Λ by
a monodirectional incoherent pumping to supply population to level |1i and then make it possible to gain spontaneously generated coherence. The Rabi frequencies of
the strong and the probe coherent fields are described by
Ωc = Ec · d12 /2~ and Ωp = Ep · d13 /2~, respectively. d13
and d12 denote the corresponding induced dipole moments
with θ representing the angle between them. Since the
existence of SGC effect depends on the nonorthogonality
of the two induced moments, we have to consider an polarization arrangement as shown in Fig. 1(b). In this case,
the probe and the coupling field are perpendicular to the
two induced dipole moments d13 and d12 , respectively, i.e.,
the strong coherent field Ec acts only upon the transition
|1i ↔ |2i; the weak coherent field Ep acts only upon the
transition |1i ↔ |3i. In order to have SGC effect in an
atom system, two conditions need to be fulfilled. There
are two near-degenerate energy levels, and the transition dipole moments are not perpendicular to each other.
In nature , however, it is difficult to find an atomic system which meet these conditions. The only experiment is
that Hui-Rong Xia et al. reported their first experimental
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Xin Yang, Dong Yan, Qianqian Bao, Yan Zhang , Cuili Cui
observation of the spontaneous emission cancellation via
quantum interference [45]. Although it is difficult to find
an atomic system that meet the above conditions, under
the dressed-states researchers could stimulate the condition of SGC effect by coupling field [46]. And 2γ2 (2γ1 )
being the spontaneous decay rate from the upper level to
the level |2i (|3i). In the interaction representation, the
Hamitonian of the system under the electric-dipole approximation and rotating-wave approximation is given by
HI = ~ ∆p |1i h1| + ∆ |2i h2|
−
Ωc |1i h2| + Ωp |1i h3| + h.c. ,
(1)
where ∆c = ω12 −ωc , ∆p = ω13 −ωp are the single-photon
detunings of the two corresponding coupling fields, while
∆ = ∆p − ∆c is the two-photon detuning. For convenient,
we set ~ as unity.
In accordance with the master equation of motion for the
density operator in an arbitrary multi-level system and
the Weisskopf-Wigner theory of spontaneous emission, we
can arrive at the density-matrix equations for the densitymatrix elements ρij . Generally, features of the systems are
not controlled by the phase of the applied fields without
the SGC effect. However, previous work have shown that
the SGC effect can modify the response of the atom-photon
entanglement [39, 47] and therefore we have to regard
Rabi frequencies as complex parameters. Supposing that
the phases of the probe and the coupling fields are φp and
φc , we can rewrite the density-matrix equations using two
real numbers Gp and Gc , where Ωp = Gp eiφp and Ωc =
Gc eiφc . Note that there are parallel variations on densitymatrix element by defining the relative phase φ = φp −φc
and replacing ρij with σij , where σii = ρii , σ13 = ρ13 eiφp ,
σ12 = ρ12 eiφc and σ23 = ρ23 eiφ are real parameters. Then
we retrieve the density-matrix equations which are similar
to the form before adaption as follows:
∂t σ22 = 2γ2 σ11 + iGc σ12 − iGc σ21 ,
spontaneously emission passageways, i.e., the SGC term.
The parameter η represents the existence of the SGC effect. Only when the energy spacing between the level
|2i and level |3i is very narrow will the SGC effect have
to be taken into account, then η = 1, otherwise η = 0.
Here we only consider the situation that η = 1. We use
cos θ = d12 · d13 / (|d12 | |d13 |) to denote this cross coupling
coefficient.
Generally, a composite system starting from a disentangled pure state can be described by its density operator
derived from the tensor product space. In the limit of weak
probe, considering perturbation theory, we arrive at the
zero order appromixmation until the first order approximation of the steady-state solutions of all density-matrix
elements as follows:
Λ (γ1 + γ2 ) Gc2
D
iGp Gc {∆c Λσq − (γ1 + γ2 ) [Gc σp + iΛσq ]}
+
,
2D
Gp
γ1 (1)
(1)
σ33 = σ11 + i σp ,
Λ
2Λ
Gp
γ1 (1)
(1)
σ22 = 1 − 1 +
σ11 − i σp ,
Λ
2Λ
h
i
σ11 =
(1)
iGp (Λ − i∆) σs − i GDc γ2 Λ (∆c + iγ1 + iγ2 )
=
γ1 + γ2 + Λ − i∆p (Λ − i∆) + Gc2
2
(1)
σ31
iγφ∗ Gc σ11
,
γ1 + γ2 + Λ − i∆p (Λ − i∆) + Gc2
(1)
−
iGp Gc2 γφ (γ1 + γ2 + Λ) Λ (γ1 + γ2 )
(i∆c + γ1 + γ2 )[(γ1 + γ2 + Λ) (Λ − i∆c ) + Gc2 ]D
i
h
(1)
G
iGc −1 + 2 + γΛ1 σ11 + i 2Λp σp
,
−
i∆c + γ1 + γ2
∆p + i (γ1 + γ2 + Λ) (1) Gp
(1)
σ32 =
σ31 +
σs ,
Gc
Gc
h
i
D = (γ1 + γ2 ) (2Λ + γ1 ) Gc2 + Λγ2 (γ1 + γ2 )2 + ∆c2 ,
√
γφ = 2 γ1 γ2 cos θηeiφ .
(3)
σ12 =
(1)
∂t σ33 = 2γ1 σ11 − 2Λσ33 + iGp σ13 − iGp σ31 ,
∂t σ12 = − (γ1 + γ2 + i∆c ) σ12 + iGp σ32 − iGc (σ11 − σ22 ) ,
∂t σ13 = − γ1 + γ2 + Λ+i∆p σ13
+ iGc σ23 − iGp (σ11 − σ33 ) ,
√
∂t σ23 = − (Λ + i∆) σ23 + 2 γ1 γ2 η cos θσ11 eiφ
+ iGc σ13 − iGp σ21 ,
(2)
where the density-matrix element in the equations are
P
constrained by σij = σji∗ and
i σii = 1. Here only
√
the item 2 γ1 γ2 η cos θσ11 eiφ includes the relative phase
caused by adaption and it stands for the quantum inerference effect caused by the cross coupling between the two
where σ13 − σ31 = σp , σ23 − σ32 = σq , σ11 − σ33 = σs
(1)
(1)
[31]. It is vital to note that only the elements σ31 and σ32
are relative phase-dependent, whereas others are phaseindependent. Obviously, each expression of the steadystate analytical solutions is related to the intensity of the
SGC effect.
In thermodynamics, entropy is a fundamental physical
quantity describing the degree of randomness of the system. Thus we can represent the entanglement adopting
the reduced quantum entropy. At present, due to the different investigation perspectives, several definitions are
admitted, such as the partial entropy of entanglement [48],
(0)
(0)
(0)
(0)
(0)
(0)
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The influence of spontaneously generated coherence on atom-photon entanglement in a Λ-type system with an incoherent pump
the relative entropy of entanglement [49], entanglement of
formation [50], entanglement of distillation [51], etc. Here
we adopt the definition of the partial entropy of entanglement. For an arbitrary system, which is composed of two or
more entangled components, quantum entanglement state
implicates that we can get the information of one part if
we know the information of the other, i.e., the partial density operators equal to the value of tracing over the other.
For a bipartite quantum pure state (|ψiAB ) composed by
two subsystems A and B, the DEM of the system can be
described by the entropy of entanglement
E(|ψiAB ) = S (ρA ) = −T r(ρA ln ρA ),
j
X
f λj |ji hj|. It is worth to
j
note that λj denotes the corresponding eigenvalue of the
reduced density matrices rather than the eigenvalue of
the Hamitonion. Then we can retrieve the definition of
the entanglement and the partial entropy of an atom-field
system as below:
3
X
E(|ψiAB ) = S (ρA ) = − λj ln λj ,
Results and discussion
In this section, first let us analyse and discuss the character of the steady state of the system, and then get a
conclusion about the influence of the SGC effect on the entropy. Afterwards, we extend the range of research to the
unsteady state (transient state) and reach several analogical conclusions, which can support the previous arguments.
There is an analysis of the following drawings and figures
for the behavior of the system with a summary of the results. To keep things simple, all parameters are reduced
to dimensionless units through scaling by γ1 = γ2 = γ
and all drawings are plotted in the unit of γ.
(4)
where ρA is the reduced density matrix of the subsystem A and S (ρA ) is the von Neumann entropy [16] of ρA .
As the total system consisting of subsystems A and B
starts at a pure state, the entropies of the two subsystems are always equal. Then S (ρA ) = S (ρB ), E(|ψiAB )
can be defined as the entropy of either subsystem. Similarly, the above conclusion is suitable for the case between
atoms and applied fields in our Λ-type system. Because
of the Hermiticity of matrix ρA , we can always choose a
set of orthonormal basis {|ji} that can diagonalize ρA .
In theX
representation of {|ji}, ρA can be expanded as
ρA =
λj |ji hj| whose functions satisfy the corresponding relation as f (ρA ) =
3.
(5)
j=1
which will be convenient for computing.
Apparently, for a direct product pure state in the form of
|ψiA ⊗ |ψiB , the degree of dynamical randomness is at a
minimum, and its total quantum entropy equals to zero.
But for a entangled state (unseparable state), the system has a nonzero quantum entropy. It has been shown
that a declining partial entropy stands for the case that
each component tends to a pure state and becomes disentangled. To the contrary, a growing one stands for that
they tend to lose their individuality and become entangled
[39, 45].
In order to make the following analysis and discussion
clearer and easier to understand, we divide the parameters to which entropy is sensitive into two categories.
They are, respectively, the angles including the cross coupling angle and the relative phase, and the intensities including the relative Rabi frequency and the rate of the
incoherent pumping. Below each category will be covered
individually.
First, we consider the effect of SGC on phase control of
the entropy for the steady state. We select the parameters
as mentioned in the caption of Fig. 2 to ensure that there
is no population inversion between two energy levels and
that weak probe and strong coherent field approximation
are satisfied, which result in more visible SGC effect. In
Fig. 2(a), the values of entropy that have the opposite values of cos θ are essentially superimposed and they lightly
decrease at ∆p = ±Gc [31]. In this case we can say that
the SGC effect does harm to the entropy considerably at
∆p = ±Gc (about 0.1), but lightly at two-photon resonance frequency (about 0.01). Then we have some comparison with the probe gain coefficient in Fig. 2(c). When
the SGC term is fixed to real number (eiπ = −1), a gain
peak and an equivalent absolute value of absorption peak
appear in a range of ∆p = −Gc and ∆p = +Gc , but their
values of the entropy are equal. That is because gain
(absorption) rely heavily on cos θ, and they rest with coherent items of the density matrices. The entropy is only
relevant to the incoherent items according to the dependence of eigen values on the diagonal items. We find out
that the cross coupling angle could give entropy a macrotuning around ∆p = ±Gc and a micro-tuning at ∆p = 0
with the SGC term fixed to real, and the entropy is not
sensitive to the difference between absorption and gain.
On the side, the values of entropy are minimum when the
gain and the absorption turn to zero. Because the case
without gain or absorption amounts to a transparent system that the interaction between a photon and an atom
is the weakest and this results in a minimalization of the
entropy.
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Figure 2.
(a) and (b) The steady state entropy as a function of the detuning ∆p for considering the SGC effect with various values of cross coupling
angles and relative phases. (c) and (d) The probe gain as a function of the probe detuning ∆p for various values of the cross coupling
angle θ and the relative phase φ. In (a) and (c), θ = π4 (black solid), θ = π2 (blue dashed), θ = 3π
4 (red dotted). Comparing with (c),
which based on same parameter conditions and legend as (a), (a) shows the correlation between the steady state entropy and the
cross coupling angle θ when the relative phase is fixed (φ = π). Other fitting parameters: γ1 = γ2 = γ, ∆c = 0, Gp = 0.1 × γ sin θ,
Gc = 10.0 × γ sin θ, Λ = 0.5γ. In (b) and (d), φ = π/2 (black solid), φ = π (blue dashed), φ = 3π/2 (red dotted). (d) adopts the same
paremeters and legend as (b). The employing parameters are the same as in (a) besides θ = π/4.
Then let us move on to Figs. 2(b) and (d). The behaviors of entropy are quite different at ∆p = ±Gc with
vaious relative phases. A symmetry (asymmetry) can be
achieved when the SGC term is real (imaginary) by carefully contrasting in Fig. 2(b). In addition, we get a relatively large minimum value of the entropy corresponding
to where we get gain, and a relatively small one corresponding to where we get absorption. Therefore it can be
held that when the SGC effect exists with the SGC term
being imaginary, gain could enlarge the entropy, while
absorption could reduce it at ∆p = ±Gc .
Thus, the cross coupling angle θ and the relative phase
φ are able to regulate and control the steady-state entropy S at certain range of modulation periods. Their
regulation ability is powerful around ∆p = ±Gc , but powerless at two-photon resonant state. That is, the entropy
is sensitive to these two parameters at gain and absorption peaks, but insensitive at two-photon resonance frequency, and the relative phases could be distinguished by
comparing the behaviors of gain and absorption.
When two fields resonantly act on the atom system, we
arrive at Fig. 3 that corresponds to the condition ∆p = 0 in
Fig. 2. In Figs. 3(a) and (c), according to the dependence
of the steady-state entropy upon the incoherent pump Λ,
each profile has a similar tendency and shape, which is
starting at zero and stabilizing at a certain value after
they reach their respective saturation state. In particular,
Λ = 0 implies that a photon is disentangled with an atom.
By this time the system returns into a typical Λ-system,
and when ∆p = ∆c = 0 the system exhibits the electromagnetically induced transparency phenomenon that
results in the non-interaction between the photonic and
atomic subsystems. In this case the behavior of the entropy is under direct control of Λ in the range of 0 and
γ, but the control abilily disappears when the system is
saturated (i.e. Λ γ). Therefore the control of the rate
of the incoherent field on the entropy is restricted.
As shown in Figs. 3(b) and (d), both of the gain coefficient and absorption coefficient can be regulated by the
relative phase φ, and initial entropies are zero and each
of them has a approximately linear rise and ultimately
reaches each saturation state. With respect to the saturation effect, we checked a wide range of the rates of
the incoherent pump and the results displayed the existing of the saturation effect. When Λ = γ, the equilibrium
of rates between the incoherent pump and the decay relaxation appropriately results in the strongest interaction
between a photon and an atom, which corresponds to the
maximum of entropy. But a larger rate of incoherent pump
does not mean a growing value of the entropy. The main
role of the incoherent pump is to make the ground state
atom excited and supply atoms continuously to the level
|1i. When Λ increases further, the interaction between
the two subsystems weakens gradually until the entropy
of entanglement reaches a steady value, which means that
the system will enter a saturation state. Additionally, the
maximum degree of excitation of the ground state by the
incoherent pump corresponds to the extremum of the gain
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The influence of spontaneously generated coherence on atom-photon entanglement in a Λ-type system with an incoherent pump
Figure 3.
(a) and (b) The steady state entropy as a function of the rate of the incoherent pumping Λ for considering the SGC effect with various
values of cross coupling angle θ and the relative phase φ at two-photon resonance frequency. (c) and (d) The probe gain as a function
of Λ for various values of the cross coupling angle θ and the relative phase φ at two-photon resonance frequency. In (a) and (c),
θ = π4 (black solid), θ = π2 (blue dashed), θ = 3π
4 (red dotted) with φ fixed to π. In (b) and (d), φ = π/2 (black solid), φ = π (blue
dashed), φ = 3π/2 (red dotted) with θ fixed to π/4. Similarly, (a) and (c) are based on same parameter conditions and legend, and they
shows the correlation between the gain-absorption coefficient Im(σ31 ) and the steady state entropy with the rate of incoherent pump 2Λ.
So do (b) and (d). Useing parameters are γ1 = γ2 = γ, ∆c = ∆p = 0, Gp = 0.1 × γ sin θ, Gc = 10.0 × γ sin θ.
or absorption of system. Consequently the behaviors of
the entropy and the gain-absorption coefficient versus the
incoherent pump follow similar trends. It is noteworthy
that our Λ-type system with an incoherent pump avoid
the disentanglement phenomena, comparing to the V-Type
one. The introduction of an incoherent pump makes the
atom-photon system remain a high entangled state, which
makes it easier for entanglement generation and control.
From the above, we researched part of the parameters
introduced by the SGC effect, which would exert influence
on the steady state entropy. In Mohammad Abazari and
collegues’ article, which dealt with a three-level V-type
system [39], each of the matrix elements is inclusive of the
additional term resulted from the SGC effect, therefore
the spontaneously emission coherence have a far greater
impact on the entropy and the degree of the entanglement.
However, for the Λ-type system, only a few expressions
are inclusive of the SGC effect. Specifically, the addition
items induced by SGC only exist in the expressions of
(1)
(1)
σ31 and σ32 under the steady-state condition among the
density-matrix equations. Below we would further verify
our conclusion by analyzing and discussing the transient
case. At this time, we focus on the dynamic behaviors of
system before getting into the steady state. And we are
more concerned with the impact on the dynamic behaviors
of the system by the weak probe approximation, which is
reflected by the ralative Rabi frequency ratio, and the rate
of the incoherent pump.
In Fig. 4, we depict the time evolution of the atomic quantum entropy and the probe gain under two resonant fields
with different cross coupling angles and relative phases,
respectively. By comparing with Fig. 2, we find that the
instantaneous process can reach a steady state, which
is exactly the saturation state at the steady-state condition. In addition, the stable values of the entropy and gain
are basically in accordance with the results of the steady
state.
With various relative Rabi frequency ratios and rates of
the incoherent pump, the quantum entropies evolve from
analogously linear growth to maximum, and finally tend
to a steady state. The ascent stages of the behaviors of
the entropy overlap with various relative Rabi frequancy
ratios in the left column of Fig. 5. During the initial moments the photon is absorbed rapidly by the system, which
promptly results in the growing value of the entropy until the maximum. With time evolution, when G < 10 the
probe gain tends to zero, which results in the DEM finally tending to zero. To some extent, since there being
neither absorption nor gain means that there is no interaction or the interaction is extremely weak between the
two subsystems. Therefore the entanglement is not xistent. When G ≥ 10, the weak probe and strong coherent
field approximation are satisfied, and the effects of SGC
are obvious. At this time the probe is amplified and the
interaction between the atom and the photon is strong,
and because of the saturation effect the entropy tends to
stability. We find out that the relative Rabi frequency
ratio does not influence the rising stage of the entropy,
but can influence the equilibrium value of entropy at twophoton resonance state. In addition, on the steady state,
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Figure 4.
(a) and (b) The time evolution of the quantum entropy under two resonant fields with different cross coupling angles and relative phases,
respectively. (c) and (d) Time evolution of the probe gain. (c) adopt the same paremeters and legend as (a). So do (b) and (d). Using
parameters are the same to Fig. 2 with ∆p = 0.
Figure 5.
(a) and (b) Time evolution of the quantum entropy under two resonant fields with different relative Rabi frequancies and incoherent
pumps. (c) and (d) Time evolution of the probe gain under two resonant fields with different relative Rabi frequancies and incoherent
pump. In (a) and (c), G = 1 (black solid), G = 10 (blue dashed), G = 100 (red dotted) with Λ = 0.5γ; In (b) and (d), Λ = 0.0γ (blace
solid), Λ = 0.5γ (blue dashed), Λ = 1.0γ (red dotted) with G = 100. Other parameters are choosen as those in Fig. 3.
the Λ-type system remains high entangled state, at least
we can say the system is entangled with arbitrary relative
Rabi frequency G, regardless of the initial point.
In the right column Λ = 0 stands for non-incoherent
pump. Here initially the probe is absorbed to a certain
extent, which corresponds to a certain degree of entanglement. While the entanglement fades away along with
the absorption tending to zero, which corresponds to noninteraction. When Λ 6= 0 we introduce an incoherent pump
into the system, and the probe gain that we get makes it
possible to develop entanglement. That is, the incoherent
pump can not only provide population by exploiting transition but also impact the entanglement. When the incoherent pump exists, we arrive at an opposite conclusion
to the case with various G that the incoherent pump can
influence the rising stage of the entropy including rising
speed and maximum, but can rarely influence the equilibrium value of entropy at two-photon resonance state.
In other words, G has a long-time impact on the entropy
and Λ has a short-time impact on it without destroying
the entanglement. Besides, by examining a wide range
of Λ (to Λ = 100γ) we find that not larger rate of the
incoherent pump benefits much to the entropy. Instead,
excessive strong incoherent pump may destroy the DEM.
This proves the statement again that a saturation state
can be achieved on the steady state in Fig. 3.
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The influence of spontaneously generated coherence on atom-photon entanglement in a Λ-type system with an incoherent pump
4.
Conclusion
In summary, the effect of the SGC on the DEM between an
atom and a photon of the coupling laser field in a normal
Λ-system coupled with an incoherent pump has been analyzed quantitatively. We find that the parameters introduced by the SGC effect may influence the entropy at different levels, but these influences are all limited. Firstly,
when the relative phase is fixed to a real number, the cross
coupling angle can bring down the entropy under singlephoton resonance condition especially around ∆p = ±Gc ,
while the ability to control the entropy is weakened under
two-photon resonance condition. Similarly, the entropy is
phase-sensitive at ∆p = ±Gc with larger entropy corresponding to gain and smaller one corresponding to absorption, while the entropy is phase-insensitive at ∆p = 0
when the SGC term is imaginary.
When 0 ≤ Λ ≤ γ the dependence of the entropy on the
rate of the incoherent pump exists without destroying the
entanglement, where Λ = 0 denotes that EIT comes into
being and that the interaction between the photonic and
atomic subsystems vanished, and Λ = γ corresponds to
the maximum of the interaction. In addition, there is a
short-time dependence of the entropy on the incoherent
pump. Instead, there is a long-time dependence of the
entropy on the ratio of the relative Rabi frequency, which
is decided by the satisfaction of the weak probe and strong
coherent field approximation conditions.
Acknowledgements
This work is supported by the financial support from the
National Nature Science Foundation of China (Grants
No. 11204019, No. 11247005, and No. 1104112), the
National Basic Research Program of China (Grant No.
2011CB921603), the Science Foudation for Youths of Jilin
Province (Grant No. 201201140), the Scientific Foundation of the Education Department of Jilin Province (Grant
No. 2013280), and the Fundamental Research Funds for
the Central Universities (12QNJJ006).
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