Synergetic Innovation Center for Quantum Effects and Application, Key Laboratory of Low-dimensional Quantum Structures and Quantum Control of Ministry of Education, School of Physics and Electronics, Hunan Normal University, Changsha 410081, China
† Corresponding author. E-mail:
mffang@hunnu.edu.cn
1. IntroductionThe precision measurement of observable quantities in quantum physical experiments is greatly restricted by quantum noises (quantum uncertainty[1]) of quantum systems. Hence, breaking through the restriction of quantum noises and realizing quantum physical experiments with ultra-low quantum noise is a strong expectation for physical scientists. As we know, the suppression of quantum noises in quantum systems via squeezing effects is one of the most outstanding achievements in modern quantum optics over the last few years. Therefore, many squeezing effects have been investigated, such as the atomic squeezing. It should be pointed out that quantum information entropy is a accurate measure of the atomic squeezing, and one of our authors has defined the entropy squeezing (ES) for a two-level atom by using quantum information entropy according to entropic uncertainty relation based on two conjugate observables,[2] which overcomes the limitation of the definition of squeezing via the variance of the system observable according to Heisenberg uncertainty relation. For more than a decade, the entropy squeezing (ES) properties of quantum system have been investigated widely. Such as, Yu and Fang have studied the steady and optimal entropy squeezing of a two-level atom with quantum-jump-based feedback and classical driving in a dissipative cavity and provided a scheme of generating steady and optimal entropy squeezing in the two-level system;[3] Wang has investigated sustained optimal entropy squeezing of a two-level atom via non-Hermitian operation and provided a scheme for generating the sustained optimal entropy squeezing of the atom via non-Hermitian operation;[4] Xiao has studied the entropy squeezing of a two-level system by detuning in non-Markovian environments and shown that the atomic entropic squeezing can be protected for a long time when both the non-Markovian effect and detuning are present simultaneously;[5] the entropy squeezing of a two-level atom in Jaynes–Commings model has been reported widely;[6] Zou has studied the quantum entropic uncertainty relation and entanglement witness in the two-atom system coupling with the non-Markovian environments.[7,8] However, these results are obtained only in the case of two-level atom, the entropy squeezing properties of three-level atom have not yet been involved until now.
On the other hand, as we know, three-level system has more obvious advantages than two-level system for the representation of quantum information,[9,10] the key distribution scheme of a three-level entanglement system is more effective than the two-level system for preventing eavesdropping in cryptography,[11] and three-level quantum systems have more obvious quantum nonlocality than the two-level system.[12] Therefore, three-level system with ultra-low quantum noise is a more useful information resource, which means that the study of entropic squeezing of three-level system becomes very meaningful in quantum information processing. In this paper, we study the entropy squeezing of three-level atom interacting with a single-mode field. The outline of this paper is as follows. In Section 2, we give a general definition of entropy squeezing for three-level atom by using quantum information entropy according to entropic uncertainty relation of three-level atom based on two conjugate observables. In Section 3, the reduced density matrix is obtained, and the calculation formalism of entropy is derived for a cascade three-level atom. We analyze the properties of entropy squeezing of a cascade three-level atom by using numerical calculation in Section 4, while in Section 5 a brief summary is presented.
2. The general definition of entropy squeezing for a three-level atomSince the entropy squeezing of three-level atom has not been discussed before, we first give a general definition of the entropy squeezing for a three-level atom. A three-level atom can be described by dipole operators[13,14]
with
where
is the conjugate transpose of
,
, and
are dimensionless constants. The spin operators can satisfy the commutation
and the uncertainty relation is given by
where the variance
, and
is the expectation of the observable
Sz.
For three complementary observables
, (
) of a three-level atom, the corresponding information entropies can be defined as[15]
where
,
, and
is an eigenvector of the operator
. Alberto
et al.
[16] derived entropic uncertainty relations for three observables systems of dimensions
d = 3
This inequality can be also written as
where
We define the squeezing of the atom by Eq. (
6). The fluctuations in the component
are said to be squeezed in entropy if the information entropy
satisfies the condition
Equation (
8) is the central result of this paper. From Eq. (
8) we find that the optimal value the entropy squeezing factor
is
.
For the convenience of the following discussion, we also give the definition of variance squeezing based on the Heisenberg uncertainty relation. When the variance in the component
satisfies the condition
or
Fluctuations in the component
are said to be squeezed.
We are interested in the case that equation (10) is trivial when
(as
) and fails to provide any useful information while the equation (8) is non-trivial and can provide a useful information.
3. The reduced density matrix and entropic calculation formalism of a cascade three-level atomIn this section, we consider a cascade three-level atom interacting with a single-mode field. The transition frequencies between
and
,
, and
are respectively related to a single-mode field with a frequency ω, and the transition between
and
is electrical dipole forbidden, that is
in Eq. (2). The Hamiltonian of the interaction between the atom and the field is given by
with
where
a and
are the annihilation and creation operators of the field with the frequency
ω,
is the eigenenergy of level
of the atom,
gj (
j = 1,2) is a coupling constant of the atom and single-mode field.
and
(
α = 1,2,3) correspond to the Fermi annihilation and creation operators of the atomic energy level
α, respectively, which follows the rules
However, for the sake of simplicity, we consider the resonance condition between the three-level atom and the field, namely,
. In the interaction picture, the Hamiltonian is given by
where
. We suppose that the initially field is the coherent state
with
,
, and the initially atom is the coherent superposition state of three levels. If we assume at time
t = 0 the three-level atom and the field are decoupled, the initial state vector of system can be written as
For the initial condition Eq. (16), the solution of the Schrodinger equation in interaction picture is given by
where
with
According to Eq. (
17), the reduced density matrix of the three-level cascade atom can be expressed as
where
,
,
are the atomic populations in energy level
,
, and
, respectively.
In order to obtain the entropic calculation formalism of the spin operators of the cascade three-level atom, we give the concrete form of the spin operators as
Using Eqs. (4), (20) and (21), we obtain the entropic calculation formalism of the spin operators Sx, Sy, Sz of the cascaded three-level atom
where
, (
,
i = 1, 2, 3), and
Employing the reduced operator given by Eq. (20) and the entropic calculation formalism of the spin operators given by Eqs. (22)–(25), we investigate the properties of the entropy squeezing of the cascade three-level atom interacting with a single-mode field in the next section.
4. Numerical results and discussionIn this section, we examine numerical results for the atomic entropy squeezing. For comparison, we also consider the atomic variance squeezing and atomic inversion.
The time evolution of the squeezing factors
,
,
, and
are shown in Fig. 1, while the atomic inversion
is plotted in Fig. 4(a), for the atom initially in the excited state (θ = 0, φ = π/2) and the field in a coherent state with average photon number
and phase β = 0. We can see from Fig. 1(a) and Fig. 1(c) that both
and
predict no squeezing in the variable Sx when the atom is initially in the excited state, while Figure 1(b) and 1(d) display that
and
exhibit entropy squeezing only in initial time of the time evolution, which means that, the cascade three-level atom initially in excited state can not generate entropy squeezing except for it in initial time. In addition, from Fig. 4(a), it can be seen that in collapse region of atomic inversion,
evolves to zero. Therefore, Heisenberg uncertainty relation shows triviality, and variance squeezing can not reflect accurately the quantum fluctuations of the observable variables of three-level atom in collapse region of atomic inversion.
Figure 2 displays the entropy and variance squeezing for the cascade three-level atom initially in the superposition state of
and
with θ = π/4, φ = 0, and the field is the coherent state with the average photon number
, while the atomic inversion
is plotted in Fig. 4(b). We see from Figs. 2(a) and 2(c) that both
and
predict no squeezing in the variable Sx, while Figure 2(b) and 2(d) show that there are great differences between
and
: the former shows entropy squeezing in certain time ranges whilst the latter predicts no variance squeezing at all. This result stems from the fact that,
, as we see in Fig. 2(d), does not exhibit any variance squeezing since the atomic inversion
evolves to values very close to zero, showing in the Fig 4(b). In this case, the value of
has no particular significance and the information provided by Heisenberg uncertainty relation is not helpful.
The entropy and variance squeezing factors are presented in Fig. 3 for the atom is initially in the coherent superposition state with θ = π/3,
and the field is the coherent state with
. It can be seen that from Figs. 3(a) and 3(b) that there are alternate entropy squeezing in the atomic variable components Sx and Sy, whereas
and
, as illustrated in Figs. 3(c) and 3(d), display no variance squeezing. This result also stems from the fact that the atomic inversion
evolves to values very close to zero, showing in the Fig. 4(c), and the variance squeezing can not provide any useful squeezing information.
From Figs. 1, 2, and 4, it can be seen that, the both squeezing factor and atom inversion exhibit violent oscillation with high frequency in some time interval in time evolution. The emergence of this phenomenon is a result generated by the superposition of Rabi oscillation with different frequency of the cascade three-level atom under coherent field driving.[13]
It is worth pointing out that, the results above obtained for the entropy squeezing for a cascade three-level atom interacting with a single-model field is based on a universally adapted spin operator given by Eq. (21), but if different form spin operator is adapted, the regularity of entropy squeezing for the cascade three-level atom is different according to the independence of squeezing definition for the different operator.[13]
5. ConclusionIn conclusion, we have studied the entropy squeezing properties for a cascade three-level atom interacting with a single-model field. First, a general definition of entropy squeezing of a three-level atom is given by using quantum information entropy according to entropic uncertainty relation of three-level atom. Secondly, the reduced density matrix is obtained, and the calculation formalism of entropy is derived for a cascade three-level atom. Finally, we analyze the properties of entropy squeezing of a three-level cascade atom by using numerical calculation. Our results show that, a cascaded three-level atom interacting with a single-model field can generate obvious entropy squeezing effect via choosing appropriate superposition state of three-level atom. Our results are meaningful for preparing three-level system information resources with ultra-low quantum noise.
