Quantum properties of the binomial state

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Physics, Mechanics & Astronomy

*Corresponding author (email: renxuezao@swust.edu.cn)

• Research Paper • May 2010 Vol.53 No.5: 864–869

doi: 10.1007/s11433-010-0182-2

Quantum properties of the binomial state field interacting with a

two-level atom

JIANG DaoLai, REN XueZao

*

, CONG HongLu & LI Lei

School of Science, Southwest University of Science and Technology, Mianyang 621010, China

Received May 27, 2009; accepted September 14, 2009

By the method of coherent-state orthogonalization expansion, the atomic inversion and the anti-bunching effect in a two-level atomic system interacting with binomial state field are studied under the Jaynes-Cummings model without rotating wave ap-proximation. The influence of the parameter and the detuning on the anti-bunching effect are discussed, and the anti-bunching effect is also discussed under the conditions of strong coupling. Our studies show that the second order coherence degrees are quite different between the situations of with and without-rotating wave approximation. For the latter when the detuning in-creases, the duration of the bunching effect increases at the beginning and then dein-creases, and finally the light field displays anti-bunching effect completely.

coherent state orthogonalization expansion, binomial state field, Jaynes-Cummings model without-rotating wave ap-proximation, atomic inversion

PACS: 42.50.Gy, 42.50.Hz, 42.50.Md

1 Introduction

Jaynes-Cummings model (JCM) [1], the fundamental system in the area of quantum optics, is widely used in quantum optics such as quantum computation and quantum entan-glement and so on [2–5]. Through research, many interesting phenomena have been found in this system such as atomic inversion revival-collapse phenomenon (RCP), and anti- bunching effect. In 1985, Stoler, Saleh and Teich put forward the conception of binomial state [6]. In 1987, Datton found that the binomial state could occur in free electric laser [7]. Recently, Italian R. Lo Franco and his cooperators generated various generalized binomial states in the experiment [8–10]. Furthermore, people made a great number of studies on the quantum properties of the binomial field [11–14]. For in-stance, reference [14] researched the quantum effect of the

two level atom interacting with the binomial field, and ref-

erence [15] worked on the quantum entanglement of the three-level atom interacting with the binomial field.

The rotating wave approximation (RWA) is widely used in quantum optics because of its analytical solution , but the RWA does not fit the cases of strong coupling, strong cavity field or large detuning. Recently, people have found that Josephson qubit, Cooper-Pair box (CPB) and ion trap is similar to the atom in a cavity [16], but the coupling strength is much greater than the atomic system, so the without-RWA terms ought to be taken into account. According to the Stoler’s work, we can get the definition of the binomial state [2]:

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where M is the maximum photon number, and the mean photon number is n=ηM. In the cases of η=0 or 1, the state is reduced to the Vacuum state and Fock state M

respectively. Actually, from ref. [17] we know that the photon exhibits the sub-Poissonian statistics and the anti- bunching effect in the binominal state.

This paper, through the method of coherent-state or-thogonalization expansion [18], has discussed the atomic inversion and the anti-bunching effect in the two-level atomic system interacting with the binomial state field.

2 Model and solution

The Hamilton of a single mode field interacting with a two-level atom in the without-RWA is

(

)

a (a+)is the annihilation (creation) operator denoting the

cavity mode, ω0 and Ω are the frequencies of the cavity

mode and the atomic transition, and g is the atom-field cou-pling constant parameter.

Using the following transition to eq. (2):

(

)

The stationary wave function of eq. (4) can be described as follows:

1 e 2 g .

φ φ

= + (5)

Putting eqs. (4) and (5) into the Schrodinger equation:

,

With new bose operators [18–24] being introduced,

0

n in the displaced Fock state, constitute the

two groups the new normalization basis on, and n_A and

Putting eqs. (11)–(13) into eqs. (9), (10), and left

multi-plying , obtain 2N+2 equations. Solving these equations, we can obtain the eigenvalues _Ei

and eigenvectors { },i n

c { }i n

(3)

(i=1"2N+2).

Assuming the cavity field is initially in the binomial state and the atomic state in the excited state E we have

where fi is the undetermined coefficient. According to eqs.

(1) and (3), we have tively, yields

2 2

3 Atomic inversion

From eq. (25), we can easily obtain the population of the

E

The atomic inversion is

( ) 2 1.

W t = P− (27)

Figure 1 shows the time evolution of the atomic inversion for various η, and M =30, ω Ω₀ =1, g/Ω=0.1. The solid line is the result of the without-RWA, and the dotted line is the result of the RWA. Because of n =ηM, the mean photon number increases and the light field becomes stronger with the increase of η when M is a constant value. For the case η=0.01, the cavity field is weak, and the results between the without-RWA (solid line) and the RWA (dotted line) fit well as shown in Figure 1(a). As we increase the value of η, the mean photon number increases, so the binomial field becomes stronger and the RCP appears. In the comparison with Figure 1(b)–(d), we find in the collapse regime, the atomic inversion collapse completely in RWA (dotted line). Unlike the RWA, the atomic inversion in the without-RWA (solid line) can not collapse completely, but shows small zigzag-shaped oscillation in the collapse regime. The difference between the two approximations is mainly due to the without-RWA terms, which means that the with-out-RWA terms can not be ignored when the light field is strong enough. A comparison with Figure 1(b)–(d) indicates that the small zigzag-shaped oscillation of the atomic inver-sion becomes larger with the increase of η in the collapse regime. This means that with the increase of η, the mean photon number increases, the binomial field becomes stronger and the contribution of the without-RWA terms becomes noticeable, which leads to the difference between the RWA and the without-RWA.

4 Anti-bunching effect

The anti-bunching effect and bunching effect can be de-scribed by second-order coherence degree, which is given by

2 2 the anti-bunching effect for 2

( ) 1.

G t < It is a critical state of the anti-bunching effect and the bunching effect when

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Figure 1 The time evolution of atomic inversion with various η, where M=30, ω0/Ω=1, g/Ω=0.1. The solid line and the dotted line represent the solution of

Figure 2 shows the time evolution of the second order co-herence degree, with M=30, ω0/Ω=1, g/Ω=0.1, The solid line is the result of the without-RWA, and the dotted line is the result of the RWA. Firstly, Figure 2 shows the difference between the RWA and the without-RWA due to the effect of the without-RWA terms. For the case η =0.01 (Figure

2(a)), the duration of the bunching effect (G t2( )>1) is longer than that of anti-bunching effect 2

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Figure 2 The time evolution of the second order coherence degree, where M=30, ω0/Ω=1, g/Ω=0.1. The solid line and the dotted line represent the solution of the without-RWA and the RWA, respectively. (a) η=0.01; (b) η=0.1; (c) η=0.5; (d) η=0.9999.

Figure 3 The time evolution of the second order coherence degree, where M=30, η=0.1, g/Ω=0.1. (a) ω0/Ω=1; (b) ω0/Ω=1.3; (c) ω0/Ω=1.6; (d) ω0/Ω=10.

with the increase of η, and the light field can not display anti-bunching effect completely although η is large enough

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increases with the increase of η, and the light field displays anti-bunching effect completely when the η is large enough. In summary, we can find that the light field displays bunch-ing effect more easily than the anti-bunchbunch-ing effect when the atom-field coupling is strong enough, and the two approxi-mations are quite different shown in Figure 2.

Figure 3 shows the time evolution of the second order coherence degree for various Δ, where Δ ω Ω Ω=( ₀− ) / ,

M=30, η=0.1, g/Ω=0.1. Firstly, the amplitude of the second order coherence degree decreases with the increase of the detuning. Secondly, the collapse and revival of the second order coherence degree appears, and the collapse regime can not collapse completely due to the without-RWA terms. This means that the without-RWA terms can’t be ignored when the detuning is large enough, and this phenomenon is dif-ferent from that of the RWA. Thirdly, the duration of the bunching effect increases at the beginning and then de-creases with the increase of detuning, and finally displays anti-bunching effect completely which is shown in Figure 3.

5 Conclusion

This paper has studied the quantum properties of the bino-mial state light field interacting with a two-level atom. From the study of atomic inversion, we find in the collapse regime the atomic inversion collapse completely in the RWA with the increase of η, but in the without-RWA it can not collapse completely and demonstrates quick oscillations. This means that the difference between the two approximations becomes larger due to the without-RWA terms. For the anti-bunching effect, when η is small, the bunching effect and the anti-bunching effect appears alternately, the duration of the anti-bunching effect increases with the increase of η, and finally displays anti-bunching effect completely in the RWA. For the without-RWA, the bunching effect becomes weaker with the increase of η, and can not display anti-bunching effect completely when the coupling is large enough. When

η is a constant, the duration of the bunching effect increases at the beginning and then decreases with the increase of detuning, finally becomes a constant value.

This work was supported by the National Natural Science Foundation of China (Grant No. 1097602/A06).

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