Department of Physics, Wuhuan University, Wuhuan 430072,. P.R. China. § Support Center, Nanchang Telecommunication Bureau, Nanchang. 330003, P.R. ...
journal of modern optics , 1996, vol . 43, no . 2, 337± 345
Numerical study of squeezing in the Jaynes-Cummings model with and without the rotating wave approximation RUI-HUA XIE² ³ , TIE-JUN ZHOU²
and QIN RAO§
² Department of Physics, Nanjing University, Nanjing 210008, P.R. China ³ Department of Physics, Wuhuan University, Wuhuan 430072, P.R. China § Support Center, Nanchang Telecommunication Bureau, Nanchang 330003, P.R. China (Received 26 December 1994; revision received 16 August 1995 ) Abstract. Dipole squeezing in the Jaynes± Cummings model with and without the rotating wave approximation (RWA) has been numerically studied when restricted to the following initial conditions: the ® eld in the superposition of the vacuum and one-photon state, and the atom in the ground state. Numerical results show that even if under the condition for which the RWA is considered to be valid, there is a signi® cant eŒ ect caused by the virtual-photon ® eld on the dipole squeezing; in the cases of su ciently strong coupling of the atom± ® eld interaction, the interference between the virtual-photon processes and the real-photon processes quenches the dipole squeezing.
1.
Introduction The theoretical model consisting of a single two-level atom interacting with a single-mode quantized ® eld, known as the Jaynes± Cummings (JC) model [ 1] in which the rotating wave approximation (RWA) is explicitly used, exhibits many very interesting rich dynamical features [2] for atom± ® eld interactions, and become practical since some of these features were observed experimentally [ 2] . Moreover, it has attracted renewed interest in the investigation of the quantum properties of the atom± ® eld interaction [ 2] . If the counter-rotating wave terms are included, the JC model cannot be solved by the usual techniques, since the eigenstates of the Hamiltonian cannot be found in closed form. However, much work has been done in recent years on this model with the counter-rotating wave term included; the ® rst solution to include this term was due to Graham and HoÈ hnerbach [3] in which a numerical method was used. It has been shown that the counter-rotating wave terms are the source of the Lamb shift [ 4] and are essential in order to ensure causality in atom± ® eld coupling systems [5] ; even if under the condition for which the RWA is considered to be valid, there is a signi® cant eŒ ect caused by the virtual-photon ® eld (counter-rotating wave terms) on the atomic inversion [6] . Several perturbative approaches have been presented for the JC model of optical resonance with the counter-rotating terms included [ 7] ; the eŒ ect of the virtual-photon ® eld causes a frequency shift of the radiation ® eld [ 8]. The semiclassical JC model outside the RWA is non-integrable and can lead to classical chaos under certain conditions [ 9] . As a `physical quantum chaotic system’ , the JC model without the RWA has been studied to investigate its quantum properties, 0950 ± 0340/ 96 $12´00
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such as the statistical properties of the energy spectrum [ 3, 10, 11] and the exact dynamics of the occupation probabilities of the two levels [ 3] , and so on. All the studies have suggested to us the signi® cance of further investigation of the quantum properties of the JC model with the counter-rotating wave terms included. As we know, the squeezing of the radiation ® eld has been a subject of considerable theoretical and experimental study in recent years [ 12± 14] due to its potential application in optical communications [15] and weak signal corrections [ 16] ; the squeezing of the ¯ uctuations of the atomic dipole variables has also recently attracted increased attention, and many schemes for producing dipole squeezing have been suggested, one of them being the JC model [17± 20] . Moreover, it has been shown that the squeezed atoms radiate a squeezed ® eld [ 14, 17, 21] and there is a striking symmetry between ® eld and atomic squeezing in the vacuum ® eld JC model [19] , re¯ ecting very well the relationship between the squeezing of the atom and that of the ® eld. Interestingly, in recent years, a new class of squeezed states known as the superposition squeezed states have been theoretically formulated [ 17, 22, 23] , which can exhibit the squeezed ¯ uctuations of the ® eld and atoms by a simple implementation of the linear superposition principle of quantum mechanics. These studies have given us helpful insights into the physical interpretation of the non-classical features of atom± ® eld interactions; however, not much work has been done in investigating the squeezing properties of the JC model with the counter-rotating wave terms included. As we know, in recent years Lais and Steimie [ 24] have numerically shown using a continued fraction method that even for moderate coupling constants of the atom± ® eld interaction, the counter-rotating terms quench the squeezing of the ® eld predicted with the RWA. Peng and Li [25] have found numerically using perturbation theory that the counter-rotating term increases the squeezing of the radiation ® eld. In this paper, we shall investigate dipole squeezing using the numerical method (truncating the in® nite matrix to in® nite order) [3] in the JC model without the RWA, and shall see that the eŒ ect of the virtual-photon ® eld may increase or decrease dipole squeezing under certain initial conditions in the cases in which the RWA is considered to be valid, and quenches dipole squeezing in the cases of su ciently strong atom± ® eld coupling. 2.
Theoretical model and numerical method The Hamiltonian for a system consisting of a single two-level atom interacting with a single-mode quantized radiation ® eld via one-photon transition processes can be written as (i.e. the JC model without the RWA) H = W a+ a + w Sz + G(a+ + a)(S + S+ ), Å
(1)
where Sz and S6 are operators of atomic inversion and transition, respectively, which obey the commutation relation
[ Sz , S6 ] = ± S6 ;
[ S+ , SÅ ] = 2Sz ,
(2)
w
+
is the atomic transition frequency, a and a are the creation and annihilation operators of the ® eld mode with frequency W , which obey the commutation relation
[a, a+ ] = 1,
(3)
and G is the atom-® eld coupling constant. If the RWA is explicitly used (i.e. for the cases satisfying |W ± w | « w and G « w [ 3]), then the Hamiltonian can be
Squeezing in the J± C model
339
written as (i.e. the well-known JC model) +
+
H = W a a + w Sz + G(a S + aS+ ).
(4)
Å
Throughout we use ò = c = 1 and take W = 1 except Special notes in our numerical calculation; for simplicity, in this paper, we only consider the case where the ® eld is resonant with the atomic transition frequency (i.e. W = w ). As usual [ 13, 17], in order to investigate the squeezing properties of the atom, we consider two Hermitian conjugate operators Sx and Sy , given by Sx = 21(S+ + S );
Sy =
Å
1 2i
(S+ ± S ).
(5)
Å
In fact, Sx and Sy correspond to the in-phase and out-of-phase components of the amplitude of the atomic polarization [ 13] , respectively, and obey the commutation relation
[Sx , Sy] = i Sz .
(6)
Correspondingly, the Heisenberg uncertainty relation is given by (D Sx )2(D Sy )2 >
¬ S
1 4
2
z
(7)
,
where (D Sx ) = Sx ±
¬
2
2
¬ S; 2
x
(D Sy ) = Sy ± 2
¬ ¬ 2
Sy
. 2
(8)
The ¯ uctuations in the components Sx or Sy of the atomic dipole are said to be squeezed [13] if the above variance in Sx or Sy satis® es the condition (D Sx )2 < 21| Sz |,
¬
(D Sy )3 < 21| Sz | ,
¬
or
(9)
which, for convenience, may be written as F1 = (D Sx ) ± 2
1 2
¬
| Sz | < 0
or
F2 = (D Sy ) ± 2
1 2
¬
| Sz | < 0.
(10)
In our numerical work, we take |m, n as the basis, where Sz |m, n = m|m, n , m = ± 1/ 2 and a+ a|m, n = n|m, n (n = 1, 2, 3, . . .), then, we can obtain the eigenstate |f i and the energy eigenvalue Ei (i = 1, 2, . . . , M ) of the Hamiltonian (1) or (4) by truncation of the in® nite tridiagonal matrix to ® nite order [ 3]. As in [ 17] , we assume that the initial state |y (0) of the system is prepared in the ground state of the atom and the superposition of the vacuum and one-photon state of the ® eld, i.e.
|y (0) = a |± 21, 0 + b |± 21, 1 ,
(11)
where |a |2 + | b |2 = 1. We can now obtain the expectation value of the observable variable Sl (l = x, y, z) at the time t as follows:
¬ S= ¬ l
¬
M
=
å
i j=1 ’
¬
¬
y (t)|Sl |y (t) = y (0)| exp (i Ht)Sl exp (± iHt)|y (0)
¬
y (0)|f
i
¬
f j |y (0)¬ f i |Sl |f
j
exp [±
i (Ej ± Ei )t] .
(12)
2 S2 x = Sy = 1/ 4 can be easily shown, and ® nally we can obtain the two quantities F1 and F2 by which we can investigate the dipole squeezing.
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3.
Study of dipole squeezing in the JC model with the RWA: analytical and numerical results Using the initial condition (11), Wodkiewicz et al. [ 17] analytically investigated the squeezing of the ® eld in the JC model with the RWA. In order to see the eŒ ects of the virtual-photon ® eld on dipole squeezing, in this section we study analytically and numerically dipole squeezing in the JC model with the RWA (i.e. the model equation (4)). Using the general evolution law of this model [25] and the initial condition (11), we ® nally obtain the time evolution of F1 and F2 as follows: F1(t) = 41 ± |a |2| b |2 sin2 (Gt) sin2 (d f ) ±
|1 ± 2| b |2 sin2 (Gt)| ,
(13)
|1 ± 2| b |2 sin2 (Gt)| .
(14)
1 4
F2(t) = 41 ± |a |2| b |2 sin2 (Gt) cos2 (d f ) ±
1 4
Here d f is the relative diŒ erence between the vacuum state and one-photon state of the ® eld. Now, we analyse the dipole squeezing in detail in the JC model with the RWA. For simplicity, we may take d f = p / 2, and then we have F1(t) = 41 ± |a |2| b |2 sin2 (Gt) ± F2(t) = 41 ±
|1 ± 2| b |2 sin2 (Gt)| ,
1 4
|1 ± 2| b |2 sin2 (Gt)| .
1 4
(15) (16)
Clearly, F2 > 0, which means that the ¯ uctuations in Sy cannot be squeezed; however, F1 can be less than zero, and so the squeezing in Sx can be exhibited. We can demonstrate this in detail as follows: when b = 1, | b | = 1, a = 1 and |a | = 1, and obviously Sx or Sy cannot be squeezed; when Ö 2/ 2 < | b | < 1, we can verify that F1 < 0 during the following time t: 2kp + B 2G
