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Generation of higher-order atomic dipole squeezing in a high-Q micromaser cavity. (V). A linear superposition of states
Physica A 315 (2002) 386 – 410
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Generation of higher-order atomic dipole squeezing in a high-Q micromaser cavity. (V). A linear superposition of states Rui-Hua Xiea;∗ , Qin Raob a Department
b Department
of Chemistry, Queen’s University, Kingston, Ont. Canada K7L 3N6 of Engineering Physics, Queen’s University, Kingston, Ont. Canada K7L 3N6 Received 25 February 2002
Abstract In our preceding paper, we investigated the generation of higher-order atomic dipole squeezing (HOADS) in a nondegenerate two-photon Jaynes–Cummings model in the absence of Stark shift. In this paper, we continue to study HOADS in this model but with the inclusion of Stark shift. Based on the linear superposition principle of quantum mechanics, we discuss the cases where the atom or the two-mode 8elds are initially prepared in a superposition of states. We demonstrate that the essential condition for generating HOADS is that the initial state of the system should include phase information, and HOADS cannot appear in atomic spontaneous radiation processes. The Stark shift brings time-dependent phase information into emitted photons and has a signi8cant e;ect (constructively and destructively) on HOADS. It is found that weakly coupling of the intermediate virtual state to the ground state of the atom would lead to much more squeezing and collapse-revival phenomena. The relation between HOADS and second-order c 2002 Elsevier Science B.V. All rights reserved. atomic dipole squeezing is also discussed. PACS: 42.50.Dv; 42.50.Lc; 32.80.−t Keywords: Dipole; Squeezed state; Superposition state; Quantum ?uctuation; Uncertainty relation; Stark shift; Two-photon micromaser
1. Introduction The Heisenberg uncertainty principle, proposed by Heisenberg [1] in 1926, is a fundamental, general principle of nature. Although it is of no consequence for ∗
Corresponding author. Present address: National Institute of Standards and Technology, 100 Bureau Drive, Mail Stop 8423, Gaithersburg, MD 20899-8423, USA. Tel.: +1-301-9755159; Fax: +1-301-9901350. E-mail address: [email protected] (R.-H. Xie). c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 1 0 - 5
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everyday, macroscopic bodies, it plays an important role in dealing with our microscopic world (atom, molecule, etc.). In 1927, Kennard [2] showed us the 8rst example of nonclassical states, named squeezed state, in which the quantum ?uctuations in a dynamical observable may be reduced below the standard quantum limit at the expense of increased ?uctuations in its canonical conjugated one without violating the Heisenberg uncertain relation. In 1956, PlebaNnski [3] made an important contribution to the theory of squeezed states. To the best of our knowledge, the exciting and constructive experimental results of the 8rst successful generation and detection of squeezed states were reported by Slusher et al. [4], Shelby et al. [5] and Wu et al. [6] in the middle of the 1980s. Ever since then, this kind of nonclassical states has been extensively studied, both theoretically and experimentally [7–24], due to their potential applications in high-resolution laser spectroscopy measurement, optical communication, gravity wave detection and quantum information theory (for example, quantum teleporation, cryptography, dense coding, power-recycled interferometer, and quantum nondemolition measurement). It has been shown that a lot of nonlinear optical systems could generate squeezed states for the 8eld and atom, for example, in four-wave mixing, two-photon laser, parametric ampli8ers, resonance ?uorescence, cooperative Dicke system and Rydberg atom maser, and a general relationship between 8eld and atomic dipole squeezing has even been established under di;erent initial conditions for both the 8eld and atom [10,15,17–19]. Meanwhile, in quantum optics and laser physics, the rapid development of techniques for making higher-order correlation measurements has led to increased interest in generating higher-order squeezed states, in which higher-order quantum ?uctuations in one quadrature of the 8eld or atomic dipole could be reduced without violating higher-order uncertainty relation. It has been shown that the generation of higher-order squeezed states for the radiation 8eld could be predicted in a number of nonlinear optical processes [25 –30] which could exhibit second-order squeezing. In 1997, Xie et al. [31] introduced the concept of higher-order atomic dipole squeezing (HOADS) and applied it to high-Q micromaser cavities, discussing thoroughly the important connection of HOADS with the second-order 8eld and atomic dipole squeezing. Moreover, Xie and Smith Jr. [32,33] have discussed the e;ects of a nonlinear one-photon process and dynamical Stark shift in two-photon processes on HOADS. It is found that the nonlinear one-photon processes in?uence the stability, period, width, pit depth and position of HOADS, and the e;ect of the Stark shift may lead to a strong HOADS in the revival region of the atomic inversion [32,33]. Very recently, assuming that the radiation 8elds are initially prepared in an uncorrelated coherent states, Xie and Smith Jr. [34] investigated HOADS in a nondegenerated two-photon Jaynes–Cummings model (NTPJCM) in the absence of Stark shift and demonstrated that a greater amount of HOADS could be achieved in the resonant cases. These extensive studies have made an important contribution to the theory of higher-order squeezed states and provided us an approach to extracting information eOciently from an optical signal by higher-order correlation measurements. It is known that the linear superposition principle is one of the most fundamental features in quantum mechanics and has been applied to all kinds of 8elds of physical science ever since the birth of quantum mechanics. Interference of quantum amplitudes and quantum beats are amongst the simplest examples of phenomena resulting from this
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principle. Recently, the coherent control approach has been successfully demonstrated, both theoretically and experimentally [35 –39], for example, in molecular hyperpolarizability, electron distribution excited in a metal, the energy and angular distribution of autoionized electrons, spontaneous emission near a photonic band edge, optical phonon emission rates and optical gain from electronic intersubband transition in semiconductors, optical dynamics in semiconductor microcavities, the polarization of an optical 8eld, the total ionization yield in a two-colour ionization process, photocurrent generation in bulk semiconductors, light absorption and terahertz radiations in semiconductor nanostructures, quantum chaotic di;usion, unimolecular breakdown reaction and reactive scattering. Although the physical systems for those cases may be di;erent, the essence of their coherent control schemes lies in two key steps: (i) create a superposition state composed of several basis states; (ii) control the relative phase of these basis, which means that the linear superposition principle plays a key role in the rapidly developing 8eld of coherent control. Also the linear superposition principle has led to an extensive study of superposition squeezed states of the 8eld and atoms, which were theoretically formulated by Wodkiewicz et al. [10] and developed by Xie et al. [17–19]. These studies have provided helpful revelations for physical interpretations of the nonclassical features of atom–8eld interactions from another angle. In this paper, we demonstrate the generation of HOADS in the modi8ed NTPJCM when the atom or two 8eld modes are initially prepared in a superposition of states. This paper is organized as follows. In Section 2, we describe the theoretical model of the modi8ed NTPJCM, in which the Stark shift is included, and derive some basic formalisms which are related to the two-photon interaction with a two-level atom in a high-Q cavity. In Section 3, assuming that the atom or two 8eld modes are initially prepared in a superposition of states, we investigate the generation of HOADS in the modi8ed NTPJCM, where the e;ect of the Stark shift on HOADS and the relation between HOADS and second-order ADS (SOADS) are discussed in detail. A summary is given in the last section. 2. Backgrounds of theoretical model It is known that a two-photon process plays an important role in atomic systems because of the high degree of correlation between the emitted photons [40]. One extension of the Jaynes–Cummings model (JCM) is the well-known two-photon JCM [41]. Over the past 20 years, the rapid development of two-photon micromasers [42], especially, the realization of the 8rst single-mode two-photon maser in the laboratory by employing Rydberg atoms in a high-Q superconducting micromaser cavity [43], has drawn a great deal of attention in the two-photon JCM and much work [44,45] has been done. To our knowledge, there are several signi8cant schemes for a two-mode two-photon maser with Rydberg atoms in a high-Q cavity system, which were proposed in order to advance one step further. To our interest, a remarkable feature of these two-mode two-photon masers is that one mode can be used to modulate, amplify, or control the output of the other mode. In this sense, the Rydberg atom plays a key role, which is similar to a coupler which correlates the two modes of the light and thus changes the entire photon statistics in a cavity. Correspondingly,
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NTPJCM was born in the world of quantum optics and laser physics and has attracted increased attention over these years. It is known that previous attempts tried to obtain lasing through two-mode two-photon-mediated transitions, but met with only limited success. Nevertheless, the advent of additional techniques [46 – 48] for demonstrating cavity QED e;ects has considerably brightened the prospects of achieving this goal and thus a number of dynamical aspects of NTPJCM were studied [49 –52]. Certainly, these studies would be complemented by a further discussion of a number of untouched and important issues in this area, for example, the generation of HOADS, which is actually the purpose of our present paper. In our preceding work [34], we investigated the HOADS in the NTPJCM but in a much idealized manner (i.e., the transition between the two atomic levels with the same parity is directly connected by photon pairs and hence the e;ective Hamiltonian approach ignores the Stark shifts caused by the virtual states). Puri and Bullough [53] have shown that the consideration of the Stark shift would change the quantum dynamics of a degenerated two-photon JCM (DTPJCM). Also, Xie and Smith Jr. [33] pointed out that the e;ect of the Stark shift may lead to a strong HOADS in the revived region of the atomic inversion in the DTPJCM. In addition, Gou [54], Mahmood and Ashraf [55], D’souza and Jayarao [56] have convinced us that the dynamical Stark e;ects will also change the quantum dynamics of the two-mode two-photon JCM (i.e., NTPJCM) in a more general way than those of DTPJCM. In the following, we brie?y introduce the modi8ed NTPJCM. We consider a two-level atom interacting with two modes of the radiation 8eld in a lossless resonant cavity, and denote ! as the natural transition frequency between the upper level state |+ and the lower level state |− of the two-level atom and j as the frequency of mode j of the radiation 8eld. It is known that some intermediate virtual states |i, which are assumed to be coupled to the upper and lower states of the two-level atom by dipole-allowed transition, are involved in the two-photon processes. We assume that 1 is about several times 2 and the two 8elds are tuned as near one-photon resonance as possible but still o; the one-photon linewidth, i.e., the detuning 1 and 2 are larger than but as close as the linewidth of |i. For the exact two-photon resonance 1 = −2 = and by the method of adiabatic elimination of the virtual state |i [53,57], Gou [54] obtained the e;ective Hamiltonian within the rotating-wave approximation H = H0 + HI
(1)
H0 = 1 a†1 a1 + 2 a†2 a2 + !Sz ;
(2)
with
† † † HI = 2 S+ S− a+ (3) 2 a2 + 1 S− S+ a1 a1 + (a1 a2 S− + a1 a2 S+ ) ; where = 1 2 . Here Sz and S± are operators of the atomic pseudospin inversion and transition, respectively, and satisfy the commutation [14]
[S+ ; S− ] = 2Sz ;
(4)
[Sz ; S± ] = ±S± ;
(5)
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and ! is the transition frequency for the atom. a†i and ai are the creation and annihilation operators for the photons of mode i with the frequency i , respectively, which obey the boson operators’ commutation relations [ai ; a†j ] = ij :
(6)
is the coupling constant between the atom and the radiation 8eld. A Stark shift is caused by the intermediate level |i, and the corresponding parameter 1 and 2 are 1 = 12 =, 2 = 22 = and = 1 2 =, where 1 and 2 are the coupling strengths of the intermediate level with the lower and upper levels of the two-level atom, respectively. Throughout we employ the units of ˝ = c = 1. We denote |n as the Fock state of the radiation 8eld. Using the standard bare-state procedure introduced by Yoo and Eberly [58] and developed by Xie et al. [59], we are able to calculate the expectation value of any operator in the modi8ed NTPJCM. In an ideal cavity, the two-level atom absorbs and emits one pair of photons, and the two bases in the bare-state representation are |+; n1 ; n2 and |−; n1 + 1; n2 + 1, respectively. Thus, the total Hamiltonian given by Eq. (1) can be written as 1 n1 + 2 n2 + !=2 + 2 n2
(n1 + 1)(n2 + 1) H= :
(n1 + 1)(n2 + 1) 1 (n1 + 1) + 2 (n2 + 1) − !=2 + 1 (n1 + 1) (7) Obviously, the eigenvalues of the above Hamiltonian are + = 1 n1 + 2 n2 + 12 ! + 12 2 n2 + 12 1 (n1 + 1) + (n1 ; n2 ) ;
(8)
− = 1 n1 + 2 n2 + 12 ! + 12 2 n2 + 12 1 (n1 + 1) − (n1 ; n2 ) ;
(9)
respectively, where (n1 ; n2 ) = (; n1 ; n2 ) + (n1 + 1)(n2 + 1) ; (; n1 ; n2 ) =
[(n1 + 1) − n2 ]2 4
(10) (11)
and = 1 = 2 . The two-photon Rabi frequency is written as (n1 ; n2 ) = (n1 ; n2 ) : Then, the time evolution operator U (n1 ; n2 ; t) of our two-mode system is E+ (t) (t) U (n1 ; n2 ; t) = ; (t) E− (t) where
1 n2 − (n1 + 1) exp(−i+ t) + √ 2 2 (n1 ; n2 ) 1 n2 − (n1 + 1) exp(−i− t) ; + − √ 2 2 (n1 ; n2 )
E+ (t) =
(12)
(13)
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1 n2 − (n1 + 1) − √ exp(−i+ t) E− (t) = 2 2 (n1 ; n2 ) 1 n2 − (n1 + 1) exp(−i− t) ; + + √ 2 2 (n1 ; n2 )
(t) =
(n1 + 1)(n2 + 1) [exp(−i+ t) − exp(−i− t)] : 2(n1 ; n2 )
(14)
Moreover, the density operator of our system at time t can be calculated with an arbitrary initial condition (t = 0) by (t) = U (t)† (t = 0)U (t) :
(15)
Therefore, the expectation value of any physical operator O can be arrived at through O(t) = Tr[(t)O(0)] :
(16)
3. Hoads and superposition states 3.1. Concept of HOADS In order to investigate the squeezing properties of the atomic dipole variables, we follow the standard procedure of de8ning the slowly varying operators [14] Sx = 12 [S+ exp(−i!t) + S− exp(i!t)] ;
(17)
1 (18) [S+ exp(−i!t) − S− exp(i!t)] ; 2i where Sx and Sy , in fact, correspond to the dispersive and absorptive components of the slowly varying atomic dipole [60], respectively. One can easily show that the above operators obey the commutation relation Sy =
[Sx ; Sy ] = iSz :
(19)
Correspondingly, we found the higher-order uncertainty relation [31] concretely given by (SSx )P (SSy )P ¿ 14 |[(SSx )P=2 ; (SSy )P=2 ]|2 ;
(20)
where (SSj )P = 2−P +
P
(CPk − CPk−1 )2k−P Sj k
(j = x; y) ;
(21)
k=2;4;::: P=2−1
[(SSx )P=2 ; (SSy )P=2 ] = iSz
m; n
m n CP=2 CP=2 22−P+m+n Sx m Sy n ;
(22)
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where P=2 ¿ 1 is an integer. If P=2 is even (odd), then m and n are odd (even) numbers. It is convenient that we de8ne the following functions: Hj (P) = (SSj )P − 12 |[(SSx )P=2 ; (SSy )P=2 ]| (j = x; y) :
(23)
Then, higher-order quantum ?uctuations in the component Sj of the dipole are squeezed if Hj ¡ 0 (j = x; y). This is the general de8nition of HOADS [31]. In our previous work, we have shown that sixth-order ADS could be exhibited in high-Q micromaser cavities. Especially, with the de8nition of two slowly varying Hermitian quadrature operators of the 8eld mode j [15] h1 = 12 [aj exp(ij t) + a†j exp(−ij t)] ;
(24)
1 [aj exp(ij t) − a†j exp(−ij t)] ; (25) 2i it has been shown that the second-order quantum ?uctuations in h1 (or h2 ) are related to those in Sy (or Sx ) [10,15,17–19], and the second-order quantum ?uctuations in Sy (or Sx ) are related to the sixth-order quantum ?uctuations in Sx (or Sy ) [31], based on which a general relation between 8eld and atomic dipole squeezing could be established. In this paper, taking the sixth-order (P = 6) quantum ?uctuations in one component of the dipole as an example, we investigate HOADS in the NTPJCM when the atom or two-mode 8elds are initially prepared in a superposition of states. As demonstrated in our previous work [34], if the 8elds are initially in a coherent state, the dynamics of the NTPJCM is not so simple and the main problem is the appearance of in8nite sums over the photon number n. However, in this paper, we will see that the NTPJCM is exactly solvable in the rotating-wave approximation when the 8eld is initially in a Fock state, showing the quantum Rabi oscillations. h2 =
3.2. Atom in a superposition of states In this part, we assume that the two-level atom is initially prepared in a superposition of its upper and lower level states |atom = c1 |− + c2 |+
(26)
and the two 8eld modes are in a vacuum state |0. Then, using Eqs. (13) – (16), we arrive at the expectation values of dipole components, Sx and Sy , and atomic inversion Sz , i.e., Sx = 1 − |c2 |2 |c2 |{cos(()) cos( + +)) − 1 − , sin(()) sin( + +))} ;
Sy =
(27) 1 − |c2 |2 |c2 |{cos(()) sin( + +)) +
1 − , sin(()) cos( + +))} ; (28)
√ Sz = |c2 |2 [1 − , sin2 (())] − 12 ; (29) where + = (1 − ,=2)= ,(, − 1) (note: + = 0 if = 0), , = 4=( + 4), ( = 1 + =4, ) = t is the scaled interaction time, and is the relative phase between the upper and
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lower level states of the two-level atom. Our calculated results Eqs. (27) – (29) are in agreement with those obtained by Mahmood and Ashraf [55]. Substituting Eqs. (27) – (29) into Eq. (23), we can get function Hj (j=x; y), based on which we can investigate the squeezing properties of higher-order quantum ?uctuations in the components of atomic dipole in the NTPJCM with the inclusion of the Stark shift. Obviously, from Eqs. (27) and (28) we see that when c1 =0 or c2 =0, there is no phase information and no squeezing appears at all. Hence, it shows that no squeezing occurs when the atom, which is coupled to the vacuum 8eld, is initially in the excited (c2 = 0) or ground (c1 = 0) state. Also it implies that squeezing results from the quantum interference. In the following, we report our results by properly choosing an initial superposition state for the two-level atom. In Fig. 1(a) (b) (c), we show 3D contour plots for function Hy (P = 6), relative phase and superposition parameter |c2 |2 at 8xed interaction time ) = - for Stark parameter = 0; 0:25; 1, respectively. In the absence of Stark shift (i.e., = 0), as shown in Fig. 1(a), we 8nd that the sixth-order quantum ?uctuations in Sy could be reduced below its higher-order lower limit around the relative phase =-=2; 3-=2 when the parameter |c2 |2 is in the range of 0 ¡ |c2 |2 ¡ 0:3 and 0:7 ¡ |c2 |2 ¡ 1, and small squeezing region appears around = 0; -; 2- in the range of 0:4 ¡ |c2 |2 ¡ 0:6. If the Stark shift is included, as shown in Fig. 1(b) for a small Stark parameter = 0:25, we see that the squeezing region in the phase space is shifted down but the positions of the squeezing pits at |c2 |2 ≈ 0:3; 0:7 do not move greatly, and the small squeezing region is enlarged and also shifted down. To our interest, when the two levels of the atom are equally coupled to the virtual state (i.e., = 1), as shown in Fig. 1(c), the squeezing regions for 0 ¡ |c2 |2 ¡ 0:4 move to = 0; -; 2- but those for 0:7 ¡ |c2 |2 ¡ 1 disappear totally, and highly squeezed states are generated around = 3-=2; -=2 for 0:25 ¡ |c2 |2 ¡ 0:75. In our previous work [31], we have demonstrated that the sixth-order quantum ?uctuations in Sy are related to the second-order ones in Sx . For comparison, Fig. 2(a) (b) (c) also present 3D contour plots for function Hx (P = 2), and |c2 |2 at ) = for = 0; 0:25; 1, respectively. In general, the results for the second-order cases are similar to the sixth-order one. However, there are some distinct di;erences. Comparing Fig. 2(a) with Fig. 1(a), we see that the squeezing region in the parameter |c2 |2 space for the second-order cases seems to cover all values except |c2 |2 = 0; 0:5; 1, and the squeezing regions in the relative phase space for the second-order cases are a little bigger than those in the sixth-order ones. Surely, there exist addition squeezing regions for the sixth-order cases, which are absent in the second-order cases, especially for a large parameter (comparing Fig. 1(b) and (c) with Fig. 2(b) and (c), respectively). Next, we examine the dynamics of the squeeze states. From the experimental point of view, the dynamics of long time scale is not practical. Hence, for experimentally relevant time systems, we only consider the case of short time scale in this paper. In detail, Fig. 3(a) (b) (c) show the time evolution of function Hy (P = 6) and Hx (P = 2) for = 0; 0:25; 1, respectively. In the absence of the Stark shift, as shown in Fig. 3(a), we 8nd an interesting case where both HOADS and SOADS with squeeze period )s = - and squeeze duration )d = - can appear at almost all the time except at ) = (k + 1=2)- and both squeeze pits occur at the same time. This means that there
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Fig. 1. The 3D contour plot of function R = 16Hy (P = 6), relative phase and parameter |c2 |2 at 8xed time ) = - for di;erent values of parameter : (a) = 0, where the solid line for R ¡ 0 in the range of −0:05 to −0:15 with the interval of −0:05, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.2– 0.8 with the interval of 0.2, (b) = 0:25, the others are the same as case (a), (c) = 1, where the solid line for R ¡ 0 in the range of −0:05 to −0:93 with the interval of −0:08, the others the same as case (a).
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Fig. 2. The 3D contour plot of function Q = 4Hx (P = 2), relative phase and parameter |c2 |2 at 8xed time ) = - for di;erent values of parameter : (a) = 0 where the solid line for Q ¡ 0 in the range of −0:05 to −0:2 with the interval of −0:05, the dotted line for Q = 0 and the dotted–dashed line for Q ¿ 0 in the range of 0.2– 0.8 with the interval of 0.2, (b) = 0:25, where the solid line for Q ¡ 0 in the range of −0:02 to −0:2 with the interval of −0:01, the others the same as case (a), (c) = 1, where the solid line for Q ¡ 0 in the range of −0:02 to −0:03 with the interval of −0:01 and the others are the same as case (a).
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Fig. 3. Time evolution of the function R = 16Hy (P = 6) (solid line) and Q = 4Hx (P = 2) (dotted line) for = -=2 and |c2 |2 = 0:1 for di;erent values of parameter : (a) = 0, (b) = 0:25, (c) = 1, and (d) = 0:01.
is a direct corresponding relation between HOADS and SOADS. As the Stark shift is switched on, as shown in Fig. 3(b) and (c), we 8nd that the squeeze duration is decreased and the squeezing pattern is no longer so periodical as in Fig. 3(a). At certain time, HOADS can appear but SOADS cannot.
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397
From above analysis, we conclude that an initial superposition of atom states could bring initial phase information into the system and lead to the generation of HOADS. At 8xed interaction time, it is found that additional high-order squeezing regions, which are absent in the second-order case, could be observed due to the Stark shift. As for the dynamics of squeezing, Stark shift has a destructive e;ect on the squeezing. In general, HOADS and SOADS could be generated at the same time with the choice of the same superposition parameters. 3.3. Field in a superposition of states In this part, we turn our attention to the case that each 8eld mode is initially prepared in a superposition of a vacuum state |0 and a single-photon state |1 |mode i = fi1 |0 + fi2 |1 ;
(30)
where i = 1; 2 and the atom is in its ground state |−. Using Eqs. (13) – (16), we obtain the expectation values of atomic dipole components Sx and Sy and atomic inversion Sz : (31) Sx = |f12 | |f22 | (1 − |f12 |2 )(1 − |f22 |2 ), sin(()) sin(51 + 52 + +)) ; (32) Sy = |f12 | |f22 | (1 − |f12 |2 )(1 − |f22 |2 ), sin(()) cos(51 + 52 + +)) ; Sz = |f12 |2 |f22 |2 sin2 (()) −
1 2
;
(33)
where 51 and 52 are the relative phases for 8eld modes 1 and 2, respectively. Once more, above calculated results are in agreement with those obtained by Mahmood and Ashraf [55]. Substituting Eqs. (31) – (33) into Eq. (23), we can get function Hj (j=x; y), based on which we can investigate the squeezing properties of higher-order quantum ?uctuations in the components of atomic dipole in the modi8ed NTPJCM with the inclusion of the Stark shift. It is worthwhile to point out that when any of the superposition coeOcients fij is equal to zero (i.e., fij = 0; i; j = 1; 2), we have Hl ¿ 0(l = x; y). This demonstrates that no HOADS appears when either of two 8eld modes, which are coupled to the atom, is initially in a pure Fock state. In other words, the initial 8eld mode should include phase information and thus HOADS may occur. Also it implies that the squeezing comes from quantum interference. In the following, we report our results by properly choosing a superposition state for each 8eld mode. In Fig. 4(a) (b) (c), we show 3D contour plots for function Hx (P = 6), relative phase 51 and 8eld strength parameter |f12 |2 at 52 =0, |f22 |2 =0:1 and 8xed interaction ) = -=2 for parameter = 0; 0:25; 1, respectively. Fig. 4(a) shows that HOADS is generated around 51 =0; -; 2- for 0 ¡ |f12 |2 6 0:75 when the Stark shift is not included. However, in the presence of the Stark shift, as shown in Fig. 4(b) and (c), we 8nd that the squeezing regions in the relative phase 51 space are shifted down but those in the parameter |f12 |2 space do not change greatly. Fig. 5(a) (b) (c) also present 3D contour plots for function Hy (P=2), 51 and |f12 |2 with the same values of other parameters as those in Fig. 4(a) (b) (c), respectively. The numerical results for the second-order cases are similar to the sixth-order one’s.
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Fig. 4. The 3D contour plot of function R = 16Hx (P = 6), relative phase 51 and parameter |f12 |2 at 52 = 0, |f22 |2 = 0:1 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for R ¡ 0 in the range of −0:001 to −0:036 with the interval of −0:005, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.13 with the interval of 0.02, (b) = 0:25, the others are the same as case (a), (c) = 1 where the solid line for R ¡ 0 in the range of −0:001 to −0:026 with the interval of −0:005, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.1 with the interval of 0.02.
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399
Fig. 5. The 3D contour plot of function Q = 4Hy (P = 2), relative phase 51 and parameter |f12 |2 at 52 = 0, |f22 |2 = 0:1 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for Q ¡ 0 in the range of −0:001 to −0:009 with the interval of −0:002, the dotted line for Q = 0 and the dotted–dashed line for Q ¿ 0 in the range of 0.01– 0.19 with the interval of 0.02, (b) = 0:25, the others are the same as case (a), (c) = 1 where the solid line for Q ¡ 0 in the range of −0:001 to −0:005 with the interval of −0:002, the others are the same as case (a).
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However, comparing Fig. 5 with Fig. 4, we notice that the squeezing region in the parameter |f12 |2 space for the sixth-order case is bigger than that of the second-order case, and the second-order squeezing is much more sensitive to the e;ect of Stark shift. In Fig. 6(a) and (b), we show 3D contour plots for function Hx (P = 6), |f12 |2 and |f22 |2 at 51 = -, 52 = 0 and 8xed interaction ) = -=2 for parameter = 0; 0:25, respectively. It is seen from Fig. 6(a) that squeezing appears in the range of 0 ¡ |f12 |2 ¡ 0:8 and 0 ¡ |f22 |2 ¡ 0:8 and a squeezing pit is observed at |f12 |2 = |f22 |2 ≈ 0:3 when the Stark shift is not included. Then, Fig. 6(b) shows that the Stark shift leads to a decreased squeezing region. Especially, when = 1, the squeezing region disappears totally. Similar results are obtained for the second-order cases which are shown in Fig. 7(a) and (b). Comparing Fig. 7 with Fig. 6, we 8nd that the region of HOADS is bigger than that of SOADS. In Fig. 8(a) and (b), we show 3D contour plots for function Hx (P = 6), 51 and 52 at |f12 |2 = 0:35, |f22 |2 = 0:1 and ) = -=2 for parameter = 0; 0:25, respectively. In the absence of Stark shift, as shown in Fig. 8(a), HOADS could be generated in several small band regions in the phase space. Fig. 8(b) shows that those squeezing bands are shifted and narrowed due to the e;ect of Stark shift. Similar results are obtained for the second-order cases shown in Fig. 9(a) and (b). Comparing Fig. 9 with Fig. 8, we see that the squeezing band for HOADS is wider than that of SOADS. Finally, Fig. 10(a) (b) (c) show the time evolution of function Hx (P = 6) and Hy (P =2) at 51 =-, 52 =0, |f12 |2 =0:35 and |f22 |2 =0:1 for =0; 0:25; 1, respectively. In the absence of the Stark shift, as shown in Fig. 10(a), both HOADS and SOADS with squeeze period )s = - and squeeze duration )d = - could appear at almost all the time except at ) = k-, and squeeze pits for both cases occur at the same time. There is a direct corresponding relation between HOADS and SOADS. However, as the Stark shift is switched on, as shown in Fig. 10(b) and (c), we 8nd that the squeeze duration is decreased and the squeezing pattern is no longer so periodical as in Fig. 10(a). At certain interaction time, HOADS can appear but SOADS cannot (especially, for the case of =1, as shown in Fig. 10(c), HOADS is still generated but SOADS disappears almost totally with the time evolution of the system). According to above numerical analysis, we conclude that an initial superposition of 8eld states could also lead to the generation of HOADS. The underlying mechanism lies in the phase information provided in the initial superposition state. At 8xed interaction time, it is found that high-order squeezing regions are bigger than those of the second-order cases, and the Stark shift has a destructive e;ect on the dynamics of squeezing. In general, HOADS and SOADS could be generated at the same time with the choice of the same superposition parameters. However, for some cases, HOADS occurs but SOADS could not. 3.4. Discussion From the two-photon Rabi frequency (n1 ; n2 ) given by Eq. (12), we 8nd that the term (; n1 ; n2 ) given by Eq. (11) would not vanish for general two-mode 8eld states due to the arbitrariness of photon number n1 and n2 presented. Hence, in this sense, we cannot ignore the Stark shift, even when the two-levels are equally coupled
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Fig. 6. The 3D contour plot of function R = 16Hx (P = 6), |f12 |2 and |f22 |2 at 51 = -, 52 = 0 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for R ¡ 0 in the range of −0:001 to −0:061 with the interval of −0:005, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.1– 0.2 with the interval of 0.1, (b) = 0:25 where the solid line for R ¡ 0 in the range of −0:001 to −0:016 with the interval of −0:005, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.29 with the interval of 0.04.
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Fig. 7. The 3D contour plot of function Q = 4Hy (P = 2), |f12 |2 and |f22 |2 at 51 = -, 52 = 0 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for Q ¡ 0 in the range of −0:001 to −0:021 with the interval of −0:005, the dotted line for Q = 0 and the dotted–dashed line for Q ¿ 0 in the range of 0.05 – 0.85 with the interval of 0.2, (b) = 0:25 where the solid line for Q ¡ 0 in the range of −0:001 to −0:007 with the interval of −0:002, the dotted line for Q = 0, the others the same as case (a).
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Fig. 8. The 3D contour plot of function R = 16Hx (P = 6), 51 and 52 for |f12 |2 = 0:35, |f22 |2 = 0:1 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for R ¡ 0 in the range of −0:001 to −0:031 with the interval of −0:01, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.13 with the interval of 0.04, (b) = 0:25 where the solid line for R ¡ 0 in the range of −0:001 to −0:033 with the interval of −0:008, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.09 with the interval of 0.04.
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Fig. 9. The 3D contour plot of function Q = 4Hy (P = 2), 51 and 52 for |f12 |2 = 0:35, |f22 |2 = 0:1 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for Q ¡ 0 in the range of −0:001 to −0:011 with the interval of −0:005, the dotted line for Q = 0 and the dotted–dashed line for Q ¿ 0 in the range of 0.01– 0.05 with the interval of 0.04, (b) = 0:25 where the solid line for Q ¡ 0 in the range of −0:001 to −0:007 with the interval of −0:002, the dotted line for Q = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.06 with the interval of 0.01.
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Fig. 10. Time evolution of the function R = 16Hx (P = 6) (solid line) and Q = 4Hy (P = 2) (dotted line) for 51 = -, 52 = 0, |f12 |2 = 0:35 and |f22 |2 = 0:1 for di;erent values of : (a) = 0, (b) = 0:25, (c) = 1, and (d) = 0:01.
to the virtual state (i.e., = 1). This can be clearly re?ected from our results. From Eqs. (27) – (29) and Eqs. (32) – (34), we see that there are terms of cos(()), sin(()), sin( + +)), cos( + +)), sin(51 + 52 + +)) and cos(51 + 52 + +)). For a small Stark parameter 1, ( ≈ 1, function cos(()) and sin(()) would exhibit a periodic behavior with a period of 2-, while the parameter + behaves as −1=2 and the
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time-dependent phase term +) in functions sin( + +)), cos( + +)), sin(51 + 52 + +)) and cos(51 + 52 + +)) would result in rapid oscillations. This can be clearly seen from Fig. 3(d) and Fig. 10(d) for = 0:01. As expected, we observe periodic collapse and revivals of Rabi oscillation with a period of -. This periodicity of the squeezing is due to a commensurate Rabi frequency. For larger parameter (for example, = 1) as shown above, squeezing region is shifted along the phase axis and the rapid oscillations broaden greatly. Of course, it is possible to weaken the Stark e;ects. For example, if = 1, then we have ( = 1; n1 ; n2 ) = (n1 − n2 + 1)2 =4. Obviously, if there exists some kinds of correlation between the two-mode 8elds, for example, the di;erence between their photon numbers is 8xed at a constant, i.e., n1 − n2 = k (k, an invariant integer), then the term would be 8xed at = (k + 1)2 =4 which is intensity independent. Moreover, if we choose the case of k = −1, we see that the term would vanish. Certainly, this is true only for physical quantities which are not sensitive to the phase. For phase-sensitive quantities, the induced phase due to the Stark shift could not disappear. For example, the two components Sx and Sy of the atomic dipole, as shown in Eqs. (27) and (28) and Eqs. (31) and (32), have an induced phase +) due to the Stark shift with the time evolution of the system. Hence, we can say that the presence of Stark shift brings phase information into emitted photons and has a signi8cant e;ect (both constructively and destructively) on the squeezing. In other words, with the time evolution of the system, the squeezing does not depend strongly on the initial phase information for the atom–8eld system and the phase sensitivity seems to result from the optical Stark shift (Surely, as discussed before, the state of the system should have initial phase information, otherwise there is no squeezing at all). In addition, photon pairs are emitted or absorbed simultaneously, thus leading to the conclusion that these photons are highly correlated. Hence, it would be interesting to examine the cases of n1 − n2 = k, for example, the squeezed vacuum state (k = 0), SU(1,1) coherent state, and pair coherent state, which are perfect correlated 8eld states. Discussion for these states will be presented in our another paper. Based on above numerical results, we 8nd that the squeezing is strongly a;ected by the intensity of the initial 8elds. As the 8eld intensity is increased, the squeezing decreases (see Figs. 4 –7). Actually, this e;ect can be understood since the presence of the second mode leads to an extra source of noise. Also, our above results show that the squeezing depends on the Stark parameter . It seems that weakly coupling of the intermediate virtual state to the ground state of the atom would lead to much more squeezing and its collapse-revival phenomena as shown in above numerical results. We 8nd that when the 8eld, initially in a vacuum state, couples with either only the ground or the excited state of the atom, there is no squeezing. Since atomic spontaneous radiation results from the interaction of the vacuum 8eld with the excited state of the atom, HOADS cannot appear in a spontaneous radiation process even though a photon pair is emitted. In above numerical results, we have discussed the relation between HOADS and SOADS in detail for every situation. In general, both HOADS and SOADS can appear at the same time for the same initial condition. Certainly, after a full comparison, we found that the Stark shift has much more destructive e;ects on SOADS than on HOADS. Even for some initial superposition states, the Stark shift leads to a strong
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HOADS, i.e., there exists a constructive e;ect on HOADS (see Figs. 1(c) and 10(c) for the Stark parameter = 1). Actually, in this paper, our discussion is still restricted to an ideal case, i.e., without considering the cavity damping. In the realistic situation, a high but 8nite cavity Q indeed causes some inevitable e;ects of photon dissipation in the cavity. For example, Gou [49] demonstrated the e;ects of cavity damping which will signi8cantly attenuate the amplitudes of revivals of the probability of 8nding the atom in the excited state unless the cavity Q is much smaller than the coupling strength of atom–8eld interaction. Certainly, it would be signi8cant and interesting to consider the e;ect of damping on HOADS. In fact, our calculated results show that the damping has a destructive e;ect on HOADS. Detailed results will be reported in our another paper. It should be mentioned that there are several signi8cant schemes for realizing a two-mode two-photon micromaser in a high-Q cavity, for example, those illustrated in Refs. [49 –51]. Certainly, these schemes could be helpful for realizing the system introduced in Section 2 and further understanding the HOADS predicted in this paper. Finally, we want to point out that involving the process of detection would be much better for us to understand the origin of quantum interference, which results in the squeezing observed in this paper, as far as the experiment measurement is concerned. We consider the general form of an initial superposition state |7 = A1 |91 + A2 |92 ;
(34)
which implies that we do not know exact information about the state in which the system would exist at the time of observation. This is the essence of quantum interference [61] that we learnt from the text book of quantum mechanics. Before the system is observed, it should pass through a detector [62], whose state is changed di;erently when interacting with the two superposed states. For convenience, as in Ref. [63], we de8ne |D0 as the state of the detector before interaction, and assume |Dl be the state after it interacts with the state |9l (l = 1; 2). Then, the interaction will lead to the following combined state |9 for the observed system and the detector: |9 = A1 |91 |D1 + A2 |92 |D2 ;
(35)
and the concrete results of experiment measurement on the observed system can be determined by the following density operator of the system by tracing over the detector states system = |A1 |2 |91 91 | + |A2 |2 |92 92 | + (A1 A∗2 |91 92 | + c:c:) ;
(36)
where = D1 |D2 is the overlap between the states, in which the detector is left as a result of its interaction with the two superposed states. Then, we can obtain the expectation value of a system operator O O = |A1 |2 91 |O|91 + |A2 |2 92 |O|92 + 2R[A1 A∗2 92 |O|91 ] :
(37)
Obvious, because of the process of detection, the last interference term shown above contains an important factor whose absolute value is less than or equal to one. Thus, the process of detection may reduce the contribution of the interference term shown above. If = 0, it is the well-known fact addressed in quantum mechanics [61] that the
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two superposed states are evidently distinguishable and the interference term vanishes (in other words, the quantum interference is lost if we know by which path the system arrived at the detector [63]). Surely, at some times, the loss of interference due to the process of detection may be attributed to a change in the observed part of the superposed states brought about by their interaction with the detector [63]. The details of the change so brought about depend upon the particular situations at hand [63]. 4. Summary In summary, we analyzed a nondegenerate two-photon JCM, where a two-level atom interacts with a two-mode 8eld in a high-Q micromaser cavity and the Stark shift is included. Taking the sixth-order ?uctuations in one part of the atomic dipole as an example and assuming that the atom or two 8eld modes are initially prepared in a superposition of states, we examined HOADS in detail, including the discussion of the e;ects of the Stark shift. It is found that the NTPJCM could generate HOADS if we properly choose the parameters for an initial superposition state of the atom or the 8eld modes. The underlying mechanism lies in the phase information provided in a superposition state. Meanwhile, we have found that the Stark shift brings time-dependent phase information into emitted photons and has a signi8cant e;ect (constructively and destructively) on HOADS (for example, weakly coupling for the intermediate virtual state to the ground state of the atom would lead to much more squeezing and its collapse-revival phenomena). In addition, the relation between SOADS and HOADS is examined. It is found that the second-order quantum ?uctuations in the dispersive (absorptive) part of the atomic dipole and the sixth-order ones in the absorptive (dispersive) parts of the atomic dipole may go below their own quantum limit at the same time, and there are additional cases that HOADS can appear but SOADS is absent. Acknowledgements We would like to thank Jianing Colin for some important contribution and constructive conversation and encouragement. One of us (Q.R.) acknowledges the Reinhart Fellowship provided by Queen’s University for 8nancial support. References [1] [2] [3] [4] [5]
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www.elsevier.com/locate/physa
Generation of higher-order atomic dipole squeezing in a high-Q micromaser cavity. (V). A linear superposition of states Rui-Hua Xiea;∗ , Qin Raob a Department
b Department
of Chemistry, Queen’s University, Kingston, Ont. Canada K7L 3N6 of Engineering Physics, Queen’s University, Kingston, Ont. Canada K7L 3N6 Received 25 February 2002
Abstract In our preceding paper, we investigated the generation of higher-order atomic dipole squeezing (HOADS) in a nondegenerate two-photon Jaynes–Cummings model in the absence of Stark shift. In this paper, we continue to study HOADS in this model but with the inclusion of Stark shift. Based on the linear superposition principle of quantum mechanics, we discuss the cases where the atom or the two-mode 8elds are initially prepared in a superposition of states. We demonstrate that the essential condition for generating HOADS is that the initial state of the system should include phase information, and HOADS cannot appear in atomic spontaneous radiation processes. The Stark shift brings time-dependent phase information into emitted photons and has a signi8cant e;ect (constructively and destructively) on HOADS. It is found that weakly coupling of the intermediate virtual state to the ground state of the atom would lead to much more squeezing and collapse-revival phenomena. The relation between HOADS and second-order c 2002 Elsevier Science B.V. All rights reserved. atomic dipole squeezing is also discussed. PACS: 42.50.Dv; 42.50.Lc; 32.80.−t Keywords: Dipole; Squeezed state; Superposition state; Quantum ?uctuation; Uncertainty relation; Stark shift; Two-photon micromaser
1. Introduction The Heisenberg uncertainty principle, proposed by Heisenberg [1] in 1926, is a fundamental, general principle of nature. Although it is of no consequence for ∗
Corresponding author. Present address: National Institute of Standards and Technology, 100 Bureau Drive, Mail Stop 8423, Gaithersburg, MD 20899-8423, USA. Tel.: +1-301-9755159; Fax: +1-301-9901350. E-mail address: [email protected] (R.-H. Xie). c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 0 1 0 - 5
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everyday, macroscopic bodies, it plays an important role in dealing with our microscopic world (atom, molecule, etc.). In 1927, Kennard [2] showed us the 8rst example of nonclassical states, named squeezed state, in which the quantum ?uctuations in a dynamical observable may be reduced below the standard quantum limit at the expense of increased ?uctuations in its canonical conjugated one without violating the Heisenberg uncertain relation. In 1956, PlebaNnski [3] made an important contribution to the theory of squeezed states. To the best of our knowledge, the exciting and constructive experimental results of the 8rst successful generation and detection of squeezed states were reported by Slusher et al. [4], Shelby et al. [5] and Wu et al. [6] in the middle of the 1980s. Ever since then, this kind of nonclassical states has been extensively studied, both theoretically and experimentally [7–24], due to their potential applications in high-resolution laser spectroscopy measurement, optical communication, gravity wave detection and quantum information theory (for example, quantum teleporation, cryptography, dense coding, power-recycled interferometer, and quantum nondemolition measurement). It has been shown that a lot of nonlinear optical systems could generate squeezed states for the 8eld and atom, for example, in four-wave mixing, two-photon laser, parametric ampli8ers, resonance ?uorescence, cooperative Dicke system and Rydberg atom maser, and a general relationship between 8eld and atomic dipole squeezing has even been established under di;erent initial conditions for both the 8eld and atom [10,15,17–19]. Meanwhile, in quantum optics and laser physics, the rapid development of techniques for making higher-order correlation measurements has led to increased interest in generating higher-order squeezed states, in which higher-order quantum ?uctuations in one quadrature of the 8eld or atomic dipole could be reduced without violating higher-order uncertainty relation. It has been shown that the generation of higher-order squeezed states for the radiation 8eld could be predicted in a number of nonlinear optical processes [25 –30] which could exhibit second-order squeezing. In 1997, Xie et al. [31] introduced the concept of higher-order atomic dipole squeezing (HOADS) and applied it to high-Q micromaser cavities, discussing thoroughly the important connection of HOADS with the second-order 8eld and atomic dipole squeezing. Moreover, Xie and Smith Jr. [32,33] have discussed the e;ects of a nonlinear one-photon process and dynamical Stark shift in two-photon processes on HOADS. It is found that the nonlinear one-photon processes in?uence the stability, period, width, pit depth and position of HOADS, and the e;ect of the Stark shift may lead to a strong HOADS in the revival region of the atomic inversion [32,33]. Very recently, assuming that the radiation 8elds are initially prepared in an uncorrelated coherent states, Xie and Smith Jr. [34] investigated HOADS in a nondegenerated two-photon Jaynes–Cummings model (NTPJCM) in the absence of Stark shift and demonstrated that a greater amount of HOADS could be achieved in the resonant cases. These extensive studies have made an important contribution to the theory of higher-order squeezed states and provided us an approach to extracting information eOciently from an optical signal by higher-order correlation measurements. It is known that the linear superposition principle is one of the most fundamental features in quantum mechanics and has been applied to all kinds of 8elds of physical science ever since the birth of quantum mechanics. Interference of quantum amplitudes and quantum beats are amongst the simplest examples of phenomena resulting from this
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principle. Recently, the coherent control approach has been successfully demonstrated, both theoretically and experimentally [35 –39], for example, in molecular hyperpolarizability, electron distribution excited in a metal, the energy and angular distribution of autoionized electrons, spontaneous emission near a photonic band edge, optical phonon emission rates and optical gain from electronic intersubband transition in semiconductors, optical dynamics in semiconductor microcavities, the polarization of an optical 8eld, the total ionization yield in a two-colour ionization process, photocurrent generation in bulk semiconductors, light absorption and terahertz radiations in semiconductor nanostructures, quantum chaotic di;usion, unimolecular breakdown reaction and reactive scattering. Although the physical systems for those cases may be di;erent, the essence of their coherent control schemes lies in two key steps: (i) create a superposition state composed of several basis states; (ii) control the relative phase of these basis, which means that the linear superposition principle plays a key role in the rapidly developing 8eld of coherent control. Also the linear superposition principle has led to an extensive study of superposition squeezed states of the 8eld and atoms, which were theoretically formulated by Wodkiewicz et al. [10] and developed by Xie et al. [17–19]. These studies have provided helpful revelations for physical interpretations of the nonclassical features of atom–8eld interactions from another angle. In this paper, we demonstrate the generation of HOADS in the modi8ed NTPJCM when the atom or two 8eld modes are initially prepared in a superposition of states. This paper is organized as follows. In Section 2, we describe the theoretical model of the modi8ed NTPJCM, in which the Stark shift is included, and derive some basic formalisms which are related to the two-photon interaction with a two-level atom in a high-Q cavity. In Section 3, assuming that the atom or two 8eld modes are initially prepared in a superposition of states, we investigate the generation of HOADS in the modi8ed NTPJCM, where the e;ect of the Stark shift on HOADS and the relation between HOADS and second-order ADS (SOADS) are discussed in detail. A summary is given in the last section. 2. Backgrounds of theoretical model It is known that a two-photon process plays an important role in atomic systems because of the high degree of correlation between the emitted photons [40]. One extension of the Jaynes–Cummings model (JCM) is the well-known two-photon JCM [41]. Over the past 20 years, the rapid development of two-photon micromasers [42], especially, the realization of the 8rst single-mode two-photon maser in the laboratory by employing Rydberg atoms in a high-Q superconducting micromaser cavity [43], has drawn a great deal of attention in the two-photon JCM and much work [44,45] has been done. To our knowledge, there are several signi8cant schemes for a two-mode two-photon maser with Rydberg atoms in a high-Q cavity system, which were proposed in order to advance one step further. To our interest, a remarkable feature of these two-mode two-photon masers is that one mode can be used to modulate, amplify, or control the output of the other mode. In this sense, the Rydberg atom plays a key role, which is similar to a coupler which correlates the two modes of the light and thus changes the entire photon statistics in a cavity. Correspondingly,
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NTPJCM was born in the world of quantum optics and laser physics and has attracted increased attention over these years. It is known that previous attempts tried to obtain lasing through two-mode two-photon-mediated transitions, but met with only limited success. Nevertheless, the advent of additional techniques [46 – 48] for demonstrating cavity QED e;ects has considerably brightened the prospects of achieving this goal and thus a number of dynamical aspects of NTPJCM were studied [49 –52]. Certainly, these studies would be complemented by a further discussion of a number of untouched and important issues in this area, for example, the generation of HOADS, which is actually the purpose of our present paper. In our preceding work [34], we investigated the HOADS in the NTPJCM but in a much idealized manner (i.e., the transition between the two atomic levels with the same parity is directly connected by photon pairs and hence the e;ective Hamiltonian approach ignores the Stark shifts caused by the virtual states). Puri and Bullough [53] have shown that the consideration of the Stark shift would change the quantum dynamics of a degenerated two-photon JCM (DTPJCM). Also, Xie and Smith Jr. [33] pointed out that the e;ect of the Stark shift may lead to a strong HOADS in the revived region of the atomic inversion in the DTPJCM. In addition, Gou [54], Mahmood and Ashraf [55], D’souza and Jayarao [56] have convinced us that the dynamical Stark e;ects will also change the quantum dynamics of the two-mode two-photon JCM (i.e., NTPJCM) in a more general way than those of DTPJCM. In the following, we brie?y introduce the modi8ed NTPJCM. We consider a two-level atom interacting with two modes of the radiation 8eld in a lossless resonant cavity, and denote ! as the natural transition frequency between the upper level state |+ and the lower level state |− of the two-level atom and j as the frequency of mode j of the radiation 8eld. It is known that some intermediate virtual states |i, which are assumed to be coupled to the upper and lower states of the two-level atom by dipole-allowed transition, are involved in the two-photon processes. We assume that 1 is about several times 2 and the two 8elds are tuned as near one-photon resonance as possible but still o; the one-photon linewidth, i.e., the detuning 1 and 2 are larger than but as close as the linewidth of |i. For the exact two-photon resonance 1 = −2 = and by the method of adiabatic elimination of the virtual state |i [53,57], Gou [54] obtained the e;ective Hamiltonian within the rotating-wave approximation H = H0 + HI
(1)
H0 = 1 a†1 a1 + 2 a†2 a2 + !Sz ;
(2)
with
† † † HI = 2 S+ S− a+ (3) 2 a2 + 1 S− S+ a1 a1 + (a1 a2 S− + a1 a2 S+ ) ; where = 1 2 . Here Sz and S± are operators of the atomic pseudospin inversion and transition, respectively, and satisfy the commutation [14]
[S+ ; S− ] = 2Sz ;
(4)
[Sz ; S± ] = ±S± ;
(5)
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and ! is the transition frequency for the atom. a†i and ai are the creation and annihilation operators for the photons of mode i with the frequency i , respectively, which obey the boson operators’ commutation relations [ai ; a†j ] = ij :
(6)
is the coupling constant between the atom and the radiation 8eld. A Stark shift is caused by the intermediate level |i, and the corresponding parameter 1 and 2 are 1 = 12 =, 2 = 22 = and = 1 2 =, where 1 and 2 are the coupling strengths of the intermediate level with the lower and upper levels of the two-level atom, respectively. Throughout we employ the units of ˝ = c = 1. We denote |n as the Fock state of the radiation 8eld. Using the standard bare-state procedure introduced by Yoo and Eberly [58] and developed by Xie et al. [59], we are able to calculate the expectation value of any operator in the modi8ed NTPJCM. In an ideal cavity, the two-level atom absorbs and emits one pair of photons, and the two bases in the bare-state representation are |+; n1 ; n2 and |−; n1 + 1; n2 + 1, respectively. Thus, the total Hamiltonian given by Eq. (1) can be written as 1 n1 + 2 n2 + !=2 + 2 n2
(n1 + 1)(n2 + 1) H= :
(n1 + 1)(n2 + 1) 1 (n1 + 1) + 2 (n2 + 1) − !=2 + 1 (n1 + 1) (7) Obviously, the eigenvalues of the above Hamiltonian are + = 1 n1 + 2 n2 + 12 ! + 12 2 n2 + 12 1 (n1 + 1) + (n1 ; n2 ) ;
(8)
− = 1 n1 + 2 n2 + 12 ! + 12 2 n2 + 12 1 (n1 + 1) − (n1 ; n2 ) ;
(9)
respectively, where (n1 ; n2 ) = (; n1 ; n2 ) + (n1 + 1)(n2 + 1) ; (; n1 ; n2 ) =
[(n1 + 1) − n2 ]2 4
(10) (11)
and = 1 = 2 . The two-photon Rabi frequency is written as (n1 ; n2 ) = (n1 ; n2 ) : Then, the time evolution operator U (n1 ; n2 ; t) of our two-mode system is E+ (t) (t) U (n1 ; n2 ; t) = ; (t) E− (t) where
1 n2 − (n1 + 1) exp(−i+ t) + √ 2 2 (n1 ; n2 ) 1 n2 − (n1 + 1) exp(−i− t) ; + − √ 2 2 (n1 ; n2 )
E+ (t) =
(12)
(13)
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1 n2 − (n1 + 1) − √ exp(−i+ t) E− (t) = 2 2 (n1 ; n2 ) 1 n2 − (n1 + 1) exp(−i− t) ; + + √ 2 2 (n1 ; n2 )
(t) =
(n1 + 1)(n2 + 1) [exp(−i+ t) − exp(−i− t)] : 2(n1 ; n2 )
(14)
Moreover, the density operator of our system at time t can be calculated with an arbitrary initial condition (t = 0) by (t) = U (t)† (t = 0)U (t) :
(15)
Therefore, the expectation value of any physical operator O can be arrived at through O(t) = Tr[(t)O(0)] :
(16)
3. Hoads and superposition states 3.1. Concept of HOADS In order to investigate the squeezing properties of the atomic dipole variables, we follow the standard procedure of de8ning the slowly varying operators [14] Sx = 12 [S+ exp(−i!t) + S− exp(i!t)] ;
(17)
1 (18) [S+ exp(−i!t) − S− exp(i!t)] ; 2i where Sx and Sy , in fact, correspond to the dispersive and absorptive components of the slowly varying atomic dipole [60], respectively. One can easily show that the above operators obey the commutation relation Sy =
[Sx ; Sy ] = iSz :
(19)
Correspondingly, we found the higher-order uncertainty relation [31] concretely given by (SSx )P (SSy )P ¿ 14 |[(SSx )P=2 ; (SSy )P=2 ]|2 ;
(20)
where (SSj )P = 2−P +
P
(CPk − CPk−1 )2k−P Sj k
(j = x; y) ;
(21)
k=2;4;::: P=2−1
[(SSx )P=2 ; (SSy )P=2 ] = iSz
m; n
m n CP=2 CP=2 22−P+m+n Sx m Sy n ;
(22)
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where P=2 ¿ 1 is an integer. If P=2 is even (odd), then m and n are odd (even) numbers. It is convenient that we de8ne the following functions: Hj (P) = (SSj )P − 12 |[(SSx )P=2 ; (SSy )P=2 ]| (j = x; y) :
(23)
Then, higher-order quantum ?uctuations in the component Sj of the dipole are squeezed if Hj ¡ 0 (j = x; y). This is the general de8nition of HOADS [31]. In our previous work, we have shown that sixth-order ADS could be exhibited in high-Q micromaser cavities. Especially, with the de8nition of two slowly varying Hermitian quadrature operators of the 8eld mode j [15] h1 = 12 [aj exp(ij t) + a†j exp(−ij t)] ;
(24)
1 [aj exp(ij t) − a†j exp(−ij t)] ; (25) 2i it has been shown that the second-order quantum ?uctuations in h1 (or h2 ) are related to those in Sy (or Sx ) [10,15,17–19], and the second-order quantum ?uctuations in Sy (or Sx ) are related to the sixth-order quantum ?uctuations in Sx (or Sy ) [31], based on which a general relation between 8eld and atomic dipole squeezing could be established. In this paper, taking the sixth-order (P = 6) quantum ?uctuations in one component of the dipole as an example, we investigate HOADS in the NTPJCM when the atom or two-mode 8elds are initially prepared in a superposition of states. As demonstrated in our previous work [34], if the 8elds are initially in a coherent state, the dynamics of the NTPJCM is not so simple and the main problem is the appearance of in8nite sums over the photon number n. However, in this paper, we will see that the NTPJCM is exactly solvable in the rotating-wave approximation when the 8eld is initially in a Fock state, showing the quantum Rabi oscillations. h2 =
3.2. Atom in a superposition of states In this part, we assume that the two-level atom is initially prepared in a superposition of its upper and lower level states |atom = c1 |− + c2 |+
(26)
and the two 8eld modes are in a vacuum state |0. Then, using Eqs. (13) – (16), we arrive at the expectation values of dipole components, Sx and Sy , and atomic inversion Sz , i.e., Sx = 1 − |c2 |2 |c2 |{cos(()) cos( + +)) − 1 − , sin(()) sin( + +))} ;
Sy =
(27) 1 − |c2 |2 |c2 |{cos(()) sin( + +)) +
1 − , sin(()) cos( + +))} ; (28)
√ Sz = |c2 |2 [1 − , sin2 (())] − 12 ; (29) where + = (1 − ,=2)= ,(, − 1) (note: + = 0 if = 0), , = 4=( + 4), ( = 1 + =4, ) = t is the scaled interaction time, and is the relative phase between the upper and
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lower level states of the two-level atom. Our calculated results Eqs. (27) – (29) are in agreement with those obtained by Mahmood and Ashraf [55]. Substituting Eqs. (27) – (29) into Eq. (23), we can get function Hj (j=x; y), based on which we can investigate the squeezing properties of higher-order quantum ?uctuations in the components of atomic dipole in the NTPJCM with the inclusion of the Stark shift. Obviously, from Eqs. (27) and (28) we see that when c1 =0 or c2 =0, there is no phase information and no squeezing appears at all. Hence, it shows that no squeezing occurs when the atom, which is coupled to the vacuum 8eld, is initially in the excited (c2 = 0) or ground (c1 = 0) state. Also it implies that squeezing results from the quantum interference. In the following, we report our results by properly choosing an initial superposition state for the two-level atom. In Fig. 1(a) (b) (c), we show 3D contour plots for function Hy (P = 6), relative phase and superposition parameter |c2 |2 at 8xed interaction time ) = - for Stark parameter = 0; 0:25; 1, respectively. In the absence of Stark shift (i.e., = 0), as shown in Fig. 1(a), we 8nd that the sixth-order quantum ?uctuations in Sy could be reduced below its higher-order lower limit around the relative phase =-=2; 3-=2 when the parameter |c2 |2 is in the range of 0 ¡ |c2 |2 ¡ 0:3 and 0:7 ¡ |c2 |2 ¡ 1, and small squeezing region appears around = 0; -; 2- in the range of 0:4 ¡ |c2 |2 ¡ 0:6. If the Stark shift is included, as shown in Fig. 1(b) for a small Stark parameter = 0:25, we see that the squeezing region in the phase space is shifted down but the positions of the squeezing pits at |c2 |2 ≈ 0:3; 0:7 do not move greatly, and the small squeezing region is enlarged and also shifted down. To our interest, when the two levels of the atom are equally coupled to the virtual state (i.e., = 1), as shown in Fig. 1(c), the squeezing regions for 0 ¡ |c2 |2 ¡ 0:4 move to = 0; -; 2- but those for 0:7 ¡ |c2 |2 ¡ 1 disappear totally, and highly squeezed states are generated around = 3-=2; -=2 for 0:25 ¡ |c2 |2 ¡ 0:75. In our previous work [31], we have demonstrated that the sixth-order quantum ?uctuations in Sy are related to the second-order ones in Sx . For comparison, Fig. 2(a) (b) (c) also present 3D contour plots for function Hx (P = 2), and |c2 |2 at ) = for = 0; 0:25; 1, respectively. In general, the results for the second-order cases are similar to the sixth-order one. However, there are some distinct di;erences. Comparing Fig. 2(a) with Fig. 1(a), we see that the squeezing region in the parameter |c2 |2 space for the second-order cases seems to cover all values except |c2 |2 = 0; 0:5; 1, and the squeezing regions in the relative phase space for the second-order cases are a little bigger than those in the sixth-order ones. Surely, there exist addition squeezing regions for the sixth-order cases, which are absent in the second-order cases, especially for a large parameter (comparing Fig. 1(b) and (c) with Fig. 2(b) and (c), respectively). Next, we examine the dynamics of the squeeze states. From the experimental point of view, the dynamics of long time scale is not practical. Hence, for experimentally relevant time systems, we only consider the case of short time scale in this paper. In detail, Fig. 3(a) (b) (c) show the time evolution of function Hy (P = 6) and Hx (P = 2) for = 0; 0:25; 1, respectively. In the absence of the Stark shift, as shown in Fig. 3(a), we 8nd an interesting case where both HOADS and SOADS with squeeze period )s = - and squeeze duration )d = - can appear at almost all the time except at ) = (k + 1=2)- and both squeeze pits occur at the same time. This means that there
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Fig. 1. The 3D contour plot of function R = 16Hy (P = 6), relative phase and parameter |c2 |2 at 8xed time ) = - for di;erent values of parameter : (a) = 0, where the solid line for R ¡ 0 in the range of −0:05 to −0:15 with the interval of −0:05, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.2– 0.8 with the interval of 0.2, (b) = 0:25, the others are the same as case (a), (c) = 1, where the solid line for R ¡ 0 in the range of −0:05 to −0:93 with the interval of −0:08, the others the same as case (a).
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395
Fig. 2. The 3D contour plot of function Q = 4Hx (P = 2), relative phase and parameter |c2 |2 at 8xed time ) = - for di;erent values of parameter : (a) = 0 where the solid line for Q ¡ 0 in the range of −0:05 to −0:2 with the interval of −0:05, the dotted line for Q = 0 and the dotted–dashed line for Q ¿ 0 in the range of 0.2– 0.8 with the interval of 0.2, (b) = 0:25, where the solid line for Q ¡ 0 in the range of −0:02 to −0:2 with the interval of −0:01, the others the same as case (a), (c) = 1, where the solid line for Q ¡ 0 in the range of −0:02 to −0:03 with the interval of −0:01 and the others are the same as case (a).
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Fig. 3. Time evolution of the function R = 16Hy (P = 6) (solid line) and Q = 4Hx (P = 2) (dotted line) for = -=2 and |c2 |2 = 0:1 for di;erent values of parameter : (a) = 0, (b) = 0:25, (c) = 1, and (d) = 0:01.
is a direct corresponding relation between HOADS and SOADS. As the Stark shift is switched on, as shown in Fig. 3(b) and (c), we 8nd that the squeeze duration is decreased and the squeezing pattern is no longer so periodical as in Fig. 3(a). At certain time, HOADS can appear but SOADS cannot.
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397
From above analysis, we conclude that an initial superposition of atom states could bring initial phase information into the system and lead to the generation of HOADS. At 8xed interaction time, it is found that additional high-order squeezing regions, which are absent in the second-order case, could be observed due to the Stark shift. As for the dynamics of squeezing, Stark shift has a destructive e;ect on the squeezing. In general, HOADS and SOADS could be generated at the same time with the choice of the same superposition parameters. 3.3. Field in a superposition of states In this part, we turn our attention to the case that each 8eld mode is initially prepared in a superposition of a vacuum state |0 and a single-photon state |1 |mode i = fi1 |0 + fi2 |1 ;
(30)
where i = 1; 2 and the atom is in its ground state |−. Using Eqs. (13) – (16), we obtain the expectation values of atomic dipole components Sx and Sy and atomic inversion Sz : (31) Sx = |f12 | |f22 | (1 − |f12 |2 )(1 − |f22 |2 ), sin(()) sin(51 + 52 + +)) ; (32) Sy = |f12 | |f22 | (1 − |f12 |2 )(1 − |f22 |2 ), sin(()) cos(51 + 52 + +)) ; Sz = |f12 |2 |f22 |2 sin2 (()) −
1 2
;
(33)
where 51 and 52 are the relative phases for 8eld modes 1 and 2, respectively. Once more, above calculated results are in agreement with those obtained by Mahmood and Ashraf [55]. Substituting Eqs. (31) – (33) into Eq. (23), we can get function Hj (j=x; y), based on which we can investigate the squeezing properties of higher-order quantum ?uctuations in the components of atomic dipole in the modi8ed NTPJCM with the inclusion of the Stark shift. It is worthwhile to point out that when any of the superposition coeOcients fij is equal to zero (i.e., fij = 0; i; j = 1; 2), we have Hl ¿ 0(l = x; y). This demonstrates that no HOADS appears when either of two 8eld modes, which are coupled to the atom, is initially in a pure Fock state. In other words, the initial 8eld mode should include phase information and thus HOADS may occur. Also it implies that the squeezing comes from quantum interference. In the following, we report our results by properly choosing a superposition state for each 8eld mode. In Fig. 4(a) (b) (c), we show 3D contour plots for function Hx (P = 6), relative phase 51 and 8eld strength parameter |f12 |2 at 52 =0, |f22 |2 =0:1 and 8xed interaction ) = -=2 for parameter = 0; 0:25; 1, respectively. Fig. 4(a) shows that HOADS is generated around 51 =0; -; 2- for 0 ¡ |f12 |2 6 0:75 when the Stark shift is not included. However, in the presence of the Stark shift, as shown in Fig. 4(b) and (c), we 8nd that the squeezing regions in the relative phase 51 space are shifted down but those in the parameter |f12 |2 space do not change greatly. Fig. 5(a) (b) (c) also present 3D contour plots for function Hy (P=2), 51 and |f12 |2 with the same values of other parameters as those in Fig. 4(a) (b) (c), respectively. The numerical results for the second-order cases are similar to the sixth-order one’s.
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Fig. 4. The 3D contour plot of function R = 16Hx (P = 6), relative phase 51 and parameter |f12 |2 at 52 = 0, |f22 |2 = 0:1 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for R ¡ 0 in the range of −0:001 to −0:036 with the interval of −0:005, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.13 with the interval of 0.02, (b) = 0:25, the others are the same as case (a), (c) = 1 where the solid line for R ¡ 0 in the range of −0:001 to −0:026 with the interval of −0:005, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.1 with the interval of 0.02.
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399
Fig. 5. The 3D contour plot of function Q = 4Hy (P = 2), relative phase 51 and parameter |f12 |2 at 52 = 0, |f22 |2 = 0:1 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for Q ¡ 0 in the range of −0:001 to −0:009 with the interval of −0:002, the dotted line for Q = 0 and the dotted–dashed line for Q ¿ 0 in the range of 0.01– 0.19 with the interval of 0.02, (b) = 0:25, the others are the same as case (a), (c) = 1 where the solid line for Q ¡ 0 in the range of −0:001 to −0:005 with the interval of −0:002, the others are the same as case (a).
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However, comparing Fig. 5 with Fig. 4, we notice that the squeezing region in the parameter |f12 |2 space for the sixth-order case is bigger than that of the second-order case, and the second-order squeezing is much more sensitive to the e;ect of Stark shift. In Fig. 6(a) and (b), we show 3D contour plots for function Hx (P = 6), |f12 |2 and |f22 |2 at 51 = -, 52 = 0 and 8xed interaction ) = -=2 for parameter = 0; 0:25, respectively. It is seen from Fig. 6(a) that squeezing appears in the range of 0 ¡ |f12 |2 ¡ 0:8 and 0 ¡ |f22 |2 ¡ 0:8 and a squeezing pit is observed at |f12 |2 = |f22 |2 ≈ 0:3 when the Stark shift is not included. Then, Fig. 6(b) shows that the Stark shift leads to a decreased squeezing region. Especially, when = 1, the squeezing region disappears totally. Similar results are obtained for the second-order cases which are shown in Fig. 7(a) and (b). Comparing Fig. 7 with Fig. 6, we 8nd that the region of HOADS is bigger than that of SOADS. In Fig. 8(a) and (b), we show 3D contour plots for function Hx (P = 6), 51 and 52 at |f12 |2 = 0:35, |f22 |2 = 0:1 and ) = -=2 for parameter = 0; 0:25, respectively. In the absence of Stark shift, as shown in Fig. 8(a), HOADS could be generated in several small band regions in the phase space. Fig. 8(b) shows that those squeezing bands are shifted and narrowed due to the e;ect of Stark shift. Similar results are obtained for the second-order cases shown in Fig. 9(a) and (b). Comparing Fig. 9 with Fig. 8, we see that the squeezing band for HOADS is wider than that of SOADS. Finally, Fig. 10(a) (b) (c) show the time evolution of function Hx (P = 6) and Hy (P =2) at 51 =-, 52 =0, |f12 |2 =0:35 and |f22 |2 =0:1 for =0; 0:25; 1, respectively. In the absence of the Stark shift, as shown in Fig. 10(a), both HOADS and SOADS with squeeze period )s = - and squeeze duration )d = - could appear at almost all the time except at ) = k-, and squeeze pits for both cases occur at the same time. There is a direct corresponding relation between HOADS and SOADS. However, as the Stark shift is switched on, as shown in Fig. 10(b) and (c), we 8nd that the squeeze duration is decreased and the squeezing pattern is no longer so periodical as in Fig. 10(a). At certain interaction time, HOADS can appear but SOADS cannot (especially, for the case of =1, as shown in Fig. 10(c), HOADS is still generated but SOADS disappears almost totally with the time evolution of the system). According to above numerical analysis, we conclude that an initial superposition of 8eld states could also lead to the generation of HOADS. The underlying mechanism lies in the phase information provided in the initial superposition state. At 8xed interaction time, it is found that high-order squeezing regions are bigger than those of the second-order cases, and the Stark shift has a destructive e;ect on the dynamics of squeezing. In general, HOADS and SOADS could be generated at the same time with the choice of the same superposition parameters. However, for some cases, HOADS occurs but SOADS could not. 3.4. Discussion From the two-photon Rabi frequency (n1 ; n2 ) given by Eq. (12), we 8nd that the term (; n1 ; n2 ) given by Eq. (11) would not vanish for general two-mode 8eld states due to the arbitrariness of photon number n1 and n2 presented. Hence, in this sense, we cannot ignore the Stark shift, even when the two-levels are equally coupled
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401
Fig. 6. The 3D contour plot of function R = 16Hx (P = 6), |f12 |2 and |f22 |2 at 51 = -, 52 = 0 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for R ¡ 0 in the range of −0:001 to −0:061 with the interval of −0:005, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.1– 0.2 with the interval of 0.1, (b) = 0:25 where the solid line for R ¡ 0 in the range of −0:001 to −0:016 with the interval of −0:005, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.29 with the interval of 0.04.
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Fig. 7. The 3D contour plot of function Q = 4Hy (P = 2), |f12 |2 and |f22 |2 at 51 = -, 52 = 0 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for Q ¡ 0 in the range of −0:001 to −0:021 with the interval of −0:005, the dotted line for Q = 0 and the dotted–dashed line for Q ¿ 0 in the range of 0.05 – 0.85 with the interval of 0.2, (b) = 0:25 where the solid line for Q ¡ 0 in the range of −0:001 to −0:007 with the interval of −0:002, the dotted line for Q = 0, the others the same as case (a).
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403
Fig. 8. The 3D contour plot of function R = 16Hx (P = 6), 51 and 52 for |f12 |2 = 0:35, |f22 |2 = 0:1 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for R ¡ 0 in the range of −0:001 to −0:031 with the interval of −0:01, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.13 with the interval of 0.04, (b) = 0:25 where the solid line for R ¡ 0 in the range of −0:001 to −0:033 with the interval of −0:008, the dotted line for R = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.09 with the interval of 0.04.
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Fig. 9. The 3D contour plot of function Q = 4Hy (P = 2), 51 and 52 for |f12 |2 = 0:35, |f22 |2 = 0:1 and ) = -=2 for di;erent values of parameter : (a) = 0 where the solid line for Q ¡ 0 in the range of −0:001 to −0:011 with the interval of −0:005, the dotted line for Q = 0 and the dotted–dashed line for Q ¿ 0 in the range of 0.01– 0.05 with the interval of 0.04, (b) = 0:25 where the solid line for Q ¡ 0 in the range of −0:001 to −0:007 with the interval of −0:002, the dotted line for Q = 0 and the dotted–dashed line for R ¿ 0 in the range of 0.01– 0.06 with the interval of 0.01.
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Fig. 10. Time evolution of the function R = 16Hx (P = 6) (solid line) and Q = 4Hy (P = 2) (dotted line) for 51 = -, 52 = 0, |f12 |2 = 0:35 and |f22 |2 = 0:1 for di;erent values of : (a) = 0, (b) = 0:25, (c) = 1, and (d) = 0:01.
to the virtual state (i.e., = 1). This can be clearly re?ected from our results. From Eqs. (27) – (29) and Eqs. (32) – (34), we see that there are terms of cos(()), sin(()), sin( + +)), cos( + +)), sin(51 + 52 + +)) and cos(51 + 52 + +)). For a small Stark parameter 1, ( ≈ 1, function cos(()) and sin(()) would exhibit a periodic behavior with a period of 2-, while the parameter + behaves as −1=2 and the
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time-dependent phase term +) in functions sin( + +)), cos( + +)), sin(51 + 52 + +)) and cos(51 + 52 + +)) would result in rapid oscillations. This can be clearly seen from Fig. 3(d) and Fig. 10(d) for = 0:01. As expected, we observe periodic collapse and revivals of Rabi oscillation with a period of -. This periodicity of the squeezing is due to a commensurate Rabi frequency. For larger parameter (for example, = 1) as shown above, squeezing region is shifted along the phase axis and the rapid oscillations broaden greatly. Of course, it is possible to weaken the Stark e;ects. For example, if = 1, then we have ( = 1; n1 ; n2 ) = (n1 − n2 + 1)2 =4. Obviously, if there exists some kinds of correlation between the two-mode 8elds, for example, the di;erence between their photon numbers is 8xed at a constant, i.e., n1 − n2 = k (k, an invariant integer), then the term would be 8xed at = (k + 1)2 =4 which is intensity independent. Moreover, if we choose the case of k = −1, we see that the term would vanish. Certainly, this is true only for physical quantities which are not sensitive to the phase. For phase-sensitive quantities, the induced phase due to the Stark shift could not disappear. For example, the two components Sx and Sy of the atomic dipole, as shown in Eqs. (27) and (28) and Eqs. (31) and (32), have an induced phase +) due to the Stark shift with the time evolution of the system. Hence, we can say that the presence of Stark shift brings phase information into emitted photons and has a signi8cant e;ect (both constructively and destructively) on the squeezing. In other words, with the time evolution of the system, the squeezing does not depend strongly on the initial phase information for the atom–8eld system and the phase sensitivity seems to result from the optical Stark shift (Surely, as discussed before, the state of the system should have initial phase information, otherwise there is no squeezing at all). In addition, photon pairs are emitted or absorbed simultaneously, thus leading to the conclusion that these photons are highly correlated. Hence, it would be interesting to examine the cases of n1 − n2 = k, for example, the squeezed vacuum state (k = 0), SU(1,1) coherent state, and pair coherent state, which are perfect correlated 8eld states. Discussion for these states will be presented in our another paper. Based on above numerical results, we 8nd that the squeezing is strongly a;ected by the intensity of the initial 8elds. As the 8eld intensity is increased, the squeezing decreases (see Figs. 4 –7). Actually, this e;ect can be understood since the presence of the second mode leads to an extra source of noise. Also, our above results show that the squeezing depends on the Stark parameter . It seems that weakly coupling of the intermediate virtual state to the ground state of the atom would lead to much more squeezing and its collapse-revival phenomena as shown in above numerical results. We 8nd that when the 8eld, initially in a vacuum state, couples with either only the ground or the excited state of the atom, there is no squeezing. Since atomic spontaneous radiation results from the interaction of the vacuum 8eld with the excited state of the atom, HOADS cannot appear in a spontaneous radiation process even though a photon pair is emitted. In above numerical results, we have discussed the relation between HOADS and SOADS in detail for every situation. In general, both HOADS and SOADS can appear at the same time for the same initial condition. Certainly, after a full comparison, we found that the Stark shift has much more destructive e;ects on SOADS than on HOADS. Even for some initial superposition states, the Stark shift leads to a strong
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HOADS, i.e., there exists a constructive e;ect on HOADS (see Figs. 1(c) and 10(c) for the Stark parameter = 1). Actually, in this paper, our discussion is still restricted to an ideal case, i.e., without considering the cavity damping. In the realistic situation, a high but 8nite cavity Q indeed causes some inevitable e;ects of photon dissipation in the cavity. For example, Gou [49] demonstrated the e;ects of cavity damping which will signi8cantly attenuate the amplitudes of revivals of the probability of 8nding the atom in the excited state unless the cavity Q is much smaller than the coupling strength of atom–8eld interaction. Certainly, it would be signi8cant and interesting to consider the e;ect of damping on HOADS. In fact, our calculated results show that the damping has a destructive e;ect on HOADS. Detailed results will be reported in our another paper. It should be mentioned that there are several signi8cant schemes for realizing a two-mode two-photon micromaser in a high-Q cavity, for example, those illustrated in Refs. [49 –51]. Certainly, these schemes could be helpful for realizing the system introduced in Section 2 and further understanding the HOADS predicted in this paper. Finally, we want to point out that involving the process of detection would be much better for us to understand the origin of quantum interference, which results in the squeezing observed in this paper, as far as the experiment measurement is concerned. We consider the general form of an initial superposition state |7 = A1 |91 + A2 |92 ;
(34)
which implies that we do not know exact information about the state in which the system would exist at the time of observation. This is the essence of quantum interference [61] that we learnt from the text book of quantum mechanics. Before the system is observed, it should pass through a detector [62], whose state is changed di;erently when interacting with the two superposed states. For convenience, as in Ref. [63], we de8ne |D0 as the state of the detector before interaction, and assume |Dl be the state after it interacts with the state |9l (l = 1; 2). Then, the interaction will lead to the following combined state |9 for the observed system and the detector: |9 = A1 |91 |D1 + A2 |92 |D2 ;
(35)
and the concrete results of experiment measurement on the observed system can be determined by the following density operator of the system by tracing over the detector states system = |A1 |2 |91 91 | + |A2 |2 |92 92 | + (A1 A∗2 |91 92 | + c:c:) ;
(36)
where = D1 |D2 is the overlap between the states, in which the detector is left as a result of its interaction with the two superposed states. Then, we can obtain the expectation value of a system operator O O = |A1 |2 91 |O|91 + |A2 |2 92 |O|92 + 2R[A1 A∗2 92 |O|91 ] :
(37)
Obvious, because of the process of detection, the last interference term shown above contains an important factor whose absolute value is less than or equal to one. Thus, the process of detection may reduce the contribution of the interference term shown above. If = 0, it is the well-known fact addressed in quantum mechanics [61] that the
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two superposed states are evidently distinguishable and the interference term vanishes (in other words, the quantum interference is lost if we know by which path the system arrived at the detector [63]). Surely, at some times, the loss of interference due to the process of detection may be attributed to a change in the observed part of the superposed states brought about by their interaction with the detector [63]. The details of the change so brought about depend upon the particular situations at hand [63]. 4. Summary In summary, we analyzed a nondegenerate two-photon JCM, where a two-level atom interacts with a two-mode 8eld in a high-Q micromaser cavity and the Stark shift is included. Taking the sixth-order ?uctuations in one part of the atomic dipole as an example and assuming that the atom or two 8eld modes are initially prepared in a superposition of states, we examined HOADS in detail, including the discussion of the e;ects of the Stark shift. It is found that the NTPJCM could generate HOADS if we properly choose the parameters for an initial superposition state of the atom or the 8eld modes. The underlying mechanism lies in the phase information provided in a superposition state. Meanwhile, we have found that the Stark shift brings time-dependent phase information into emitted photons and has a signi8cant e;ect (constructively and destructively) on HOADS (for example, weakly coupling for the intermediate virtual state to the ground state of the atom would lead to much more squeezing and its collapse-revival phenomena). In addition, the relation between SOADS and HOADS is examined. It is found that the second-order quantum ?uctuations in the dispersive (absorptive) part of the atomic dipole and the sixth-order ones in the absorptive (dispersive) parts of the atomic dipole may go below their own quantum limit at the same time, and there are additional cases that HOADS can appear but SOADS is absent. Acknowledgements We would like to thank Jianing Colin for some important contribution and constructive conversation and encouragement. One of us (Q.R.) acknowledges the Reinhart Fellowship provided by Queen’s University for 8nancial support. References [1] [2] [3] [4] [5]
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