aa r X i v : . [ qu a n t - ph ] O c t Linear atomic quantum coupler
Faisal A. A. El-Orany ∗ and Wahiddin M. R. B. † Department of Mathematics and computer Science,Faculty of Science, Suez Canal University 41522, Ismailia,Egypt; Cyberspace Security Laboratory, MIMOS Berhad,Technology Park Malaysia, 57000 Kuala Lumpur, Malaysia Cyberspace Security Laboratory, MIMOS Berhad,Technology Park Malaysia, 57000 Kuala Lumpur, Malaysia (Dated: October 30, 2018)In this paper, we develop the notion of the linear atomic quantum coupler. This deviceconsists of two modes propagating into two waveguides, each of them includes a localizedand/or a trapped atom. These waveguides are placed close enough to allow exchangingenergy between them via evanescent waves. Each mode interacts with the atom in the samewaveguide in the standard way, i.e. as the Jaynes-Cummings model (JCM), and with theatom-mode in the second waveguide via evanescent wave. We present the Hamiltonian forthe system and deduce the exact form for the wavefunction. We investigate the atomicinversions and the second-order correlation function. In contrast to the conventional linearcoupler, the atomic quantum coupler is able to generate nonclassical effects. The atomicinversions can exhibit long revival-collapse phenomenon as well as subsidiary revivals basedon the competition among the switching mechanisms in the system. Finally, under certainconditions, the system can yield the results of the two-mode JCM.
PACS numbers: 42.50.Dv,42.50.-p
I. INTRODUCTION
Quantum directional coupler is a device composed of two (or more) waveguides placed closeenough to allow exchanging energy between them via evanescent waves [1]. The rate of flow ofthe exchanged energy can be controlled by the device design and the intensity of the input flux aswell. The outgoing fields from the coupler can be examined in the standard ways to observe thenonclassical effects. Quite recently, this device has attracted much attention in the framework of theoptics communication and quantum computing networks [2], which require data transmission and ∗ Electronic address: el˙[email protected]; [email protected] † Electronic address: [email protected] ultra-high-speed data processing [3]. Furthermore, the directional coupler has been experimentallyimplemented, e.g. in planar structures [4], dual optical fibres [5] and certain organic polymers [6].For more details related to the quantum properties of the fields in the directional couplers thereader can consult the review paper [7] and the references therein.The interaction between the radiation field and the matter (, i.e. atom), namely, Jaynes-Cummings model (JCM) [8], is an important topic in the quantum optics and quantum informationtheories [9]. The simplest form of the JCM is the two-level atom interacting with the single-modeof the radiation field. The JCM is a rich source for the nonclassical effects, e.g. the revival-collapsephenomenon (RCP) [10], sub-Poissonian statistics and squeezing [11]. Furthermore, the JCM hasbeen experimentally implemented by various means, e.g. one-atom mazer [12], the NMR refocusing[13], a Rydberg atom in a superconducting cavity [14], the trapped ion [15] and the micromaser [16].Various extensions to the JCM have been reported including the two two-level atoms interactingwith the radiation field(s) [17, 18].The trapped atoms or molecules are promising systems for quantum information processing andcommunications [19]. They can serve as convenient and robust quantum memories for photons,providing thereby an interface between static and flying qubits [20]. The subject of coupling coldatoms to the radiation field sustained by an optical waveguide has already appeared in variouscontexts. For example, hollow optical glass fibers were used to guide atoms over long distances[21], especially, employing red detuned light field filling out the hollow core [22, 23]. Substratebased atom waveguide can also be realized by using guided two-color evanescent light fields [24].Moreover, the coupling of atomic dipoles to the evanescent field of tapered optical fibers has beendemonstrated in [25, 26]. In this respect the optical nanofibers can manipulate and probe single-atom fluorescence. Moreover, it has been suggested that using a two-color evanescent light fieldaround a subwavelength-diameter fiber traps and guides atoms. The optical fiber carries a red-detuned light and a blue-detuned light, with both modes far from resonance. When both input lightfields are circularly polarized, a set of trapping minima of the total potential in the transverse planeappears as a ring around the fiber. This design allows confinement of atoms to a cylindrical shellaround the fiber [27]. Additionally, it has been shown that sub-wavelength diameter optical fiberscan be used to detect, spectroscopically investigate, and mechanically manipulate extremely smallsamples of cold atoms. In particular, on resonance, as little as two atoms on average, coupled to theevanescent field surrounding the fiber, already absorbed 20 of the total power transmitted throughthe fiber. By optically trapping one or more atoms around such fibers [28], it should become possibleto deterministically couple the atoms to the guided fiber mode and to even mediate a couplingbetween two simultaneously trapped atoms [29]. This leads to a number of applications, e.g., in thecontext of quantum information processing, high precision measurements, single-photon generationin optical fiber or EIT-based parametric four-wave mixing [30] using a few atoms around opticalnanofibers. Inspired by these facts we develop here the notion of the atomic quantum coupler(AQC), for which the interaction mechanisms inside the waveguides and between the waveguidesdepend on both the atomic and bosonic systems. These mechanisms are more complicated thanthose in the JCM, as we shall show shortly. For the AQC we show that the atomic inversionscan exhibit long revival-collapse phenomenon as well as subsidiary-revival patterns based on theswitching mechanisms in the system. Furthermore, under certain conditions, the system can givethe results of the two-mode JCM. Also, the system is able to generate nonclassical effects. It isworth mentioning that the inclusion of one atom in one of the ports of the non-linear coupler hasbeen considered in [31]. Nevertheless, the solution of the equations of motion there is obtained bythe rotation of axes, which does not give complete information on the system.We restrict the study in this paper to the development of the Hamiltonian model, its dynamicalwavefunction and how does it work. These issues are discussed in section II. Additionally, insection III, we study two quantities, namely, the atomic inversions and the second-order correlationfunctions.
II. MODEL FORMALISM AND ITS WAVEFUNCTION
In this section we describe the linear directional atomic quantum coupler (AQC) and deriveits wavefunction. Also we discuss some basic differences between this device and the conventionaldirectional coupler [7]. Thus it is reasonable to shed some light on the linear directional coupler,which is described by the following Hamiltonian [7]:ˆ H ~ = X j =1 ω j ˆ a † j ˆ a j + λ (ˆ a ˆ a † + ˆ a † ˆ a ) , (1)where ˆ a (ˆ a † ) and ˆ a (ˆ a † ) are the annihilation (creation) operators of the first and the secondmodes in the first and the second waveguides with the frequencies ω and ω ; λ is the couplingconstant between the waveguides. Basically this device operates as a quantum switcher since it canswitch the nonclassical effects as well as the intensities of the modes propagating inside one of thewaveguides to the other [32]. In other words, it can not generate nonclassical effects by itself. Forsome reason that will be clear shortly, we calculate the mean-photon numbers for the Hamiltonian FIG. 1: Scheme of realization of the Hamiltonian (3). It is composed from two optical waveguides (yellowcolor). The circles in these waveguides denote the localized and/or trapped atoms. Mode 1 (2) pumped by,e.g., laser sources propagates along the first (second) waveguide and interacts with the first (second) atomvia the coupling constant λ ( λ ). The interaction between the first and the second waveguide occurs viathe evanescent wave with the coupling constant λ . The outgoing fields from the coupler can be measuredin the standard ways, e.g., using photon detectors. (1) when the two modes are in the states | α, i . Thus we arrive at: h ˆ a † ( T )ˆ a ( T ) i = | α | cos ( T ) , h ˆ a † ( T )ˆ a ( T ) i = | α | sin ( T ) , (2)where T = λt . These equations indicate strong switching mechanism in the linear coupler, wherethe intensity | α | in the first waveguide has been completely switched to the other one. Moreover,the mean-photon numbers can not exhibit the RCP.Now we are in a position to develop the AQC, which is the main object of the paper. The atomiccoupler consists of two waveguides, each of which includes a localized and/or a trapped atom. Thewavegudies are placed close enough to each other to allow interchanging energy between them. Thetwo atoms (in the different waveguides) are located very adjacent to each other. In each waveguideone mode propagates along and interacts with the atom inside in a standard way as the JCM. Theatom-mode in each waveguide interacts with the other one via the evanescent wave. The fieldsexited from the coupler can be examined as single or compound modes by means of homodynedetection to observe the squeezing of vacuum fluctuations, or by means of a set of photodetectorsto measure photon antibunching and sub-Poissonian photon statistics in the standard ways. Thescheme for the AQC is depicted in Fig. 1. From this figure and in the framework of the rotatingwave approximation (RWA) the Hamiltonian describing the AQC can be expressed as: ˆ H ~ = ˆ H + ˆ H I , ˆ H = P j =0 ω j ˆ a † j ˆ a j + ω a (ˆ σ (1) z + ˆ σ (2) z ) , ˆ H I = P j =1 λ j (ˆ a j ˆ σ ( j )+ + ˆ a † j ˆ σ ( j ) − ) + λ (ˆ a ˆ a † ˆ σ (1)+ ˆ σ (2) − + ˆ a † ˆ a ˆ σ (1) − ˆ σ (2)+ ) , (3)where ˆ H and ˆ H I are the free and the interaction parts of the Hamiltonian, ˆ σ ( j ) ± and ˆ σ ( j ) z are thePauli spin operators of the j th atom ( j = 1 , a j (ˆ a † j ) is the annihilation (creation) operator ofthe j th-mode with the frequency ω j and ω a is the atomic transition frequency (we consider thatthe frequencies of the two atoms are equal) and λ ( λ ) is the atom-field coupling constant in thefirst (second) waveguide in the framework of the JCM. The derivation of the JCM Hamiltonian iswell known, e.g. [33]. The interaction between the modes in the two waveguides occurs through theevanescent wave with the coupling constant λ . This term is the only one, which is conservative andcan execute switching between the two waveguides. Thus it plays an essential role in the behaviorof the AQC. We should stress that the switching mechanism occurs through the two JCMs (in thetwo waveguides) and can be obtained by applying the RWA in each individual waveguide. In otherwords, the quantity λ (ˆ a ˆ σ (1) − ˆ a † ˆ σ (2)+ + ˆ a † ˆ σ (1)+ ˆ a ˆ σ (2) − ) is nonconservative and hence it is cancelledout. Finally, the treatment of the switching mechanism in (3) is related to the notion of coupler,however, the existence of atoms in the waveguides has been taken into account. In (3) the treatmentis considered only at the moment when the two fields interacting with atoms in the waveguides.Also when we treat the atoms (fields) classically the Hamiltonian (3) tends to that of the lineardirectional coupler (two-atom interaction).The interaction of two two-level atoms with the two modes has been considered in the opticalcavity earlier [18, 34, 35], however, in the sense different from that presented above. For instance,as a sum of two separate Jaynes-Cummings Hamiltonians to investigate the entanglement [34] aswell as the entanglement transfer from a bipartite continuous-variable (CV) system to a pair oflocalized qubits [35]. Also, the quantum properties of the system of two two-level atoms interactingwith the two nondegenerate cavity modes when the atoms and the field are initially in the atomicsuperposition states and the pair-coherent state has been investigated in [18].Next, we evaluate the wave function for the Hamiltonian (3). We assume that the two modesand atoms are initially prepared in the coherent states | α, β i and in the excited atomic states | e , e i , respectively. For resonance case 2 ω a = ω + ω one can easily prove that [ ˆ H , ˆ H I ] = 0.Under these conditions, the dynamical wave function describing the system can be expressed as: | Ψ( t ) i = ∞ P n,m =0 C n,m [ X ( t, n, m ) | e , e , n, m i + X ( t, n, m ) | e , g , n, m + 1 i + X ( t, n, m ) | g , e , n + 1 , m i + X ( t, n, m ) | g , g , n + 1 , m + 1 i ] ,C n,m = exp( − | α | − | β | ) α n β m √ n ! m ! , (4)where | g i stands for atomic ground state. From the Schr¨odinger equation we obtain the followingsystem of differential equations: i ˙ X ( t, n, m ) = λ √ m + 1 X ( t, n, m ) + λ √ n + 1 X ( t, n, m ) ,i ˙ X ( t, n, m ) = λ √ m + 1 X ( t, n, m ) + λ p ( n + 1)( m + 1) X ( t, n, m ) + λ √ n + 1 X ( t, n, m ) ,i ˙ X ( t, n, m ) = λ √ n + 1 X ( t, n, m ) + λ p ( n + 1)( m + 1) X ( t, n, m ) + λ √ m + 1 X ( t, n, m ) ,i ˙ X ( t, n, m ) = λ √ n + 1 X ( t, n, m ) + λ √ m + 1 X ( t, n, m ) , (5)where the superscript ” . ” means differentiation w.r.t. time. In the following, we give only thedetails related to the solution of the coefficient X ( t, n, m ), where the others can be similarlytreated. Differentiating the first and last equations in (5) and re-substitute by the others weobtain:( ˆ D + A n,m ) X ( t, n, m ) = − ( iλ c D + c ) X ( t, n, m ) , ( ˆ D + A n,m ) X ( t, n, m ) = − ( iλ c D + c ) X ( t, n, m ) , ˆ D = ddt , A n,m = λ ( n + 1) + λ ( m + 1) , c = 2 λ λ p ( n + 1)( m + 1) , c = λ p ( n + 1)( m + 1) . (6)From (6) one can easily obtain:( ˆ D + A n,m ) X ( t, n, m ) = ( iλ c D + c ) X ( t, n, m ) . (7)This equation can be easily solved. By means of the initial conditions stated above the exact formsof the coefficients X j can be expressed as: X ( t, n, m ) = exp( i t c ) h cos( t Ω − ) − i c − sin( t Ω − ) i + exp( − i t c ) h cos( t Ω + ) + i c + sin( t Ω + ) i ,X ( t, n, m ) = − i √ m +12 c [ A n,m − λ λ λ ] n exp( i t c ) h ( c − c ) λ ( m + 1) + (2 A n,m − c ) (cid:16) λ c − λ λ ( n + 1) c (cid:17)i × sin( t Ω − )Ω − + exp( − i t c ) h ( c + 2 c ) λ ( m + 1) − (2 A n,m + c ) (cid:16) λ c − λ λ ( n + 1) c (cid:17)i sin( t Ω + )Ω + o ,X ( t, n, m ) = − i √ n +12 c [ A n,m − λ λ λ ] n exp( i t c ) h ( c − c ) λ ( n + 1) + (2 A n,m − c ) (cid:16) λ c − λ λ ( m + 1) c (cid:17)i × sin( t Ω − )Ω − + exp( − i t c ) h ( c + 2 c ) λ ( n + 1) − (2 A n,m + c ) (cid:16) λ c − λ λ ( m + 1) c (cid:17)i sin( t Ω + )Ω + o ,X ( t, n, m ) = exp( i t c ) h − cos( t Ω − ) + i c − sin( t Ω − ) i + exp( − i t c ) h cos( t Ω + ) + i c + sin( t Ω + ) i , (8)where Ω ± = 12 q λ ( n + 1)( m + 1) + 4( λ √ n + 1 ± λ √ m + 1) . (9)It is obvious that the Rabi oscillation in the AQC is more complicated than that of the JCM. Fromthe solution (8) different limits can be checked. For instance, when ( λ , λ ) → (0 ,
0) ( λ → λ , λ ) → (0 ,
0) the system reduces to a simple form, which is in a good correspondencewith the conventional coupler (1). Nevertheless, the device, in this case, is a rich source for thenonclassical effects. This depends on the types of initial atomic states and can be explained asfollows: (i) The atoms are initially prepared in | e , e i . In this case the system reduces to the darkstate, where ˆ H int | e , e i = 0. These states do not evolve in time. This property has been exploitedin the quantum clock synchronization [36]. (ii) The atoms are initially prepared in | e , g i . Thedynamical state of the system takes the form: | Ψ( T ) i = ∞ P n,m =0 C n,m h cos[ T p ( n + 1)( m + 1)] | e , g , n, m + 1 i− i sin[ T p ( n + 1)( m + 1)] | g , e , n + 1 , m i i , (10)where T = λ t . The expression (10) reveals that the behavior of the radiation fields is typicallythat of the two-mode single-atom JCM [37]. Finally, when the two atoms are initially in the Bellstate [ | e , g i + | g , e i ] / √ T -1.0-0.50.00.51.0 < σ Ζ ( ) ( Τ ) > (a) < σ Ζ ( ) ( Τ ) > (b)
400 440 480 520T-0.10-0.050.000.050.100.15 < σ Ζ ( ) ( Τ ) > (b) < σ Ζ ( ) ( Τ ) > (c) < σ Ζ ( ) ( Τ ) > (d) FIG. 2: Evolution of the h ˆ σ (1) z ( t ) i against the interaction time T = λ t with ( α, β ) = (5 ,
5) for ( λ , λ ) = (1 , , .
6) (b), (2 ,
3) (c) and (1 ,
1) (d). | Ψ( T ) i = √ ∞ P n,m =0 C n,m exp[ − iT p ( n + 1)( m + 1)] [ | e , g , n, m + 1 i + | g , e , n + 1 , m i ] . (11)It is evident that the system exhibits atomic trapping, i.e. h ˆ σ (1) z ( T ) i = h ˆ σ (2) z ( T ) i = 0. Furthermore,the system is able to generate nonclassical effects, in particular, in the quantities, which dependon the off-diagonal elements of the density matrix such as squeezing (we have checked this fact).Now, we comment on the switching mechanism in the AQC. For the sake of comparison, wesubstitute β = 0 in relations (4)–(8) and calculate the mean-photon numbers as: h ˆ a † ( T )ˆ a ( T ) i = | α | + ∞ P n =0 | C n, | [ | X ( T, n, | + | X ( T, n, | ] , h ˆ a † ( T )ˆ a ( T ) i = ∞ P n =0 | C n, | [ | X ( T, n, | + | X ( T, n, | ] , (12)whre T = tλ . From these equations it is obvious that the intensity of the mode in the firstwaveguide cannot be switched to the other one. This is in a clear contrast with the linear directionalcoupler (compare (2) and (12)). This behavior is related to the nature of the atom-field interactionmechanism, which is close to the classic Lee model of quantum field theory. Moreover, this behavioris still valid even if the interaction between the modes and the atoms in the same waveguide isneglected, i.e. λ = λ = 0. In this case, expressions (12) exhibit the well-known RCP of thestandard JCM [10]. The final remark, AQC is able to switch the nonclassical effects from onewaveguide to another based on the values of the interaction parameters. This is remarkable from(12), where the mean-photon number in the second waveguide h ˆ a † ( T )ˆ a ( T ) i can exhibit RCP eventhough the second mode is initially in vacuum state. On the other hand, assume that the modein the first waveguide is initially prepared in the even coherent state, which can exhibit squeezing,while the second mode is still in vacuum state. In this case, the density matrix of the second modetakes the form:ˆ ρ = ∞ X n =0 | C n, | n [ | X ( T, n, | + | X ( T, n, | ] | ih | + [ | X ( T, n, | + | X ( T, n, | ] | ih | o , (13)where | C n | is the photon-number distribution of the even coherent sate. From (13), squeezingcannot be switched to the second mode. Nevertheless, if the second mode is prepared in thecoherent state, it can exhibit squeezing. In this case, the source of the nonclassical effects could bethe switching mechanism between the waveguides or the nature of the atom-field interaction.Now, we use above relations to investigate the atomic inversions and second-order correlationfunctions in the following section. For the sake of simplicity we consider α and β to be real. III. ATOMIC INVERSIONS AND SECOND-ORDER CORRELATION FUNCTION
Atomic inversion of the standard JCM is well known in quantum optics by exhibiting RCP. TheRCP has a nonclassical origin and reflects the nature of the statistics of the radiation field. Theevolution of the atomic inversion has been realized via, e.g., the one-atom mazer [12] and usingtechnique similar to that of the NMR refocusing [13]. In this section we investigate the behavior0 T -0.005-0.003-0.0010.0010.003 g ( ) ( T ) (a) T -0.002-0.0010.0000.0010.002 g ( ) ( T ) (b) T -0.003-0.002-0.0010.0000.0010.002 g ( ) ( T ) (c) T -0.004-0.0020.0000.002 g ( ) ( T ) (d) FIG. 3: Evolution of the single-mode second-order correlation function as indicated against the interactiontime T = λ t with ( λ , λ ) = (1 ,
0) (a), (1 , .
6) (b), (2 ,
3) (c)–(d). of the AQC by studying the evolution of the atomic inversions and the second-order correlationfunctions. As the system includes two atoms we have two types of the atomic inversion, namely,single atomic inversion and total atomic inversion h ˆ σ z ( T ) i = [ h ˆ σ (1) z ( T ) i + h ˆ σ (2) z ( T ) i ]. From (4) onecan obtain the following expressions: h ˆ σ (1) z ( T ) i = ∞ P n,m =0 | C n,m | [ | X ( T, n, m ) | + | X ( T, n, m ) | − | X ( T, n, m ) | − | X ( T, n, m ) | ] , h ˆ σ (2) z ( T ) i = ∞ P n,m =0 | C n,m | [ | X ( T, n, m ) | − | X ( T, n, m ) | + | X ( T, n, m ) | − | X ( T, n, m ) | ] , h ˆ σ z ( T ) i = ∞ P n,m =0 | C n,m | [ | X ( T, n, m ) | − | X ( T, n, m ) | ] . (14)1As we mentioned in the preceding section the conventional directional coupler cannot exhibit RCPin the evolution of the mean-photon numbers. Nevertheless, the standard JCM can exhibit RCPprovided that the photon-number distribution of the initial field has a smooth envelope. Similarconclusion has been reported to the two-atom single-mode JCM [17]. For the AQC we have foundwhen α = β and λ j = 0 the different types of the atomic inversions (14) provide quite similarbehaviors. It seems that the contributions of the coherence coefficients X , X are comparable.Moreover, one can easily prove when λ = 0 and λ = λ the atomic inversions reduce to that ofthe standard JCM (see Fig. 2(a)). It is worth reminding that for the standard JCM the revivalpatterns occur in the atomic inversion over certain period of the interaction time afterward theyinterfere providing chaotic behavior. Additionally, the revival time is connected with the amplitude α through the relation T r = 2 π √ ¯ n ≃ π | α | [10]. We proceed, for λ = λ they provide differentforms of the revival patterns. Here we restrict the attention to the atomic inversion of the firstatom (see Figs. 2(b)-(d) for the given values of the interaction parameters). We study three casesbased on the relationship between the strength of the switching mechanisms in and between thewaveguides, namely, λ < λ j , λ = λ j , λ > λ j . Comparisons between Figs. 2(b)-(d) and Fig.2(a) are instructive. From Fig. 2(b) one can observe that the atomic inversion, after the zeroand first revival patterns, exhibits long series of the subsidiary-revival patterns (see the inset inFig. 2(b)). This behavior is completely different from that of the JCM. This indicates that thenonclassical effects generated by this device can sustain for an interaction time longer than that ofthe JCM. It is worth mentioning that the subsidiary-revival patterns have been observed for theJCM against the squeezed coherent state [38]. This has been explained in relation to the photon-number distribution of the initial states. More illustratively, the photon-number distributions ofthe squeezed states exhibit many peaks structure, each of which gives its own revival patterns inthe evolution of the atomic inversion. These patterns interfere with each other to produce thesesubsidiary-revival patterns. Nevertheless, for the system under consideration the occurrence ofthese patterns is related to the switching mechanism between the waveguides (compare Figs. 2(a)and (b)). This mechanism reflects itself in very complicated Rabi oscillations Ω ± as well as in thedouble summations in the atomic inversions formulae (14). Fig. 2(c) presents the case when thecoupling constants are different. It is obvious that the RCP is still remarkable and the subsidiaryrevivals are smoothly washed out compared to those in Fig. 2(b). Generally, we have found when λ ≥ λ = λ the atomic inversion exhibits long RCP (see Fig. 2(d)). Above information indicatesthat the switching mechanism between the waveguides plays an important role in the behaviorof the AQC. Actually, we have found difficulties in giving mathematical treatment for the RCP2presented by the AQC since the Rabi oscillation is rather complicated.Now we draw the attention to the second-order correlation functions for the single-mode case,which is defined as: g (2) j ( t ) = h ˆ a † j ( t )ˆ a j ( t ) ih ˆ a † j ( t )ˆ a j ( t ) i − , j = 1 , , (15)where g (2) j ( t ) = 0 for Poissonian statistics (standard case), g (2) j ( t ) < g (2) j ( t ) > g (2)1 ( t ) <
0. Furthermore, thebasic features of the dynamics are still similar to those of the atomic inversion. Fig. 3(a) presentsthe well-known shape of the second-order function of the standard JCM. When the switchingmechanism between the waveguides is involved the long RCP is dominant in the evolution of the g (2) j ( t ). Nevertheless, the shape of this phenomenon is quite different from that in the correspondingatomic inversion (compare Fig. 2(b) to Fig. 3(b)). For instance, the revival times in the twoquantities are different. Also, the number of the subsidiary revivals in the atomic inversion isgreater than that in the corresponding g (2)1 ( t ). In contrast to the atomic inversions, g (2)1 ( t ) and g (2)2 ( t ) can provide different behavior for the same values of the interaction parameters. This factcan be realized by comparing Fig. 3(c) to (d).In conclusion, in this paper we have developed, for the first time, the notion of the AQC. We haveexplained how does it work. Also we have derived the exact solution for the equations of motion.In contrast to the conventional coupler the AQC can generate nonclassical effects. Nevertheless,the switching mechanism in the former is more effective than that in the latter. Furthermore, thebehavior of the AQC is sensitive to the types of the initial atomic states. We have shown that thesystem can give the results of the two-mode JCM under certain conditions. Additionally, we havediscussed the evolution of the atomic inversions and second-order correlation functions. These twoquantities can exhibit RCP, long RCP and long subsidiary-revival patterns based on the values ofthe coupling constants. Second-order correlation function can exhibit long-lived nonclassical effects.From the information given in the Introduction one can realize that the AQC is in the reach of thecurrent technology. Also it may be of interest in the framework of quantum information.3 Acknowledgement
The authors would like to thank Professor Jan Peˇrina for the interesting discussion.
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