Linear and nonlinear quantum Zeno and anti-Zeno effects in a nonlinear optical coupler
aa r X i v : . [ qu a n t - ph ] N ov Linear and nonlinear quantum Zeno and anti-Zeno effects in a nonlinear optical coupler
Kishore Thapliyal a , Anirban Pathak a, ∗ , and Jan ˇPerina b,c a Jaypee Institute of Information Technology, A-10, Sector-62, Noida, UP-201307, India b RCPTM, Joint Laboratory of Optics of Palacky University and Institute of Physics of Academy of Science of the Czech Republic,Faculty of Science, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic c Department of Optics, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
Quantum Zeno and anti-Zeno effects are studied in a symmetric nonlinear optical coupler, which is composedof two nonlinear ( χ (2) ) waveguides that are interacting with each other via the evanescent waves. Both thewaveguides operate under second harmonic generation. However, to study quantum Zeno and anti-Zeno effectsone of them is considered as the system and the other one is considered as the probe. Considering all thefields involved as weak, a completely quantum mechanical description is provided, and the analytic solutionsof Heisenberg’s equations of motion for all the field modes are obtained using a perturbative technique. Photonnumber statistics of the second harmonic mode of the system is shown to depend on the presence of the probe,and this dependence is considered as quantum Zeno and anti-Zeno effects. Further, it is established that as aspecial case of the momentum operator for χ (2) − χ (2) symmetric coupler we can obtain momentum operatorof χ (2) − χ (1) asymmetric coupler with linear ( χ (1) ) waveguide as the probe, and in such a particular case, theexpressions obtained for Zeno and anti-Zeno effects with nonlinear probe (which we referred to as nonlinearquantum Zeno and anti-Zeno effects) may be reduced to the corresponding expressions with linear probe (whichwe referred to as the linear quantum Zeno and anti-Zeno effects). Linear and nonlinear quantum Zeno and anti-Zeno effects are rigorously investigated, and it is established that in the stimulated case, we may switch betweenquantum Zeno and anti-Zeno effects just by controlling the phase of the second harmonic mode of the systemor probe. PACS numbers:Keywords:
I. INTRODUCTION
In the 5th century BC, the Greek philosopher Zeno of Eleaintroduced a set of paradoxes of motion. These paradoxes,which are now known as Zeno’s paradoxes, were unsolved forlong, and they fascinated mathematicians, logicians, physi-cists and other creative minds since their introduction. In therecent past, a quantum analogue of Zeno’s paradoxes has beenstudied intensively. Specifically, in the late 50s and early 60sof the 20th century, Khalfin studied nonexponential decay ofunstable atoms [1]. Later on, in 1977, Misra and Sudarshan[2] showed that under continuous measurement, an unstableparticle will never be found to decay, and in analogy with clas-sical Zeno’s paradox they named this phenomenon as
Zeno’squantum paradox . Quantum Zeno effect (QZE) in the originalformulation refers to the inhibition of the temporal evolutionof a system on continuous measurement [2], while quantumanti-Zeno effect (QAZE) or inverse Zeno effect refers the en-hancement of the evolution instead of the inhibition (see Refs.[3–5] for the reviews). Here, it is important to note that usu-ally quantum Zeno effect is viewed as a process, which is as-sociated with the repeated projective measurement. This isonly a specific manifestation of quantum Zeno effect. In fact,it can be manifested in a few equivalent ways [6]. One suchmanifestation of quantum Zeno effect is a process in whichcontinuous interaction between the system and probe leads ∗ Email: [email protected], Phone: +91 9717066494 to quantum Zeno effect. Here, we aim to study the contin-uous interaction type manifestation of quantum Zeno effect ina symmetric nonlinear optical coupler, which is made of twononlinear waveguides with χ (2) nonlinearity, and each of thewaveguides is operating under second harmonic generation.As we are describing an optical coupler, the waveguides arecoupled with each other. More precisely, these two waveg-uides interact with each other through the evanescent waves.We consider one of the waveguides as the probe and the otherone as the system. In what follows, we will show that thebeauty of the symmetric nonlinear optical coupler ( χ (2) − χ (2) coupler) studied here is that the results obtained for this cou-pler can be directly reduced to the corresponding results foran asymmetric nonlinear optical coupler where the probe islinear ( χ (1) ) and the system is nonlinear ( χ (2) ) . We have re-ported quantum Zeno and anti-Zeno effects in both the sym-metric ( χ (2) − χ (2) ) nonlinear optical coupler and asymmetric( χ (2) − χ (1) ) nonlinear optical coupler. Following an earlierwork [7], we refer to the Zeno effect observed due to the con-tinuous interaction of a nonlinear (cid:0) χ (2) (cid:1) probe as the nonlin-ear Zeno effect. Similarly, the Zeno effect observed due to thecontinuous interaction of a linear (cid:0) χ (1) (cid:1) probe is referred toas the linear Zeno effect.Optical couplers can be prepared easily and several excitingapplications of the optical couplers have been reported in therecent past (see [8–10] and references therein). Consequently,it is no wonder that quantum Zeno and anti-Zeno effects havebeen investigated in various types of optical couplers [11–15].Specifically, quantum Zeno and anti-Zeno effects were shownin Raman and Brillouin scattering using a ( χ (3) − χ (1) ) asym-metric nonlinear optical coupler [12]; their existence was alsoshown in the χ (2) − χ (1) optical couplers [13], χ (2) − χ (2) optical couplers [14], etc. In all these studies, it was alwaysconsidered that one of the mode in the nonlinear waveguide(this waveguide is considered as the system) is coupled withthe auxiliary mode in a (non)linear waveguide (this waveg-uide is considered as the probe). Actually, the auxiliary modeacts as the probe since its coupling with the system imple-ments continuous observation on the evolution of the system(nonlinear waveguide) and changes the photon statistics of theother modes (which are not coupled to the probe mode) of thenonlinear waveguide. Quantum Zeno and anti-Zeno effectshave also been investigated in optical systems other than cou-plers, such as in parametric down-conversion [16–18], para-metric down conversion with losses [19], an arrangement ofbeam splitters [20], etc. In these studies on quantum Zeno ef-fect in optical systems, often the pump mode has been consid-ered strong, and thus the complexity of a completely quantummechanical treatment has been circumvented. Keeping this inmind, here we plan to use a completely quantum mechanicaldescription of the coupler.Initially, interest in quantum Zeno effect was theoreticaland purely academic in nature, but with time quantum Zenoeffect has been experimentally realized by several groups us-ing different techniques [21–23]. Not only that, several in-teresting applications of quantum Zeno effect have also beenproposed [21, 24–27]. Specifically, in Refs. [26, 27], it wasestablished that the quantum Zeno effect may be used to in-crease the resolution of absorption tomography. A few of theproposed applications have also been experimentally realized.For example, Kwait et al. implemented high-efficiency quan-tum interrogation measurement using quantum Zeno effect[21]. Until recently, all the investigations related to the quan-tum Zeno effect were restricted to the microscopic world. Re-cently, in a very interesting work, it has been extended to themacroscopic world by showing the evidence for the existenceof quantum Zeno effect for large black holes [28]. Possibilityof observing the macroscopic Zeno effect was also studied inthe context of stationary flows in nonlinear waveguides withlocalized dissipation [29]. The interest in quantum Zeno ef-fect has recently been amplified with the advent of variousprotocols of quantum communication that are based on quan-tum Zeno effect. Specifically, in Ref. [25], a counterfactualprotocol of direct quantum communication was proposed us-ing chained quantum Zeno effect, and in Ref. [24], the sameeffect is used to propose a scheme for counterfactual quantumcomputation. In the past, a proposal for quantum computingwas made using an environment induced quantum Zeno ef-fect to confine the dynamics in a decoherence-free subspace[30]. Recently, quantum Zeno effect has also been used toreduce communication complexity [31]. These applicationsof quantum Zeno effect and easy production of optical cou-plers motivated us to systematically investigate the possibilityof observing quantum Zeno and anti-Zeno effects in a sym-metric nonlinear coupler which is not studied earlier using acompletely quantum description.To investigate the existence of quantum Zeno and anti-Zenoeffects in the optical coupler of our interest, we have ob-tained closed form analytic expressions for the spatial evolu- tion of the different field operators using the Sen-Mandal per-turbative approach [8, 9, 32], which is known to produce bet-ter results compared to the usual short-length approximationmethod [33]. Actually, in sharp contrast to the short lengthapproximated solutions, the solutions of Heisenberg’s equa-tions of motion obtained using the Sen-Mandal approach arenot restricted by length. This is why we use Sen-Mandal per-turbative approach and a completely quantum mechanical de-scription of the coupler for our investigation. In the past, pho-ton statistics and dynamics of the symmetric coupler of ourinterest was studied by some of the present authors with anassumption that both the second harmonic modes are strong[34]. The assumption circumvented the use of completelyquantum mechanical description. Further, the system has alsobeen used to model a beam splitter with second order nonlin-earity [35]. Present investigation, not only revealed the ex-istence of nonlinear quantum Zeno and anti-Zeno effects italso established the existence of linear Zeno and anti-Zenoeffects. The study also showed that switching between quan-tum Zeno and anti-Zeno effects is possible by varying phase-mismatches.The rest of the paper is organized as follows. In SectionII, we briefly describe the momentum operator for the sym-metric nonlinear optical coupler and the method used here toobtain the analytic expressions of the spatial evolution of thefield operators of various modes. However, the detailed so-lution obtained here is shown in the Appendix A. In SectionIII, the existence of quantum Zeno and anti-Zeno effects aresystematically investigated. Finally the paper is concluded inSection IV. II. SYSTEM AND SOLUTION
Momentum operator of a symmetric nonlinear optical cou-pler, prepared by combining two nonlinear (quadratic) waveg-uides operated by second harmonic generation (as shown inFig. 1 a), in interaction picture is given by [36] G sym = ~ ka b † + ~ Γ a a a † exp( i ∆ k a z )+ ~ Γ b b b † exp( i ∆ k b z ) + H . c . , (1)where the annihilation (creation) operators a i ( a † i ) and b i ( b † i ) correspond to the field operators in two nonlinear waveg-uides. Here, a ( k a ) and a ( k a ) denote annihilation oper-ators (wave vectors) for fundamental and second harmonicmodes, respectively, in one waveguide. Similarly, b ( k b ) and b ( k b ) represent annihilation operators (wave vectors)for fundamental and second harmonic modes, respectively, insecond waveguide. Further, H . c . stands for the Hermitian con-jugate; ∆ k j = | k j − k j | refers to the phase mismatch be-tween the fundamental and second harmonic beams; the pa-rameters k and Γ j denote the linear and nonlinear couplingconstants, respectively, where j ∈ { a, b } . The momentum op-erator described above is completely quantum mechanical inthe sense that all the modes involved in the process are con-sidered weak and treated quantum mechanically. Thus, weconsider pump as weak and note that Γ j ≪ k as k and Γ j areproportional to the linear ( χ (1) ) and nonlinear ( χ (2) ) suscep-tibilities, respectively and usually χ (2) /χ (1) ≃ − .For the study of quantum Zeno and anti-Zeno effects in asystem we need a system momentum operator with a contin-uous interaction with a probe. In this particular system, thesymmetric nonlinear optical coupler, we can consider that thesystem, which is described by G sys = ~ Γ b b b † exp( i ∆ k b z ) +H . c . , is in continuous interaction with the probe, which is de-scribed by G probe = ~ ka b † + ~ Γ a a a † exp( i ∆ k a z ) + H . c .. Here, the probe itself is considered to be nonlinear. Further,if we take Γ a = 0 in Eq. (1), i.e., if we consider probe to belinear, we obtain [8, 9, 32] G asym = ~ kab † + ~ Γ b b † exp( i ∆ kz ) + H . c . , (2)which is the momentum operator of an asymmetric nonlinearoptical coupler in the interaction picture.The spatial evolution of various modes involved in the mo-mentum operators (1-2) can be obtained as the simultaneoussolutions of the Heisenberg’s equations of motion correspond-ing to each mode. However, for the complex systems, such asconsidered here, the closed form analytic solutions are pos-sible only by using some perturbative methods. Here, weuse Sen-Mandal perturbative method [8, 9, 13, 32], whichhas already been shown to be superior to the frequently usedshort-length/time method [12, 33]. In fact, the solutions ob-tained using the short-length perturbative technique can be ob-tained as a limiting case of a solution obtained using Sen-Mandal approach. Specifically, it may be obtained by ne-glecting higher power terms in length, from the solutions ofthe Sen-Mandal method. Actually, in Sen-Mandal approach,potential solutions of the Heisenberg equations of motion for different field modes are systematically constructed. The as-sumed solution for the evolution of annihilation operator ofa field mode is constructed in such a way that it contains allthe possible higher power terms in length, but higher powerterms in weak coupling constant are neglected (for detail see[8, 9, 13, 32]). As the solutions obtained using Sen-Mandalmethod is applicable to relatively large interaction lengthsand it contains several higher power terms that are neglectedin conventional short-length approach, it provides more ac-curate solution compared to the short-length solutions. Fur-ther, the solution obtained using Sen-Mandal method is of-ten found successful in detecting nonclassical character of aphysical system that are not detected by conventional short-length/time solution [12, 33]. Keeping these facts in mind,here we use Sen-Mandal method to obtain spatial evolutionof all the field operators involved in (1). However, the closedform analytic expresseion for the spatial evolution in b mode,i.e., b ( z ) is provided in Appendix A. In the Appendix A, werestrict our description to b ( z ) as only the coefficeints appearin the analytic expression of b ( z ) appear in the expressionsof linear and nonlinear Zeno parameters. Specifically, Heisen-berg’s equations of motion for different field modes involvedin the momentum operator (1) are obtained in Eq. (A.1) in Ap-pendix A. The closed form analytic expressions for the spatialevolution of all the field modes up to quadratic terms in non-linear coupling constants ( Γ j ) are subsequently obtained usingSen-Mandal perturbation method (cf. Suuplementary materialfor all modes and Appendix A for b modes). In what follows,we use the expression of b ( z ) provided in Appendix A to in-vestigate the linear and nonlinear Zeno and anti-Zeno effectsin the optical couplers. a (z), a (z)a (0), a (0)b (0), b (0) b (z), b (z) a(z)a(0)b (0), b (0) b (z), b (z) L L (cid:1) (2) (cid:1) (1) (cid:1) (2) (cid:1) (2) (a) (b)
Figure 1: (Color online) Schematic diagrams of (a) a symmetric and (b) an asymmetric nonlinear optical coupler of interaction length L ina codirectional propagation of different field modes involved. The symmetric coupler is prepared by combining two nonlinear (quadratic)waveguides operating under second harmonic generation, and in the asymmetric coupler one nonlinear waveguide of the symmetric coupler isreplaced by a linear waveguide. III. LINEAR AND NONLINEAR QUANTUM ZENO ANDANTI-ZENO EFFECTS
Being consistent with the theme of the present work, thepresence of quantum Zeno and anti-Zeno effects with a non- linear probe corresponds to the nonlinear quantum Zeno andanti-Zeno effects. Similarly, a linear probe will give the linearquantum Zeno and anti-Zeno effects. Further, it has alreadybeen mentioned in Section (1), that analytical expressions forZeno parameter for a linear probe can be obtained as the lim-iting cases of the expressions obtained for nonlinear probe byneglecting the nonlinearity present in the probe [37]. Quitesimilar analogue of nonlinear and linear quantum Zeno andanti-Zeno effects were also discussed in the recent past [7, 38]in other physical systems.
A. Number operator and Zeno parameter
The analytic expression of the number operator for the sec-ond harmonic field mode in the system waveguide, i.e., b mode using Eq. (A.2) in Appendix A is given by N b ( z ) = b † ( z ) b ( z )= b † (0) b (0) + | l | b † (0) b (0) + | l | a † (0) a (0) b † (0) b (0) + | l | a † (0) a (0) + h l b † (0) b (0)+ l b † (0) b (0) a (0) + l b † (0) a (0) + l ∗ l a (0) b † (0) b (0) + l ∗ l a (0) b † (0) + l ∗ l a † (0) a (0) b † (0)+ l b † (0) b (0) b † (0) b (0) + l b † (0) b (0) + l a (0) b † (0) b † (0) b (0) + l a † (0) b (0) b † (0) b (0)+ l a † (0) a (0) b † (0) b (0) + l a † (0) a (0) b (0) b † (0) + l a † (0) a (0) a (0) b † (0) + l a (0) b † (0)+ l a (0) b † (0) b (0) b † (0) + l a (0) a (0) b † (0) b † (0) + H . c . i , (3)where the functional form of coefficients l i is given in Eq.(A.3) in Appendix A.Without any loss of generality, we considered the initialstate being a multimode coherent state | α i| β i| γ i| δ i , which isthe product of four single mode coherent states | α i , | β i , | γ i ,and | δ i . Here, | α i , | β i , | γ i , and | δ i are the eigenkets of the an-nihilation operators for the corresponding field modes, i.e., a , b , a , and b , respectively. For example, b (0) | α i| β i| γ i| δ i = β | α i| β i| γ i| δ i and a (0) | α i| β i| γ i| δ i = α | α i| β i| γ i| δ i , where | α | , | β | , | γ | , and | δ | are the initial number of photons inthe field modes a , b , a , and b , respectively. The symmet-ric nonlinear optical coupler and its approximated special caseof the asymmetric nonlinear optical coupler can operate undertwo conditions: spontaneous and stimulated. In the sponta-neous (stimulated) case, initially, i.e., at t = 0 , there is nophoton (non-zero number of photons) in the second-harmonicmode of the system, whereas average photon numbers in theother modes are non-zero.Following earlier works of some of the present authors [11–15], the effect of the presence of the probe mode on the photonstatistics of the second harmonic mode of the system is inves-tigated using Zeno parameter ( ∆ N Z ), which is defined as ∆ N Z = h N X ( z ) i − h N X ( z ) i k =0 . (4)The Zeno parameter is a measure of the effect caused onthe evolution of the photon statistics of the system (obtainedfor mode X ) due to its interaction with the probe. It can beinferred from Eq. (4) that the negative values of the Zenoparameter signify the continuous measurement via the probeinhibited the evolution of mode X by decreasing the photongeneration in that particular mode, which demonstrate pres-ence of the quantum Zeno effect. On the other hand, the posi-tive values of the Zeno parameter correspond to enhancementof the photon generation due to coupling with auxiliary modein the probe. This is the signature of the presence of quantumanti-Zeno effect. (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3) Figure 2: (Color online) (a) The variation in linear Zeno parameterwith rescaled interaction length for Γ b k = 10 − , ∆ k b k = 10 − with α = 5 , β = 3 , and δ = 1 and -1 in stimulated case (smooth anddashed lines) and δ = 0 in spontaneous case (dot-dashed line). (b) Inspontaneous case, α = 10 with β = 3 , , and 7 for smooth, dashedand dot-dashed lines, respectively. (c) α = 10 , β = 8 , and δ = − , , and 4 for smooth, dashed and dot-dashed lines, respectively. For the system of our interest, the symmetric nonlinear op-tical coupler, using the analytic expression of the photon num-ber operator in Eq. (3), the Zeno parameter can be calculatedfor second harmonic mode of the system waveguide as ∆ N NZ = (cid:16) | l | − | p | (cid:17) | β | + | l | | α | | β | + | l | | α | + (cid:2) ( l − p ) β δ ∗ + l αβδ ∗ + l α δ ∗ + l ∗ l | β | αβ ∗ + l ∗ l α β ∗ + l ∗ l | α | αβ ∗ + ( l − p ) | β | | δ | + ( l − p ) | δ | + l | δ | αβ ∗ + l | δ | α ∗ β + l | α | | δ | + l α ∗ βγδ ∗ + l | α | γδ ∗ + l γδ ∗ + l | β | γδ ∗ + l αβ ∗ γδ ∗ + c . c . ] , (5)where p = − Γ b G ∗− ∆ k b ,p = 2 p = − | Γ b | ( G ∗− + i ∆ k b z ) (∆ k b ) . (6)Here, p i s are obtained by taking k = 0 in corresponding l i sin Eq. (A.3) in Appendix A. All the remaining p ′ i s vanishes inthe absence of the probe. The subscript N Z in the Zeno pa-rameter corresponds to the physical situation where a nonlin-ear probe is used, i.e., “nonlinear Zeno” effect is investigated.Thus ∆ N NZ can be referred to as the nonlinear Zeno param-eter. Similarly ∆ N LZ will denote linear Zeno parameter, i.e.,Zeno parameter for a physical situation where linear probe isused.It is easy to obtain Zeno parameter for the spontaneous case.Specifically, in the spontaneous case, i.e., in absence of anyphoton in the second harmonic mode of the system at t = 0 (or considering δ = 0 at t = 0) , the analytic expression ofthe nonlinear Zeno parameter can be obtained from Eq. (5) bykeeping the δ independent terms as (∆ N NZ ) δ =0 = (cid:16) | l | − | p | (cid:17) | β | + | l | | α | | β | + | l | | α | + h l ∗ l | β | αβ ∗ + l ∗ l α β ∗ + l ∗ l | α | αβ ∗ + c . c . i . (7)In Section II, we have already mentioned that the mo-mentum operator for an asymmetric nonlinear optical coupler( χ (2) − χ (1) ) can be obtained by just neglecting the nonlinearcoupling term in one of the nonlinear waveguides present inthe symmetric nonlinear coupler ( χ (2) − χ (2) ) studied here.Thus, we may consider the probe in the nonlinear Zeno pa-rameter obtained in Eq. (5) to be linear by taking Γ a = 0 .This is how we can obtain the expression for linear Zeno pa-rameter. Thus, the Zeno parameter of the asymmetric nonlin-ear optical coupler characterized by Eq. (2) can be obtainedas ∆ N LZ = (cid:16) | l | − | p | (cid:17) | β | + | l | | α | | β | + | l | | α | + (cid:2) ( l − p ) β δ ∗ + l αβδ ∗ + l α δ ∗ + l ∗ l | β | αβ ∗ + l ∗ l α β ∗ + l ∗ l | α | αβ ∗ + ( l − p ) | β | | δ | + ( l − p ) | δ | + l | δ | αβ ∗ + l | δ | α ∗ β + l | α | | δ | + c . c . i . (8) Further, if we neglect all the terms beyond linear power innonlinear coupling constant Γ b in Eq. (8), we find the resultobtained here matches exactly with the result reported in Ref.[13]. Interestingly, the analytic expressions of the nonlinearand linear Zeno parameters have the same expressions in thespontaneous case. It can also be checked that the expressionobtained in Eq. (7) vanishes, if we neglect all the terms be-yond linear powers in the nonlinear coupling constant. This isalso consistent with the earlier result [13]. (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3) Figure 3: (Color online) (a) Nonlinear Zeno parameter with Γ a k = Γ b k = 10 − , ∆ k a k = 1 . × − , ∆ k b k = 1 . × − with α =5 , β = 3 , and γ = 2 , δ = 1 and -1 in stimulated case (smooth anddashed lines) and γ = − , δ = 1 and -1 in dot-dashed and dottedlines. (b) α = 10 , γ = 3 , δ = 1 with β = 3 , , and 7 for smooth,dashed and dot-dashed lines, respectively. (c) shows the change innonlinear Zeno parameter with changing the phase of α or β by anamount of π for α = β = 6 , and γ = 3 , δ = 2 . It is also observedthat the change in phase of α is equivalent to change in phase of β . B. Variation of Zeno parameter with different variables
The analytic expressions obtained for both nonlinear andlinear Zeno parameters depend on various parameters, suchas photon numbers and phases of different field modes, lin-ear and nonlinear coupling, interaction length and phase mis-match between fundamental and second harmonic modes inthe nonlinear waveguides. However, in the spontaneous case,system shows quantum anti-Zeno effect initially which even-tually goes towards quantum Zeno effect with increase inrescaled interaction length. This behavior of the linear Zenoparameter in the spontaneous case is further elaborated in Fig.2 b, where it can be observed that as the number of photons inthe linear mode of the system waveguide becomes compara-ble to the photon numbers in the probe mode, quantum Zenoeffect is prominent. A similar effect on photon numbers isobserved even in the stimulated case in Fig. 2 c, where thetransition to quantum Zeno effect with increasing rescaled in-teraction length is more dominating than in Fig. 2 a. Further,in the stimulated case, a transition between linear quantumZeno and anti-Zeno effects can be obtained by controlling thephase of second harmonic mode of the system waveguide asillustrated in Fig. 2 a. However, with an increase in num- ber of photons in fundamental mode of the system waveguideshown in Fig. 2 c, this nature disappears gradually due to itsdominant effect to tend towards quantum Zeno effect.To illustrate the variation of the nonlinear Zeno parameterwith the rescaled interaction length of the coupler in Fig. 3,we have considered specific values of all the remaining pa-rameters. As in the case of linear Zeno parameter (cf. Fig.2) nonlinear Zeno parameter also shows dependence on thephases of both second harmonic modes involved in the sym-metric coupler. Specifically, Fig. 3 a illustrates that the changein phase of the second harmonic mode of the system createssome changes in the photon statistics which causes a transitionbetween quantum Zeno and anti-Zeno effects. This becomemore dominant with the change of phase of second harmonicmode of the probe as well. Similarly, Fig. 3 b establishes ananalogous fact for nonlinear Zeno parameter as in Fig. 2 bfor linear Zeno parameter, i.e., when the photon numbers inthe linear modes of both the waveguides are comparable thenquantum Zeno effect prevails. Fig. 3 c shows that by changingthe phase of α by π (i.e., transforming α to − α ) has a similareffect as changing the phase of β by the same amount. Inter-estingly, this kind of nature can be attributed to the symmetrypresent in the symmetric coupler studied here. (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3)(cid:1)(cid:6)(cid:3) Figure 4: (Color online) (a) Linear Zeno parameter in the spontaneous case for ∆ k b k = 10 − with α = 5 , β = 3 for Γ b k = 10 − (cid:0) × − (cid:1) in smooth blue (dashed red) line. (b) A similar observation in the stimulated case with δ = 1 and all the remaining values same as (a). In (c)and (d), the effect of phase mismatch in the spontaneous and stimulated (with δ = 1 ) cases of linear Zeno parameter is shown, respectively.The remaining parameters are Γ b k = 10 − with ∆ k b k = 10 − (cid:0) − (cid:1) in the smooth blue (dashed red) line. The explicit dependence of the linear and nonlinear Zenoparameters on the remaining parameters, such as nonlinearcoupling constants and phase mismatches, of both system andprobe waveguides is illustrated in Figs. 4 and 5, respectively.Specifically, Figs. 4 a-b show the variation in the linear Zenoparameter for two values of nonlinear coupling constant of thesystem in spontaneous and stimulated cases, respectively. Asimilar study is shown in Figs. 5 a-b for the nonlinear Zenoparameter with two values of nonlinear coupling constants ofthe system and probe waveguides, respectively. All the casesdemonstrate that with increase in the nonlinear coupling of the system a dominant oscillatory nature is observed. While in-crease in the nonlinear coupling of the probe waveguide showspreference for quantum anti-Zeno effect.The phase mismatch between fundamental and second har-monic modes of the system (probe) waveguide has negligibleeffect on the linear (nonlinear) Zeno parameter in the sponta-neous (stimulated) case as depicted in Fig. 4 c (5 c). Whilea similar observation for linear (nonlinear) Zeno parametershown in Fig. 4 d (5 d) for the stimulated case with phasemismatch in the system waveguide exhibits a transition fromquantum Zeno effect to quantum anti-Zeno effect. (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3)(cid:1)(cid:6)(cid:3)
Figure 5: (Color online) The dependence of nonlinear Zeno parameter on the nonlinear coupling constant is depicted in (a) and (b). In (a) thenonlinear Zeno parameter is shown with rescaled interaction length for Γ a k = 10 − , ∆ k a k = 1 . × − , ∆ k b k = 10 − with α = 5 , β =3 , γ = 2 , and δ = 1 . The smooth (blue) and dashed (red) lines correspond to Γ b k = 10 − and Γ b k = 5 × − , respectively. Similarly, in (b)the smooth (blue) and dashed (red) lines correspond to Γ a k = 10 − and Γ a k = 5 × − , respectively with Γ b k = 10 − and all the remainingvalues as in (a). In (c) the nonlinear Zeno parameter is shown in the smooth blue (dashed red) line with Γ a k = Γ b k = 10 − , ∆ k b k = 10 − for ∆ k a k = 1 . × − (cid:0) . × − (cid:1) . Similarly, in (d) the nonlinear Zeno parameter is shown in the smooth blue (dashed red) line with ∆ k a k = 1 . × − for ∆ k b k = 10 − (cid:0) − (cid:1) . The dependence of both linear and nonlinear Zeno parameterson the linear coupling can be observed with interaction lengthin Fig. 6 a and b, respectively. With the particular choice ofvalues for other parameters, both Zeno parameters show quan- tum Zeno effect. However, as observed in Fig. 2 quantumanti-Zeno effect can be illustrated here by just controlling thephase of the second harmonic mode in the system. Though anincrease in the effect of the presence of the probe in the photonstatistics of the second harmonic mode with increasing inter-action length and linear coupling can be observed from thefigure. This dominance of the effect of the probe is oscillatoryin nature and gives a ripple like structure in Fig. 6. (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:3)
Figure 6: (Color online) The variation in (a) linear and (b) nonlinearZeno parameter with linear coupling constant and interaction lengthare shown for Γ a k = Γ b k = 10 − , ∆ k a k = 1 . × − , ∆ k b k = 10 − with α = 6 , β = 4 , γ = 2 , and δ = 1 . The effect of change in phase mismatch between funda-mental and second harmonic modes explored in Figs. 4 c-dand 5 c-d is further illustrated in Fig. 7. The phase mis-match between the fundamental and second harmonic modesin the system waveguide has evident effect on the linear Zenoparameter only in the small mismatch region for the sponta-neous case (cf. Fig. 7 a). Whereas an increase in the ini-tial number of photons in the second harmonic mode of thesystem, i.e., in the stimulated case, changes the photon statis-tics drastically and both quantum Zeno and anti-Zeno effects,with continuous switching between them, have been observedin Fig. 7 b. The corresponding plot for nonlinear Zeno pa-rameter shows a quite similar behavior in Fig. 7 c with slightchanges in the photon statistics due to the presence of the sec-ond harmonic mode in the probe. A similar study for the effectof phase mismatch between the fundamental and second har-monic modes of the probe on nonlinear Zeno parameter showsample amount of variation only for small mismatch and be-comes almost constant for larger values of phase mismatch. These features of quantum Zeno and anti-Zeno effects canbe further illustrated using contour plots as shown in Fig. 8,where the values of different parameters are the same as thoseused in Fig. 7. The contour plots can be drawn to clearly showthe regions of quantum Zeno and anti-Zeno effects (withoutreferring to the magnitude of the Zeno parameter) as shownin Fig. 8 a and b, where the blue regions correspond to thequantum Zeno effect while the yellow regions correspond tothe quantum anti-Zeno effect in the linear Zeno case. Thecontour plots can also be drawn to illustrate the depth of Zenoparameter for both the effects as illustrated in Fig. 8 c and dfor nonlinear Zeno parameter.Fig. 9 demonstrates the nature of linear Zeno parameterwith changes in the number of photons in the linear modes ofboth the waveguides. Here, it can be seen that with increasein photon numbers in probe mode quantum anti-Zeno effectis preferred while with increasing the intensity in the linearmode of system waveguide it tends towards quantum Zenoeffect.
IV. CONCLUSIONS
Linear and nonlinear quantum Zeno and anti-Zeno effectsin a symmetric and an asymmetric nonlinear optical couplersare rigorously investigated in the present work. The investiga-tion is performed using linear and nonlinear Zeno parameters,which are introduced in this paper in analogy with that of theZeno parameter introduced in Ref. [13]. Closed form analyticexpressions for both linear and nonlinear Zeno parameters areobtained here using Sen-Mandal perturbative method. Subse-quently, variation of the Zeno and anti-Zeno parameters withrespect to various quantities are investigated and the same isillustrated in Figs. 2-9. The investigation led to several in-teresting observations. For example, we have observed thatthe analytic expressions obtained for both linear and nonlinearZeno parameters are the same for the spontaneous case. Fur-ther, in the spontaneous case, it is observed that the transitionfrom the quantum anti-Zeno effect to quantum Zeno effect canbe achieved by increasing the intensity of the radiation fieldin the linear mode of the system waveguide (cf. Fig. 2 b).Similarly, a switching between the linear (nonlinear) quantumZeno and anti-Zeno effects is also observed in the stimulatedcase. However, it is observed that this switching can be ob-tained just by controlling the phase of the second harmonicmode in the system waveguide in the linear case (cf. Figs. 2a and c) and by controlling the phase of the nonlinear modesof both the waveguides (cf. Figs. 3 a and c). Here we maynote that the change in phase of the linear mode of the probeis equivalent to change in phase of the linear mode of the non-linear system waveguide. This kind of nature can be attributedto the symmetry present in the system, which is evident evenin the system momentum operator (cf. Fig. 3 c). In fact, ingeneral, we have observed that increase in the intensity of theprobe leads to increase the quantum anti-Zeno effect, whilewith the increase in the intensity of the linear mode of the sys-tem waveguide the quantum Zeno effect is more prominent(cf. Figs. 2 b and c and Fig. 9). (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3)(cid:1)(cid:6)(cid:3)
Figure 7: (Color online) The variation in linear Zeno parameter with phase mismatch between fundamental and second harmonic modes insystem waveguide and rescaled interaction length in (a) spontaneous and (b) stimulated cases are shown for Γ b k = 10 − with α = 6 , β = 4 , and δ = 0 and 1 in (a) and (b), respectively. (c) shows the dependence of the nonlinear Zeno parameter on the phase mismatch betweenfundamental and second harmonic modes in system waveguide and rescaled interaction length for Γ a k = 10 − , ∆ k a k = 1 . × − and γ = 2 , δ = 1 with all remaining values same as (a) and (b). In (d), the effect of phase mismatch between fundamental and second harmonicmodes in probe waveguide and rescaled interaction length on the nonlinear Zeno parameter are shown for ∆ k b k = 10 − with all the remainingvalues as (c). (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3)(cid:1)(cid:5)(cid:3)(cid:1)(cid:6)(cid:3) Figure 8: (Color online) Variation in linear and nonlinear Zeno parameters shown in three dimensional plots in Fig. 7 (a)-(d) are illustrated viaequivalent contour plots. Here, all the four contour plots corresponding to Fig. 7 (a)-(d) are are obtained using the same parameters. In (a) and(b), the yellow regions illustrate the region for quantum anti-Zeno effect while the blue regions correspond to quantum Zeno effect. In (c) and(d), along with the regions of Zeno and anti-Zeno effects, variation of magnitude of Zeno parameters are also shown with different colors (seethe color bars in right side of the figures). (cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:3) Figure 9: (Color online) The variation in linear Zeno parameterin (a) spontaneous and (b) stimulated cases with photon numbersof linear modes in both the waveguides ( α and β ) are shown for Γ b k = 10 − , ∆ k b k = 10 − with γ = α and δ = β after rescaledinteraction length kz = 1 . A similar behavior to the linear Zeno pa-rameter is observed for nonlinear Zeno parameter in the stimulatedcase. For the smaller values of the linear coupling constant, aconsiderable amount of variation in the photon number statis-tics is observed through the linear Zeno parameter. This vari-ation is observed to fade away as ripples with increasing inter-action length for higher values of linear coupling constant (cf.Fig. 6 a). Similar but more prominent nature is observed inthe nonlinear case (cf. Fig. 6 b). Similarly, we have observedthat with the increase in phase mismatch between fundamen-tal and second harmonic modes in the system waveguide, a transition from quantum Zeno effect to quantum anti-Zeno ef-fect occurs. The change in photon number statistics of thenonlinear waveguide is more prominent in the stimulated casecompared to that in the spontaneous case of linear and nonlin-ear Zeno parameters, respectively (cf. Fig. 7).In brief, possibility of observing Zeno and anti-Zeno effectsis rigorously investigated in symmetric and asymmetric non-linear optical couplers, which are experimentally realizableat ease. For completely quantum description of the primaryphysical system (i.e., symmetric nonlinear optical coupler),appropriate use of a perturbative technique which is known toperform better than short-length method, reducibility of theresults obtained for symmetric nonlinear optical coupler tothat of asymmetric nonlinear optical coupler, easy experimen-tal realizability of the physical systems, etc., provide an edgeto this work over the existing works on Zeno effects in opti-cal coupler, where usually the use of complete quantum de-scription is circumvented by considering one or more modesas strong and/or short length method is used to reduce com-putational difficulty. The approach adopted here is also verygeneral and can be easily extended to the study of other op-tical couplers and other quantum optical systems having thesimilar structure of momentum operators or Hamiltonian as isused here. We conclude the work with an expectation that theexperimentalists will find this work interesting for an experi-mental verification and it could be possible to find its appli-cability in some of the recently proposed Zeno-effect-basedschemes for quantum computation and communication.
Appendix A: Solution of Heisenberg’s equations of motion
The solution of the momentum operator given in Eq. (1)using Sen-Mandal perturbative approach can be obtained oncewe write the Heisenberg’s equations of motion for all the fieldmodes involved, which are obtained as da dz = ik ∗ b + 2 i Γ ∗ a a † a exp ( − i ∆ k a z ) , db dz = ika + 2 i Γ ∗ b b † b exp ( − i ∆ k b z ) , da dz = i Γ a a exp ( i ∆ k a z ) , db dz = i Γ b b exp ( i ∆ k b z ) . (A.1)Now, the evolution the all the field modes can be assumed upto quadratic terms in nonlinear coupling constants Γ i in theform1 a ( z ) = f a (0) + f b (0) + f a † (0) a (0) + f a (0) b † (0) + f b † (0) b (0) + f a † (0) b (0)+ f a (0) a † (0) a (0) + f a † (0) a (0) + f a † (0) a (0) b (0) + f a † (0) a (0) b (0)+ f a (0) b † (0) b (0) + f a † (0) b (0) + f a (0) b † (0) + f b † (0) b (0)+ f a (0) b (0) b † (0) + f a (0) a (0) b † (0) + f a (0) a † (0) b (0)+ f a † (0) b (0) b (0) + f b (0) b † (0) b (0) + f b † (0) b (0) + f a † (0) b (0)+ f a (0) b † (0) b (0) + f a (0) b † (0) b (0) + f a † (0) a (0) b (0) + f a (0) b † (0)+ f a † (0) a (0) ,b ( z ) = g a (0) + g b (0) + g a † (0) a (0) + g a (0) b † (0) + g b † (0) b (0) + g a † (0) b (0)+ g a (0) a † (0) a (0) + g a † (0) a (0) + g a † (0) a (0) b (0) + g a † (0) a (0) b (0)+ g a (0) b † (0) b (0) + g a † (0) b (0) + g a (0) b † (0) + g b † (0) b (0)+ g a (0) b (0) b † (0) + g a (0) a (0) b † (0) + g a (0) a † (0) b (0)+ g a † (0) b (0) b (0) + g b (0) b † (0) b (0) + g b † (0) b (0) + g a † (0) b (0)+ g a (0) b † (0) b (0) + g a (0) b † (0) b (0) + g a † (0) a (0) b (0) + g a (0) b † (0)+ g a † (0) a (0) ,a ( z ) = h a (0) + h a (0) + h b (0) a (0) + h b (0) + h a † (0) a (0) a (0) + h a (0)+ h a † (0) a (0) b (0) + h a (0) a (0) b † (0) + h a (0) b † (0) b (0) + h a (0) b † (0) b (0)+ h b † (0) b (0) b (0) + h b (0) + h a † (0) a (0) b (0) + h a † (0) b (0) b (0) ,b ( z ) = l b (0) + l b (0) + l b (0) a (0) + l a (0) + l b † (0) b (0) b (0) + l b (0)+ l a (0) b † (0) b (0) + l a † (0) b (0) b (0) + l a † (0) a (0) b (0) + l a † (0) a (0) b (0)+ l a † (0) a (0) a (0) + l a (0) + l a (0) b † (0) b (0) + l a (0) a (0) b † (0) . (A.2)All the f i , g i , h i , and l i can be obtained using the assumedsolution (A.2) for different field modes in the coupled differ-ential equations in Eq. (A.1) with the boundary conditions forall F ( z = 0) = 1 , where F ∈ { f, g, h, l } . The closed formanalytic solutions given in Eq. (A.2) contain various coeffi-cients, for example, l = 1 ,l = − Γ b G ∗ b − k b + iC b (cid:2) | k | (cid:0) G ∗ b + − (cid:1) sin 2 | k | z − i ∆ k b × (cid:0) − (cid:0) G ∗ b + − (cid:1) cos 2 | k | z (cid:1)(cid:3) ,l = − C b | k | [ i ∆ k b ( G ∗ b + − ) sin 2 | k | z +2 | k | ( − ( G ∗ b + − ) cos 2 | k | z )] k ∗ ,l = Γ b | k | G ∗ b − k ∗ ∆ k b + iC b | k | k ∗ (cid:2) | k | (cid:0) G ∗ b + − (cid:1) sin 2 | k | z − i ∆ k b × (cid:0) − (cid:0) G ∗ b + − (cid:1) cos 2 | k | z (cid:1)(cid:3) ,l = | C b | | k | ∆ k b (cid:2) − | k | (cid:0) G ∗ b − + i ∆ k b z (cid:1) − i | k | ∆ k b G ∗ b − × sin 2 | k | z + 6 i | k | ∆ k b G ∗ b − sin 2 | k | z − i ∆ k b sin 2 | k | z + 4 | k | ∆ k b (cid:0) cos 2 | k | z − G ∗ b − + 3 i ∆ k b z (cid:1) + ∆ k b × | k | (cid:0)(cid:0) − G ∗ b − (cid:1) cos 2 | k | z − − G ∗ b − − i ∆ k b z (cid:1)(cid:3) ,l = | C b | ∆ k b (cid:2) − | k | (cid:0) G ∗ b − + i ∆ k b z (cid:1) − i | k | ∆ k b × (cid:0) G ∗ b + − (cid:1) sin 2 | k | z + 4 | k | ∆ k b × (cid:0) cos 2 | k | z (cid:0) G ∗ b + − (cid:1) + 2 G ∗ b − − i ∆ k b z (cid:1) + ∆ k b (cid:0) cos 2 | k | z (cid:0) G ∗ b + − (cid:1) − − G ∗ b − − i ∆ k b z (cid:1)(cid:3) ,l = | C b | k ∗ ∆ k b (cid:2) k b sin | k | z + 4 i | k | ∆ k b sin 2 | k | z − i | k | ∆ k b (cid:0) G ∗ b − − (cid:1) sin 2 | k | z + 8 | k | (cid:0) − G ∗ b − cos 2 | k | z + G ∗ b − − i ∆ k b z (cid:1) + 2 | k | ∆ k b (cid:0) G ∗ b − cos 2 | k | z − G ∗ b − + i ∆ k b z (cid:1)(cid:3) , l = − | C b | k ∆ k b (cid:2) k b sin | k | z − i | k | ∆ k b sin 2 | k | z + 4 i | k | ∆ k b (cid:0) G ∗ b − − (cid:1) sin 2 | k | z + 8 | k | (cid:0) − G ∗ b − cos 2 | k | z + G ∗ b − + i ∆ k b z (cid:1) + 2 | k | × ∆ k b (cid:0)(cid:0) G ∗ b + + 2 (cid:1) cos 2 | k | z − G ∗ b − − − i ∆ k b z (cid:1)(cid:3) ,l = | C b | | k | ∆ k b (cid:2) − | k | (cid:0) G ∗ b − + i ∆ k b z (cid:1) + (cid:8) i | k | ∆ k b G ∗ b − − i | k | ∆ k b (cid:0) G ∗ b + + 2 (cid:1) + i ∆ k b (cid:9) sin 2 | k | z + 4 | k | × (cid:0)(cid:0) − G ∗ b − (cid:1) cos 2 | k | z − G ∗ b − + 3 i ∆ k b z (cid:1) × ∆ k b + | k | ∆ k b (cos 2 | k | z − − i ∆ k b z ) (cid:3) ,l = C ab | k | ( G ab + − k ∗ (cid:2) i | k | ∆ k b ∆ k ab sin 2 | k | z (cid:8) | k | × (cid:0) ∆ k b (cid:0) G ∗ a − − (cid:1) − ∆ k a (cid:0) G ∗ a − + 1 (cid:1) + ∆ k b (cid:1) − (cid:0) ∆ k b (∆ k b − k a ) + (cid:0) ∆ k ab − k a ∆ k b (cid:1) × (cid:0) G ∗ a − − (cid:1)(cid:1)(cid:9) − i ∆ k a ∆ k b ∆ k ab (cid:0) G ∗ a + − (cid:1) × sin 2 | k | z + 16 | k | (cid:0) − ∆ k b ∆ k ab G ∗ a − cos 2 | k | z + (cid:0) ∆ k a (cid:0) G ∗ ab − − (cid:1) + ∆ k ab (cid:0) G ∗ a + − (cid:1) + ∆ k b (cid:1) × ∆ k a ) − | k | { ∆ k b ∆ k ab cos 2 | k | z × (cid:0) ∆ k a (cid:0) G ∗ a + − (cid:1) − ∆ k a ∆ k b G ∗ a + − ∆ k b G ∗ a − (cid:1) + ∆ k a (cid:0)(cid:0) ∆ k b ∆ k ab + ∆ k ab (cid:1) (cid:0) G ∗ a + − (cid:1) + ∆ k b + ∆ k b ∆ k ab − (cid:0) G ∗ ab + − (cid:1) × (cid:0) k a ∆ k b − k a ∆ k b + ∆ k a + 2∆ k b (cid:1)(cid:1)(cid:9) − | k | ∆ k a ∆ k b ∆ k ab (cid:0) ∆ k b ∆ k ab (cid:0) G ∗ a − − (cid:1) + 2∆ k a (cid:0) G ∗ ab + − (cid:1) − ∆ k b + cos 2 | k | z (cid:0) − ∆ k b + (cid:0) ∆ k a ∆ k b − k a + ∆ k b (cid:1) (cid:0) G ∗ a + − (cid:1)(cid:1)(cid:1)(cid:3) , l = − C ab k ( G ab + − k ∗ (cid:2) | k | (cid:0) ∆ k a (cid:0) G ∗ ab − − (cid:1) + ∆ k b + ∆ k ab (cid:0) G ∗ a + − (cid:1)(cid:1) + 16 i ∆ k b ∆ k ab | k | G ∗ a − sin 2 | k | z − k a ∆ k b ∆ k ab (cid:0) G ∗ a + − (cid:1) sin | k | z + { i ∆ k b ∆ k ab × | k | (cid:0) − ∆ k a ∆ k b G ∗ a + − ∆ k b G ∗ a − + ∆ k a (cid:0) G ∗ a + − (cid:1)(cid:1) − i ∆ k a (cid:0) ∆ k b + (cid:0) k a − ∆ k a ∆ k b − ∆ k b (cid:1) (cid:0) G ∗ a + − (cid:1)(cid:1) × ∆ k b ∆ k ab | k | (cid:9) sin 2 | k | z − | k | { ∆ k b ∆ k ab cos 2 | k | z × (cid:0) ∆ k b G ∗ a − − ∆ k a (cid:0) G ∗ a − + 1 (cid:1)(cid:1) − ∆ k a (cid:0) G ∗ ab + − (cid:1) × (cid:0) ∆ k a ∆ k b + 3∆ k ab (cid:1) + ∆ k b (cid:0) ∆ k ab + ∆ k b (cid:1) + (cid:0) G ∗ a + − (cid:0) − k a ∆ k b + 4∆ k a ∆ k b + 3∆ k a − k b (cid:1)(cid:9) + 2∆ k ab | k | (cid:8) ∆ k b (cid:0) ∆ k b (∆ k b − k a ) + (cid:0) G ∗ a + − (cid:1) × (cid:0) ∆ k ab − k a ∆ k b (cid:1)(cid:1) cos 2 | k | z + ∆ k a (cid:0) G ∗ ab − + 1 (cid:1) × (2∆ k ab − ∆ k b ) + ∆ k b ∆ k ab + (cid:0) G ∗ a + − (cid:1) × (cid:0) k a ∆ k ab − k a ∆ k b + 3∆ k a ∆ k b + ∆ k b (cid:1)(cid:9)(cid:3) ,l = − C ab k ( G ab + − ( | k | − ∆ k ab ) k ∗ (cid:2) | k | (cid:0) ∆ k a (cid:0) G ∗ ab − − (cid:1) + ∆ k ab (cid:0) G ∗ a + − (cid:1) + ∆ k b (cid:1) + ∆ k a ∆ k b ∆ k ab sin | k | z × (cid:0) G ∗ a + − (cid:1) − | k | (cid:8) ∆ k a ∆ k ab (cid:0) G ∗ a + − (cid:1) cos 2 | k | z × ∆ k b + ∆ k ab (cid:0) ∆ k a + ∆ k b (cid:1) (cid:0) G ∗ a + − (cid:1) + ∆ k a × (cid:0) G ∗ a − − (cid:1) ∆ k b (cid:9) + i ∆ k a ∆ k b | k | (cid:0) ∆ k a − ∆ k b (cid:1) × (cid:0) G ∗ a + − (cid:1) sin 2 | k | z (cid:3) ,l = − C ab k | k | ( G ab + − k ∗ (cid:2) | k | (cid:0) ∆ k a (cid:0) G ∗ ab − − (cid:1) + ∆ k b + ∆ k ab (cid:0) G ∗ a + − (cid:1)(cid:1) − i ∆ k b ∆ k ab | k | sin 2 | k | z × (cid:8) | k | G ∗ a − − ∆ k a ∆ k b (cid:0) G ∗ a + − (cid:1) + ∆ k a G ∗ a + − ∆ k b G ∗ a − (cid:9) − i ∆ k a ∆ k b ∆ k ab sin 2 | k | z × (cid:0) − ∆ k b + ∆ k ab (cid:0) G ∗ a + − (cid:1)(cid:1) − | k | { ∆ k b ∆ k ab × (cid:0) − ∆ k b G ∗ a − + ∆ k a (cid:0) G ∗ a + + 1 (cid:1)(cid:1) cos 2 | k | z − ∆ k ab (cid:0) G ∗ ab + − (cid:1) (cid:0) ∆ k a + ∆ k b ∆ k ab (cid:1) + (cid:0) ∆ k b + ∆ k ab (cid:0) G ∗ a + − (cid:1)(cid:1) (cid:0) ∆ k b + ∆ k ab (cid:1)(cid:9) − k b ∆ k ab | k | (cid:8) cos 2 | k | z (cid:0) − ∆ k b (∆ k a + ∆ k ab ) − ∆ k ab (∆ k a + ∆ k b ) (cid:0) G ∗ a + − (cid:1)(cid:1) − ∆ k b ∆ k ab × (cid:0) G ∗ a + − (cid:1) + ∆ k a (cid:0) G ∗ ab + − (cid:1) − ∆ k b ∆ k ab (cid:9)(cid:3) ,l = − C ab k ( G ab + − k ∗ (cid:2) k a ∆ k b ∆ k ab ( − ∆ k b + ∆ k ab × (cid:0) G ∗ a + − (cid:1)(cid:1) + 2 i ∆ k b ∆ k ab | k | sin 2 | k | z (cid:8) − | k | × (cid:0) − ∆ k b G ∗ a − + ∆ k a (cid:0) G ∗ a + + 1 (cid:1)(cid:1) + (∆ k a + ∆ k ab ) × ∆ k b + ∆ k ab (∆ k a + ∆ k b ) (cid:0) G ∗ a + − (cid:1)(cid:9) + 16 | k | × (cid:8) − ∆ k b ∆ k ab G ∗ a − cos 2 | k | z + ∆ k a (cid:0)(cid:0) G ∗ a + − (cid:1) × ∆ k ab + (cid:0) (∆ k b − ∆ k ab ) (cid:0) G ∗ ab + − (cid:1) − ∆ k b (cid:1)(cid:9) − | k | (cid:8) ∆ k b ∆ k ab (cid:0) − ∆ k a ∆ k b (cid:0) G ∗ a + − (cid:1) − ∆ k b × G ∗ a − + ∆ k a G ∗ a + (cid:1) cos 2 | k | z + ∆ k a ((∆ k b − ∆ k ab ) × ∆ k a (cid:0) G ∗ ab + − (cid:1) + (cid:0) ∆ k b + ∆ k ab (cid:1) ( − ∆ k b + ∆ k ab (cid:0) G ∗ a + − (cid:1)(cid:1)(cid:1)(cid:9)(cid:3) (A.3) with G i ± = (1 ± exp( − i ∆ k i z )) for i ∈{ a, b, ab } , and ∆ k ab = ∆ k a − ∆ k b . Also, C a = Γ a [4 | k | − ∆ k a ] , C b = Γ b [ | k | − ∆ k b ] and C ab = Γ ∗ a Γ b ∆ k a ∆ k b ∆ k ab [4 | k | − ∆ k a ] [ | k | − ∆ k b ][ | k | − ∆ k ab ] . Acknowledgement:
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