Quantum properties of the three-mode squeezed operator: triply concurrent parametric amplifiers
Faisal A. A. El-Orany, Azeddine Messikh, Gharib S. Mahmoud, Wahiddin M. R. B
aa r X i v : . [ qu a n t - ph ] O c t Quantum properties of the three-mode squeezed operator: triply concurrentparametric amplifiers
Faisal A A El-Orany , Azeddine Messikh , Gharib S Mahmoud , Wahiddin M. R. B.
1, 2 Cyberspace Security Laboratory, MIMOS Berhad,Technology Park Malaysia, 57000 Kuala Lumpur, Malaysia International Islamic University Malaysia,P.O. Box 10, 50728 Kuala Lumpur, Malaysia (Dated: November 1, 2018)In this paper, we study the quantum properties of the three-mode squeezed operator. Thisoperator is constructed from the optical parametric oscillator based on the three concurrent χ (2) nonlinearities. We give a complete treatment for this operator including the symmetricand asymmetric nonlinearities cases. The action of the operator on the number and coherentstates are studied in the framework of squeezing, second-order correlation function, Cauchy-Schwartz inequality and single-mode quasiprobability function. The nonclassical effects areremarkable in all these quantities. We show that the nonclassical effects generated by theasymmetric case–for certain values of the system parameters–are greater than those of thesymmetric one. This reflects the important role for the asymmetry in the system. Moreover,the system can generate different types of the Schr¨odinger-cat states. PACS numbers: 42.50.Dv,42.50.-p
I. INTRODUCTION
Squeezed light fulfils the uncertainty relation and has less noise than the coherent light inthe one of the field quadratures. With the development of quantum information, squeezed stateshave become very important tool in providing efficient techniques for the encoding and decodingprocesses in the quantum cryptography [1]. For instance, the two-mode squeezed states are of greatinterest in the framework of continuous-variable protocol (CVP) [2]. In the CVP the quantum keydistribution goes as follows. The two-mode squeezed source–such as parametric down conversion–emits two fields: one is distributed to Alice (A) and the other to Bob (B). Alice and Bob randomlychoose to measure one of two conjugate field quadrature amplitudes. The correlation between theresults of the same quadrature measurements on Alice’s and Bob’s side increases by increasing thevalues of the squeezing parameter. Through a public classical channel the users communicate theirchoices for the measurements. They keep only the results when both of them measure the samequadrature and hence the key is generated.The generation of the multiparities squeezed entangled states is an essential issue in the multi-party communication including a quantum teleportation network [3], telecloning [4], and controlleddense coding [5]. The N -mode CV entangled states have been generated by combining N single-mode squeezed light in appropriately coupled beam splitters [3]. The three-mode CV entangledstates have taken much interest in the literatures, e.g., [3, 6, 7, 8, 9]. For instance, it has beentheoretically shown that tripartite entanglement with different wavelengths can be generated bycascaded nonlinear interaction in an optical parametric oscillator cavity with parametric downconversion and sum-frequency generation [6]. Also, the three-mode CV states have been generatedby three concurrent χ (2) nonlinearities [7]. This has been experimentally verified by the observa-tion of the triply coincident nonlinearities in periodically poled KT iOP O [8]. More details aboutthis issue will be given in the next section. Furthermore, the comparison between the tripartiteentanglement in the three concurrent nonlinearities and in the three independent squeezed statesmixed on the beam splitters [3], in the framework of the van Loock-Furusawa inequalities [10], hasbeen performed in [7]. It is worth mentioning that the generation of macroscopic and spatiallyseparated three-mode entangled light for triply coupled χ (3) Kerr coupler inside a pumped opticalcavity has been discussed in [9]. It has been shown that the bright three-mode squeezing and fullinseparable entanglement can be established inside and outside the cavity.Since the early days of the quantum optics squeezing is connected with what is so called squeezedoperator. There have been different forms of this operator in the literatures, e.g. [11, 12, 13, 14,15, 16, 17]. For example, degenerate and non-degenerate parametric amplifiers are sources of thesingle-mode [11] and the two-mode [12] squeezing, respectively. The quantum properties of thethree-mode squeezed operator (TMS), which is constructed from two parametric amplifiers andone frequency converter, have been demonstrated in [14]. This operator can be represented by the SU (1 ,
1) Lie algebra generators [14, 15]. Additionally, it can be generated–under certain condition–from bulk nonlinear crystal in which three dynamical modes are injected by three beams. Anotherpossibility for the realization is the nonlinear directional coupler which is composed of two opticalwaveguides fabricated from some nonlinear material described by the quadratic susceptibility χ (2) .Finally, the mathematical treatments for particular type of the n -mode squeezed operator againstvacuum states are given in [16].In this paper, we treat the three concurrent parametric amplifiers given in [7, 8] as three-mode squeezed operator. We quantitatively investigate the nonclassical effects associated with thisoperator when acting on the three-mode coherent and number states. For these states we inves-tigate squeezing, second-order correlation function, Cauchy-Schwartz inequality and single-modequasiprobability functions. This investigation includes the symmetric (equal nonlinearities) andasymmetric (non-equal nonlinearities) cases. In the previous studies the entanglement of the sym-metric case only has been discussed [7, 8]. The investigation in the current paper is motivated bythe importance of the three concurrent parametric amplifiers in the quantum information research[3, 7, 8]. Additionally, quantifying the nonclassical effects in the quantum systems is of fundamen-tal interest in its own right. We prepare the paper in the following order. In section 2 we constructthe operator and write down its Bogoliubov transformations. In sections 3 and 4 we study thequadrature squeezing, the second-order correlation function as well as the Cauchy-Schwartz in-equality, respectively. In section 5 we investigate the single-mode quasiprobability functions. Themain results are summarized in section 6. II. OPERATOR FORMALISM
In this section we present the operator formalism for the optical parametric oscillator based onthe three concurrent χ (2) nonlinearities. We follow the technique given in [7, 8] to construct theHamiltonian of the system. In this regard, we consider three modes injected into a nonlinear crystal,whose susceptibility is χ (2) , to form three output beams at frequencies ω , ω , ω . The interactionsare selected to couple distinct polarizations. Assuming that x is the axis of the propagation withinthe crystal. The mode ˆ b is pumped at frequency and polarization ( ω + ω , y ) to produce themodes ˆ a ( ω , z ) and ˆ a ( ω , y ). The mode ˆ b is pumped at ( ω + ω , y ) to produce the modes ˆ a and ˆ a ( ω , z ). Eventually, the mode ˆ b is pumped at (2 ω , z ) to produce the modes ˆ a and ˆ a . Thescheme for this interaction can be found in [7, 8]. The interaction Hamiltonian for this concurrenttriple nonlinearity takes the form [7, 8]:ˆ H int = i ~ ( χ ˆ b ˆ a † ˆ a † + χ ˆ b ˆ a † ˆ a † + χ ˆ b ˆ a † ˆ a † ) + h . c ., (1)where χ j , j = 1 , , , represent the effective nonlinearities and h.c. stands for the hermitian conju-gate. The unitary operator associated with (1) is:ˆ U ( t ) = exp − it ˆ H int ~ ! . (2)In the undepleted pump approximation we set r j = χ j h ˆ b j (0) i as real parameters. Now we obtainthe requested squeezed operator as:ˆ S ( r ) = exp[ r (ˆ a ˆ a − ˆ a † ˆ a † ) + r (ˆ a ˆ a − ˆ a † ˆ a † ) + r (ˆ a ˆ a − ˆ a † ˆ a † )] , (3)where ( r ) = ( r , r , r ). Throughout this paper, the symmetric case means r = r = r = r ,otherwise we have an asymmetric case. It is evident that three disentangled state can be entangledunder the action of this operator. This operator provides the following Bogoliubov transformations:ˆ S † ( r )ˆ a j ˆ S ( r ) = f ( j )1 ˆ a + f ( j )2 ˆ a † + g ( j )1 ˆ a + g ( j )2 ˆ a † + h ( j )1 ˆ a + h ( j )2 ˆ a † , j = 1 , , f ( j ) j ′ , g ( j ) j ′ , h ( j ) j ′ , j ′ = 1 , r , r , r . The formulae ofthese functions for the asymmetric case are rather lengthy. Nevertheless, we write down only herethe explicit forms for the symmetric case as [7]: f (1)1 = [2 cosh( r ) + cosh(2 r )] , f (1)2 = [2 sinh( r ) − sinh(2 r )] ,g (1)1 = [ − cosh( r ) + cosh(2 r )] , g (1)2 = − [sinh( r ) + sinh(2 r )] ,g (1)1 = h (1)1 = f (2)1 = h (2)1 = f (3)1 = g (3)1 ,f (1)1 = g (2)1 = h (3)1 , f (1)2 = g (2)2 = h (3)2 ,g (1)2 = h (1)2 = f (2)2 = h (2)2 = f (3)2 = g (3)2 . (5)Relations (4) and (5) will be frequently used in the paper. For the symmetric case, the entanglementhas been already studied in terms of the van Loock-Furusawa measure [7]. It has been shown thatthe larger the value of r , the greater the quantity of the entanglement in the tripartite. Moreover,the tripartite CV entangled state created tends towards GHZ state in the limit of infinite squeezing,but is analogous to a W state for finite squeezing [18]. In this paper we give an investigation forthe entanglement of the asymmetric case from different point of view. This is based on the factthat the entanglement between different components in the system is a direct consequence of theoccurrence of the nonclassical effects in their compound quantities and vice versa. We show that theasymmetric case can provide amounts of the nonclassical effects and/or entanglement greater thanthose of the symmetric case. Thus the asymmetry in the triply concurrent parametric amplifiersis important.The investigation of the operator (3) will be given through the three-mode squeezed coherentand number states having the forms: | ψ n i = ˆ S ( r ) | n , n , n i , | ψ c i = ˆ S ( r ) | α , α , α i . (6)Three-mode squeezed vacuum states can be obtained by simply setting n j = 0 or α j = 0 in theabove expressions. In the following sections we study the quantum properties for the states (6) ingreater details. III. QUADRATURE SQUEEZING
Squeezing is an important phenomenon in the quantum theory, which can reflect the correlationin the compound systems very well. Precisely, squeezing can occur in combination of the quantummechanical systems even if the single systems are not themselves squeezed. In this regard thenonclassicality of the system is a direct consequence of the entanglement. Squeezed light canbe measured by the homodyne detector, in which the signal is superimposed on a strong coherentbeam of the local oscillator. Additionally, squeezing has many applications in various areas, e.g., inquantum optics, optics communication, quantum information theory, etc [19]. Thus investigatingsqueezing for the quantum mechanical systems is an essential subject in the quantum theory. Inthis section we demonstrate different types of squeezing for the three-mode squeezed vacuum states(6). To do so we define two quadratures ˆ X and ˆ Y , which denote the real (electric) and imaginary(magnetic) parts, respectively, of the radiation field as:ˆ X = [ˆ a + ˆ a † + c (ˆ a + ˆ a † ) + c (ˆ a + ˆ a † )] , ˆ Y = i [ˆ a − ˆ a † + c (ˆ a − ˆ a † ) + c (ˆ a − ˆ a † )] , (7)where c , c are c -numbers take the values 0 or 1 to yield single-mode, two-mode and three-modesqueezing. These two operators, ˆ X and ˆ Y , satisfy the following commutation relation:[ ˆ X, ˆ Y ] = iC, (8)where C = (1 + c + c ) /
2. It is said that the system is able to generate squeezing in the x - or y -quadrature if S x = 2 h (∆ ˆ X ) i − CC < , or (9) S y = 2 h (∆ ˆ Y ) i − CC < , where h (∆ ˆ X ) i = h ˆ X i − h ˆ X i is the variance. Maximum squeezing occurs when S x = − S y = − S x = c + c ) { (1 + c + c )[2 exp(2 r ) + exp( − r ) −
3] + 2( c + c + c c )[exp( − r ) − exp(2 r )] } ,S y = c + c ) { (1 + c + c )[2 exp( − r ) + exp(4 r ) −
3] + 2( c + c + c c )[exp(4 r ) − exp( − r )] } . (10)For the single-mode case, c = c = 0, the expressions (10) reduce to: S x = [2 exp(2 r ) + exp( − r ) − ,S y = [2 exp( − r ) + exp(4 r ) − . (11)It is evident that the system cannot generate single-mode squeezing. This fact is valid for theasymmetric case, too. For the two-mode case, c = 1 , c = 0, i.e. first-second mode squeezing, weobtain S x = [exp(2 r ) + 2 exp( − r ) − ,S y = [exp( − r ) + 2 exp(4 r ) − . (12)Squeezing can be generated in the x -component only with a maximum value at r = ln(2) /
3. Alsothe maximum squeezing, i.e. S x = −
1, cannot be established in this case. This is in contrastwith the two-mode squeezed operator [12] for which S x = − r . Roughly speaking, thequantum correlation in this system decreases the squeezing, which can be involved in the one ofthe bipartites. Finally, for three-mode case, c = c = 1, we have S x = exp( − r ) − , S y = exp(4 r ) − . (13)Squeezing can be generated in the x -component only for r >
0. Squeezing reaches its maximumvalue for large values of r . The origin of the occurrence squeezing in (13) is in the strong correlationamong the components of the system. Moreover, the amount of the produced squeezing is two (four)times greater than that of the two-mode [12] (single-mode [11]) squeezed operator for certain valuesof r .In Figs. 1(a) and (b) we plot the squeezing parameter S x against r for two- and three-modesqueezing, respectively. We found that squeezing is not remarkable in S y . The solid curve is plottedfor the symmetric case. The two-mode squeezing is given for the first-second mode system. We startthe discussion with the two-mode case (Fig. 1(a)). From the solid curve, squeezing is graduallygenerated as r increases providing its maximum value S x = − .
206 at r = ln(2) / . S x ≥
0, at r ≥ ln(1 + √ / . S x = − r , r S x (a) r S x (b) FIG. 1: Two-mode (a) and three-mode (b) squeezing against r for three-mode squeezed vacuum states.Solid, dashed and dotted curves are given for ( r , r ) = ( r , r ) , (0 . , .
2) and (0 . , . then rapidly decreases and vanishes (see the dashed and dotted curves). This can be understoodas follows. When the values of r and r are relatively small, the main contribution in the systemis related to the first parametric amplifier. Thus the system behaves as the conventional two-modesqueezed operator for a certain range of r . This remark is noticeable when we compare the dottedcurve to the dashed one. Generally, when the values of r and r increase, the degradation of thesqueezing increases, too. Furthermore, in the range of r for which S x = −
1, the entanglementin the bipartite (1 ,
2) is maximum, however, this is not the case for the other bipartities. This isconnected with the fact: the quantum entanglement cannot be equally distributed among manydifferent objects in the system. Comparison among different curves in Fig. 1(a) shows that theasymmetric case can provide amounts of squeezing much greater than those of the symmetric case.Now, we draw the attention to the three-mode squeezing, which is displayed in Fig. 1(b). Forthe symmetric case, S x exhibits squeezing for r >
0, which monotonically increases providingmaximum value for large r , as we discussed above. This is in a good agreement with the factthat the symmetric case exhibits genuine tripartite entanglement for large values of r [21]. Forthe asymmetric case, the curves show initially squeezing, which reaches its maximum by increasing r , then it gradually decreases and vanishes for large values of r . The greater the values of r , r the higher the values of the maximum squeezing in S x and the shorter the range of r over whichsqueezing occurs (compare dotted and dashed curves in Fig. 1(b)). This situation is the inverseof that of the two-mode case (compare Fig. 1(a) to (b)). Trivial remark, for small values of r theamounts of squeezing produced by the symmetric case are smaller than those by the asymmetricone. We can conclude that for the asymmetric case the entanglement in the tripartite may bedestroyed for large r , where S x >
0. Of course this is sensitive to the values of r , r . Conversely,the amounts of the entanglement between different bipartites in the system for the asymmetriccase can be much greater than those of the symmetric one for particular choice of the parameters.The final remark, the amounts of squeezing produced by the operator (3) are greater than thosegenerated by the TMS [14]. IV. SECOND-ORDER CORRELATION FUNCTION AND CAUCHY-SCHWARTZINEQUALITY
In this section we study the second-order correlation function and the Cauchy-Schwartz inequal-ity for the states (6). These two quantities are useful for getting information on the correlationsbetween different components in the system. In contrast to the quadrature squeezing, these quan-tities are not phase dependent and are therefore related to the particle nature of the field. We startwith the single-mode second-order correlation function, which for the j th mode is defined as: g (2) j (0) = h ˆ a † j ˆ a j ih ˆ a † j ˆ a j i − , (14)where g (2) j (0) = 0 for Poissonian statistics (standard case), g (2) j (0) < g (2) j (0) > g (2)1 (0) of the three-mode squeezed number states. Form (4) and (6), onecan obtain the following moments: r g ( ) (a) r g ( ) ( ) (b) FIG. 2: Second-order correlation of the first mode against r . (a) ( n , n , n ) = (1 , , r , r ) = ( r , r )solid curve, (0 . , .
2) dashed curve, and (0 . , .
6) dotted curve. (b) ( r , r ) = ( r , r ), ( n , n , n ) = (1 , , , ,
0) dashed curve. h ˆ a † ˆ a i = n f (1)21 + ( n + 1) f (1)22 + n g (1)21 + ( n + 1) g (1)22 + n h (1)21 + ( n + 1) h (1)22 , h ˆ a † ˆ a i = n ( n − f (1)41 + ( n + 1)( n + 2) f (1)42 + (2 n + 1) f (1)21 f (1)22 + n ( n − g (1)41 + ( n + 1)( n + 2) g (1)42 + (2 n + 1) g (1)21 g (1)22 + n ( n − h (1)41 + ( n + 1)( n + 2) h (1)42 + (2 n + 1) h (1)21 h (1)22 +(2 n + 1) f (1)1 f (1)2 [2(2 n + 1) g (1)1 g (1)2 + (2 n + 1) h (1)1 h (1)2 ]+(2 n + 1) h (1)1 h (1)2 [2(2 n + 1) g (1)1 g (1)2 + (2 n + 1) f (1)1 f (1)2 ]+4[ n f (1)21 + ( n + 1) f (1)22 ][ n g (1)21 + ( n + 1) g (1)22 + n h (1)21 + ( n + 1) h (1)22 ]+4[ n g (1)21 + ( n + 1) g (1)22 ][ n h (1)21 + ( n + 1) h (1)22 ] . (15)By means of (14) and (15), the quantity g (2)1 (0) is depicted in Figs. 2 for given values of theparameters. In Fig. 2(a) we present the role of the squeezing parameters r j on the behavior ofthe g (2)1 (0). From the solid curve (, i.e. symmetric case), one can observe that the maximum sub-Piossonian statistics occur for relatively small values of r j . In this case, the system tends to the0Fock state | i , which is a pure nonclassical state. As the values of r increase, the nonclassicalitymonotonically decreases and completely vanishes around r ≃ .
3. Comparison among the curvesin Fig. 2(a) shows when the values of ( r , r ) increase, the amounts of the sub-Poissonian statisticsinherited in the first mode decrease. This is connected with the nature of the operator (3), inwhich the behavior of the single-mode undergoes an amplification process caused by the variousdown-conversions involved in the system. Furthermore, particular values of the asymmetry canenlarge the range of nonclassicality (compare the solid curve to the dashed one). Now, we drawthe attention to the Fig. 2(b), which is given to the symmetric case . From this figure one realizeshow can obtain sub-Poissonian statistics from a particular mode as an output from the operator(3). Precisely, this mode should be initially prepared in the nonclassical state. We can analyticallyprove this fact by substituting n = 0 , n = n = n into (14) and (15). After minor algebra, wearrive at: h ˆ a † ˆ a i − h ˆ a † ˆ a i = f (1)42 + 2 n ( n − g (1)41 + 2( n + 1)( n + 2) g (1)42 +2 n ( n + 1) g (1)21 g (1)22 + [ f (1)1 f (1)2 + 2(2 n + 1) g (1)1 g (1)2 ] +4 f (1)22 [ ng (1)21 + ( n + 1) g (1)22 ] ≥ . (16)The final remark, the comparison between the solid curves in Figs. 2(a) and (b) shows that thenonclassical range of r in (b) is greater than that in (a). In other words, to enhance the sub-Poissonian statistics in a certain mode, the other modes have to be prepared in states close to theclassical ones. r V j k (a) r V j k (b) r V j k (c) FIG. 3: The parameter V jk against r for the coherent state with α = α = α = 1, where ( j, k ) = (1 , ,
3) (b) and (2 ,
3) (c). Additionally, ( r , r ) = ( r , r ) solid curve, (0 . , .
2) dashed curve and (0 . , . The violation of the classical inequalities has verified the quantum theory. Among these in-equalities is the Cauchy-Schwarz inequality [24], which its violation provides information on theintermodal correlations in the system. The first observation of this violation was obtained by1Clauser, who used an atomic two-photon cascade system [25]. More recently, strong violations us-ing four-wave mixing have been adopted in [26, 27]. In addition, a frequency analysis has been usedto infer the violation of this inequality over a limited frequency regime [28]. The Cauchy-Schwarzinequality is V jk ≤
0, where V jk has the form: V jk = q h ˆ a † j ˆ a j ih ˆ a † k ˆ a k ih ˆ a † j ˆ a j ˆ a † k ˆ a k i − . (17)Occurrence of the negative values in V jk means that the intermodal correlation is larger than thecorrelation among the photons in the same mode. This indicates a strong deviation from theclassical Cauchy-Schwarz inequality. This is related to the quantum mechanical features, whichinclude pseudodistributions instead of the true ones. In this respect, the Glauber-Sudarshan P function possesses strong quantum properties [29]. r V j k FIG. 4: The parameter V jk against r for the Fock-state case with n = n = n = 1, where ( j, k ) = (1 , r , r ) = ( r , r ) solid curve, (0 . , .
2) dashed curve and (0 . , .
6) dotted curve.
We have found that the Cauchy-Schwartz inequality can be violated for both coherent- andnumber-state cases. The expressions of the different quantities in (17) are too lengthy but straight-forward and hence we don’t present them here. We start with the three-mode squeezed coherentstates. Information about them is shown in Figs. 3(a), (b) and (c), for given values of the systemparameters. The negative values are remarkable in most of the curves, reflecting the deviationfrom the classical inequality. For the symmetric case the nonclassical correlation is remarkablefor r ≥ .
4, which increases gradually till r ≃ . V jk ≃
0, for large r . For the asymmetric case, the2deviation from the classical inequality is obvious, which may be smaller or greater than those inthe symmetric one based on the competition among different nonlinearities in the system r j . Inother words, e.g., the correlation between modes 1 and 2 is much stronger than that between theothers only when r > r , r . This can be understood from the structure of the operator (3).Comparing this behavior to that of the two-mode squeezing given in the preceding section one canconclude that a large amount of squeezing does not imply large violation of the inequality [30]. Aswe mentioned before: the entanglement is a direct consequence of the occurrence of the nonclas-sical effects. As a result of this, the behavior of the two-mode squeezing and V jk may provide atype of contradiction. Precisely, the bipartite can be entangled (non-entangled) with respect to,say, two-mode squeezing ( V jk ). This supports the fact that these two quantities provide only asufficient condition for entanglement. Considering both of them we may obtain conditions closerto the necessary and sufficient condition. A study about this controversial issue has been alreadydiscussed for the entanglement in a parametric converter [20], where different entanglement criterialeaded to different results.In Fig. 4 we plot the parameter V jk for the Fock-state case. From this figure the deviationfrom the classical V jk is quite remarkable. For the symmetric case, maximum deviation occurs in V , for r = 0, monotonically decreases as r evolving and vanishes at r ≃
1. Comparison amongdifferent curves in this figure shows that the asymmetry can enlarge the range of r over whichthe deviation of the inequality occurs. Similar behavior has been observed for V , and V , (wehave checked this fact). In conclusion, the violation of the classical inequalities provides an explicitevidence of the quantum nature of intermodal correlation between modes. This is not surprising,as the entanglement is a pure quantum mechanical phenomenon that requires a certain degree ofnonclassicality either in the initial state or in the process that governs the system. V. QUASIPROBABILITY DISTRIBUTION FUNCTION
Quasiprobability distribution functions, namely, Husimi function ( Q ), Wigner function ( W ), andGlauber P functions, are very important since they can give a global description of the nonclassicaleffects in the quantum systems. These functions can be measured by various means, e.g. photoncounting experiments [31], using simple experiments similar to that used in the cavity (QED) andion traps [32, 33], and homodyne tomography [34]. For the system under consideration, we focusthe attention here on the single-mode case, say, the first mode for the three-mode squeezed numberstates (6). We start with the s − parameterized characteristic function C ( ζ, s ), which is defined as:3 C ( ζ, s ) = Tr[ˆ ρ exp( ζ ˆ a † − ζ ∗ ˆ a + s | ζ | )] , (18)where ˆ ρ is the density matrix of the system under consideration and s is a parameter taking thevalues 0 , , − C ( ζ, s ) = exp[ − | υ | − | υ | − | υ | + s | ζ | ] × L n ( | υ | )L n ( | υ | )L n ( | υ | ) , (19)where υ = ζf (1)1 − ζ ∗ f (1)2 , υ = ζg (1)1 − ζ ∗ g (1)2 , υ = ζh (1)1 − ζ ∗ h (1)2 (20)and L γk ( . ) is the associated Laguerre polynomial having the form:L γk ( x ) = k X l =0 ( γ + k )!( − x ) l ( γ + l )!( k − l )! l ! . (21)The s -parameterized quasiprobability distribution functions are defined as W ( z, s ) = π − Z d ζC ( ζ, s ) exp( zζ ∗ − ζz ∗ ) , (22)where z = x + iy and s = 0 , , − W, P, Q functions, respectively. On substi-tuting (19) into (22) and applying the method of the differentiation under the sign of integrationwe can obtain the following expression: W ( z, s ) = π { n ,n ,n } P j ′ ,j,k =0 { j ′ ,j + k } P l ,l =0 n j ′ n k n j j ′ l j + kl ( − j + j ′ + k j ! j ′ ! k ! × ( f (1)21 + f (1)22 ) l ( g (1)21 + g (1)22 ) l ( f (1)1 f (1)2 ) j ′ − l ( g (1)1 g (1)2 ) j + k − l ∂ l l ∂b l l | b =0 ∂ j + j ′ + k − l − l ∂b j + j ′ + k − l − l | b =0 × √ K exp[ − K ( B | z | + ( z + z ∗ )(Λ + b )] , (23)where Λ = f (1)21 + f (1)22 + g (1)21 + g (1)22 + h (1)21 + h (1)22 , Λ = f (1)1 f (1)2 + g (1)1 g (1)2 + h (1)1 h (1)2 ,B = (Λ − s ) − b , K = B − (Λ + b ) . (24)4The correlation between the modes in the system can be realized in W ( z, s ) as cross terms, e.g. inΛ . This can give a qualitative information about the entanglement in the system. For the three-mode squeezed vacuum states (, i.e., n = n = n = 0) the W function (23) can be expressedas: W ( x, y, s ) = 1 π p ϑ + ϑ − exp[ − x ϑ + − y ϑ − ] , (25)where ϑ + = 2 h (∆ ˆ X ) i − s , = [( f (1)1 + f (1)2 ) + ( g (1)1 + g (1)2 ) + ( h (1)1 + h (1)2 ) ] − s ,ϑ − = 2 h (∆ ˆ Y ) i − s , = [( f (1)1 − f (1)2 ) + ( g (1)1 − g (1)2 ) + ( h (1)1 − h (1)2 ) ] − s . (26)From (25) and (26) it is evident that the quasidistributions are Gaussians, narrowed in the y direction and expanded in the x direction. Nevertheless, this does not mean squeezing is availablein this mode. Actually, this behavior represents the thermal squeezed light, which, in this case,is a super-classical light. Precisely, with s = 0 the W function exhibits stretched contour, whosearea is broader than that of the coherent light. In this regard the phase distribution and thephoton-number distribution associated with the single-mode case exhibits a single-peak structurefor all values of r j . This peak is broader than that of the coherent state, which has the same mean-photon number. Actually, this is a quite common property for the multimode squeezed operators[11, 12, 13, 14, 15, 16, 17]. Thus the single-mode vacuum or coherent states, as outputs fromthe three concurrent amplifiers described by (3) are not nonclassical states. This agrees with theinformation given in the sections 3 and 4.Now we consider two cases: ( n , n , n ) = (0 , , n ) and ( n , n , n ) = ( n , , n = n = 0 in (23) and after minor algebra we arrive at: W ( x, y, s ) = ( − n π q ( Λ1 − s ) − Λ ( η − ϑ + ) n exp[ − x ϑ − − y ϑ + ] × n P m =0 (cid:16) ϑ + η + ϑ − η − (cid:17) m L − m [( η + + ϑ − η + ϑ − ) x ]L − n − m [( η − + ϑ + η − ϑ + ) y ] , (27)5where η ± = ( h (1)1 ± h (1)2 ) − ϑ ∓ . (28)One can easily check when r = r = r = 0 the W function (27) reduces to that of the vacuumstate. The form (27) includes Laguerre polynomial, which is well known in the literature byproviding nonclassical effects in the phase space. This indicates that the nonclasssical effects canbe transferred from one mode to another under the action of the operator (3). Of course theamount of the transferred data depends on the values of the squeezing parameters r j . −10−50 x−5 −4−2.5 −30.0 −2 5−1 y2.5 0 15.0 27.5 3 4 5 10 (a) r W ( , ) (b) FIG. 5: The Wigner function of the first mode (a) and the evolution of the phase space origin of the Wignerfunction (b). In (a) we use ( r , r , r , n , n , n ) = (1 . , . , . , , , r , r , n , n , n ) =( r , r , , ,
1) solid curve, (0 . , . , , ,
1) dashed curve, and (0 . , . , , ,
0) dotted curve.
The second case ( n , ,
0) has the same expression (27) with the following transformations: n → n , η ± = ( f (1)1 ± f (1)2 ) − ϑ ∓ . (29)Now we prove that this W function tends to that of the number state when r = r = r = 0. Inthis case, the transformations (29) tend to: η ± = 1 + s , ϑ ± = 1 − s . (30)6 (a) (b)(c) (d) FIG. 6: The Wigner function of the first mode when ( r , r , r , n , n , n ) = (0 . , . , . , , ,
0) (a),(0 . , . , . , , ,
1) (b), (0 . , . , , , ,
1) (c) and (0 . , . , , , ,
2) (d).
Substituting these variables in the expression (27) (with n → n ) we obtain: W ( x, y, s ) = − n π (1+ s ) n (1 − s ) n exp[ − x + y )1 − s ] n P m =0 L − m [ x − s ]L − n − m [ y − s ]= − n π (1+ s ) n (1 − s ) n exp[ − x + y )1 − s ]L n [ x + y )1 − s ] , (31)In (31) the transition from the first line to the second one has been done using the identity: m X n =0 L τ n ( x )L τ m − n ( y ) = L τ + τ +1 m ( x + y ) . (32)The expression (31) is the s -quasprobability distribution for the number state, e.g. [35]. We7conclude this part by writing down the form of the W function at the phase space origin, whichis a sensitive point for the occurrence of the nonclassical effects. Moreover, it can simply givevisualization about the behavior of the system. Additionally, this point can be measured by thephoton counting method [31]. From (27) we have: W (0 , , s ) = ( − n π p ϑ + ϑ − ( η − ϑ + + η + ϑ − ) n . (33)It is obvious that the Winger function exhibits negative values at the phase space origin only when n is an odd number.In Figs. 5 and 6 we plot the W functions for the given values of the system parameters. We startthe discussion with the symmetric case. In Fig. 5(a) we use ( n , n , n ) = (0 , ,
1) meaning thatthe mode under consideration is in the vacuum state. Thus for r j = 0 the W function exhibits thesingle-peak-Gaussian structure with a center at the phase space origin. When r j = 0 this behavioris completely changed, where one can observe a lot of the nonclassical features, e.g. negative values,multipeak structure and stretching contour (see Fig. 5(a)). This indicates that the nonclassicaleffects can be transferred from one mode to the other under the action of the operator (3). InFig. 5(b) we plot the ”evolution” of the W function given by (33) against the parameter r ( r )for the symmetric (asymmetric) case. The aim of this figure is to estimate the exact value ofthe nonlinearity r ( r ) for which the nonclassical effects maximally occur and/or transfer fromcertain mode to the other. For the symmetric case, this occurs at r = 1 .
2, while the nonclassicalityis completely washed out at r = 3. Now, we draw the attention to the asymmetric case which isplotted in Figs. 6. Fig. 6(a) gives information on the case ( n , n , n ) = (1 , , W functionof the Fock state | i is well known in the literatures by having inverted peak in phase space withmaximum negative values. This is related to that this state provides maximum sub-Poissonianstatistics. Under the action of the operator (3) these negative values are reduced and the two-peak structure is started to be constructed. This indicates that the system is able to generateparticular types of the Schr¨odinger-cat states by controlling the system parameters. This is reallyobvious in Figs. 6(b)–(d), which are given to the cases n = n = 0 , n = 1 ,
2. For instance, fromFig. 6(b) the W function provides a two-peak structure. Nevertheless, by increasing the valuesof the parameter r , the W function exhibits the two Gaussian peaks and inverted negative peakin-between indicating the occurrence of the interference in phase space (see Fig. 6(c)). This shapeis similar to that of the odd-coherent state. Additionally, the Fig. 6(d), in which n = 2, providesthe well-known shape of the W function of the even coherent state. Generally, the even and theodd Schr¨odinger-cat states have nearly identical classical components (, i.e. the positive peaks)8and only differ in the sign of their quantum interferences. These are interesting results, whichshow that by controlling the nonlinearity of the system and preparing a certain mode in the Fockstate | i or | i one can generate cat states. It is worth mentioning that the Fock state | n i can beprepared with very high efficiency according to the recent experiments [37]. Similar results havebeen obtained from the codirectional three-mode Kerr nonlinear coupler [36]. Furthermore, quiterecently the construction of the cat state trapped in the cavity in which several photons survivelong enough to be repeatedly measured is given in [38]. In this technique, the atoms crossing thecavity one by one are used to obtain information about the field. We proceed, we have noted thatthe W function of the cases n = n = 0 , n = 1 , r j . Now, we draw the attention to the dashed anddotted curves in the Fig. 5(b). These curves provide information on the evolution of the W (0 , r for the case of Fig. 6(a) and (c), respectively. From the dotted curve, i.e. the modeunder consideration is in the Fock state, the W (0 ,
0) exhibits the maximum negativity at r = 0,which monotonically decreases and completely vanishes at r = 4. This shows for how long thenonclassicality inherited in the first mode survives based on the intensity of the third amplifier.On the other hand, form the dashed curve, i.e. the mode under consideration is in the vacuumstate, the W (0 ,
0) exhibits negative values for r ≥
1, increases rapidly to show maximum around r = 2, reduces gradually and vanishes for r ≥
5. This range of negativity is greater than thatof the dotted curve. This is in a good agrement with the behavior of the second-order correlationfunction. Finally, comparison among the curves in Fig. 5(b) confirms the fact: for certain valuesof the system parameters, the asymmetric case can provide nonclassical effects greater than thoseof the symmetric one.
VI. CONCLUSION
In this paper we have studied the three-mode squeezed operator, which can be implementedfrom the triply coincident nonlinearities in periodically poled
KT iOP O . The action of this oper-ator on the three-mode coherent and number states is demonstrated. We have studied quadraturesqueezing, second-order correlation function, Cauchy-Schwartz inequality and quasiprobability dis-tribution function. The obtained results can be summarized as follows. Generally, the single-modevacuum or coherent states, as outputs from the three concurrent amplifiers are not nonclassicalstates. The system can exhibit two-mode and three-mode squeezing. The amount of the two-modesqueezing generated by the asymmetric case is much greater than that of the symmetric case.9Three-mode squeezed coherent (number) states cannot (can) exhibit sub-Poissonian statistics. Toobtain maximum sub-Poissonian statistics from a particular mode, under the action of the op-erator (3), it must be prepared in the nonclassical state and the other modes in states close tothe classical ones. We have found that the Cauchy-Schwartz inequality can be violated for bothcoherent states and number states. The origin in the violation is in the strong quantum correla-tion among different modes. For the Fock-state case, the asymmetry in the system enhances therange of nonlinearities for which V jk is nonclassical compared to that of the symmetric one. Inthe framework of the quasiprobability distribution we have shown that the nonclassical effects canbe transferred from one mode to another under the action of the operator (3). The amount oftransferred nonclassicality is sensitive to the values of the squeezing parameters. Interestingly, thesystem can generate particular types of the Schr¨odinger-cat states for certain values of the systemparameters. Generally, we have found that the nonclassical effects generated by the operator (3)are greater than those obtained from the operator TMS [14]. Finally, the asymmetry in the threeconcurrent nonlinearities process is important for obtaining significant nonclassical effects. References [1] Hillery M 2000
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