Revival-collapse phenomenon in the quadrature squeezing of the multiphoton Jaynes-Cummings model with the binomial states
aa r X i v : . [ qu a n t - ph ] A p r Revival-collapse phenomenon in the quadrature squeezing of themultiphoton Jaynes-Cummings model with the binomial states
Faisal A. A. El-Orany ∗ Department of Mathematics and computer Science,Faculty of Science, Suez Canal University 41522, Ismailia, Egypt (Dated: November 11, 2018)In this paper we study the interaction between two-level atom and quantizedsingle-mode field, namely, Jaynes-Cummings model (JCM). The field and the atomare initially prepared in the binomial state and the excited atomic state, respectively.For this system we prove that the revival-collapse phenomenon exhibited in theatomic inversion of the standard JCM can be numerically (naturally) manifested inthe evolution of the squeezing factor of the three-photon (standard) JCM providedthat the initial photon-number distribution of the radiation has a smooth envelope.
PACS numbers: 42.50Dv,42.60.Gd
I. INTRODUCTION
Jaynes-Cummings model (JCM) [1] is one of the fundamental systems in the quantumoptics. The simplest form of the JCM is the single quantized mode interacting with the two-level atom. Various phenomena have been realized for this system such as revival-collapsephenomenon (RCP) in the evolution of the atomic inversion [2], sub-Poissonian statistics andsqueezing, e.g., [3]. Actually, the RCP represents the most important phenomena reportedto this model since it manifests the granular nature of the initial field distribution [4] aswell as the strong entanglement between the radiation field and the atom. The RCP hasbeen observed via the one-atom mazer [5] and also using technique similar to that of theNMR refocusing [6]. It is worth mentioning that the RCP has been also remarked in theevolution of different quantities in the nonlinear optics such as the mean-photon number of ∗ Electronic address: el˙[email protected] the Kerr nonlinear coupler [7] and the photon-number distribution of the single-mode [8] andtwo-mode [9] squeezed coherent states with complex squeeze and displacement parameters.In the latter two cases the RCP occurs in the photon-number domain rather than in theinteraction time domain.There is an important class of nonclassical states, namely, binomial states. The binomialstate (BS) is an intermediate state between the Fock state and the coherent state [10], i.e. itis linear combination of Fock states weighted by binomial distributions. The BS can exhibitmany nonclassical effects, e.g., squeezing, sub-Poissonian statistics and negative values inthe Wigner function [11]. Recently, the BSs have been proposed as reference field states inschemes measuring the canonical phase of quantum electromagnetic fields [12]. The BS canbe generated by a classical current interacting with two quantized radiation fields [10, 13] aswell as via quantum state engineering [14]. In the latter technique the resonant interactionof N ′ consecutive two-level atoms with the cavity initially prepared in its vacuum state isconstructed and the desired cavity field state can be obtained from total state reductionby performing measurement on the atoms coming out of the cavity [15]. Quite recently,an efficient scheme for generating and detecting two-photon generalized binomial state in asingle-mode high- Q cavity is described in [16]. Evolution of the BS with the JCM has beeninvestigated for the single-photon JCM [17], the two-photon JCM [18] and the single-photonKerr-nonlinear JCM [19]. The object of these studies is to investigate the construction ofdifferent phenomena when the field evolves gradually from the Fock state to the coherentstate. It is worth referring that the evolution of the BS with the JCM can provide differentbehaviors than those with the coherent state, e.g., under certain conditions the evolutionof the atomic inversion related to the BS exhibits a very steady beat phenomenon similarto that found in the classical physics [19]. The superposition of the BS (SBS) has beendeveloped [20, 21] as: | M, η i ǫ = M X n =0 C Mn ( η, ǫ ) | n i , (1)where the coefficient C Mn ( η, ǫ ) takes the form C Mn ( η, ǫ ) = λ ǫ q M !( M − n )! n ! η n (1 − | η | ) [ M − n ] [1 + ( − n ǫ ] , | λ ǫ | − = 1 + ǫ + 2(1 − | η | ) M ǫ, (2)where M is a positive integer, 0 < | η | ≤ ǫ is a parameter taking one of the values 0 , − η to be real. In the limiting cases ( ǫ, η ) → (0 ,
1) and ( η, M ) → (0 , ∞ )such that M η = α the state (1) reduces to the Fock state | M i and the superposition ofthe coherent state | α i ǫ [22], respectively.There is another type of the SBS, which is called the phased generalized binomial state[21]. This type of state is represented by the superposition of the even or odd binomialstates. As an example we give the definition of the orthogonal-even binomial state as | M, η i e = A [ M/ X n =0 C M n ( η, | n i , (3)where C M n ( η,
0) can be obtained from (2) and A is the normalization constant having theform A = 41 + (1 − | η | ) M + 2Re(1 − | η | + i | η | ) M . (4)Using appropriate limit the state (3) tends to the orthogonal-even coherent state [23]. It isworth mentioning that the common property of the binomial states is that the probabilityof detecting m quanta when m > M is zero. From the above information one can realizethat the SBS is one of the most generalized states in quantum optics.Recently, for the JCM it has been shown that there is a relationship between the atomicinversion and the quadrature squeezing [24, 25]. More illustratively, for particular typeof initial states, e.g. l -photon coherent states, the squeezing factors can naturally providecomplete information on the corresponding atomic inversion. Nevertheless, for the initialcoherent state it has been numerically shown that the evolution of the quadrature squeezingof the three-photon JCM reflects the RCP involved in the atomic inversion of the standard,i.e. the single-photon, JCM. These relations have been obtained based on the fact that for theinitial l -photon coherent state and coherent state the harmonic approximation is applicable.In this paper we show that these relations exist also for any arbitrary initial field statesprovided that their photon-number distributions have smooth envelopes. In doing so westudy the evolution of the JCM with the SBS. For this system we obtain various interestingresults. For instance, we show that the relations between the atomic inversion and thequadrature squeezing are sensitive to the interference in phase space. Additionally, the odd N th-order squeezing of the standard JCM with the even-orthogonal binomial state exhibitsRCP as that of the corresponding atomic inversion. The motivation of these relations isthat the RCP exhibited in the evolution of the atomic inversion can be measured by thehomodyne detectors [26]. This is supported by the recent developments in the cavity QEDin which the homodyne detector technique has been applied to the single Rydberg atom andone-photon field for studying the field-phase evolution of the JCM [27].We construct the paper in the following order: In section 2 we give the basic relationsand equations related to the system under consideration. In sections 3 and 4 we investigatenaturally and numerically the occurrence of the RCP in the higher-order squeezing. Insection 5 we summarize the main results. II. BASIC EQUATIONS AND RELATIONS
In this section we give the basic relations and equations, which will be used in the paper.Specifically, we develop the Hamiltonian of the system and its wavefunction as well as thedefinition of the quadrature squeezing. Also we shed light on the relation between thephoton-number distribution and the atomic inversion.The Hamiltonian controlling the interaction between the two-level atom and the k th-photon single-mode field in the rotating wave approximation is [28]:ˆ H ~ = ω ˆ a † ˆ a + 12 ω a ˆ σ z + λ (ˆ a k ˆ σ + + ˆ a † k ˆ σ − ) , (5)where ˆ σ ± and ˆ σ z are the Pauli spin operators; ˆ a (ˆ a † ) is the annihilation (creation) oper-ator denoting the cavity mode, ω and ω a are the frequencies of the cavity mode and theatomic transition, respectively; λ is the atom-field coupling constant and k is the transitionparameter.We consider that the field and atom are initially prepared in the SBS (1) and the excitedatomic state | + i , respectively. Also we restrict the investigation to the exact resonance case.Under these conditions the dynamical state of the system can be expressed as | Ψ( T ) i = M X n =0 C Mn ( η, ǫ ) [cos( T ν n,k ) | + , n i − i sin( T ν n,k ) |− , n + k i ] , (6)where T = λt, ν n,k = q ( n + k )! n ! and |−i denotes the ground atomic state. The atomic inversionassociated with (6) is h σ z ( T ) i = M X n =0 P ( n ) cos(2 T ν n,k ) , (7) FIG. 1: The P ( m ) against m for the BS. (a) η = 0 . M = 50 (long-dashed curve), 100 (short-dashed curve) and 370 (solid curve). (b) ( η, M ) = (0 . , . , . , ǫ, η, M ) =(1 , . , h σ z ( T ) i against the scaled time T for k = 1 when the field is initiallyprepared in the SBS with different values of η, M and ǫ . (a) ( η, M, ǫ ) = (0 . , ,
0) (curve A) and(0 . , ,
0) (curve B), whereas the curve C is given for the initial orthogonal-even binomial statewith (
M, η ) = (370 , . M = 200 and ( η, ǫ ) = (0 . ,
1) (curve A), (0 . ,
0) (curve B) and (0 . , , , where P ( n ) = | C Mn ( η, ǫ ) | . To understand the relation between P ( m ) , h σ z ( T ) i and quadra-ture squeezing we plot P ( m ) and h σ z ( T ) i in Figs. 1 and 2, respectively, for the given valuesof the parameters. Fig. 1(a) gives the development of the binomial state to the coherentstate. This is obvious from the solid curve in Fig. 1(a) as well as the curve A in the Fig.2(a), which represents the RCP of the coherent state with α = p M η = √ . P ( m ) exhibits smooth envelope the h σ z ( T ) i provides the RCP. Moreover,the interference in phase space manifests itself as two times revival patterns in the evolutionof the h σ z ( T ) i compared to those related to the BS (compare the curves A and B in Fig.2(b)). Also from the dashed curve in Fig. 1(b) the maximum value of the P ( m ) is close to m ≃ ¯ n = h ˆ a † (0)ˆ a (0) i . For the future purpose we have plotted curve C in Fig. 2(a) for the h σ z ( T ) i of the initial orthogonal-even BS.The different moments of the operators ˆ a † and ˆ a for the state (6) can be evaluated as h ˆ a † s ( T )ˆ a s ( T ) i = M − s P n =0 (cid:0) C Mn + s ( η, ǫ ) (cid:1) ∗ C Mn + s ( η, ǫ ) (cid:20) cos( T ν n + s ,k ) cos( T ν n + s ,k ) √ ( n + s )!( n + s )! n ! + sin( T ν n + s ,k ) sin( T ν n + s ,k ) √ ( n + k + s )!( n + k + s )!( n + k )! (cid:21) , (8)where s and s are positive integers and M < s . Finally, the N th-order quadraturesqueezing operators are defined by ˆ X N = (ˆ a N + ˆ a † N ) , ˆ Y N = i (ˆ a N − ˆ a † N ), where N is apositive integer. The squeezing factors associated with the ˆ X N and ˆ Y N can be, respectively,expressed as [29]: F N ( T ) = h ˆ a † N ( T )ˆ a N ( T ) i + Re h ˆ a N ( T ) i − h ˆ a N ( T ) i ) ,S N ( T ) = h ˆ a † N ( T )ˆ a N ( T ) i − Re h ˆ a N ( T ) i − h ˆ a N ( T ) i ) . (9)Now we are in a position to investigate the relation between the atomic inversion of thestandard, i.e. k = 1, JCM denoting by h σ z ( T ) i k =1 and the quadrature squeezing. This willbe done in the following sections. III. NATURAL APPROACH
Natural approach is based on the fact: the quantity h σ z ( T ) i + h ˆ a † ( T )ˆ a ( T ) i is a con-stant of motion and hence h σ z ( T ) i and h ˆ a † ( T )ˆ a ( T ) i can carry information on each others[24]. Furthermore, this approach can be generalized to find a relation between h σ z ( T ) i and h ˆ a † N ( T )ˆ a N ( T ) i , where N is a positive integer, as we show shortly. The discussion will berestricted to the case k = 1. Now the question is that for the JCM which type of thebinomial states making the N th-order squeezing factors provide complete information onthe corresponding atomic inversion? The answer to this question can be realized from (9).Precisely, when there is a type of binomial states satisfying simultaneously the conditions: h ˆ a N ( T ) i = 0 , h ˆ a N ( T ) i = 0 . (10)In this case the squeezing factors reduce to h ˆ a † N ( T )ˆ a N ( T ) i , which can be connected withthe corresponding h σ z ( T ) i . As an example of these states is the orthogonal-even binomialstates (3) provided that the squeezing order N is odd integer, i.e. 2 N + 1. For this case onecan easily check that the conditions (10) are fulfilled and hence the squeezing factors reduceto F N +1 ( T ) = h ˆ a † N +1 ( T )ˆ a N +1 ( T ) i = h ˆ a † N +1 (0)ˆ a N +1 (0) i + ( N + ) h ˆ a † N (0)ˆ a N (0) i− ( N + ) A M/ P n =0 | C M n | n )!(4 n − N )! cos(2 T ν n, ) . (11)Using suitable limits for the summation in (11) and by means of the following relation M ! = ( M − N )! M N N − Y j =0 (1 − jM ) (12)we arrive at F N +1 ( T ) = h ˆ a † N +1 (0)ˆ a N +1 (0) i + ( N + ) h ˆ a † N (0)ˆ a N (0) i− ( N + ) | η | N M N " N − Q j =0 (1 − jM ) A M/ − N ] P n =0 | C M − N n | cos(2 T ν n +2 N, ) . (13)For finite (large) values of N ( M ) with 0 < η <
1, i.e. the P ( m ) has smooth envelope, wecan use the substitutions ν n +2 N, ≃ ν n, and | C M − N n | ≃ | C M n | and hence the expression(13) can be modified to give the rescaled squeezing factor W N ( T )(= h σ z ( T ) i k =1 ) throughthe relation: W N ( T ) = 2 h ˆ a † N +1 (0)ˆ a N +1 (0) i + (2 N + 1) h ˆ a † N (0)ˆ a N (0) i − F N +1 ( T )(2 N + 1) h ˆ a † N (0)ˆ a N (0) i b , (14) FIG. 3: The third-order rescaled squeezing factor given by (14) against the scaled time T , whenthe field is initially in the orthogonal-even BS with ( M, η ) = (370 , . where the subscript b in the denominator means that the quantity h ˆ a † N (0)ˆ a N (0) i is relatedto the BS. Now we are in a position to check the validity of the (14). Thus we plot (14) inFig. 3 for the third-order squeezing and the given values of the interaction parameters. Weshould stress that in Fig. 3 we have used the explicit form for F N +1 given by (11). Thecomparison between the curve C in Fig. 2(a) and Fig. 3 demonstrates our conclusion: forparticular type of binomial states the squeezing factor can provide complete information onthe corresponding atomic inversion. The origin in this is that the expressions of the h σ z ( T ) i and h ˆ a † N ( T )ˆ a N ( T ) i depend on the diagonal elements of the density matrix of the systemunder consideration. IV. NUMERICAL APPROACH
In this section we study the possibility of obtaining information on the h σ z ( T ) i k =1 fromthe squeezing factors of the k th-photon JCM, when the field is initially prepared in the SBS.Our object is to find the value of the transition parameter k ( k >
2) for which one or bothof the squeezing factors produce RCP as that involved in the h σ z ( T ) i k =1 [24]. From (9) theRCP can likely occur in the F N ( T ) ( S N ( T )) only when Re h ˆ a N ( T ) i = 0 (Im h ˆ a N ( T ) i = 0)since these quantities are squared, i.e. they destroy the RCP if it exists. According tothis fact the occurrence of the RCP in F N or in S N depends on the values of the ǫ and N .Moreover, for k > h ˆ a † N ( T )ˆ a N ( T ) i exhibits chaotic behavior and hence wecan use h ˆ a † N ( T )ˆ a N ( T ) i ≃ h ˆ a † N (0)ˆ a N (0) i . From this discussion we can conclude that if thesqueezing factors exhibit RCP this will be related to the quantity Re h ˆ a N ( T ) i . Thus wetreat this quantity in a greater details. From (8) and after minor algebra we arrive at h ˆ a N ( T ) i = | η | N M N (1 −| η | ) N M P n =0 | C Mn ( η, ǫ ) | (s N Q j =1 (1 − ( n +2 N − j ) M ) ) × h cos( T ν n +2 N,k ) cos(
T ν n,k ) + s N − Q j =0 (1+ ( k +2 N − j ) n )(1+ N − jn ) sin( T ν n +2 N,k ) sin(
T ν n,k ) i . (15)In (15) we have extended the upper limit of the summation from M − N to M using thefact l ! = −∞ when l < h σ z ( T ) i k =1 . Moreover, we assume that M >> N , 0 < η < n is very large. Therefore,the quantity in the square root in the second line of (15) tends to unity. Additionally, for ǫ = 0 , i the P ( n ) exhibits smooth envelope and then the terms contributing effectively tothe summation in (15) are those close to n ≃ ¯ n = M | η | . In this case the quantity in thecurely curves in (15) can be simplified as vuut N Y j =1 (1 − ( n + 2 N − j ) M ) = vuut N Y j =1 (1 − | η | − (2 N − j ) M ) ≃ (cid:0) − | η | (cid:1) N , (16)where we have considered ϑ/M → ϑ is a finite c-number and M >> ϑ . On the otherhand, when ǫ = 1, say, the P ( n ) exhibits oscillatory behavior with maximum value around n ≃ ¯ n (see the dashed curve in the Fig. 1(b)) and we arrive at0 FIG. 4: The rescaled squeezing factors Q ( T ) and Q ( T ) as indicated against the scaled time T for different values of η, M and ǫ . (a) ( η, M, ǫ ) = (0 . , ,
0) (curve A) and (0 . , ,
0) (curveB). (b) M = 200 and ( η, ǫ ) = (0 . ,
1) (curve A), (0 . ,
0) (curve B) and (0 . ,
0) (curve C). (c)( η, M, ǫ ) = (0 . , ,
0) (curve A) and (0 . , ,
0) (curve B). The curves in (b) and (c) are shiftedfrom the bottom by 0 , , ,
2, respectively, whereas in (a) are shifted by 0 , vuut N Y j =1 (1 − ( n + 2 N − j ) M ) ≃ (cid:16) − ¯ nM (cid:17) N = (cid:18) − | η | (1 − z M − )1 + z M (cid:19) N = (1 −| η | ) N (cid:18) z M − z M (cid:19) N , (17)where we have used the mean-photon number of the even binomial states as [20]:¯ n = | η | M (1 − z M − )(1 − z M ) , z = 1 − | η | . (18)It is evident that | z | < < η < z M − ≃ M is very large. Thus theresult given by (16) is valid for all values of ǫ and hence the expression (15) reduces to h ˆ a N ( T ) i ≃ | η | N M N M X n =0 | C Mn ( η, ǫ ) | cos[ T ( ν n +2 N,k − ν n,k )] . (19)Comparison between (7) (i.e. h σ z ( T ) i k =1 ) and (19) shows that the two expressions canprovide similar dynamical behavior only when the arguments of the cosines are comparable.This is regardless of the different scales resulting from the pre-factor M | η | N in (19). Theproportionality factor µ N , say, which makes the dynamical behaviors in the two expressions1similar, can be evaluated from the following relation µ N = ν n +2 N,k − ν n,k √ n +1 , = n k s k Q j =1 (1+ jn ) " N Q j =1 ( n + k + j ) − N Q j =1 ( n + j ) n N + 12 √ n s N Q j =1 (1+ jn ) "s N Q j =1 (1+ k + jn )+ s N Q j =1 (1+ jn ) . (20)It is worth recalling that ¯ n is very large, P ( n ) exhibits smooth envelope, i.e. n ≃ ¯ n , and thesqueezing-order N is finite. Therefore, by applying the Taylor expansion for different squareroots in (20) we obtain [25]: µ N ≃
14 [2
N k ¯ n k − + ¯ n k − ( ... ) + ¯ n k − ( ... ) + ... ] . (21)From (21) it is evident that the RCP can occur in the squeezing factor only when k = 3and hence µ N = N . In this case we have neglected such type of terms ¯ n − , ¯ n − , ..., where¯ n is very large. From the above investigation one can realize that the N th-order rescaledsqueezing factor, which can give complete information on the h σ z ( T ) i k =1 , is Q N ( T ) = h ˆ n (0) i Nb − V N ( T ) h ˆ n (0) i Nb , (22)where V N ( T ) = S N ( T N ) for ǫ = 0 ,F N ( T N ) for ǫ = i,S N ( T N ) = F N ( T N ) for ǫ = ± , N = 2 m ′ + 1 ,S N ( T N ) for ǫ = ± , N = 2 m ′ (23)and m ′ is a positive integer. In the derivation of the formula (22) we have considered thatthe mean-photon numbers of the BS and the SBS are the same. This is correct for 0 < η < n . It is worth mentioning that the formula (22) is valid for the initial superpositionof the coherent states, too. Now we check the validity of (22) by plotting Figs. 4(a)–(c)for the given values of the interaction parameters. The comparison between the curves inFigs. 4(a)-(b) and the corresponding ones in the Figs. 2 leads to the following fact: when0 < η < n is large, regardless of the values of ǫ , the Q ( T ) copies well with the2 h σ z ( T ) i k =1 . Nevertheless, when ¯ n is relatively small (with P ( m ) has a smooth envelope) theRCP can be established in Q ( T ), but the overall behavior could be different from that ofthe h σ z ( T ) i k =1 . This result is obvious when we compare the curves A in Fig. 2(a) and Fig.4(a), where one can observe | Q ( T ) | >
1. Fig. 4(c) is given for the higher-order squeezing.The comparison between the curve A in this figure and the curve C in Fig. 4(b) leadsto that the normal squeezing can provide better information on the h σ z ( T ) i k =1 than theamplitude-squared squeezing. Nevertheless, the information obtained from the higher-ordersqueezing can be improved by increasing the value of the ¯ n (compare the curves A and B inFig. 4(c)). V. CONCLUSION
In this paper we have shown that for the JCM there is a relationship between the quadra-ture squeezing and the atomic inversion provided that the initial photon-number distributionexhibits smooth envelope. This fact has been proved using one of the most general quantumstate, namely, the superposition of the binomial states. Precisely, we have shown that forparticular types of the initial binomial states the N th-order squeezing factor can naturallygive complete information on the corresponding atomic inversion. Also we have numericallyshown that the N th-order squeezing factor of the three-photon JCM can provide completeinformation on the h σ z ( T ) i k =1 . These relations exist only when the P ( m ) exhibits smoothenvelope and ¯ n is large. Moreover, as the squeezing order N increases the values of the ¯ n have to be increased for getting better information from the Q N ( T ) on the atomic inversion.Finally, the results obtained in this paper are valid also when the field is initially preparedin the cat states. Acknowledgement
The author would like to thank the Abdus Salam International Centre for TheoreticalPhysiscs, Strada Costiers, 11 34014 Trieste Italy for the hospitality and financial support3under the system of associateship, where a part of this work is done. [1] Jaynes E T and Cummings F W 1963 Proc. IEEE Phys. Rev. Lett. Phys. Rev. A J. Phys. A Phys. Rep.
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