Some Features on Entropy Squeezing for Two-Level System with a New Nonlinear Coherent State

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1 Joural of Quatum Iformatio Sciece, 4, 4, 7-8 Published Olie March 4 i SciRes. Some Features o Etropy Squeezig for Two-Level System with a New Noliear Coheret State Salama I. Ali, Sayed Abdel-Khalek Mathematics Departmet, Faculty of Sciece, Al-Azher Uiversity, Cairo, Egypt Mathematics Departmet, Faculty of Sciece, Sohag Uiversity, Sohag, Egypt salama5laser@yahoo.com Received 3 Jauary 4; revised February 4; accepted March 4 Copyright 4 by authors ad Scietific Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY. Abstract Etropy squeezig is a importat feature i performig differet tasks i quatum iformatio processig such as quatum cryptography ad superdese codig. These quatum iformatio tasks deped o fidig the states i which squeezig ca be created. I this article, a ew feature o etropy squeezig for a two level system with a class of oliear coheret state (NCS is observed. A iterestig result o the compariso betwee the coheret state (CS ad NCS is eplored. The ifluece of the Lamb-Dick parameter i both absece ad presece of the Kerr medium is eamied. A rich feature of etropy squeezig i the case of NCS, which is observed to describe the motio of the trapped io, has bee obtaied. Keywords Etropy Squeezig; Kerr-Like Medium; Noliear Coheret State. Itroductio It is kow that quatum etagled state plays a importat role i the fields of quatum iformatio theory as well as quatum teleportatio ad computatio. The etropy which automatically icludes all momets of the desity operator has bee show to be a very useful operatioal measure of the purity of the quatum state. The most importat ad iterestig work to uderstad relatio betwee etropy ad iformatio was doe by Shao [], who itroduced the etropy (Shao etropy ito commuicatios theory. Recetly it has bee show that oliear coheret states are useful i the descriptio of the motio of a trapped io ad various o-classical properties of such states have also bee studied []. We ote that i Refs [] ad [3] oliear coheret states How to cite this paper: Ali, S.I. ad Abdel-Khalek, S. (4 Some Features o Etropy Squeezig for Two-Level System with a New Noliear Coheret State. Joural of Quatum Iformatio Sciece, 4,

2 have bee defied as the right eigestate of a geeralized aihilatio operator  (which emerges from the Hamiltoia describig the dyamics this is because i the case of oliear algebras the commutator AA, is ot a costat or a liear fuctio of the geerators of the algebra but oliear i the i the geerators. As a cosequece it is difficult to obtai a eplicit form of oliear coheret state costructed via the displacemet operator techique. The oliear coheret state (NCS was itroduced as a ew state of the source of the coheret field to describe some of the o-classical properties like squeezig ad Sub-Poissia behavior [4]. There are some previous studies o the etropy squeezig of a two-level atom, such as oe photo trasitio [5], the oliear Kerr medium [6] ad degeerate two-photo process [7]. The authors of these papers have focused oly o the iitial coheret state of the field. Recetly, much attetio has bee draw to squeezig i a esemble of atoms illumiated with light, ivolvig quatum oise ad atomic spi polarizatio measuremet [8], ad quatum-cotrolled few-photo states geerated by squeezed atoms [9]. These studies of atomic squeezig are based o the Heiseberg ucertaity relatio (HUR, which is regarded as the stadard limitatio o measuremets of quatum fluctuatios. HUR is formulated i terms of the variaces or stadard deviatios of the system observable. As a alterative to the HUR, Hirschma [] studied quatum ucertaity by usig quatum etropy theory, ad obtaied a etropic ucertaity relatio for positio ad mometum which ca overcome the limitatios of the HUR. The etropy squeezig ad variace squeezig for the etagled sate of a sigle two-level atom iteractig with a sigle electromagetic field i a squeezed vacuum a broad badwidth are studied []. Also, the similarities ad differeces of both reservoirs for the two differet models have bee eplaied through some calculatios, such as the atomic iversio ad the vo Neuma etropy. It is to be oted that cosiderig a mode structure plays a role like the squeezig parameter i the case of a squeezed vacuum reservoir. Also, the etropy squeezig of a two-level atom drive by a strog classical field ad damped ito a modeled reservoir with o-flat desity of modes has bee ivestigated []. O the other had, the dyamics of a sigle atom etropy squeezig of the two-qubit system, i the presece of local squeezed reservoirs, has bee discussed. Our aim i the preset paper is to ivestigate the etropy squeezig of a two level atom whe the iitial state of the field is take to be NCS ad discuss differet features of etropy squeezig i the case of NCS. These features are coected with the Lamb-Dick parameter. Here we also eamie the ifluece of a oliear medium ad the detuig parameter o the squeezig parameter of the atomic operators S, S y ad S z. The orgaizatio of the paper is arraged as follows. I Sectio, we preset a brief review of the Hamiltoia model ad give a eact epressio for the desity matri ρ ( t. I Sectio 3, we employ the desity matri to ivestigate the properties of the etropy squeezig. Fially, we give our discussio i Sectio 4.. Dyamics of Oe Photo JCM I this paper, we focus our attetio to a quatum optical model, where a sigle two-level atom via oe photo process, iteracts with a sigle quatized cavity mode of the radiatio field. The the Hamiltoia of the above system of iterest may be writte as H ω aa ω S κa a + a S + as, ( = f + A z + γ ( +? where ω f is the field frequecy, ω A is the atomic frequecy, â ad â are the aihilatio ad the crea- tio operators for the mode of the cavity field satisfyig aa, = ad Ŝ ± ad S z are the atomic spi operators defied by S z = ( ( S = ad S = + We deoted by ad the upper ad lower states of the atom, respectively ad γ is the effective couplig costat. Also, we deoted by κ the dispersive part of the third-order oliearity of the Kerr-like medium, with the detuig parameter = ωa ωf. Therefore, we employ the uites of =. The effective Hamiltoia ca be writte as H = H + H where, eff I 7

3 ( H I ( z ( aa S z ( a S as H = ω f aa+ S, + H I = γ Sz + γκa a + γ + + S. I. Ali, S. Abdel-Khalek For coveiece, we take, χ = γκ, δ = γ ad assume that the atom is iitially i the superpositio state A ( = A + A. Also, we assume that the field is iitially i the ew oliear coheret state (NCS, ψ = α = B, (4 where, B( is the distributio fuctio of the NCS. The NCS is defied as α D d = (3 = i.e., it is a coheret state (CS correspodig to the secod algebra [3]. Oe ca write α i the followig form [4]-[7] [3] α d α = N, d =! where N is a ormalizatio costat, which ca be determied from the coditio αα = ad is give d by d = ad the coefficiets { } N d! αα = = While d = f!, it is clear that for differet choices of the oliearity fuctio, we shall get differet oliear coheret states. I the preset case we choose a oliearity fuctio, which has bee used i the descriptio of the motio of a trapped io [4]. = ( η ( + ( η where η is kow as the Lamb-Dick parameter ad. f L L L m are geeralized Lagurre polyomials give by i ( η m m+ i L ( η = i= + i i! (5 Clearly f ( = whe η = ad i this case oliearity coheret states become the stadard coheret states. However, whe η oliearity starts developig with degree of depedig o the magitude of η [4]. The the iitial state of the atom-field couplig system reads as ψ ( = F{ A, + A, } (6 = where, B =, A = cos( ad A = si ( ep( iφ. Here [, π] deotes the iitial co- = herece of the two-level atom ad ϕ [, π] is the relative phase betwee the upper ad lower states of the two-level atom. Thus the iitial desity operator of the system is give by ρ( = ρa ( ρ( where, F ρ ( = α α ad ρ ( = ψ ( ψ (, ρ ( describes the iitial values for the field-atom desity F A A A operator. At ay time t > the solutio of the Schrödiger equatio d ψ i dt ( t for the state vector ψ ( t with the iitial coditio (6 is = I ( t = H ψ (7 {,,,, } ψ ( t = B ψ ( t + ψ ( t (8 73

4 ad the desity matri of system is ρ t = ψ t, ψ mt,,, m+ ψ t, ψ mt,,, m = m= where the coefficiet ψ ( t, ad ( t, + ψ t, ψ mt,,, m+ ψ t, ψ mt,,, m ψ are give by si ϒ si ϒ ψ (, t = ep( iγtχ A B cos i ( χ ia B( ϒ + + Ω Ω si ϒ si ϒ ψ (, t = ep iγtχ( A B cos ϒ + i χ( ia B Ω Ω (9 ( where, ϒ = γ tω, Ω = χ + + with Ω is the Rabi frequecy which depeds o the detuig parameter ad oliear medium parameter. 3. Atomic Iversio We maily devote the preset sectio to cosiderig the atomic iversio, from which the pheomeo of collapses ad revivals ca be observed. However, we shall first itroduce some epressios for the probability amplitude. The epressios ψ ( t, ad ψ ( t, represet the probabilities that at time t, the field has photos preset ad the atom is i level ad respectively. The probability P( t, that there is photos i the field at time t is therefore obtaied by takig the trace over the atomic states, i.e., where (, (, ψ (, ψ (, P t = t + t ( P is the probability that there are photos preset the field at time t =, which is give for a NCS for the field by (, B P = ( Aother importat quatity oe may cosider is the atomic iversio W( t, which is related to the probability amplitudes ψ ( t, ad ψ ( t, by the epressio W t t t (3 = = ψ (, ψ (, Thus from Equatio (3 ad after some rearragemets, we ca obtai si cos ϒ = cos ( χ si cos ϒ + ϒ + ( χ( si ϒ W t B = Ω Ω γ + si ϒ si + γ si ϒ cos + B( B( + = Ω Ω si si ϒ si ϒ B B( γ B B( siφ B B( γ = Ω Ω si ϒ si ϒ δ χ χ cosφ Ω Ω [ ] B B( ( (4 74

5 4. Etropy Squeezig I this paper, we use the Heiseberg ucertaity relatio (HUR to study the squeezig iformatio etropy. It has bee poited out that the Heiseberg ucertaity relatio (HUR caot give sufficiet iformatio o the atomic squeezig i some cases [3]. For istace, for a two-level atom, characterized by Pauli operator S, S y ad S z, the ucertaity relatio is give by S Sy Sz where the Pauli operators S, S y ad S z, satisfyig the commutatio S, i S y = Sz ad α α α S = S S. I this way, the fluctuatio i the compoet squeezed if Ŝ α satisfies the coditio Ŝα of the atomic dipole is said to be ( S z V Sα = Sα <, α = or y (5 A optimal etropic ucertaity relatio for sets of N + complemetary observables with o-degeerate eigevalues i a eve N-dimesioal Hilbert space has bee recetly ivestigated usig quatum etropy theory [5]. It takes the form where ( H S α l + + l +, (6 N + N N N N α = H S α represets the iformatio etropy of the variable Ŝα. The aim of this paper is to use etropy ucertaity relatio EUR (6 as a geeral criterio for the squeezig i terms of iformatio etropy for a two-level atom i the Jayes-Cummig model with oe-photo process i a o-liear Kerr medium. The probability distributio for N possible outcomes of measuremets for a arbitrary quatum state of a operators Ŝ α is ( Pi S α = Ψ α i ρ Ψ α i, where Ψ αi is a eigevector of the operator Ŝα such that S α Ψ αi = λ Ψ,,,,,,, i i yzi N α α α = =. The correspodig Shao iformatio etropies are the defied as N ( ( ( i i H S = P S l P S, =, y, z. (7 α α α α i= To obtai the Shao iformatio etropies of the atomic operators S, S y ad S z for a two-level atom, with ρ t, thus we have the followig epressio, so that N =, oe ca use the reduced atomic desity operator ( α α α α α H S = ρ t l ρ t ρ t l ρ t, α yz,, + + = ( H S = Re( ρ t l Re( ρ ( t Re( ρ ( t l Re ( ρ ( t, + + ( y H S = Im ( ρ t l Im ( ρ ( t Im ( ρ ( t l Im ( ρ ( t, + + ( z where ρ ( t, ρ ( t,re ρ ( t ad Im ( ( t ( (8 (9 H S = ρ t l ρ t ρ t l ρ t, ( ρ are give by 75

6 ρ si si ϒ + ϒ = ϒ + = + + Ω = Ω ( t cos B cos ( χ si B( + si ϒ si ϒ si ( B B( cos si ( φ χ + ϒ cos( φ = Ω Ω B ( γ si ϒ cos Ω ( ρ si ϒ si ϒ = ϒ + + = Ω = Ω ( t si B cos χ( cos B( si ϒ si ϒ + si ( { B B( } cos si ( φ χ( ϒ cos( φ = Ω Ω = { R v ( } Re ρ ( t = U ( t, cos γt( R R + U t, si γt R R (3 = { R v ( } Im ρ ( t = U ( t, si γt( R R U t, cos γt R R (4 δ U si R, t = X cosφ + Ysi φ B + B + B + si ϒ si ϒ cosφ ΩΩ si ϒsi ϒ cos + δ si ϒsi ϒ si + B B( B B( + ΩΩ ΩΩ U si V, t = Xsiφ Y cosφ B B + B X + si ϒ si ϒ siφ ΩΩ si ϒ cos ϒ cos + si ϒsi ϒ si + B B( B B( + Ω Ω δδ si ϒ si ϒ δ si ϒ cos ϒ δ si ϒ cos ϒ = cos ϒ cos ϒ Y = + ΩΩ Ω Ω Sice the ucertaity relatio of the etropy ca be used as a geeral criterio for the squeezig of a atom, therefore for a two-level atom. For N =, we have H( S α l, ad the Shao iformatio etropies of the operators S, S, S will satisfy the iequality y z H( S ( ( H Sy H Sz l. I other words, if we defie δ H( S ep ( α = H Sα, the we ca write H( S ( ( H Sy H Sz 4. It is evidet that δ H( S α = correspods to the atom beig i a pure state ad H( S α S, t Sy t ad S z ( t, where δ H( S ad H( Sy ( (5 (6 (7 + + (7 δ δ δ (8 δ = correspods to the atom beig i a mied state. The EUR (6 shows the impossibility of simultaeously havig complete iforδ respectively measure matio about the observables 76

7 the ucertaities of the polarizatio compoets S ad S y. Now, we defie the squeezig of the atom usig EUR (6, amed squeezig etropy [5]. The fluctuatio of the compoet S α ( α = or y of the atom dipole are said to be squeezed i etropy if the iformatio etropy H( S α ( t of Ŝ α satisfies the coditio E( S ( α = δh Sα <, α =, y (9 δ H S what follows we shall cosider the effect of detuig parameter ad the oliear Kerr like medium o the dyamical behavior of the squeezig etropy of the system uder cosideratio. 5. Numerical Computatio O the basis of the aalytical solutio preseted i the previous sectio, we shall study umerically the depe-, W t, for various parameters of the dece of the ( ( y ( z E S E S etropy squeezig ad the atomic iversio oe photo model. We recall that time t has bee scaled; oe uit of time is give by the iverse of the couplig costat λ. I all our plots we take η to represet the Lamb-Dicke parameter ad the parameter α. For the case of = χ = (Figure, we ivestigate the ifluece of the mea photo umber o the etropy squeezig Figure. The time evolutio of the etropy squeezig factors E( S, ( y E S ad populatio iversio W t of a two-level atom iteractig with a sigle-mode i the ecited state = φ =, for parameter χ = =, η =.8 ad with differet values of the parameter α where α = i Figures (a,d,g, α = 3 i Figures (b,e,h ad α = 6 i Figures (c,f,i. 77

8 factor ( (, y E S E S, ad populatio iversio W t. We otice that o squeezig occurs o the atomic varia- ble S as the parameter icreased (see Figures (a-(c. But Figures (d-(f, preset that with icreasig the parameter α, the etropy squeezig o the atomic variables S y is icreased. Also the quatum revival is icreased but the collapse pheomeo is decreased as α icrease (see Figures (g-(i. These results agree with the case of coheret field but the differece betwee the coheret state ad the oliear coheret state appear o the collapse ad revival pheomea. For fied values of the detuig, the detuig parameter ad the case of absece the oliear medium parame- χ = the etropy squeezig factor E( S ad ( y ter, E S are plotted as a fuctio of the scaled time t, where we set three differet values of, i.e. =, 5,. We oticed o etropy squeezig o the atomic variable S as the detuig parameter icreases; see Figures (a, (c. Although there is great etropy squeezig o whe =, see Figure (d ad o squeezig o S y whe = 5, see Figures (c ad (f. W t > the This is because, where. H( S y H S e e E S y = (3 W t = S z Also oe ca see as the detuig parameter icreased the period of evolutio icreased. Figure 3 eplais the effect of the detuig parameter whe the atom is iitially i the superpositio state of the upper ad lower atomic states ad all the other parameters are the same as Figure. It is observed that the situatio is completely differet betwee the ecited ad superpositio states (see Figures ad 3 First we see that whe the detuig icreases, the etropy squeezig o the atomic variables S icreases. Also there is a great etropy squeezig o the S y. I order to see how the etropy squeezig iflueced by the oliear medium parameter, we set two differet values of the Kerr-medium parameter. Figures 4(a, (d ad (g show the absece of the oliear parameter but Figure. The time evolutio of the etropy squeezig factors E( S, E( S y ad populatio iversio W( t of a two-level atom iteractig with a sigle-mode i the ecited state = ϕ =, for parameter χ =, α = 6, η =.8 ad with differet values of the parameter where = i Figures (a (d (g, = 5 i Figures (b (e (h ad = i Figures (c (f (i. 78

9 Figure 3. The time evolutio of the etropy squeezig factors, W t of a two-level atom i- ( ( y E S E S ad populatio iversio teractig with a sigle-mode as Figure but = π ad φ = π 4. Figure 4. The time evolutio of the etropy squeezig factors ( (, y E S E S ad populatio iversio W t of a two-level atom iteractig with a sigle-mode i the ecited state = ϕ =, for parameter α = 6, =, η =.6 ad with differet values of the Kerr oliearity parameter χ where χ = i Figures 4(a (d (g, χ =.3 i Figures 4(b (e (h ad χ =.5 i Figures 4(c (f (i. 79

10 the other figures show the ifluece of the oliear parameter. It is remarkable that the oliear parameter leads to the followig effects, o etropy squeezig occurs o S whe χ =, but there is etropy squeezig o S y. Whe χ =.3, there is a optimal etropy squeezig o both the atomic variables S ad S y. For large values of χ as χ =.5 oe ca see there are more ad more optimal etropy squeezig o both of the atomic variables S ad S y. Also the etropy squeezig factors E( S ad E( S y are periodic fuctios, which have a period π ad π respectively (see Figures 4(c ad (f. I order to eplai the differece betwee the sources of the radiatio fields of the coheret state ad the oliear coheret state, we must set differet values of the Lamb-Diche parameter η (,.5,.9. Whe η =, the the field is iitially i the coheret state, but whe η the field is iitially i the oliear coheret state. Figures 5(a, (d ad (g show the etropy squeezig i the case of coheret state, which has bee eamied i may papers [] [5] [7]. But Figures 5(b, (e ad (h show the differece betwee the coheret ad ew oliear coheret states, where we set η =.5 while we set η =.9 i Figures 5(d, (f ad (i. Oe ca see there is strog etropy squeezig o the atomic variables S y ad there is a etropy squeezig ad the period of collapse also icreases. 6. Coclusios We have treated the etropy squeezig of a two-level atom whe the field is iitially prepared i the NCS. New results ca be eplored as follows. Whe the field is iitially i a NCS, the rich features of the etropy squeezig ca be observed, the the etropy squeezig is a good measuremet of the iformatio cocerig the case of trapped io. There is a great differece betwee the ifluece of the oliearity fuctio f(, which is used i the de- Figure 5. The time evolutio of the etropy squeezig factors E( S, E( S y ad populatio iversio W( t of a two-level atom iteractig with a sigle-mode i the ecited state = ϕ =, for parameter χ = =, α = 6 ad with differet values of the Lamb-Dicke parameter η where η = i Figures (a,d,g, η =.5 i Figures 5(b (e (h ad η =.9 i Figures 5(c (f (i. 8

11 scriptio of the motio of a trapped io through Lamb-Dicke parameter η ad Kerr medium oliearity through parameter χ. The first decreases the etropy squeezig but the secod icreases etropy squeezig. Refereces [] Shao, C.E. (948 A Mathematical Theory of Commuicatio. Bell System Techical Joural, 7, [] Obada, A.-S.F. ad Abdel-Aty, M. ( Ifluece of the Stark Shift ad Kerr-Like Medium o the Evolutio of Field Etropy ad Etaglemet i Two-Photo Processes. Acta Physica Poloica, B3, 589. [3] Steae, A. (998 Quatum Computig. Reports o Progress i Physics, 6, [4] Roy, B. ad Roy, P. ( New Noliear Coheret States ad Some of Their Noclassical Properties. arxiv: quat-ph/43. [5] Fag, M.F., Zhou, P. ad Swai, S. ( Etropy Squeezig for a Two-Level Atom. Joural of Moder Optics, 47, [6] Abdel-Aty, M., Abdel-Khalek, S. ad Obada, A.-S.F. ( Chaos, Solitos & Fractals,, 5-. [7] El-Shahat, T.M., Abdel-Khalek, S., Abdel-Aty, M. ad Obada, A.-S.F. (3 Chaos Solitos ad Fractals Aspects o Etropy Squeezig of a Two-Level Atom i a Squeezed Vacuum. Joural of Moder Optics, 8, [8] Kuzmich, A., Molmer, K. ad Polzik, E.S. (997 Spi Squeezig i a Esemble of Atoms Illumiated with Squeezed Light. Physical Review Letters, 79, [9] Saito, H. ad Ueda, M. (999 Squeezed Few-Photo States of the Field Geerated from Squeezed Atoms. Physical Review A, 59, [] Hirschma, L.L. (957 A Note o Etropy. Joural of Mathematical Physics, 79, [] El-shahat, T.M., Abdel-Khalek, S., Abdel-Aty, M. ad Obada, A.-S.F. (3 Aspects o Etropy Squeezig of a Two-Level Atom i a Squeezed Vacuum. Chaos, Solitos & Fractals, 8, [] Obada, A.-S.F., Abdel-Khalek, S., Ahmed, M.M.A. ad Abo-Kahla, D.A.M. (9 The Master Equatio for a Two- Level Atom i a Laser Field with Squeezig-Like Terms. Optics Commuicatios, 8, [3] Kares, K. (983 States, Effects od Operatios. Spriger, Berli. El-Shahat, T.M., Abdel-Khalek, S. ad Obada, A.-S.F. (5 Etropy Squeezig of a Drive Two-Level Atom i a Cavity with Ijected Squeezed Vacuum. Chaos, Solitos & Fractals, 6, [4] Matos Filho, R.L. ad Vogel, W. (996 Noliear Coheret States. Physical Review A, 54,