Entanglement of an Atom and Its Spontaneous Emission Fields via Spontaneously Generated Coherence

Transcription

1 Journl of Sciences Islmic Republic of Irn (): 7-76 () University of Tehrn ISSN Entnglement of n Atom nd Its Spontneous Emission Fields vi Spontneously Generted Coherence M. Shri * H. Noshd nd M. Mhmoudi Reserch Institute for Applied Physics nd Astronomy University of Tbriz Tbriz Islmic Republic of Irn Deprtment of Physics Znjn University P. O. Box Znjn Islmic Republic of Irn Received: 5 April / Revised: August / Accepted: 7 August Abstrct The entnglement between Λ-type three-level tom nd its spontneous emission fields is investigted. The effect of spontneously generted coherence (SGC) on entnglement between the tom nd its spontneous emission fields is then discussed. We find tht in the presence of SGC the entnglement between the tom nd its spontneous emission fields is completely phse dependent while in bsence of this coherence the phse dependence of the entnglement disppers. Moreover the degree of entnglement drmticlly chnges by the coherent superposition of the tomic sttes. Keywords: Quntum entnglement; Quntum interference; Quntum entropy Introduction It is well known tht the photon cn be creted when the tom decys from n upper level to lower one [- ]. It is lso believed tht the spontneous emission destroys coherence in n tomic system but the spontneous emission hs potentil ppliction in mny physics processes; such s high-precision mesurement lsing without inversion quntum teleporttion quntum computtion nd quntum informtion theory [4]. Spontneous emission however is used for produce tomic coherence s long s there exist two close-lying levels with non-orthogonl dipoles in n tomic system. Atomic coherence bsed on the spontneous emission is usully referred to s vcuum-induced coherence or spontneously generted coherence (SGC) [5]. Therefore spontneous decy cn produce quntum interference. This hppens in two cses one occurs when n excited stte doublet decys to single ground stte [67] nd the other ppers when single exited stte decys to lower stte doublet [5]. The effect of SGC cn substntilly modify the behvior of the system. In recent yers spontneous emission hs widely been used for vrious purposes [8 9]. For exmple the dissiption from the spontneous emission cn induce trnsient entnglement between the two toms which is essentil to implementtion of quntum protocol such s quntum computtion []. Moreover entngled light cn be creted by the dissiption from white noise of the spontneous emission [8]. The effect of quntum interference on the entnglement of pir of three-level toms hs been proposed []. Quntum entngled sttes ply n importnt role in the field of quntum informtion theory; prticulrly quntum teleporttion quntum computtion etc []. In such stte the system is inseprble nd ech component * Corresponding uthor Tel.: +98(4)9 Fx: : +98(4)475 E-mil: shri@tbrizu.c.ir 7

2 Vol. No. Spring Shri et l. J. Sci. I. R. Irn does not hve properties independent of the other component. Phoenix nd Knight [45] hve shown tht the reduced entropy is n ccurte mesure of entnglement between two components. Furthermore the evolution of the tomic (field) entropy for the threelevel tom (one nd two mode model) hs been studied [6-]. The time evolution of the field (tom) quntum entropy reflects time evolution of the degree of the entnglement of field (tom). The higher the entropy the greter the entnglement. In this rticle the entnglement between the tom nd its spontneous emission fields is studied by mens of quntum entropy. We show tht in the presence of SGC entnglement of tom nd spontneous emission strongly depends on reltive phse of driving fields. In the bsence of this coherence the phse dependence of the medium disppers. Moreover the effect of tomic prmeters such s Rbi- frequency nd frequency detuning on quntum entropy re discussed. Mterils nd Methods Consider closed three-level Λ -type tomic system with two closely lower levels nd the upper level s shown in Figure (). Two strong coherent coupling fields of frequencies ν nd ν nd couple the trnsitions respectively. The corresponding Rbi-frequencies re denoted by E. E. Ω = nd Ω =. The prmeters j ( ħ ħ j = ) denote the tomic dipole moments but E nd represent the mplitudes of the coupling fields. E The density mtrix equtions of motion in the rotting wve pproximtion nd in the rotting frme re [ ]. ( ) ρ = γ ρ i Ω ρ ρ ( ) i ( ) i ( ) ρ = γ + γ ρ + Ω ρ ρ + Ω ρ ρ ( ) ρ = γ ρ i Ω ρ ρ ( i ) i i ( ) ρ = γ + γ ρ + Ω ρ + Ω ρ ρ ( ) ρ = i ρ i Ω ρ + i Ω ρ + η γ γ ρ ρ = ( γ + γ i ) ρ + i Ω ρ i Ω ( ρ ρ ). () ω ω Here γ = ( γ = ) re the spontne- πε c πε c ous decy rtes in trnsition j ( j = ). The detuning prmeters re defined s = ν ω nd = ν ω. The term ( η γγ ρ ) represents the interference mong decy chnnels tht ppers due to. SGC. The prmeter η( = = cos θ ) denotes the lignment of the two dipole moments nd where θ is the ngle between two induced dipole moments nd s shown in Fig. (b). Since the existence of SGC effect depends on the nonorthogonlity of the two dipole moments nd so we hve to consider n rrngement where ech field cts only on one trnsition (Fig. (b)). Moreover η represents the strengths of the interference in spontneous emission. For prllel dipole moments the interference is mximum nd η = while for perpendiculr dipole moments there is no interference nd η =. Therefore η represents the existence of the SGC nd it will be zero (one) if the SGC effect is ignore (included). We note tht only for nerly degenerte lower levels i.e. ω ω the effect of SGC becomes importnt nd for lrge lower energy levels seprtion it my be dropped [ ]. In the Λ - type tomic system considered here n extr coherence term (SGC) ppers between the lower levels due to the spontneous decy from the upper level. The Rbi () (b) Figure. () Proposed level scheme. A Λ-type three-level tomic system driven by coherent fields. (b) The rrngement of field polriztion required for single field driving one trnsition if dipoles re orthogonl. 7

3 Entnglement of n Atom nd Its Spontneous Emission Fields vi frequencies re connected to prmeter η by the reltion Ω = g η nd Ω = g η. Note tht the phse ppers in the eqution through η. If we use i g = g e ϕ g = g e ϕ nd redefining the tomic i i e ϕ vrible in eqution () s ρ = ρ i ρ e ϕ i ϕ ρ = ρ = ρ e we obtin equtions for the redefined density mtrix elements ρ ij. The equtions re identicl to equtions () expect tht η is replced by e () i η η ϕ where ϕ = ϕ ϕ. We ssume tht the Λ -type threelevel tom nd the rdition-field reservoir re initilly in non-entngled pure stte. So the system is bicomponent quntum system in pure stte. For such system the reduced quntum entropy cn be used s mesure of the degree of entnglement between n tom nd its spontneous emission fields [4 5]. The reduced entropy of the tom i.e. S ( t ) cn be defined through its respective reduced-density opertor by S ( t ) = Tr( ρ ln( ρ )). () Here ρ is the reduced density opertor of the tom with the elements given in eqution () in which the Boltzmn constnt is set equls one. We cn express the Λ -type three-level tomic quntum entropy in terms of the eigenvlues λ ( t ) of reduced tomic density opertor ρ s S ( t ) = λ ( t ) ln( λ ( t )). i = Results nd Discusion In this section we numericlly clculte the entnglement between the tom nd it spontneous emission fields vi equtions () nd (). The influence of quntum interference due to SGC on entnglement of the tom nd spontneous emission fields is then discussed. We disply the quntum entropy of the Λ - type three-level tom versus the normlized time γ t in Fig. (-c) for different tomic initil sttes vrious g nd g with η = (or.99). Fig. () shows tht for g = = nd η = the tomic quntum entropy g quickly rises from zero to its mximum nd reches to fixed vlue s time increses. So the Λ -type threelevel tom nd its spontneous emission fields re strongly entngled t the stedy stte. The degree of entnglement depends on the initilly superposition of the tomic sttes. For tom initilly in upper level the degree of entnglement is lrger thn the cse the tom is initilly prepred in superposition sttes. In fct the coherent superposition of the tomic sttes leds to S (t) S (t) S (t) γ t Figure. The time evolution of the tomic quntum entropy s function of normlized time γ t () g = g = η = (b) g = g = γ η = (c) g = g = γ η =.99 other prmeters re γ =.γ γ =.γ () ρ ()= ρ ()=ρ ()=/ ρ ()=ρ ()=ρ ()=/ γ t (b) (c) ρ ()= ρ ()=ρ ()=/ ρ ()=ρ ()=ρ ()=/ ρ ()= ρ ()=ρ ()=/ ρ ()=ρ ()=ρ ()=/ γ t = =.5γ nd ϕ =.. The 7

4 Vol. No. Spring Shri et l. J. Sci. I. R. Irn.4. η=.99 η=.4. Stedy stte S.8.6 Stedy stte S φ η Figure. The stedy stte tomic quntum entropy s function of ϕ for η =.99 (solid line) nd η = (dotted line). The other prmeters re sme s Fig.. Figure 4. The stedy stte tomic quntum entropy s function of s function of η for ϕ =. The other prmeters re sme s Fig... η=.99 η=.4. η=.99 η= Stedy stte S Stedy stte S g Figure 5. The stedy stte tomic quntum entropy s function of s function of for η =.99 (solid line) nd η = (dotted line). The other prmeters re sme s Fig.. Figure 6. The stedy stte tomic quntum entropy s function of s function of g for η =.99 (solid line) nd η = (dotted line). The other prmeters re sme s Fig.. decrese the popultion of the tomic excited stte nd consequently reduction the probbility of tomic spontneous emission. This my led in reduction of entnglement between the tom nd its spontneous emission. Fig. (b c) shows tht in the presence of coupling fields i.e. g = g = γ nd η = (or.99) the stedy stte entnglement does not depend on the initilly preprtion of tom. However for η =.99 the degree of entnglement is lrger thn η = (see Fig. (b c)). Now we propose the effect of the tomic prmeters on the entnglement between the tom nd spontneous emission fields. It hs lredy been shown tht the Λ - type three-level tomic system with SGC is phse dependent [] nd phse ppers in the equtions through η. So the entnglement between the tom nd its spontneous emission fields should depend on the reltive phse between pplied fields. The phse vrition of the entnglement for different vlues of quntum interference prmeter is shown in Figure. Note tht in the bsence of quntum interference i.e. 74

5 Entnglement of n Atom nd Its Spontneous Emission Fields vi η = the entnglement of the tom nd the fields is phse independent (dotted line) while for η =.99 the entnglement substntilly chnges by the chnge of reltive phse of pplied fields (solid line). We relized tht for even multiples of π the tom nd fields re strongly entngled while for odd multiples of π tom is disentngled from the spontneous emission fields. Physiclly the chnge of phse difference between pplied fields my chnge the direction of the dipole moments: thus it chnges prmeter η. Note tht the degree of entnglement strongly depends on strength of the quntum interference η. ρ Figure 7. The stedy stte popultion distribution of upper level s function of η nd ϕ. The other prmeters re sme s Fig.. ρ g η Figure 8. The stedy stte popultion distribution of upper level s function of nd g. The other prmeters re sme s Fig.. -5 φ 5 η=.99 η= Figure 4 shows tht for ϕ = the degree of entnglement increses by incresing η. Frequency detuning hs n importnt role in cretion of the entnglement between tom nd its spontneous emission fields. We show the stedy stte entropy S ( t ) s function of in the Figure 5. It is seen tht the tom nd its spontneous emission fields re disentngled for =. But round zero detuning the degree of entnglement for η =.99 is lrger thn η =. The degree of entnglement cn lso be chnged by the Rbi-frequencies of pplied fields. The quntum entropy versus g is displyed in Figure 6. It cn lso be relized tht for η =.99 the degree of entnglement is lrger thnη =. A similr behvior cn be found for entnglement of the tom nd its spontneous emission fields by vrition g. Now we discuss the physicl mechnisms underlying the behind of the bove results. Here we plot the popultion distribution of level versus vrious tomic prmeters. Figure 7 shows the two dimensionl behvior of the popultion distribution of level versus η nd ϕ. We observe tht the peks of upper level popultion re locted in plces tht the quntum entropies re mxim (see Figs. 4 nd Fig. 7). The higher the popultion the greter the entnglement. Physiclly incresing the popultion of level cn increse the probbility of tomic spontneous emission leding to increse in entnglement. Finlly the vrition of upper level popultion versus g nd is shown in Figure 8. For = the popultion of upper level is pproximtely zero nd the spontneous emission is suppressed. In this cse the tom is disentngled from the spontneous emission fields. However for the upper level is populted leding to cretion n entnglement between the tom nd fields (see Figs. 5 6 nd 8). References. Lee H. Polynkin P. Scully M. O. nd Zhu S.Y. Quenching of spontneous emission vi quntum interference. Phys. Rev. A 55: (997).. Zhu S. Y. Chn R. C. F. nd Lee C. P. Spontneous emission from three-level tom. Phys. Rev. A 5: 7-76 (995).. Ghfoor F. Zhu S. Y. nd Zubiry M. S. Amplitude nd phse control of spontneous emission. Phys. Rev. A 6: 8-88 (). 4. Zhng H. Z. Tng S. H. nd Dong P. He J. Quntum 75

6 Vol. No. Spring Shri et l. J. Sci. I. R. Irn interference in spontneous emission of n tom embedded in double-bnd photonic crystl. Phys. Rev. A 65: (). 5. Jvninen J. Effect of Stte Superpositions Creted by Spontneous Emission on Lser-Driven Trnsitions. Europhys Lett. 7: 47-4 (99). 6. Zhu S. Y. nd Scully M. O. Spectrl Line Elimintion nd Spontneous Emission Cncelltion vi Quntum Interference. Phys. Rev. Lett. 76: 88-9 (996); Psplskis E. nd Knight P. L. Phse Control of Spontneous Emission. Phys. Rev. Lett. 8: 9-96 (989); Zhou P. nd Swin S. Ultrnrrow Spectrl Lines vi Quntum Interference. Phys. Rev. Lett. 77: (996). 7. Hrris S. E. Lsers without inversion: Interference of lifetime-brodened resonnces. Phys. Rev. Lett. 6: - 6 (989); Immoglu A. Interference of rditively brodened resonnces. Phys. Rev. A 4: 85-88(989). 8. Plenio M. B. nd Huelg S. F. Entngled Light from White Noise. Phys. Rev. Lett. 88: (). 9. Fng M. F. nd Zhu S.Y. Entnglement between Λ- type three-level tom nd its spontneous emission fields. Physic A 69: (6).. Beige A. Bose S. Brun D. Huelg S. F. Knight P. L. Plenio M. B. nd Vedrl V. Entngling Atoms nd Ions in Dissiptive Environments. Mod. J. Opt. 47: (); Cbrillo C. Circ J. I. Grci-Fernndez P. nd Zoller P. Cretion of entngled sttes of distnt toms by interference. Phys. Rev. A 59: 5- (999).. Derkcz L. nd Jkobczyk L. Quntum interference nd evolution of entnglement in system of three-level toms. Phys. Rev. A 74: --7 (6).. Cmpos Venuti L. Gimpolo S. M. Illuminti F. nd Znrdi P. Long-distnce entnglement nd quntum teleporttion in XX spin chins. Phys. Rev. A 76: (7).. Zhou Y. nd Zhng G-F. Quntum teleporttion vi two-qubit Heisenberg XXZ chin - effects of nisotropy nd mgnetic field. Eur. Phys. J. D 47: 7- (8). 4. Phoenix S. J. D. nd Knight P. L. Fluctutions nd entropy in models of quntum opticl resonnce. Ann. Phys. NY. 86: 8-47 (988). 5. Phoenix S. J. D. nd Knight P. L. Estblishment of n entngled tom-field stte in the Jynes-Cummings model. Phys. Rev. A 44: 6-69 (99). 6. Liu X. Entropy behviors nd sttisticl properties of the field intercting with Ξ-type three-level tom. Physic A 86: (). 7. Abdel- Aty M. Influence of Kerr-like medium on the evolution of field entropy nd entnglement in threelevel tom. J. Phys. B : (). 8. Abdel- Aty M. nd Obd A.-S.F. Engineering entnglement of generl three-level system intercting with correlted two-mode nonliner coherent stte. Eur. Phys. J. D : (). 9. Klychko A. Cn M.A. Ckir O. nd Shumovsky A. Persistent entnglement in three-level tomic systems. J. Opt. B 6: -7 (4).. Obd A-S. F. Eied A. A. nd Abd Al- Kder G. M. Entnglement of Generl Formlism V Type Three- Level Atom Intercting with Single-Mode Field in the Presence of Nonlinerities. Int. J. Theor. Phys.48: 8-9 (9).. Menon S. nd Agrwl G. Effects of spontneously generted coherence on the pump-probe response of Λ system. Phys. Rev. A 57: (998).. Bi Y. Guo H. Hn P. Sun H. Effects of incoherent pumping on the phse control of mplifiction without inversion in Λ system with spontneously generted coherence J. Opt. B. 7: 5-8 (5).. Mhmoudi M. Shri M. Tjlli H. J. The effects of the incoherent pumping field on the phse control of group velocity. J. Phys. B. 9: (6). 76