Coherent Control of Quantum Entropy via Quantum Interference in a Four-Level Atomic System

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1 Journal of Scences, Islamc Republc of Iran 24(2): (2013) Unversty of Tehran, ISSN Coherent Control of Quantum Entropy va Quantum Interference n a Four-Level Atomc System M. Sahra, B. Arzhang, * H. Sefoory, and P. Navaepour Research Insttute for Appled Physcs and Astronomy, Unversty of Tabrz, Tabrz, Islamc Republc of Iran Receved: 10 January 2013 / Revsed: 6 May 2013 / Accepted: 24 June 2013 Abstract The tme evaluaton of quantum entropy n a four-level double- type atomc system s theoretcally nvestgated. Quantum entanglement of the atom and ts spontaneous emsson felds s then dscussed va quantum entropy. It s found that the degree of entanglement can be ncreased by the quantum nterference nduced by spontaneous emsson. The phase dependence of the atom-feld entanglement s also presented. Keywords: Quantum entanglement; Quantum entropy; Quantum nterference; Spontaneous emsson Introducton Quantum correlaton between dfferent parts of a system leads to an mportant quantum phenomenon known as entanglement. A system consstng two components s sad to be entangled f ts quantum states cannot be descrbed by a smple product of the quantum states of the two components [1]. The measurement on one of them gves nformaton about the other component. Entanglement allows to reach a much closer relatonshp than s possble n classcal physcs. For a bcomponent system n a pure state, the reduced quantum entropy s the best tool for accurate measure of entanglement between two components [2-4]. The system s nseparable n such state, and each component does not have propertes ndependent of the other component. The hgher reduced quantum entropy, the hgher degree of entanglement. In bcomponent systems, the entanglement can be establshed between two partcles or between the partcle and the feld. The Ensten-Podolsky-Rosen (EPR) state [5] s an nterestng example of two-partcle components entanglement whch can be used n secure quantum communcaton [6]. Quantum entanglement can also be generated n a system wth multpule components [7]. Entanglement plays an essental role n quantum nformaton processng such as quantum computng [8], quantum teleportaton [9], quantum cryptography [10], and quantum communcaton [11]. The entanglement between atom and felds has wdely been dscussed due to ts potental applcatons n memory storage devces. Varous atomc systems have been proposed to nvestgate the entanglement of atom and ts spontaneous emsson felds. A theoretcal descrpton on evoluton of entanglement between the atom and feld s recently proposed by Abdalla et al. [12]. The entropy evaluaton of the feld ntensty wth V-type three level atom was proposed to reach the atom-photon entanglement. It s shown that the quantum entropy as well as atom- photon entanglement can be controlled by the ntensty of couplng feld, detunng parameter and the ntal photon number [13]. In another proposal, the effect of couplng laser feld on the entanglement of the atom and ts spontaneous emsson * Correspondng author, Tel.: +98(937) , Fax: +98(381) , E-mal: arzhang.beh@gmal.com 179

2 Vol. 24 No. 2 Sprng 2013 Sahra et al. J. Sc. I. R. Iran felds was theoretcally dscussed n two dfferent threelevel confguratons,.e. type and V-type [14]. Fang et al. [15] nvestgated the effect of coherent superposton of the atomc level on the entanglement of a type three level atom and ts spontaneous emsson felds. It s shown that the atom-photon entanglement can be controlled by the ntal coherent condton of the atomc levels. Phase control of the entanglement has also been proposed by Malnovsky et al. [16]. Atomc coherence and quantum nterference, on the other hand, are the basc mechansms for controllng the optcal propertes of the medum. Over the past few years, much attenton has been devoted to the effects of quantum nterference between multple atomc transtons pathways and ts applcatons for controllng the coherent phenomena such as electromagnetcally nduced transparency (EIT) [17]. In fact, the dscovery of EIT has opened up a new route to control the optcal propertes of atom-photon coherent nteracton [18]. It s also shown that the spontaneous emsson can be used to produce atomc coherence as long as there exst two closely-lyng levels wth non-orthogonal dpole n an atomc system. Atomc coherence based on spontaneous emsson s usually referred to as vacuum-nduced coherence or spontaneously generated coherence ( SGC )[19]. Quantum nterference nduced by spontaneous emsson, however, can modfy the response of atomc system to the appled felds [20]. Now, ntrgung queston arses that what s the effect of quantum nterference on the atom-photon entanglement. It would be nterestng to ncrease the degree of atom-photon entanglement va quantum nterference arsng among decay channels. In fact, quantum nterference nduced by spontaneous emsson has recently been employed to coherent control of quantum entropy whch can turn n quantum entanglement. The effect of quantum nterference on the entanglement of a drven V-type three-level atom and ts spontaneous emsson felds has recently been dscussed. It s predcted that n the absence of quantum nterference the atom and ts spontaneous emsson felds are always entangled. However, n the presence of quantum nterference the atom-photon entanglement wll depend on the atomc parameters [21]. In ths paper, the effect of quantum nterference on entanglement of a coherently drven four-level double-ʌ type atom and ts spontaneous emsson felds s nvestgated. The tme evoluton of the reduced atomc entropy s proposed to reach the entanglement of the atom and ts spontaneous emsson felds. The effect of couplng laser felds on entanglement of the atom and ts spontaneous emsson felds s then dscussed. It s found that the entanglement of the atom and ts spontaneous emsson felds strongly depends on the quantum nterference nduced by spontaneous emsson. In addton, we fnd that the quantum entropy (and also the atom-photon entanglement) becomes phase dependent when the nterference parameter s swtched on. Materals and Methods Consder a four-level atomc system n a double-ʌ confguraton as depcted n Fg. 1. The scheme s consstng two metastable lower levels 1, 2, and two excted levels 3, 4. The electrc-dpole allowed transtons 1 3, 2 3, and 2 4 are drven by three coherent strong laser felds, respectvely. The correspondng Rab-frequences are denoted by g E. /, g 32 E. / and g 42 E. / The other transton 1 4 s drven by a weak coherent feld, where ts Rabfrequency s gven by g 41 E. / Here, are the correspondng atomc dpole moments, whle E denotes the ampltude of coherent laser felds. The {3, 4} to spontaneous decay rates from level the level j j {1, 2} are denoted by 2 j. The dynamcs of the system s descrbed by the densty matrx equaton of moton [ H, ], (1) where H s the nteracton Hamltonan of atom and felds. The term represents the decay processes, where s the louvllan operator actng on the densty matrx. The densty matrx elements n rotatng frame and rotatng wave approxmaton are gven by: Fgure 1. The four-level double-ʌ type atomc system. The system s drven by three strong felds and a weak laser feld. 180

3 Coherent Control of Quantum Entropy va Quantum Interference n a Four-Level Atomc System g g g g , g g g g , ( ) g g * g g * ( g g g g ) e, g ( ) g g 2, ( ) g ( ) g g, g32 ( ) g 21 g, g ( ) g g,, ( ) g g g g, , where j (2) are the laser feld detunng. Here j denote the frequency deference between level and level j, and j s the frequency of couplng felds. We have further defned j 1j 2j, and our chosen level scheme mples (2 2 j ) / 2( {1,2} and j {3, 4}) are the dampng rate of the coherences on transtons j. The term ( ) ( ) n Eqs. 2 represents the quantum nterference resultng from the cross couplng between spontaneous emsson paths 3 1 and 3 2 (4 1 and 4 2 ). Note that n a V- type three- level system only for nearly degenerate upper levels the effect of quantum nterference could be appeared n equatons. However, for a gven type system only for nearly degenerate lower- levels the effect of SGC become sgnfcant, and for large lower energy level separaton t may be dropped [22, 23]. In a double type atomc system consdered here, two extra coherence terms ( SGC ) appear between the lower levels due to the spontaneous decay from the upper levels to the nearly degenerate lower levels. The dervng laser felds coupled to ths coherence allow addtonal path for the transtons from the upper levels to the lower levels and thus accounts for the nterference effects n the coherence terms gven by Eqs. 2. The parameters 1 and 2 denote the algnment of the two dpole moments, whch defne as. 32 ( 1 cos 1).. ( cos 2).. Here, represents the electrc dpole moment, 1( 2) s the angle between the two nduced dpole moments 32 ( ). Accordng to ther defnton, the algnment factor takes the value 1 for parallel dpole moments, 1 for antparallel, and 0 for orthogonal. Maxmal coherence corresponds to parallel or antparallel dpole moments, whle zero coherence corresponds to orthogonal dpole moments. These two extremes of maxmal and mnmal coherence deserve specal attenton. However, ntermedate values on the [ 1, 1] nternal are also possble. In a real experment the parameter 1 2 can be determned by the ntensty of appled felds. In fact, the coeffcent η depends on angle between two electrc dpole moments, whch can be controlled by the ntensty of appled felds. Eqs (2) show that the relatve phase appear n equaton only through 1 and 2. In fact, f we use g g e, g g e, g g e and g g e, and redefnng the atomc varables n equatons (2) as ,, e e e, e, ( 43) 12 12e and ( 4 3) ( 2 1), we obtan equatons for the reduced densty matrx elements. The equatons are dentcal to equatons (2), except that 1 and 2 replaced by 1 1 e and 2 2 e wth ( 4 3) [24]. Therefore, for a nearly degenerate lower levels the effect of SGC should be taken nto account, thus the system becomes completely phase dependent. For such a system, the reduced quantum entropy can be used as a measure of the degree of entanglement between the atom and ts spontaneous emsson felds [3, 4]. The entropy of the atom and the spontaneous emsson felds can be defned through ther respectve 181

4 Vol. 24 No. 2 Sprng 2013 Sahra et al. J. Sc. I. R. Iran reduced-densty operators by I I S () t Tr ( ln ),( a, f ). (3) The entropy of a general two-component quantum system are lnked by a remarkable theorem presented by Arak and Leb [25], whch states S () t S () t S S () t S (), t (4) a f af a f I I where Saf Tr{ ln } s the total entropy of the atom-spontaneous emsson felds system. We assume that the atom and the vacuum felds are ntally n a dsentangled pure state, so the total entropy S af of the atom-spontaneous emsson felds system s zero. One mmedate consequence of nequalty (4) s Sa() t Sf () t. Consequently, we only need to calculate the atomc quantum entropy Sa () t to dscuss the entanglement of the atom and felds. We can also express the Ʌ-type four-level atomc quantum entropy n terms of the egenvalues aj () t of reduced atomc densty operator by a f aj aj 1 4 S () t S () t ()ln( t ()) t. (5) Now, we dscuss the atom-photon entanglement only by the atomc quantum entropy Sa () t gven by equaton (3) and (5). Results and Dscusson In ths secton, we numercally calculate the entanglement between the atom and ts spontaneous emsson felds va equatons (2) and (3). The evaluaton of the quantum entropy s employed to determnng the quantum entanglement of the atom and ts spontaneous emsson felds. In partcular, the effect of quantum nterference due to the spontaneous emsson,.e. SGC, on entanglement of the atom and ts spontaneous emsson felds s then dscussed. We assume that the atomc system s ntally n superposton of upper levels 3 and 4. In addton, we choose and all fgures are plotted n unt of. Quantum entropy Sa t versus normalzed tme t s dsplayed n Fg. 2. From Fg. 2a, we observe that for g41 g42 0 (sold) and g g32 0 (dashed) the quantum entropy ncreases and fnally reaches to a constant value at the steady state as normalzed tme ncreases. In fact, for g41 g42 0 (or g g32 0 ) the medum converts to a three level Ʌ-type atomc Fgure 2. The tme evoluton of the atomc quantum entropy as a functon of normalzed tme t. The ntensty of appled felds are (a) g g 0.01, g g 0 (sold lne), g g 1.5 and g g 0 (dashed lne), and (b) g g g (dashed lne), 1.2 (dashed-dotted lne), 1.5 (sold lne), and g Other parameters are , and 0, Fgure 3. The effect of quantum nterference nduced by spontaneous emsson on quantum entropy. (a) The parameters are g41 g42 g32 1.5, g 0.01, 0 (dasheddotted), 0.5 (dashed), 1 (sold). Other parameters are , 0 and

5 Coherent Control of Quantum Entropy va Quantum Interference n a Four-Level Atomc System system, and the results are n a good agreement wth the obtaned results of ref. [15]. The effect of ntensty of couplng felds on quantum entropy s dsplayed n Fg. 2b. In ths case the degree of entanglement of the atom and ts spontaneous emsson felds depends on the ntensty of couplng laser felds. Physcally, strong couplng felds create strong atomc coherence that leads to strong correlaton between the atom and felds. Ths may explan the enhancement of entropy by the ntensty of couplng laser felds. It s well known that atomc coherence and quantum nterference are the basc mechansms for controllng the optcal propertes of the medum. On the other hand, atomc coherence nduced by spontaneous emsson,.e. SGC, substantally changes the correlaton between the atom and felds. Here, we nvestgate the effect of quantum nterference nduced by spontaneous emsson,.e. SGC, on behavor of the quantum entropy n Fg. 3. We observe that the quantum entropy ncrease by ncreasng the quantum nterference parameter. So, for the degree of entanglement of the atom and ts spontaneous emsson felds s larger than In fact, the four-level double-ʌ type atom and ts spontaneous emsson felds are strongly entangled due to the quantum nterference nduced by spontaneous emsson felds. Now, we propose the effect of the relatve phase of appled felds on entanglement between the atom and ts spontaneous emsson felds. It has already been shown that the Ʌ -type three-level atomc system wth SGC s completely phase dependent, and phase appears n equatons through parameter [20]. So, the entanglement between the atom and ts spontaneous emsson felds should depend on the relatve phase of appled felds. The phase varaton of the entanglement for dfferent values of quantum nterference s shown n Fg. 4. In the absence of quantum nterference,.e. 0, the entanglement of the atom and the felds s phase ndependent Fg. 4a, whle for 1 the entanglement substantally changes by changng the relatve phase of appled felds (Fg. 4b). The phase varaton of the quantum entropy s also dsplayed n Fg. 4c. We observe that for even multples of, the atom and the felds are strongly entangled, whle for odd multples of the degree of entanglement of the atom and spontaneous emsson felds s substantally reduced. Physcally, the change of phase dfference between appled felds may change the drecton of the dpole moments; thus t changes parameter. The parameter may drectly affect n the quantum entropes, whch mples the entanglement of the atom and ts Fgure 4. The steady state atomc quantum entropy as a functon of normalzed tme γt for (a) η=0, Δφ=0 (sold lne) and Δφ=π (dashed lne), and for (b) η=1, Δφ=0 (sold lne) and Δφ=π (dashed lne). (c) The venaton of steady state quantum entropy versus relatve phase Δφ for η=1 (sold lne), and η=0 (dashed lne). Other parameters are same as Fg. 2. Fgure 5. The steady state atomc quantum entropy as a functon of as a functon of Δ for η=0.99. Other parameters are same as Fg

6 Vol. 24 No. 2 Sprng 2013 Sahra et al. J. Sc. I. R. Iran spontaneous emsson felds. Frequently detunng has an mportant role n creaton the entanglement between atom and ts spontaneous emsson felds. The steady state entropy Sa () t as a functon of s dsplayed n Fg. 5. It can be realzed that the degree of entanglement between the atom and ts spontaneous emsson felds are substantally decrease for 0. However, around zero detunng the atom and the felds are strongly entangled. References 1. Bennett, C.H. DVncenzo, D.P. Smoln, J.A. Wootters, W.K. Mxed-state entanglement and quantum error correcton. Phys. Rev. A, 54: (1996). 2. Bennett, C.H. Bernsten, H.J. Popescu, S. Schumacher, B. Concentratng partal entanglement by local operatons. Phys. Rev. A, 53: (1996). 3. Phoenx, S.J.D. Knght, P.L. Fluctuatons and entropy n models of quantum optcal resonance. Ann. Phys. (NY). 186: (1988). 4. Phoenx, S.J.D. Knght, P.L. Establshment of an entangled atom-feld state n the Jaynes-Cummngs model. Phys. Rev. A. 44: (1991). 5. Ensten, A. Podolsky, A.B. Rosen, N. Can Quantum- Mechancal Descrpton of Physcal Realty Be Consdered Complete?. Phys. Rev. 47: (1935). 6. Bennett, C.H. Brassard, G. Quantum Cryptography, In Proceedngs of the IEEE Internatonal Conference on Computers, Systems, and Sgnal Processng. Bangalore, Inda, December; (1984). 7. Fox, M. Quantum Optcs: An Introducton, 1st ed.; Oxford Unversty Press: New York, NY, USA, (2006). 8. Benent, G. Casat, G. Strn, G. Prncples of Quantum Computaton and Informaton. World Scentfc Publshng Company: Sngapore, Volume 1 (2004). 9. Bennett, C.H. Brassard, G. Crepeau, C. Jozsa, R. Peres, A. Wootters, W.K. Phys. Rev. Lett. 70: (1993). 10. Ekert, A.K. Quantum cryptography based on Bell s theorem. Phys. Rev. Lett. 67: (1991). 11. Bouwmeester, D. Pan, J-W. Mattle, K. Ebl, M.; Wenfurter, H. Zelnger, A. Expermental Quantum Teleportaton. Nature, 390: (1997). 12. Abdalla, M.S. Abdel-Aty, M. Obada, A.S.F. Entropy and entanglement of tme dependent two-mode Jaynes- Cummngs model. Phys. A, 326: (2003). 13. Huang,Ch. Tang, L. Kong, F. Fang, J. Zhou M. Entropy evoluton of feld nteractng wth V-type three-level atom va ntensty-dependent couplng. Physca A. 368: (2006). 14. Mortezapour, A. Abed, M. Mahmoud, M. Khajehpour, M.R.H. Phase-controlled atom-photon entanglement n a three-level V-type atomc system va spontaneously generated coherence. J. Phys. B: At. Mol. Opt. Phys. 44: (2011). 15. Mao-Fa Fang, Sh-Yao Zhu. Entanglement between a Λ- type three-level atom and ts spontaneous emsson felds. Physca A. 369: (2006). 16. Malnovsky, V.S. Sola, I.R. Quantum Phase Control of Entanglement. Phys. Rev. Lett. 93: (2004). 17. Boller, K.J. Imamolu, A. Harrs, S.E. Observaton of electromagnetcally nduced transparency. Phys. Rev. Lett. 66: (1991). 18. Feld, J.E.; Hahn, K.H.; Harrs, S.E. Observaton of electromagnetcally nduced transparency n collsonally broadened lead vapor. Phys. Rev. Lett. 67: (1991). 19. Javananen. Effect of State Superpostons Created by Spontaneous Emsson on Laser-Drven Transtons. J. Europhys. Lett. 17: (1992). 20. Mahmoud, M. Sahra, M; Tajall, H. The effects of the ncoherent pumpng feld on the phase control of group velocty. J. Phys. B: At. Mol. Opt. Phys. 39: (2006). 21. Roshan Entezar, S. Dsentanglement of atom-photon va quantum nterference n drven three-level atoms. Opt. Commun. 282: (2009). 22. L. Y and Xao. M. Transent propertes of an electromagnetcally nduced transparency n three-level atoms. Opt. Lett. 20: (1995). 23. Huang. H, Zhu. S.Y and Zubary. M. S. Two-tme ntensty correlaton n a drven three-level system. Phys. Rev. A. 53: (1996). 24. Olga Kocharovskaya, Paul Mandel. Amplfcaton wthout nverson: The double-λ scheme. Phys. Rev. A. 42: (1990). 25. Arak. M, Leb. E. Entropy nequaltes. Commun. Math. Phys. 18: (1970). 184