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energies Article

Realistic Quantum Control of Energy Transfer in Photosynthetic Processes Reda M. El-Shishtawy 1 , Robert Haddon 2,3 , Saleh Al-Heniti 4 , Bahaaudin Raffah 4 , Sayed Abdel-Khalek 5, *, Kamal Berrada 6 and Yas Al-Hadeethi 4,7 1 2 3 4 5 6 7

*

Department of Chemistry, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia; [email protected] Center for Nanoscale Science and Engineering, Departments of Chemistry and Chemical and Environmental Engineering, University of California, Riverside, CA 92521-0403, USA; [email protected] Physics Department, King Abdulaziz University, Jeddah 21589, Saudi Arabia Physics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia; [email protected] (S.A.-H.); [email protected] (B.R.); [email protected] (Y.A.-H.) Mathematics Department, Faculty of Science, Taif University, Taif 009662, Saudi Arabia Department of Physics, College of Science, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 009661, Saudi Arabia; [email protected] Lithography in Devices Fabrication and Development Research Group, DSR, King Abdulaziz University, Jeddah 21589, Saudi Arabia Correspondence: [email protected]

Academic Editor: Thomas E. Amidon Received: 28 October 2016; Accepted: 6 December 2016; Published: 15 December 2016

Abstract: The occurrence of coherence phenomenon as a result of the interference of the probability amplitude terms is among the principle features of quantum mechanics concepts. Current experiments display the presence of quantum techniques whose coherence is supplied over large interval times. Specifically, photosynthetic mechanisms in light-harvesting complexes furnish oscillatory behaviors owing to quantum coherence. In this manuscript, we study the coherent quantum energy transfer for a single-excitation and nonlocal correlation in a dimer system (donor+acceptor) displayed by two-level systems (TLSs), interacting with a cavity field with a time-dependent coupling effect considering the realistic situation of coupling between each TLS and the cavity field. We analyze and explore the specific conditions which are viable with real experimental realization for the ultimate transfer of quantum energy and nonlocal quantum correlation. We show that the enhancement of the probability for a single-excitation energy transfer greatly benefits from the energy detuning, photon-number transition, classicality of the field, and the time-dependent coupling effect. We also find that the entanglement between the donor and acceptor is very sensitive to the physical parameters and it can be generated during the coherent energy transfer. Keywords: quantum effects in biology; energy transfer; dipole-dipole interaction; time-dependent coupling effect; quantum correlations

1. Introduction Aphotosynthetic light-harvesting system transforms the energy from the absorbed photons to the reaction center [1–7]. It employs light-harvesting antennae to absorb and transform solar energy to the reaction center with quantum efficiency. The transfer of the energy of light from an atom to a nearby atom can be performed by electronic energy transfer (see Figure 1), referred to as the resonance energy transfer. Several recent important applications and illustrations of energy transfer are defined, such as the increase of the spectral and spatial cross-section in photosynthetic proteins using resonance energy transfer (light-harvesting proteins), in order to capture the quantum solar Energies 2016, 9, 1063; doi:10.3390/en9121063

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are defined, such as the increase of the spectral and spatial cross-section in photosynthetic proteins using resonance energy transfer (light-harvesting proteins), in order to capture the quantum solar energy by by photonic photonic systems. systems. Usually, Usually, the the quantum quantum transport transport of of energy energy is is described described by by an an incoherent incoherent energy process where a donor atom captures the excitation energy and then transported to an acceptor atom. process where a donor atom captures the excitation energy and then transported to an acceptor atom. Here, the the electronic electronic coupling coupling elements elements are are used used to to describe the transferred energy considering the Here, describe the transferred energy considering the effect of inter-atomic interaction in the framework of dipole-dipole coupling. More recently, coherent effect of inter-atomic interaction in the framework of dipole-dipole coupling. More recently, coherent energy transfer to be periodperiod inphotosynthesis, in orderin to order produce energy transferisisprovided provided to an beinteresting an interesting inphotosynthesis, to electronic produce excitations from photosynthetic pigments and transport this excited energy to a reaction center [8–14]. electronic excitations from photosynthetic pigments and transport this excited energy to a reaction A simple mechanism is proposed to study the light-harvesting complex in a dimer system consisting center [8–14]. A simple mechanism is proposed to study the light-harvesting complex in a dimer of a donor and an of acceptor where each one is described byone a two-level quantum system. Recently, system consisting a donor and an acceptor where each is described by a two-level quantuma number of both theoretical studies have been developed and experimental tools have been conducted system. Recently, a number of both theoretical studies have been developed and experimental tools usingbeen the phenomenon of electronic coherence in the transport of the energy [15,16]. It is have conducted using the phenomenon of electronic coherence in quantum the transport of the quantum shown that exciton together with pure dephasing bydephasing stochastic fluctuations energy [15,16]. It isdelocalization, shown that exciton delocalization, together induced with pure induced by of the external environment, has been considered to improve the efficiency of quantum transfer of in stochastic fluctuations of the external environment, has been considered to improve the efficiency multichromophoric quantum systems [17,18]. quantum transfer in multichromophoric quantum systems [17,18].

Figure 1. A diagram illustrating our system. TLS1 and TLS2 are both placed inside one cavity. A donor Figure 1. A diagram illustrating our system. TLS1 and TLS2 are both placed inside one cavity. ω1 ) andωan1 ) acceptor ω 2 ) areω2modeled (with resonant frequency by two(with resonant frequencyfrequency A donor (with resonant and an acceptor (with resonant frequency ) are modeled by level atoms withwith strong dipole-dipole interaction. Realistic two-level atoms strong dipole-dipole interaction. Realisticquantum quantumcontrol controlalmost almost necessarily engagement of continuous continuous variable variable interaction interaction of of TLSs TLSs having having aa finite number of states with implies engagement coupling strength that provides the physical a “large” system with a continuous quantum state. The coupling mechanism for coherent energy transfer and entanglement generation is considered in the context of the time-dependent time-dependent coupling coupling effect effect GG (t(t ). ) . the

observation of long-lived long-lived coherence coherence in in those those quantum quantum systems systems has has been been aa topic topic of great The observation contemplation. It is shown the quantum efficiency of contemplation. shown that thatthe thequantum quantumcoherence coherenceleads leadstotoa adecrease decreaseinin the quantum efficiency thethe energy transfer and includes dephasing, which which combines the tunneling and noiseand effects leading to of energy transfer and includes dephasing, combines the tunneling noise effects a highlyto efficient quantum transfer [19–21]. Recent investigations on photosynthetic complexes leading a highly efficientenergy quantum energy transfer [19–21]. Recent investigations on photosynthetic show that the coherence resists thepopulation transfer long enough toenough impact to quantum complexes show that the coherence resists thepopulation transfer long impact transport quantum dynamics in the time in scale It has been shown in the photosynthetic quantum antenna systems, transport dynamics the[22]. time scale [22]. It hasthat been shown that in theantenna photosynthetic the quantum coherences could be coherences removed during long lifeofHowever, the quantum quantum systems, the quantum couldthe bedynamics. removed However, during thethe dynamics. the coherence, alone, will notcoherence, be sufficient to result in be ahigh quantum efficiency of the transferred energy. long lifeof the quantum alone, will not sufficient to result in ahigh quantum efficiency of Thetransferred quantum coherences play a role incontrolling the asystem dynamics and excitation populations the energy. The quantum coherences play role incontrolling thethe system dynamics and the in a givenpopulations quantum state transport). excitation in a(quantum given quantum state (quantum transport). of of thethe most promising phenomenon in different branches of science, Quantum entanglement entanglementisisone one most promising phenomenon in different branches of which exhibits quantum between different physical systems and was introduced by science, which the exhibits the correlations quantum correlations between different physical systems and was Schrodingerby[23,24], as the [23,24], characteristic trait of quantum the one and that the enforces its introduced Schrodinger as the characteristic traitmechanics, of quantumand mechanics, one that entire departure classical lines of thought [25]. These[25]. quantum correlations quantify from the enforces its entirefrom departure from classical lines of thought These quantum correlations quantify kind of quantum states of states the composite systems.systems. Nonlocal correlations in the combined system from thethe kind of the quantum of the composite Nonlocal correlations in the combined are independent of the spatial of the constituents, in orderintoorder appear a lone system are independent of the separation spatial separation of the constituents, to as appear assystem. a lone As Schrodinger mentioned, the best possible of a wholeof does not necessarily the system. As Schrodinger mentioned, the bestknowledge possible knowledge a whole does not include necessarily

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best possible knowledge of its parts [25]. Investigation of this kind of quantum correlation and its consequences for quantum measurements led to solving and understanding many physical problems, including the Bell inequalities [26,27], and verified the experimental part of the spooky action at a distance, as mentioned by Einstein. More recently, the development of quantum information theory has provided a rise in knowledge and also augmented the literature of the entanglement phenomenon, which has played a crucial role in different tasks of quantum information processing and transmission and, more recently, in quantum metrology [28–34]. The significance of this kind of correlation (entanglement) in different applications has led to analysis and investigation of high-dimensional quantum systems and brings a new role and application of these kinds of correlations in many particle quantum systems [35–47]. A measure of entanglement must be invariant under local operations and classical communication. The understanding of derivable measures of quantum entanglement are borne in mind when searching to detect entanglement in large dimensional complex systems, such as those considered in biological systems. Motivated by the above considerations, this study aims to investigate the influence of the atomic motion effect in the interactions between the pigments on energy transfer considering a model of atom-atom interaction that closely describes a realistic experimental scenario. We will present, in detail, how the time-dependent coupling effect and energy frequencies can influence the efficiency of the coherent energy transfer for a single excitation from a donor to an acceptor where each one is modeled by a two-level atom system. We can obtain physical answers to this very complex problem of quantum energy transfer considering the atomic motion effect. On the other hand, we will study the dynamics of the nonlocal correlation between the pigments during the process of the quantum energy transfer. Furthermore, we will study the dynamic behavior of the quantum variance when performing a measurement on an observable for the density matrix in pigments, which is rather significant in different tasks of quantum information and computational technologies. This article is organized as follows: in Section 2, we present the model for our system and the details of the formalism that describes the dependence of the physical parameters on the population dynamics of the coherent quantum energy transfer for a single-excitation and nonlocal correlation between the pigments; in Section 3, we discuss the main obtained results. Our aim is to highlight the dependence of the population, entanglement, and classicality of the cavity field on the energy frequencies and the time-dependent coupling effect during the time evolution. Finally, conclusions are drawn in Section 4. 2. Model of the Physical System The physical system under our investigation is a dimer system considering dipole-dipole interaction with the time-dependent coupling effect. Here, the dimer is defined by a pair of two-level systems (TLSs), which compose TLS1 as the donor system and TLS2 as the acceptor system, with energy separations ω1 and ω2 , respectively, as depicted in Figure 1. The Hamiltonian for the donor and acceptor systems can be written as:

+ ω22 σ2z + ωa† a + λ σ1+ σ2− + σ1− σ2+ 2 n l l o + G (t) ∑ a†2 σj− + σj+ a2 HTLSs =

ω1 z 2 σ1

 (1)

j =1

where λ is the dipole-dipole interaction strength between TLS1 and TLS2 and the usual Pauli operators σj+ = |e j ih g j | (raising operator for the jth two-level atoms), σj− = | g j ihe j | (lowering operator for  the jth two-level atoms), and σjz = |e j ihe j | − | g j ih g j |, where | g j i |e j i is the ground (excited) states of the jth (j = 1, 2) TLS. The coupling provided by the dipole-dipole interaction term exhibits the physical process for excitation energy transfer and the nonlocal correlation between TLS1 and TLS2. G (t) denotes the time-dependent coupling between TLSs and the cavity field and l is the photon number transition. The generalization from the constant coupling g to a coupling that evolves with

the cavity field and l is the photon number transition. The generalization from the constant the cavitygfield the evolves photonwith number generalization from the constant l isthat G(t ) , will The coupling to a and coupling time transition. provide new physical phenomena that G (t ) g coupling to a coupling that evolves with time , will provide new physical phenomena the have not been discussed before. A realization of particular interest, with respect to G(t ) , may bethat

have not been discussed before. A realization of particular interest, with respect to G(t ) , may be the time-dependent alignment or orientation of the atomic/molecular dipole moment using a laser pulse Energies 2016, 9, 1063alignment or orientation of the atomic/molecular dipole moment using a laser pulse 4 of 11 time-dependent [48–53] and the motion of the atom through the cavity. For an atom oscillating back and forth across

andcavity the motion atomtrap, through cavity.isFor an atom approximately oscillating backto and across a[48–53] narrow withinofathe square the the coupling modelled beforth sinusoidal 2 provide time will physicaltrap, phenomena that have not been discussed before.toAbe realization of aG cavity a square the coupling is modelled approximately sinusoidal (narrow t ) G=(gt),sin (t ) within [48]. new 2 particular interest, with respect to G ( t ) , may be the time-dependent alignment or orientation of the G(t )In=order g sinto(tinvestigate ) [48]. the populations of the transfer of energy in a single excitation from the atomic/molecular dipole moment using a laser pulse [48–53] and the motion of the atom through the In order to investigate the populations of theis transfer energy single state excitation the | e1  from and the donor to the acceptor, we consider that the donor initially of defined inin anaexcited cavity. For an atom oscillating back and forth across a narrow cavity within a square trap, the coupling donor to is the acceptor, consider is initially defined in an excited state | e1  and the g 2 sinusoidal :that the donor acceptor a groundwe state is modelledinapproximately to| be G (t) = g sin2 (t) [48]. | gpopulations acceptor is in atoground state the In order investigate of the transfer of energy in a single excitation from the 2 : | y12 ñ =|ise1initially g2 ñ donor to the acceptor, we consider that the donor defined in an excited state |e1 i and(2) the | y12 ñ =| e1 g 2 ñ (2) acceptor is in a ground state | g2 i: and the probability to get the acceptor in an excited state (single excitation energy transfer) at time t (2) |ψ12 i =|e1 g2 i and be: the probability to get the acceptor in an excited state (single excitation energy transfer) at time t will and be: the probability to get the acceptor in an excited state (single excitation energy transfer) at time t will will be: P(t ) = Tr r2s2+s2- (3) + P (3) (3) P((tt)) = = Tr ρr22σs2+2 σs2−2 The entanglement between the donor and acceptor can be quantified via the concurrence defined The entanglement entanglement between between the the donor donor and and acceptor acceptor can can be bequantified quantified via viathe theconcurrence concurrence defined defined The by [54]: by [54]: by [54]: C= = max{0, max{0, l λ11 −l λ22 −l λ33 −lλ44}} (4)(4) C = max{0, l1 -l2 -l3 -l4 } (4) ~andand ~defined where λλi define the eigenvalues giving ininthe decreasing order ofofρ12ρρe12ρ ρe12 isρ by: by: where define the eigenvalues giving the decreasing order is defined i 12 12 ~ ~12  ∗ where λi define the eigenvalues giving in the decreasing order of ρ12 ρ12 and ρ12 is defined by: ρe12 = σy ⊗ σy ρ12 σy ⊗ σy (5) r12 = s y Ä s y r12* s y Ä s y (5) ∗ denotes the conjugate of ρ r1212 in =the s ystandard Ä s y r12*basis s y of Äs where ρ12 two (5) y bipartite system and σy is the Pauli ∗ In order to study the nonclassical properties of the cavity field during the energy transfer Y operator. where ρ12 denotes the conjugate of ρ12 in the standard basis of two bipartite system and σ y is ∗ process,ρwe use Mandel’s parameter as a quantifier of the statistical properties and verify the presence where 12 denotes the conjugate of ρ12 in the standard basis of two bipartite system and σ y is Y operator. In order to This studyparameter the nonclassical properties the Pauli of quantumness in the cavity field. is defined as [55]: of the cavity field during the Y operator. In order to study the nonclassical properties the cavity field duringand the the Pauli energy transfer process, we use Mandel’s parameter as a quantifier ofofthe statistical properties 2 ˆi h(∆nˆ ) ias−ahnquantifier energythe transfer process, we use Mandel’s parameter ofisthe statistical properties and verify presence of quantumness in the defined as [55]: M pcavity = field. This parameter , (6) ˆ n h i verify the presence of quantumness in the cavity field. This parameter is defined as [55]:

( (

( (

) ( ) (

) )

) )

( )2 ( )

 Δnˆ field  − and nˆ h(∆nˆ )2 i corresponds to the mean-square where hnˆ i is the average photon numberM of the cavity 2 (6) , p =  Δn ˆ  −  ˆ  nˆ , whether the photon distribution of(6)  n variance. Mandel’s parameter is used to determine the = Mprecisely p ˆ  n  cavity field is sub-Poissonian (−1 ≤ M p ≤ 0), evidently being the nonclassical state, the Poissonian 2  corresponds to the nˆ 2super-Poissonian ˆ  corresponds (M p = n 0) to photon the casenumber of the of classical coherent state, Δ and (M pmean> 0). is the average the cavity field and where ˆ  Δ  n ˆ n  is the average photon number of the cavity field and corresponds to the meanwhere Mandel’s parameter can be considered as a tool to investigate the effect of the physical parameters, square variance. Mandel’s parameter is used to precisely determine whether the photon distribution which contain all information on the statistical properties of thewhether cavity field duringdistribution the energy square variance. Mandel’s parameter is( used determine the photon − 1 ≤ to Mprecisely of the cavity field is sub-Poissonian p ≤ 0 ), evidently being the nonclassical state, the transfer process. of the cavity field is sub-Poissonian ( − 1 ≤ M ≤ 0 ), evidently being the nonclassical state, the Poissonian ( M p = 0 ) corresponds to the case of pthe classical coherent state, and super-Poissonian ( 3. Results and Poissonian ( M Discussion p = 0 ) corresponds to the case of the classical coherent state, and super-Poissonian ( M p > 0 ). Mandel’s parameter can be considered as a tool to investigate the effect of the physical of the terms of thethe probability is M p >The 0 ).coherence Mandel’sintroduced parameter by canthe beinfluence considered as adifferent tool to investigate effect of amplitude the physical parameters, which contain all information on the statisticalThe properties the cavity field duringtothe one of discriminatory characters of quantum mechanics. coherentofphenomena are shown be parameters, which all information on the statistical properties of It theis cavity during the responsible forprocess. the contain oscillatory-appearingbehaviors in quantum systems. shownfield experimentally energy transfer that thetransfer light-harvesting energy process. provides oscillatory electronic dynamics and explains the nature of such an comportment 3.oscillatory Results and Discussion[4,6], showing the importance of quantum mechanics in biological functional systems it is important to treat these oscillations realistically in order to disclose and 3. Results[4]. andThus, Discussion The coherence introduced by the influence of the different terms of the probability amplitude is understand the mechanisms adopted by nature, which will lead, in the future, to inspiring new The coherence introduced influence of the different terms ofphenomena the probability amplitude is one of discriminatory charactersbyofthe quantum mechanics. The coherent are shown to be quantum technologies. one of discriminatory characters of quantum mechanics. The coherent phenomena are shown to be Let us investigate the influence of the physical parameters on the dynamic behavior of the probability P for a single-excitation energy transfer and entanglement in the absence and presence of the time-dependent coupling effect when TLSs are interacting with a continuous quantum field following the considered models, as shown in Figure 1. We plot, in Figures 2 and 3, the evolution of the probability and entanglement versus time gt for various values of the energy detuning ωd without

( ) ( )

quantum technologies. Let us investigate the influence of the physical parameters on the dynamic behavior of the probability P for a single-excitation energy transfer and entanglement in the absence and presence of the time-dependent coupling effect when TLSs are interacting with a continuous quantum field following the considered models, as shown in Figure 1. We plot, in Figures 2 and 3, the evolution of Energies 2016, 9, 1063

the probability and entanglement versus time gt for various values of the energy detuning

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ωd without

and with time-dependent effect, respectively. It can be seen that the probability and entanglement and with time-dependent effect, respectively. It can be seen that probability and entanglement experience damped oscillations during the time-evolution and the they tend to stabilize at steady experience damped oscillations during the time-evolution and they tend to stabilize at trapped steady behavior behavior in the asymptotic gt → ∞ limit, showing the donor and acceptor are by the in the asymptotic gt → ∞ limit, showing the donor and acceptor are trapped by the cavity field cavity field at this limit. We find that the value of the probability and degree of the entanglement at this limit. We find that the value of the probability and degree of the entanglement during during the process of the coherent energy transfer are very sensitive to the energy detuning of the the process of the coherent energy transfer are very sensitive to the energy detuning of the TLSs. TLSs. Interestingly, the amplitude of the probability (oscillations) decreases (enhances) with the Interestingly, the amplitude of the probability (oscillations) decreases (enhances) with the increase of increase of the detuning parameter with and without the time-dependent coupling effect. This means the detuning parameter with and without the time-dependent coupling effect. This means that when that when the strength of the energy detuning is large, the energy emitted by the donor may be the strength theone energy detuning is large, the energy by the donor may more excited more of than acceptor and leads to enhance theemitted probability oscillations. Inbe theexcited presence of than one acceptor and leads to enhance the probability oscillations. In the presence of the TLSs motion 2 the TLSs motion ( G(t ) = g sin (t ) ), the dynamic behavior of the probability and entanglement are (G (t) = g sin2 (t)), the dynamic behavior of the probability and entanglement are deeply influenced by deeply influenced by the coupling between eachfield. TLS and the cavity In general, we find that the the coupling between each TLS and the cavity In general, wefield. find that the motion effect leads motion effect leads to enhancing the probability of energy transfer and entanglement between the to enhancing the probability of energy transfer and entanglement between the donor and acceptor donor during the the other hand,exists we find that there exists a critical duringand the acceptor time-evolution. On time-evolution. the other hand, On we find that there a critical value of the energy value of the energy detuning for which the amount of correlation is maximal during the process of detuning for which the amount of correlation is maximal during the process of coherent energy transfer. coherent energy transfer. These results indicate that the improvement of the efficiency of the coherent These results indicate that the improvement of the efficiency of the coherent energy transfer for a energy transfer for a singleTLSs excitation benefits from theenergy combination of and the single excitation between greatlybetween benefits TLSs from greatly the combination of the detuning energy detuning and time-dependent coupling effect in the TLS-field interaction. time-dependent coupling effect in the TLS-field interaction.

Figure Figure 2. 2. (Color (Color Online) Online) The The probability probability and and entanglement entanglement are are plotted plotted versus versus the the dimensionless dimensionless time time forvarious variousvalues valuesof ofthe theenergy energy detuning detuning for one-photon transition in the absence of the = 11)) in gt for gt transition ((ll = the atomic motion effect G (t) = g with |α|2 = 10 . (a,c,e) display the variation of the population for a single-excitation with the time; (b,d,f) present the evolution of the concurrence of the donor-acceptor state during the time evolution. (a,b) is for ωd = 0; (c,d) is for ωd = 2; and (e,f) is for ωd = 4.

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atomic motion effect

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G(t ) = g

with | α |2 = 10 . (a,c,e) display the variation of the population for a

single-excitation with the time; (b,d,f) present the evolution of the concurrence of the donor-acceptor Energies 2016, 9, 1063

state during the time evolution. (a,b) is for

ωd = 0 ; (c,d) is for ωd = 2 ; and (e,f) is for ωd = 4 . 6 of 11

Figure 3. 3. The time for various various Figure The probability probabilityand andentanglement entanglementare areplotted plottedversus versusthe thedimensionless dimensionless timegt gt for values one-photon transition transition ((ll = in presence presence of of atomic atomic motion motion effect values of the energy detuning for one-photon = 11)) in 2 ( t )2 with α = 10 . 2(a,c,e) display the variation of the population for a single-excitation G ( t ) = g sin | | G(t ) = g sin (t ) with2 | α | = 10 . (a,c,e) display the variation of the population for a singlewith the time; (b,d,f) present the evolution of the concurrence of the donor-acceptor state during the excitation with the time; (b,d,f) present the evolution of the concurrence of the donor-acceptor state time evolution. (a,b) is for ωd = 0; (c,d) is for ωd = 2; and (e,f) is for ωd = 4. during the time evolution. (a,b) is for ωd = 0 ; (c,d) is for ωd = 2 ; and (e,f) is for ωd = 4 .

In Figures 4 and 5 we present a comparison between the dynamic behavior of the probability In Figures 4 and 5 we present a comparison the dynamic behavior of coupling the probability and entanglement in the multi-photon process withbetween and without the time-dependent effect, and entanglement in the multi-photon process with and without the time-dependent coupling effect, respectively. From the figures, we observe that the number of the photon transition in the interaction respectively. the figures, we observe that the the number of theofphoton transitionand in the interaction between TLSsFrom and cavity field significantly affects variation the probability entanglement between TLSs and cavity field significantly affects the variation of the probability and entanglement between the donor and acceptor system during the energy transfer operation. Interestingly, we find between the donor and acceptor during the energy operation. Interestingly, we find that the photon-transition numbersystem leads to a reduction in thetransfer oscillations, exhibiting a quasi-periodic that the photon-transition number leads to a reduction in the oscillations, exhibiting a quasi-periodic behavior of the probability and entanglement when the detuning energy gets close tothe resonance behavior the probability and entanglement wheneffect. the detuning energy gets close tothe resonance case in theofabsence of the time-dependent coupling case in the absence of the time-dependent coupling effect.

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Figure 4. (Color Online) The probability and entanglement are plotted versus the dimensionless time Figure 4. (Color Online) The probability and entanglement are plotted versus the dimensionless time Figure (Colorvalues Online) Theenergy probability and for entanglement plotted (versus dimensionless of the detuning two-photonare transition in the absence of gt for 4.various l = 2 )the various values of the energy detuning for for two-photon transition ( l =(l 2=) 2) in in thethe absence of gt for time gt for various values of the energy detuning two-photon transition absence for a atomic motion effect G(t ) = g with | α | 22 = 10 . (a,c,e) display the variation of the population GG(t()t)==gg with with |α| α (a,c,e)display displaythe thevariation variation of of the the population population for a atomic motion effect of atomic motion effect |2 |==10 10. .(a,c,e) single-excitation with the time; (b,d,f)present the evolution of the concurrence of the donor-acceptor single-excitation single-excitation with with the the time; time; (b,d,f)present (b,d,f)present the the evolution evolution of of the the concurrence concurrence of of the the donor-acceptor donor-acceptor 2 ;(e,f) state time evolution. evolution. (a,b) ; (c,d) is for and is (e,f) d2; = state during during the the time (a,b) is is for for ωωd d==0;0(c,d) is for ωd ω = and forisωfor =ω 4. d = 4 . state during the time evolution. (a,b) is for ωd = 0 ; (c,d) is for ωd = 2 ; and (e,f) is dfor ωd = 4 .

Figure Figure 5. 5. (Color (Color Online) Online) The The probability probability and and entanglement entanglement are are plotted plotted versus versus the the dimensionless dimensionless time time Figure 5. (Color Online) The probability and entanglement are plotted versus the dimensionless time forvarious variousvalues valuesofof the energy detuning two-photon transition the presence of l in = the 2 ) in gtfor gt the energy detuning for for two-photon transition (l = (2) presence of atomic energy detuning for two-photon transition ( l = 2 ) in the presence of gt for various values of the 2 2 motion effect t) = g G sin(t()t)=with . (a,c,e) display the variation the variation populationoffor g sin|α2|2(t=) 10with displayofthe thea atomic motionG (effect | α |22 = 10 . (a,c,e) Gtime; (t ) =(b,d,f) g sinpresent (t ) with (a,c,e) display of thethevariation of the atomic motion effect | α | = 10of. the single-excitation with the the evolution concurrence donor-acceptor population for a single-excitation with the time; (b,d,f) present the evolution of the concurrence of the state duringfor thea time evolution. (a,b) fortime; ωd =(b,d,f) 0; (c,d)present is for ωthe 2; and (e,f) is for ωd = 4. of the population single-excitation withisthe of the concurrence d =evolution donor-acceptor state during the time evolution. (a,b) is for ωd = 0 ; (c,d) is for ωd = 2 ; and (e,f) donor-acceptor state during the time evolution. (a,b) is for ωd = 0 ; (c,d) is for ωd = 2 ; and (e,f) = 4investigate . is for ω Now letd us the dynamic behavior of the statistical properties of the cavity field is for ωd = 4 .

using Mandel’s parameter during the quantum energy transfer. In order to explore the influence

Now let us investigate the dynamic behavior of the statistical properties of the cavity field using Mandel’s parameter during the quantum energy transfer. In order to explore the influence of the physical parameters on Mandel’s parameter clearly, in Figure 6, we show the variation of Mandel’s parameter with respect to different values of the energy detuning

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ω d . In order to make results 8 of 11

comparable, we have considered both cases without (Figure 6a,c,e) and with (Figure 6b,d,f) timedependent coupling effects. From the figure, we find that that the parameters ω d significantly affect of the physical parameters on Mandel’s parameter clearly, in Figure 6, we show the variation of the photonparameter distribution of the cavity where the increase (decrease) in theωparameter leads Mandel’s with respect to field, different values of the energy detuning . In orderωto d make d results comparable, we have considered both without (Figure 6a,c,e) and with (Figure 6b,d,f) to enhancing the non-classicality of the field in cases the absence (presence) of the time-dependent coupling time-dependent coupling effects. From the figure, we find that that the parameters ω significantly d verifyies the effect. Interestingly, in the resonance case ωd = 0 with G(t ) = g , Mandel’s parameter affect the photon distribution of the cavity field, where the increase (decrease) in the parameter ωd − 1 ≤ Mthe 0 for small values inequality offield timeinshowing the sub-Poissonian distribution of the p ≤non-classicality leads to enhancing of the the absence (presence) of the time-dependent coupling in the resonance case ωd = 0 with G (t)exhibiting = g, Mandel’s parameter photons effect. and MInterestingly, large, super-Poissonian p > 0 as the time becomes significantly verifyies the inequality −1 ≤ M p ≤ 0 for small values of time showing the sub-Poissonian distribution 2 G(t ) = g sinlarge, (t ) , the M p is always distribution of the for the case of significantly parameter of the photons andphotons. M p > Whereas 0 as the time becomes exhibiting super-Poissonian 2 distribution of the Whereas case of G (case t) =( ω g sin≠ (0t)),, the parameter M p is tends always negative during thephotons. time evolution. For for the the off-resonance Mandel’s parameter to d negative during the time evolution. For the off-resonance case (ωd 6= 0), Mandel’s parameter tends be near to the trivial case M = 0 as the time becomes large, reflecting that the photon number to be near to the trivial case Mpp = 0 as the time becomes large, reflecting that the photon number distribution is is Poissonian. Poissonian. distribution

Figure 6. 6. (Color (Color Online) Online) Mandel’s Mandel’s parameter parameter given given in in Equation Equation (6) (6) is is plotted plotted vs. vs. the the dimensionless dimensionless Figure for various values of the energy detuning for one-photon transition ( time l =|α1|2) =with gt time gt for various values of the energy detuning for one-photon transition (l = 1) with 10 . 2 present the variation of the quantumness in the field in the absence of the time-dependent (a,c,e) | α | = 10 . (a,c,e) present the variation of the quantumness in the field in the absence of the timeeffect G (t) = g; (b,d,f) exhibit the variation of the quantumness in the field in the presence of the dependent effect G(t ) = g ; (b,d,f) exhibit the variation of the quantumness in the field in the time-dependent effect G (t) = g sin2 (t). (a,b) is for ωd = 0; (c,d) is for ωd = 2; and (e,f) is for ωd = 4. presence of the time-dependent effect

G(t ) = g sin 2 (t ) . (a,b) is for ωd = 0 ; (c,d) is for ωd = 2

These results the quantumness in the field can be used as an indicator to enhance ωd = 4that ; and (e,f) is forindicate . and control the quantum energy transfer and entanglement between the donor and acceptor in the photosynthesis process by a proper choice of the detuning parameter and photon-number transition in the presence of the time-dependent effect during the time-evolution.

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4. Conclusions Exploring new fundamental processes responsible for controlling quantum energy transfer in photosynthesis is the crucial aim for both theoretical and experimental investigations. In summary, we have investigated the coherent quantum transfer energy for a single-excitation and nonlocal correlation in a dimer system consisting of a donor and an acceptor, where each one is described by a two-level atom in the absence and presence of the time-dependent coupling effect. We have analyzed and explored the required conditions that are feasible with real experimental realization for optimal transfer of quantum energy and generation of nonlocal quantum correlation. We have shown that the enhancement of the probability for a single-excitation transfer energy is greatly benefits from the combination of energy detuning and time-dependent coupling effect. On the other hand, we have investigated the generation of a degree of quantum nonlocal correlation in the dimer during the process of the energy transfer in terms of the physical parameters using the linear entropy as a quantifier of entanglement. Furthermore, we have studied the dynamic behavior of the quantum variance when performing a measurement on an observable for the density matrix in the pigments. Finally, an interesting relationship between the transfer probability, entanglement, and quantum variance is explored during the time evolution in terms of the physical parameters. We treat here the generation of the entanglement among the pigments during the process of energy transfer in the presence of the atomic motion effect. A study into the initial prepared entanglement among the pigments on the quantum energy transfer in the presence of the time-dependent coupling effect will make for an interesting investigation. Another interesting contribution is to the study of the population of the energy transfer and entanglement in multi-level systems interacting with external fields under the effects of the atomic motions and field parameters. Acknowledgments: This project was funded by the deanship of scientific research (DSR) at King Abdulaziz University, Jeddah, under grant No. (44-130-36-RG). The authors, therefore, acknowledge with thanks DSR technical and financial support. Author Contributions: Reda M. El-Shishtawy conceived, shared in revising the manuscript and helped with facilities. Robert Haddon, Saleh Al-Heniti, Bahaaudin Raffah, Sayed Abdel-Khalek, Kamal Berrada and Yas Al-Hadeethi shared the concept, revision, enriched the research point, conducted the theoretical calculations and writing the manuscript. All authors shared equally the revision of the final version. Conflicts of Interest: The authors declare no conflict of interest.

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