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Motion-Enhanced Quantum Entanglement in the Dynamics of Excitation Transfer *

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Motion-Enhanced Quantum Entanglement in the Dynamics of Excitation Transfer * ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 32 Number 8 August 2015 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIET Y CHIN. PHYS. LETT. Vol. 32, No. 8 (2015) 088701 Motion-Enhanced Quantum Entanglement in the Dynamics of Excitation Transfer * SONG Wei(y伟)1**, HUANG Yi-Sheng(黄怿v)2, YANG Ming(杨¶)2, CAO Zhuo-Liang(ùRû)1 1Institute for Quantum Control and Quantum Information, and School of Electronic and Information Engineering, Hefei Normal University, Hefei 230601 2School of Physics and Material Science, Anhui University, Hefei 230601 (Received 15 March 2015) We investigate the dynamics of entanglement in the excitation transfer through a model consisting of three interacting molecules coupled to environments. It is shown that the entanglement can be further enhanced if the distance between the molecules is oscillating. Our results demonstrate that the motional effect plays a constructive role on quantum entanglement in the dynamics of excitation transfer. This mechanism might provide a useful guideline for designing artificial systems to battle against decoherence. PACS: 87.15.A−, 87.15.hj, 03.67.−a DOI: 10.1088/0256-307X/32/8/088701 Usually quantum entanglement is regarded as a of quantum coherence in light harvesting efficiency. fragile resource and very sensitive to noise. Thus we In particular, Sarovar et al.[25] present numerical evi- need rigorous laboratory conditions to manufacture dences for the existence of entanglement in the FMO and maintain entanglement. It is believed that en- complex for relatively long times. The influence of tanglement cannot exist outside the laboratories, not Markovian, as well as non-Markovian noise on the dy- to mention biological systems, which are wet and hot, namics of entanglement in FMO complex have also and with extremely high levels of noise.[1] However, been analyzed by Caruso et al.[27] several recent studies have shown that entanglement The third aspect includes the investigation of en- might exist in non-equilibrium systems and survive tanglement in larger light harvesting complexes. Light for relatively long time scales at physiological temper- harvesting complex II (LHCII) is the most abundant atures in biological systems. To date, these studies photosynthetic antenna complex in plants containing can be divided into three aspects: the first aspect over 50% of the world’s chlorophyll molecules.[29;30] is focused on quantum entanglement in model sys- It is shown that LHCII also exhibits long-lived elec- tems. The first study was made by Thorwart et al.[2] tronic coherence. Ishizaki et al.[28] have investigated who demonstrate that enhanced quantum entangle- quantum entanglement in LHCII across different bi- ment in the non-Markovian dynamics of biomolecular partitions of the chlorophyll pigments. excitons. It was shown in Ref. [3] that dynamic entan- However, in previous investigations, the motional glement can be continuously generated in noisy non- effect of the molecules in the dynamics of excita- equilibrium systems. Subsequently, this phenomenon tion transfer is omitted. It is suggested that con- has been generalized to the spin gas model and non- formational motion is an important feature of molec- Markovian models.[4] Galve et al.[5] also predict that ular processes, such as protein folding or excitation nanomechanical oscillators can be entangled at much transfer in light harvesting complex. These lead higher temperatures than those previously thought to effectively time-dependent interactions, with their possible. strengths modulated by the motion of the molecular. The second aspect is quantum entanglement in In analogy to the static case, the effect of the motion Fenna–Matthews–Olson (FMO) complex of green sul- on the entanglement dynamics of excitation transfer is fur bacterium. The FMO complex is a water solu- far from being understood. Here we tackle this prob- ble complex and acts as a molecular wire to transfer lem by considering a model consisting of three inter- the excitation energy from the light harvesting an- acting molecules driven through the oscillating mo- tenna to the reaction center. Many theoretical stud- tion. Our studies demonstrate that for a wide range ies in light harvesting structures are focused on the of parameters, the average entanglement can be en- FMO due to its well characterized pigment-protein hanced if the motional effect is taken into account. In structure. Recent experiments by Engel et al.[6] and our discussion, we assume that the conformational mo- Panitchayangkoon et al.[7] have shown that the elec- tion of the molecular structure can be described classi- tronic coherence between two excitonic levels at both cally. This semi-quantal approximation holds if the in- cryogenic (77 K) and ambient (300 K) temperatures. volved molecules are too large to show their quantum These experimental studies have generated numerous behavior.[4] Motion of the individual molecules leads theoretical interests[8−24] in understanding the role to a change in dipole moments and thus also induces *Supported by the National Natural Science Foundation of China under Grant Nos 11374085, 61073048 and 11274010, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No 20113401110002, the 211 Project of Anhui University, the Anhui Provincial Natural Science Foundation under Grant No 1408085MA20, the Personnel Department of Anhui Province, and the 136 Foundation of Hefei Normal University under Grant No 2014136KJB04. **Corresponding author. Email: [email protected] © 2015 Chinese Physical Society and IOP Publishing Ltd 088701-1 CHIN. PHYS. LETT. Vol. 32, No. 8 (2015) 088701 a time-dependence of the coupling. This mechanism the form will be dominant whenever the molecules are tightly N embedded in a protein scaffold such as in the FMO X + − − + complex. For simplicity, we only consider the modu- Ldiss(휌) = Γn[−{휎n 휎n ; 휌g + 2휎n 휌휎n ]; (3) lation of coupling strength due to the change in dis- n=1 tance. According to Ref. [4], we choose the motion of + − where 휎n = jnih0j and 휎n = j0ihnj are, respectively, the molecules change their distance periodically, which the raising and lowering operators for site n, and j0i holds for small amplitudes in the harmonic regime. refers to the zero exciton state of the system. The Furthermore, we suppose that the molecules are dis- symbol fA; Bg is an anticommutator, and Γn denotes tant enough from each other such that we only con- the dissipation rate of the nth molecule. sider the nearest neighbor interactions. In natural conditions, it is reasonable to consider at most one 1.0 excitation during the excitation transfer process.[25] (a) 0.8 Under these conditions, the Hamiltonian of the chain of interacting molecules in the single-excitation 0.6 manifold can be written as 0.4 Concurrence 0.2 N N−1 X X 0.0 H = "njnihnj + Jn(jnihn + 1j + jn + 1ihnj); 1.0 n=1 n=1 (1) 0.8 where jni represents the state with the excitation at 0.6 the nth site having energy "n and all other states are 0.4 in their electronic ground state, and J is the cou- n Concurrence 0.2 pling strength between the nth and the (n + 1)th molecule. Motion of the molecules will induce a time- 0.0 dependent coupling strength and deformation of the 1.0 molecules will also lead to the change of dipolar cou- 0.8 pling. For simplicity, we suppose that the coupling 0.6 strength only changes with the relative distance be- tween the molecules. Suppose that the distance be- 0.4 Concurrence tween the molecules n and n + 1 is dn(t) = d0 − 0.2 [u (t) − u (t)] = d [1 − 2a sin(!t + 휑 )], where 0.0 n n+1 0 n n 0 1 2 3 4 5 6 7 8 un(t) denotes the position of the nth molecule, d0 t is the equilibrium distance between two neighboring Fig. 1. (Color online) Time evolutions of the entangle- molecules, and an is the individual sites’ relative am- ment of molecules 1 and 2, in the motional case (red line), plitude of oscillation. Correspondingly, the dipole- and in the static case (blue line): (a) ! = 1, a1 = 1=4, (b) dipole coupling strength between two molecules is ! = 2, a1 = 1=4, and (c) ! = 5, a1 = 1=3. [3;4] given by To model the dynamics of the excitation transfer along a chain of molecules, we introduce an additional site, the sink that resembles the reaction center in pho- J~ J 0 0 tosynthesis. It denotes an irreversible decay of exci- Jn(t) = 3 = 3 ; (2) [dn(t)] [1 − 2an sin(!t + 휑n)] tations from the site N to the last N + 1 site. The absorption of the energy from the site N to the sink ~ 3 ~ where we have defined J0 = J0=d0, and J0 contains (numbered N + 1) is modeled by a Lindblad operator the dipole moments and physical constants. Here + − + − our discussion corresponds to a given amplitude, fre- Lsink(휌) = Γs[2휎N+1휎N 휌휎N 휎N+1 + − + − quency, and phase synchronized with the propagation − f휎N 휎N+1휎N+1휎N ; 휌g]; (4) of the excitation. This assumption is reasonable due to the fact that the wave packet that describes the nu- where Γs is the absorption rate of the sink which de- clear motion has been observed to exhibit surprisingly scribes the irreversible decay of the excitations to the long coherence times in reaction center proteins.[26] sink. Thus the master equation of the density matrix Although this model is rather simple, several biologi- 휌 of the system is given by cal systems show a similar structure that agrees with 푑휌 the assumptions underlying our model. For example, = i[휌, H] + Ldiss(휌) + Lsink(휌): (5) a type of secondary structure in proteins named 훼- dt helix or FMO complex, among which an excitation To study the role of the motional effect on en- can be exchanged due to dipole-dipole couplings be- tanglement, we only consider the simplest case of a tween the molecules.
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