# Environment-Assisted Modulation of Heat Flux in a Bio-Inspired System Based on Collision Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Collision Model

- ${R}_{n}^{h}$ interacts with ${S}_{1}$
- ${S}_{1}$ interacts with ${S}_{2}$
- ${S}_{2}$ interacts with A
- A interacts with ${B}_{n}$
- ${S}_{2}$ interacts with ${S}_{3}$
- ${S}_{3}$ interacts with ${R}_{n}^{c}$

## 3. Results

#### 3.1. Effect of the System-HSE and Inter-HSE Couplings

#### 3.2. Effect of Coherence within the HSE

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Blankenship, R. Molecular Mechanisms of Photosynthesis; Wiley: Hoboken, NJ, USA, 2014. [Google Scholar]
- Kenkre, V.M.; Knox, R.S. Generalized-master-equation theory of excitation transfer. Phys. Rev. B
**1974**, 9, 5279–5290. [Google Scholar] [CrossRef] - Leegwater, J.A. Coherent versus Incoherent Energy Transfer and Trapping in Photosynthetic Antenna Complexes. J. Phys. Chem.
**1996**, 100, 14403–14409. [Google Scholar] [CrossRef] [Green Version] - Kakitani, T.; Kimura, A.; Sumi, H. Theory of Excitation Transfer in the Intermediate Coupling Case. J. Phys. Chem. B
**1999**, 103, 3720–3726. [Google Scholar] [CrossRef] - Kimura, A.; Kakitani, T.; Yamato, T. Theory of Excitation Energy Transfer in the Intermediate Coupling Case. II. Criterion for Intermediate Coupling Excitation Energy Transfer Mechanism and Application to the Photosynthetic Antenna System. J. Phys. Chem. B
**2000**, 104, 9276–9287. [Google Scholar] [CrossRef] - Ishizaki, A.; Fleming, G.R. Unified treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: Reduced hierarchy equation approach. J. Chem. Phys.
**2009**, 130, 234111. [Google Scholar] [CrossRef] [Green Version] - Kimura, A.; Kakitani, T. Advanced Theory of Excitation Energy Transfer in Dimers. J. Phys. Chem. A
**2007**, 111, 12042–12048. [Google Scholar] [CrossRef] - Kimura, A. General theory of excitation energy transfer in donor-mediator-acceptor systems. J. Chem. Phys.
**2009**, 130, 154103. [Google Scholar] [CrossRef] - Jang, S. Theory of multichromophoric coherent resonance energy transfer: A polaronic quantum master equation approach. J. Chem. Phys.
**2011**, 135, 034105. [Google Scholar] [CrossRef] - Jang, S. Theory of coherent resonance energy transfer for coherent initial condition. J. Chem. Phys.
**2009**, 131, 164101. [Google Scholar] [CrossRef] - Hossein-Nejad, H.; Olaya-Castro, A.; Scholes, G.D. Phonon-mediated path-interference in electronic energy transfer. J. Chem. Phys.
**2012**, 136, 024112. [Google Scholar] [CrossRef] - Hu, X.; Ritz, T.; Damjanović, A.; Schulten, K. Pigment Organization and Transfer of Electronic Excitation in the Photosynthetic Unit of Purple Bacteria. J. Phys. Chem. B
**1997**, 101, 3854–3871. [Google Scholar] [CrossRef] - Chenu, A.; Scholes, G.D. Coherence in Energy Transfer and Photosynthesis. Annu. Rev. Phys. Chem.
**2015**, 66, 69–96. [Google Scholar] [CrossRef] [Green Version] - Curutchet, C.; Mennucci, B. Quantum Chemical Studies of Light Harvesting. Chem. Rev.
**2017**, 117, 294–343. [Google Scholar] [CrossRef] - Mančal, T. A decade with quantum coherence: How our past became classical and the future turned quantum. Chem. Phys.
**2020**, 532, 110663. [Google Scholar] [CrossRef] - Tao, M.J.; Zhang, N.N.; Wen, P.Y.; Deng, F.G.; Ai, Q.; Long, G.L. Coherent and incoherent theories for photosynthetic energy transfer. Sci. Bull.
**2020**, 65, 318–328. [Google Scholar] [CrossRef] [Green Version] - Savikhin, S.; Buck, D.R.; Struve, W.S. Oscillating anisotropies in a bacteriochlorophyll protein: Evidence for quantum beating between exciton levels. Chem. Phys.
**1997**, 223, 303–312. [Google Scholar] [CrossRef] - Engel, G.S.; Calhoun, T.R.; Read, E.L.; Ahn, T.K.; Mančal, T.; Cheng, Y.C.; Blankenship, R.E.; Fleming, G.R. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature
**2007**, 446, 782–786. [Google Scholar] [CrossRef] - Calhoun, T.R.; Ginsberg, N.S.; Schlau-Cohen, G.S.; Cheng, Y.C.; Ballottari, M.; Bassi, R.; Fleming, G.R. Quantum Coherence Enabled Determination of the Energy Landscape in Light-Harvesting Complex II. J. Phys. Chem. B
**2009**, 113, 16291–16295. [Google Scholar] [CrossRef] - Collini, E.; Wong, C.Y.; Wilk, K.E.; Curmi, P.M.G.; Brumer, P.; Scholes, G.D. Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature. Nature
**2010**, 463, 644–647. [Google Scholar] [CrossRef] - Panitchayangkoon, G.; Hayes, D.; Fransted, K.A.; Caram, J.R.; Harel, E.; Wen, J.; Blankenship, R.E.; Engel, G.S. Long-lived quantum coherence in photosynthetic complexes at physiological temperature. Proc. Natl. Acad. Sci. USA
**2010**, 107, 12766–12770. [Google Scholar] [CrossRef] [Green Version] - Panitchayangkoon, G.; Voronine, D.V.; Abramavicius, D.; Caram, J.R.; Lewis, N.H.C.; Mukamel, S.; Engel, G.S. Direct evidence of quantum transport in photosynthetic light-harvesting complexes. Proc. Natl. Acad. Sci. USA
**2011**, 108, 20908–20912. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mohseni, M.; Rebentrost, P.; Lloyd, S.; Aspuru-Guzik, A. Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys.
**2008**, 129, 174106. [Google Scholar] [CrossRef] [Green Version] - Rebentrost, P.; Mohseni, M.; Kassal, I.; Lloyd, S.; Aspuru-Guzik, A. Environment-assisted quantum transport. New J. Phys.
**2009**, 11, 033003. [Google Scholar] [CrossRef] - Plenio, M.B.; Huelga, S.F. Dephasing-assisted transport: Quantum networks and biomolecules. New J. Phys.
**2008**, 10, 113019. [Google Scholar] [CrossRef] - Olaya-Castro, A.; Lee, C.F.; Olsen, F.F.; Johnson, N.F. Efficiency of energy transfer in a light-harvesting system under quantum coherence. Phys. Rev. B
**2008**, 78, 085115. [Google Scholar] [CrossRef] [Green Version] - Duan, H.G.; Prokhorenko, V.I.; Cogdell, R.J.; Ashraf, K.; Stevens, A.L.; Thorwart, M.; Miller, R.J.D. Nature does not rely on long-lived electronic quantum coherence for photosynthetic energy transfer. Proc. Natl. Acad. Sci. USA
**2017**, 114, 8493–8498. [Google Scholar] [CrossRef] [Green Version] - Harush, E.Z.; Dubi, Y. Do photosynthetic complexes use quantum coherence to increase their efficiency? Probably not. Sci. Adv.
**2021**, 7, eabc4631. [Google Scholar] [CrossRef] [PubMed] - Wilkins, D.M.; Dattani, N.S. Why Quantum Coherence Is Not Important in the Fenna–Matthews–Olsen Complex. J. Chem. Theory Comput.
**2015**, 11, 3411–3419. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kassal, I.; Yuen-Zhou, J.; Rahimi-Keshari, S. Does Coherence Enhance Transport in Photosynthesis? J. Phys. Chem. Lett.
**2013**, 4, 362–367. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cao, J.; Cogdell, R.J.; Coker, D.F.; Duan, H.G.; Hauer, J.; Kleinekathöfer, U.; Jansen, T.L.C.; Mančal, T.; Miller, R.J.D.; Ogilvie, J.P.; et al. Quantum biology revisited. Sci. Adv.
**2020**, 6, eaaz4888. [Google Scholar] [CrossRef] [Green Version] - Potočnik, A.; Bargerbos, A.; Schröder, F.A.Y.N.; Khan, S.A.; Collodo, M.C.; Gasparinetti, S.; Salathé, Y.; Creatore, C.; Eichler, C.; Türeci, H.E.; et al. Studying light-harvesting models with superconducting circuits. Nat. Commun.
**2018**, 9, 904. [Google Scholar] [CrossRef] [PubMed] - Kozyrev, S.V. Model of Vibrons in Quantum Photosynthesis as an Analog of a Model of Laser. Proc. Steklov Inst. Math.
**2019**, 306, 145–156. [Google Scholar] [CrossRef] - Mattiotti, F.; Brown, W.M.; Piovella, N.; Olivares, S.; Gauger, E.M.; Celardo, G.L. Bio-inspired natural sunlight-pumped lasers. New J. Phys.
**2021**, 23, 103015. [Google Scholar] [CrossRef] - Scully, M.O.; Chapin, K.R.; Dorfman, K.E.; Kim, M.B.; Svidzinsky, A. Quantum heat engine power can be increased by noise-induced coherence. Proc. Natl. Acad. Sci. USA
**2011**, 108, 15097–15100. [Google Scholar] [CrossRef] [Green Version] - Killoran, N.; Huelga, S.F.; Plenio, M.B. Enhancing light-harvesting power with coherent vibrational interactions: A quantum heat engine picture. J. Chem. Phys.
**2015**, 143, 155102. [Google Scholar] [CrossRef] - Chen, F.; Gao, Y.; Galperin, M. Molecular Heat Engines: Quantum Coherence Effects. Entropy
**2017**, 19, 472. [Google Scholar] [CrossRef] [Green Version] - Ringsmuth, A.K.; Milburn, G.J.; Stace, T.M. Multiscale photosynthetic and biomimetic excitation energy transfer. Nat. Phys.
**2012**, 8, 562–567. [Google Scholar] [CrossRef] [Green Version] - Sarovar, M.; Whaley, K.B. Design principles and fundamental trade-offs in biomimetic light harvesting. New J. Phys.
**2013**, 15, 013030. [Google Scholar] [CrossRef] - Romero, E.; Novoderezhkin, V.I.; van Grondelle, R. Quantum design of photosynthesis for bio-inspired solar-energy conversion. Nature
**2017**, 543, 355–365. [Google Scholar] [CrossRef] - Rupp, A.I.K.S.; Gruber, P. Biomimetic Groundwork for Thermal Exchange Structures Inspired by Plant Leaf Design. Biomimetics
**2019**, 4, 75. [Google Scholar] [CrossRef] [Green Version] - Scarani, V.; Ziman, M.; Štelmachovič, P.; Gisin, N.; Bužek, V. Thermalizing Quantum Machines: Dissipation and Entanglement. Phys. Rev. Lett.
**2002**, 88, 097905. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ciccarello, F. Collision models in quantum optics. Quantum Meas. Quantum Metrol.
**2017**, 4, 53–63. [Google Scholar] [CrossRef] [Green Version] - Ciccarello, F.; Palma, G.M.; Giovannetti, V. Collision-model-based approach to non-Markovian quantum dynamics. Phys. Rev. A
**2013**, 87, 040103. [Google Scholar] [CrossRef] - Lorenzo, S.; Ciccarello, F.; Palma, G.M.; Vacchini, B. Quantum Non-Markovian Piecewise Dynamics from Collision Models. Open Syst. Inf. Dyn.
**2017**, 24, 1740011. [Google Scholar] [CrossRef] [Green Version] - Lorenzo, S.; Ciccarello, F.; Palma, G.M. Composite quantum collision models. Phys. Rev. A
**2017**, 96, 032107. [Google Scholar] [CrossRef] [Green Version] - Ciccarello, F.; Lorenzo, S.; Giovannetti, V.; Palma, G.M. Quantum collision models: Open system dynamics from repeated interactions. Phys. Rep.
**2022**, 954, 1–70. [Google Scholar] [CrossRef] - Arısoy, O.; Campbell, S.; Müstecaplıoğlu, Z.E. Thermalization of Finite Many-Body Systems by a Collision Model. Entropy
**2019**, 21, 1182. [Google Scholar] [CrossRef] [Green Version] - Çakmak, B.; Pezzutto, M.; Paternostro, M.; Müstecaplıoğlu, E. Non-Markovianity, coherence, and system-environment correlations in a long-range collision model. Phys. Rev. A
**2017**, 96, 022109. [Google Scholar] [CrossRef] [Green Version] - Çakmak, B.; Campbell, S.; Vacchini, B.; Müstecaplıoğlu, Z.E.; Paternostro, M. Robust multipartite entanglement generation via a collision model. Phys. Rev. A
**2019**, 99, 012319. [Google Scholar] [CrossRef] [Green Version] - Campbell, S.; Çakmak, B.; Müstecaplıoğlu, Z.E.; Paternostro, M.; Vacchini, B. Collisional unfolding of quantum Darwinism. Phys. Rev. A
**2019**, 99, 042103. [Google Scholar] [CrossRef] [Green Version] - Cattaneo, M.; De Chiara, G.; Maniscalco, S.; Zambrini, R.; Giorgi, G.L. Collision Models Can Efficiently Simulate Any Multipartite Markovian Quantum Dynamics. Phys. Rev. Lett.
**2021**, 126, 130403. [Google Scholar] [CrossRef] [PubMed] - Chisholm, D.A.; García-Pérez, G.; Rossi, M.A.C.; Palma, G.M.; Maniscalco, S. Stochastic collision model approach to transport phenomena in quantum networks. New J. Phys.
**2021**, 23, 033031. [Google Scholar] [CrossRef] - Gallina, F.; Bruschi, M.; Fresch, B. Strategies to simulate dephasing-assisted quantum transport on digital quantum computers. New J. Phys.
**2022**, 24, 023039. [Google Scholar] [CrossRef] - Tian, F.; Zou, J.; Li, L.; Li, H.; Shao, B. Effect of Inter-System Coupling on Heat Transport in a Microscopic Collision Model. Entropy
**2021**, 23, 471. [Google Scholar] [CrossRef] - Li, Y.; Li, L. Hierarchical-environment-assisted non-Markovian and its effect on thermodynamic properties. EPJ Quantum Technol.
**2021**, 8, 9. [Google Scholar] [CrossRef] - Yu, W.L.; Li, T.; Li, H.; Zhang, Y.; Zou, J.; Wang, Y.D. Heat Modulation on Target Thermal Bath via Coherent Auxiliary Bath. Entropy
**2021**, 23, 1183. [Google Scholar] [CrossRef] [PubMed] - Levi, E.K.; Irish, E.K.; Lovett, B.W. Coherent exciton dynamics in a dissipative environment maintained by an off-resonant vibrational mode. Phys. Rev. A
**2016**, 93, 042109. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**A schematic representation of the model. The system qubits, denoted by ${S}_{1}$, ${S}_{2}$ and ${S}_{3}$, are interacting with each other and the qubits from the cold bath, hot bath and the hierarchical environment.

**Figure 2.**Heat flux to the cold bath, J, with and without interaction with the HSE for different coupling constants. The HSE are assumed to be in thermal state ($p=0$) and for blue curves we have ${g}_{a}=0$. The remaining couplings are (

**a**) ${g}_{12}=30$, ${g}_{23}=15$ and for the orange curve ${g}_{a}=20$, ${g}_{b}=40$ (

**b**) ${g}_{12}=50$, ${g}_{23}=25$ and for the orange curve ${g}_{a}=40$, ${g}_{b}=30$.

**Figure 3.**Energy level diagram for the system with and without interaction with the HSE. The interaction couplings of the system are ${g}_{12}=50$ and ${g}_{23}=20$. The system-HSE interaction is (

**a**) ${g}_{a}=40$ (

**b**) ${g}_{a}=0$.

**Figure 4.**Contour plots for the steady state heat flux ${J}_{ss}$ vs. system-HSE and inter-HSE couplings ${g}_{a}$ and ${g}_{b}$ considering no coherence in HSE ($p=0$). The inter-system couplings are ${g}_{12}=50$ and ${g}_{23}=25$.

**Figure 5.**Contour plots for the steady state heat flux, ${J}_{ss}$ vs. system-HSE coupling, ${g}_{a}$, and coherence in the HSE, p. The couplings are (

**a**) ${g}_{12}=50$, ${g}_{23}=25$, ${g}_{b}=10$ (

**b**) ${g}_{12}=50$, ${g}_{23}=25$, ${g}_{b}=30$.

**Figure 6.**Contour plots for the steady state heat flux, ${J}_{ss}$, vs inter-HSE coupling, ${g}_{b}$, and coherence in the HSE, p. The couplings are (

**a**) ${g}_{12}=50$, ${g}_{23}=25$, ${g}_{a}=20$ (

**b**) ${g}_{12}=50$, ${g}_{23}=25$, ${g}_{a}=40$.

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**MDPI and ACS Style**

Pedram, A.; Çakmak, B.; Müstecaplıoğlu, Ö.E.
Environment-Assisted Modulation of Heat Flux in a Bio-Inspired System Based on Collision Model. *Entropy* **2022**, *24*, 1162.
https://doi.org/10.3390/e24081162

**AMA Style**

Pedram A, Çakmak B, Müstecaplıoğlu ÖE.
Environment-Assisted Modulation of Heat Flux in a Bio-Inspired System Based on Collision Model. *Entropy*. 2022; 24(8):1162.
https://doi.org/10.3390/e24081162

**Chicago/Turabian Style**

Pedram, Ali, Barış Çakmak, and Özgür E. Müstecaplıoğlu.
2022. "Environment-Assisted Modulation of Heat Flux in a Bio-Inspired System Based on Collision Model" *Entropy* 24, no. 8: 1162.
https://doi.org/10.3390/e24081162