Quantum random walks in energy landscapes

1. Quantum random walks in energy landscapes Stephan Hoyer, Mohan Sarovar and K. Birgitta Whaley QuEBS 2009 Thanks: DARPA, Akihito Ishizaki, Yuan-Chung Cheng Berkeley Quantum Information and Computation Center

2. Is photosynthesis doing quantum computing? • Fleming group at Berkeley found quantum coherences in light-harvesting complexes • Comparisons to quantum algorithms reasonable? • Intuitive analogy – quantum random walk • Quantum speedup in photosynthesis? Engel et al., Nature 446, 782 (2007) Lee et al., Science 316, 1462 (2007)

3. 1 2 6 7 5 3 4 Light-harvesting complexes as quantum walks • Single exciton, tight-binding approach • Energy landscapes • Disordered – random environment • Ordered – energy funnels • Decoherence • Room temperature in a protein cage Energy Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria

4. Outline • Principles of quantum walks in energy landscapes • Coherent transport • Transport under decoherence • Applications • FMO and other light harvesting complexes

5. 1. Coherent transport in energy landscapes

6. Coherent exciton transport • Tight-binding, single particle model • Three cases for different forms of En on infinite chains with constant nearest-neighbor coupling i. En =0: quantum walk ii.Enrandom: Anderson model iii. En =nF: Bloch oscillations

7. Quantum random walk • Flat site energies: En =0 • Spatially extended eigenstates • Characteristic square root speedup over classical walks • Classical walks: • Quantum walks: • Quantum walks universal for quantum computing* Ballistic Reviews: quant-ph/0303081, quant-ph/0403120, quant-ph/0606016 *A. Childs, PRL 102, 180501 (2009)

8. Anderson localization • Randomly chosen site energies En • Destroyed symmetry leads to localized eigenstates • Exponential localization • Different for N>1 dimensional • Initial transport ballistic like a quantum walk Ballistic Localized Reviews: Anderson, Rev Mod Phys 50, 191-201 (1978) Phillips, Annu Rev of Phys Chem44, 115-44 (1993)

9. Stark localization • Linear energy bias: En =nF • Eigenstates in terms of Bessel functions • Factorial localization • Wannier-Stark ladder • Bloch oscillations with frequency F Reviews: Kolovsky and Korsch, cond-mat/0403205 Hartmann et al., New J Phys 6, 2 (2004)

10. Origins of Stark localization Allowed energies Intuition • Conservation of energy • Quantum recurrence theorem Bocchieri & Loinger, Phys. Rev. 107, 337-339 (1957)

11. Localization for arbitrary potentials • Allowed energies within En ±2Jn for arbitrary smooth potentials Serdyukova and Zakhariev, PRA 46, 58-62 (1992)

12. Stark localization with disorder • Anderson and Stark localization are complementary Lubanet al., PRB 34, 3674-77 (1986) Kolovsky, PRL 101, 190602 (2008)

13. Implications of Stark localization • Quantum particles on a lattice do not roll downhill • Energy funnels do not work • Different from continuous space wave packet • Discrete space and any variation in site energies leads to localization Discrete space Continuous space

14. 2. Transport under decoherence in energy landscapes

15. Dephasing assisted transport • Time-evolution with quantum master equation • Constant rate dephasing in the site basis (Haken-Strobl model) equivalent to continuous measurement • General result (Aspuru-Guzik and Plenio groups) • Low levels of dephasing allow for escape of the Anderson localization regime • High levels of dephasing suppress transport with the quantum Zeno effect Mohseniet al., arXiv:0805.2741, Rebentrostet al., 0807.0929, 0806.4725 Plenio and Huelga, 0807.4902, Caruso et al., 0901.4454

16. Dephasing on Stark localized cold atoms in optical lattices • Pure dephasing arises from either • Spontaneous emission* • Coupling to equally populated bath of bosons under Bose-Hubbard model† • Analytic tight-binding result: diffusive transport at long times* • Achieving conduction requires finite temperature thermal bath† • Pure dephasing is at infinite temperature *Kolovsky et al., PRA66, 053405(2002) †Ponomarev et al., PRL 96, 050404 (2006)

17. Stark localization under dephasing • Bloch oscillations fade to diffusive spreading • Holds even for quantum walks (F=0) Ballistic Diffusive Localized Kolovsky et al., PRA66, 053405 (2002)

18. Anderson model under dephasing • Anderson model transitions to diffusive transport for any dephasing rate Ballistic Diffusive Localized

19. Diffusion coefficients (as t → ∞) • No quantum speedup beyond localization length • Dephasing assisted transport is diffusive

20. 3. Applications to light-harvesting complexes

21. FMO as 1-D quantum walk Fenna-Matthews-Olson complex of green sulfur bacteria: Source 6 1 5 7 2 4 3 Trap Energies in cm-1 FMO Hamiltonian for C. Tepidum from J. Adolphs and T. Renger, Biophys J 91, 2778 (2006) FMO picture courtesy of Akihito Ishizaki

22. Dephasing noise on FMO • No quantum speedup in FMO after ∼70 fs Ballistic Diffusive Localized Saturated 6 5 4,7 3 1,2 • Simulated using full Hamiltonian, constant dephasing rate, and trapping rate at site 3 of (1 ps)-1 • Similar results hold with no trap and/or initial excitation at site 1 • Model and temperature dependent dephasing rates from Rebentrostet al., arXiv:0807.0929

23. Realistic noise on FMO • Results hold under more realistic noise models incorporating phonon dynamics in bath Ballistic Diffusive Localized Saturated 6 5 4,7 3 1,2 Unpublished data courtesy of Akihito Ishizaki Model from Ishizaki and Fleming, J. Chem. Phys. 130, 234111 (2009)

24. LH2 LH1+RC Future work: Other systems LHCII of higher plants Dendrimers and binary trees Reaction center of purple bacteria 42 sites Hu et al., Q Rev Biophys 35, 1 (2002)

25. Conclusions • Ordered or disordered site energies make coherent transport localized • Dephasing noise overcomes localization by allowing diffusion • Transport with non-constant site energies is always slower • Quantum speedup in FMO short lived (∼70 fs) • Properties of LHCs (disorder, funnel, dephasing) not suitable for scalable quantum walks