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Quantum Coherent Effects in
Photosynthesis
K. Birgitta Whaley
Mohan Sarovar
Stephan Hoyer
Akihito Ishizaki
Graham Fleming
Jahan Dawlaty
Doran Bennett
Berkeley Quantum Information
and Computation Center
Outline
1. Quantum mechanics and biology
2. Electronic energy transfer in light harvesting
step of photosynthesis
3. Non-trivial quantum effects in light
harvesting for photosynthesis
- entanglement?
- quantum computation?
4. Possible biological role for quantum
coherence in photosynthesis
1. From wave mechanics to biology: I
1914 Marcelin: potential energy surface for chemical reactions
1926 Schrödinger: wave mechanics
1927 Heisenberg, Dirac: resonance/exchange phenomena
1927 Heitler-London: quantum mechanical theory of chemical bond
1927 Born-Oppenheimer: structure of molecular energy levels
1927 Hund
molecular orbital theory
1928 Mulliken
1928 Gamow: alpha particle tunneling
1933 Journal of Chemical Physics established
1933 Bell: hydrogen tunneling in chemical reactions
1935 Einstein-Podolsky-Rosen: non-local quantum correlations
1935 Schrödinger: identifies entanglement (Verschränkung) as “the
characteristic trait of quantum mechanics”
1935 Delbrück et al.: mutation and genetic structure
1935 Eyring, Evans-Polanyi: quantum statistical theory of chemical
reaction rates (precursors - Marcelin, Rice, Herzfeld, Tolman…)
1943 Schrödinger: “What is Life” lectures
Schrödinger, entanglement & biology:
•
•
•
1935: “Entanglement” is ”the characteristic trait of
quantum mechanics, the one that enforces its entire
departure from classical lines of thought”
1943 “What is Life”: genetic structure and stability
determined by quantum nature of molecular energy
levels, energy barriers between stable configurations;
no consideration of i) tunneling, ii) quantum coherence
or entanglement in such biological processes
1943 “quantum indeterminacy plays no biologically
relevant role” in bodily events corresponding to activity
of the mind, except possibly by enhancing accidental
nature of meiosis, mutations, etc.
From wave mechanics to biology: II
Experiments:
1963 Yoshizawa-Wald: photoisomerization in primary step of vision
(1991 fsec dynamics, 2010 conical intersection) *
1966 DeVault-Chance: electron tunneling in photosynthesis
1989 Klinman et al.: hydrogen tunneling in enzyme reactions
2007 Fleming et al.: quantum coherence of electronic energy transfer
in light harvesting complexes
2010 Engel et al., Scholes et al.: quantum coherence of EET in LHCs at
ambient temperatures
Proposals:
1995 Hameroff-Penrose: quantum coherence in brain microtubules
1996 Turin: inelastic electron tunneling in olfaction *
1998 Schulten et al.: radical pair mechanism of bird navigation
(many subsequent experiments consistent with this theory)
2010 Vaziri-Plenio: quantum coherent transport in ion channels
Biological function across all time and
size scales
Developing tools for
studying biological
structure and function
at unprecedented
spatial and temporal
resolution
Can quantum
coherence be relevant
for biological
function?
A. Vaziri
2. Photosynthesis
The “light” reactions are rapid and
efficient, >95% conversion
Blue-absorbing pigments
Orange-absorbing
pigments
“Antenna”
Light-harvesting
Red-absorbing
pigments
Reaction
Center
Charge separation
…Secondary electron transfer reactions, Water
splitting, Proton transport across thylakoid membrane,
Reduction of NADP+, ATP synthesis…
Electronic energy transfer in
photosynthetic light harvesting systems
1949 Förster theory – incoherent hopping
2007 expts of Fleming et al. - how and why does
coherence play a role in EET?
-
quantum coherence in open quantum system
does transport show quantum speedup?
entanglement ?
what biological role does coherence play?
Light Harvesting Complexes
LH 1 and 2 in purple bacteria
• A bewildering variety of antennae
• All composed of densely packed molecules
LH2
LH1+RC
• The molecular aggregates are often
embedded in protein scaffolds
Bahatyrova et al., Nature 430,
1058 (2004)
PS II of higher plants, blue-green algae, cyanobacteria
LHC II
50% of green matter on earth uses PS II
Light Harvesting Apparatus of Purple Bacteria:
Peripheral LH2 complexes
capture sunlight, transfer
energy of this to internal
LH1 complex and reaction
center
LH2
Light Harvesting Complexes
LH 1 and 2 in purple bacteria
• A bewildering variety of antennae
• All composed of densely packed molecules
LH2
LH1+RC
• The molecular aggregates are often
embedded in protein scaffolds
Bahatyrova et al., Nature 430,
1058 (2004)
PS II of higher plants, blue-green algae, cyanobacteria
LHC II
50% of green matter on earth uses PS II
Chromatophore vesicle, size: ~1 – 10 nm
Extension of cytoplasmic membrane: contains
chlorophylls (chromophores), proteins, lipids, water
Quantum effects?
Entanglement?
Quantum enhanced energy transport?
Light harvesting apparatus of green sulfur bacteria
FMO: energy ‘wire’ connecting chlorosome to reaction center
well characterized system
1
2
6
7
3
5
Muh et al. PNAS,
4 104, 16862 (2007)
James Allen et al., Photosynth. Res., 75, 49 (2003)
FMO Complex: (Frenkel) excitons
Optical transitions to delocalized exciton
states
Energy
1
• prototypical LH complex
2
6
• Well characterized by pump-probe
experiments [1] and theoretical modeling [2]
7
5
• Mostly delocalized on two BChls
• Lowest energy exciton sits on BChl 3
3
4
RC
[1] A. Freiberg, S. Lin, K. Timpmann & R. E. Blankenship, J. Phys. Chem. B 101, 7211 (1997)
[2] J. Adolphs & T. Renger, Biophys. J. 91, 2778 (2006).
Energy Transfer Time Scales in FMO
• Energy transfer from one end to the other in a few ps
• From pump-probe experiments* and theoretical modeling#
*A. Freiberg, S. Lin, K. Timpmann & R. E. Blankenship, J. Phys. Chem. B 101, 7211 (1997)
#J. Adolphs & T. Renger, Biophys. J. 91, 2778 (2006).
2D Femtosecond spectroscopy
signal
t
 Signal S(3)(,T, τ) is output electric field
 Obtain S(3)(ω1,T, ω3) by double Fourier Transformations in  and t
 Retrieves correlation between absorption and emission frequencies
 Echo technique can remove inhomogeneous broadening
An Optical Analog of 2D NMR:
2D Electronic Spectroscopy
Sample
τ
T
Light
e
g
Quantum
State
P
Polarization
Photon Echo Signal
2D Electronic Correlation Spectrum
T
Emission
Frequency
τ
Fourier Transform
with Respect to
τ
Spectrally
resolved,
heterodyne
detected photon
echo.
Excitation Frequency
A 2D spectrum is a correlation map between the initial and final
coherences.
The correlation depends on the processes occurring during time T.
Amplitude
Amplitude(arb
(arbunits)
units)
Amplitude (arb units)
Amplitude (arb units)
Probe Electronic Coherence in FMO (77K)
• Oscillatory signal is completely consistent with
electronic coherence
• Electronic populations stabilize but quantum coherence
persists: no simple T1/T2 relation
• Strong evidence for coherent wavelike energy transfer
“Evidence for wavelike energy transfer through quantum coherence in photosynthetic
systems”, G.S. Engel, T.R. Calhoun, E.L. Read, T. Ahn, T. Mancal, Y.-C. Cheng,
R.E. Blankenship, and G. R. Fleming, Nature 446, 782 (2007)
Natural Light Harvesting Systems:
experimental evidence for quantum effects
Green sulfur bacteria
FMO (dynamics):
Engel et al., Nature, 446, 782 (2007) (T=77K)
Engel et al., PNAS 107, 12766 (2010) (T=277K)
Chlorosome (exciton dephasing times):
Prokhorenko et al., Biophys. J. 79, 2105 (2000)
Marine algae – phycobiliproteins
Collini et al., Nature, 463, 644 (2010) (T=294K);
Womick et al., JCP 133, 024507 (2010)
Purple bacteria
LH1/LH2 (static, dynamics):
Many studies – e.g: Monshouwer et al., Chem. Phys.
Lett. 246, 341 (1995); van Oijen et al., Science 285 400
(1999)
Reaction center (dynamics):
Lee et al., Science, 316, 1462 (2007)
Higher plants
LHC-II (dynamics):
Calhoun et al., J. Phys. Chem. B, 113, 16291 (2009)
T=77K, τc ≈ 700 fs
3. Non-trivial quantum behavior of LHCs?
• Entanglement
Nature 425, 48 (2003)
( LiHo1-yYyF4 )
Vedral, News & Views,
Nature 425, 28 (2003)
Entanglement in biological systems?
at physiological temperatures?
quantify, measure, understand in context
Can a Classical or Quasi-classical Description of Biological Reality be Considered Complete?
• Quantum information processing
Scientific American, April, 2007
Quantum speedup from coherence?
Dynamics of Light Harvesting:
• Fenna-Matthews-Olson (FMO) is prototypical LH complex
• 7 chromophores (pigments), Frenkel excitons
• Well characterized from pump-probe experiments[1] and
theoretical modeling[2].
• Closed system (excitonic) dynamics[2]:
But pigment complex is an open quantum system:
Reorganization energy:
(from exciton-phonon coupling)
Phonon relaxation time:
Reaction center trapping rate:
[1] A. Freiberg, S. Lin, K. Timpmann & R. E. Blankenship, J. Phys. Chem. B 101, 7211 (1997)
[2] J. Adolphs & T. Renger, Biophys. J. 91, 2778 (2006).
Coupled pigment-protein dynamics:
J12
Pigment j
τj
λj
ε 0j
A non-perturbative, non-Markovian treatment
Ishizaki and Fleming: JCP 130, 234111 (2009)
application to FMO PNAS 106, 17255 (2009)
A1) Linear Coupling
of Protein Environment:
Protein degrees of freedom
Typical photosynthetic Systems
A2) Environmental fluctuations are Gaussian
• Following Kubo, Tanimura
• Based on a truncated cumulant expansion
• Fluctuation-dissipation relation preserved
Coherent Energy Transfer through FMO
(a)
Antenna
(b)
(a)
Quantum
Coherence 6
5
4
77K
1
~ 700 fs
2
7
(b)
(a)
3
~ 700 fs
(b)
300K
Reaction Center
Coherences at physiological temperatures
Ishizaki & Fleming, PNAS 106, 17255 (2009)
Confirmed by recent experiments
at 277 K - Engel et al. arXIv:1001.5108
~ 350 fs
~ 350 fs
Quantum Coherent Effects in Photosynthesis
(the greening of quantum information)
x
Quantum search for
reaction center?
S. Hoyer, M. Sarovar., K. B. Whaley
New J. Phys. 12, 065041 (2010);
arXiv:0910.1847 [quant-ph]
x
Have photosynthetic systems been relying on principles of
quantum computation for millions of years?
FMO as a 1-D quantum walk
Source
6
1
5
7
2
4
3
Trap
Energies in cm-1
classical
quantum
No quantum speedup characteristic of
quantum search
Hoyer, Sarovar, Whaley: Limits of quantum speedup in photosynthetic light harvesting New
Journal of Physics 12, 065041 (2010)
classical
1D Quantum Random Walk analysis:
quantum
Ballistic
Diffusive
Localized
6
5
4,7
3
Saturated
1,2
Localization due to energy disorder and dephasing,
No quantum speedup as required for quantum computing for t > 70-100 fs,
yet coherence persists to ~700 fs
Entanglement: I
Source generates
and sends first spin to Alice,
second spin to Bob
S
Alice
Bob
Alice measures σz=+1
Bob measures σz : finds σz=-1
Bob measures σx: finds 50% σx=+1, 50% σx=-1
Alice measures σx=+1
Bob measures σx : finds σx=-1
Bell inequalities 1964: measure correlations between measurement outcomes in different
bases, quantify quantum correlations; confirmed by many experiments (1980 onwards)
Entanglement: II
• Formal definition:
Entangled <=> Not separable
Separable state:
generalization:
• 2 particle entanglement
e.g.,
von Neumann entropy
quantifies entanglement
“Best possible knowledge of the
whole does not include best
possible knowledge of its parts”
– Schrödinger, 1935
Entanglement: III
Schrodinger 1935 (German): restriction of local measurement outcomes due to quantum
correlations in a state shared between several parties - “Verschraenkung”
Schrodinger 1935 (English): “Entanglement” is
”the characteristic trait of quantum mechanics, the one that enforces its entire departure from
classical lines of thought”
Non-local correlations (Einstein “spooky action at a distance”)
Einstein, Podolsky, Rosen 1935: violates local realism
Quantum information perspective
Entanglement is a resource:
quantum communication: teleportation, key distribution...
quantum computation: enables some quantum algorithms, measurement based QC
Entanglement provides insights for strongly correlated quantum materials:
complexity of many body states, understanding of quantum phase transitions…
Quantifying entanglement in LHCs
• Entanglement = non-classical correlations between the
electronic states of separated chromophores,
provides further characterization of quantum effects
• Multi-partite, mixed-state entanglement
?
• Simplification: single excitation subspace
• in vivo conditions for FMO: single excitation enters
from baseplate
cf. mode entanglement
1. Multipartite, global entanglement measure: based on relative entropy of
entanglement [V. Vedral & M. B. Plenio, Phys. Rev. A 57, 1619 (1998)]
2. Bipartite entanglement measure:
concurrence [Hill, Wootters , Phys. Rev. Lett. 79, 5022 (1997)]
Entanglement witness
M. Sarovar, A. Ishizaki, G. R. Fleming, K. B. Whaley, arXiv:0905.3787, Nature Physics 6, 462 (2010)
Natural Light Harvesting Systems:
coherence and entanglement
Global entanglement
Chromophore-chromophore entanglement
77K
• FMO studies (7 chromophores), exciton dynamics with reduced hierarchy approach
• large amounts of global and bipartite entanglement at short times
• significant entanglement at long times (steady-state); limited by trapping dynamics.
• global entanglement accompanied by bipartite entanglement across multiple partitions
• “Long-range” entanglement across complex (~28 Å)
• first demonstration of entanglement for biological system under natural ( in vivo) conditions,
many subsequent related theoretical studies
M. Sarovar, A. Ishizaki, G. R. Fleming, K. B. Whaley, arXiv:0905.3 787, Nature Physics 6, 462 (2010)
LHC Entanglement is robust:
Sarovar et al., Nature Physics (2010)
FMO:
long lived entanglement at T=200 K seen in
all dynamical models with realistic temp. dependence;
details not captured by Markovian or infinite T models
reduced hierarchy
Redfield
Redfield secular
Monomeric subunit of LHCII
isolated from spinach
Many measures of entanglement:
negativity, log negativity, time integrated concurrence
(entanglement yield) …
Entanglement studies to date:
FMO:
Sarovar et al. arXiv:0905.3787, Nat. Phys. 6, 462 (2010)
Caruso et al. arXiv:0901.4454v2, J. Chem. Phys. 131,105106 (2009).
Caruso et al. arXiv:0912.0122, Phys. Rev. A 81, 062346 (2010)
Bradler et al., arXiv:0912.5112v2
Fassioli & Olaya-Castro, arXiv:1003.3610, NJP 12, 085006 (2010)
LHCII:
Ishizaki & Fleming, NJP 12, 055004 (2010)
Model systems:
Thorwart et al., Chem. Phys. Lett. 478, 234 (2009)
Scholak et al., arXiv:0912.3560
Log-Negativity
LHCII:
Ishizaki/Fleming NJP (2010)
Quantum entanglement
between
Chla s and Chlbs
Initial excitation:
b606
8 Chlorophyll a molecules (Chla )
6 Chlorophyll b molecules (Chlb)
Quantum entanglement in light harvesting:
1 exciton:
Mode entanglement – 1 photon distributed over 2 modes
photon non-locality (Einstein 1927)
demonstrated resource for teleportation
(Lombardi et al. 2002)
Non-local quantum correlations between electronic states:
Site 1
Site 2
Open question: Biological function or evolutionary role ??
3. Possible biological
quantum advantage from
coherence
LHC quantum transport, quantum information and
quantum control:
- First demonstration that entanglement, the most remarkable and non-classical feature
of quantum systems, is manifested in a biological structure
… exciton-exciton entanglement in larger LHC’s?
- No quantum speedup (in the quantum information sense)
- limited by localization length which is dictated by energetic gradient/disorder
- quantum speedup necessary for quantum computation
- What is the quantum advantage? Overcome local energy minima, efficiency,
robustness, unidirectionality …possible functional role of coherent ratcheting...
- Efficient transport a result of a delicate interplay of coherent and decoherent dynamics
…what is enabling feature in environment
(protein scaffold, pigment organization…)?
- some futuristic directions:
• control of natural quantum processors
• develop design rules for robust biomimetic sensors and
artificial devices for effective ‘quantum conversion’ of
sunlight into chemical energy…