Quantum decoherence of excited states of optically active biomolecules

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Quantum decoherence of excited states of
optically active biomolecules

• Ross McKenzie

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Outline

• Optically active biomolecules as complex quantum
  systems
• A minimal model quantum many-body Hamiltonian
• Spectral density for system-environment
  interaction is well characterised.
• Observing the collapse of the quantum state!
• Ref J. Gilmore and RHM, quant-ph/0609075

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Some key questions concerning biomolecular
functionality

• Which details matter?
• What role does water play?
• Do biomolecules have the optimum structure to
  exploit dynamics for their functionality?
• When is quantum dynamics (e.g., tunneling,
  coherence) necessary for functionality?

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Why should quantum physicists be interested in
biomolecules?
Photo-active biomolecules are tuneable systems at
the quantum-classical boundary

• Retinal, responsible for vision
• Single photon detector
• Quantum dynamics when the
• Born-Oppenheimer approx. breaks down
• - Entanglement of electrons nuclei
• - Effect of decoherence on Berrys phase

Photosynthetic Light harvesting complexes Quantum
coherence over large distances?
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Quantum biology at amazon.com?
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A complex quantum system Photo-active yellow
protein

• Quantum system
• Ground electronic
• excited state of
• chromophore
• Environment
• Protein
• Water bound to
• Protein
• Bulk water

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Seeking a minimal model for this quantum system
and its environment

• Must capture and give insights into essential
  physics.
• Tells us which physical parameters lead to
  qualitative changes in quantum dynamics.

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Independent boson model Hamiltonian

• Chromophore is two level system (TLS).
• The environment is modelled as an infinite bath
  of harmonic oscillators.
• Effect of environment on quantum dynamics of TLS
  is completely determined by the spectral density

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Leggetts important idea

• We dont need to know all the microscopic details
  of the environment, nor its interaction with the
  system. Only need J(? ).
• Spectral density can be determined from
  measurements of the classical dynamics.
• Most spectral densities are ohmic, i.e.,
• J(? ) ? ? for ? lt 1/t
  
• t is relaxation time of the bath.
• For a gt 1 quantum dynamics is incoherent.
• Caldeira and Leggett, Ann. Phys. (1983) Leggett,
  J. Phys. Cond. Matt. (2002).

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Quantum dynamics of TLS
TLS is initially in a coherent superposition
state uncoupled from the bath. Reduced density
matrix of TLS is
Decay of coherence
Spectral diffusion
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Collapse of the wave function

• Zurek (82), Joos and Zeh (85), Unruh (89)
• Environment causes decay of the off-diagonal
  density matrix elements (decoherence)
• Collapse occurs due to continuous
  measurement of the state of the system by the
  environment.
• What is the relevant time scale for these
  biomolecules?
• h/(kBT a) 10 fsec

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Spectral density can be extracted from ultra-fast
laser spectroscopy

• Measure the time dependence of the frequency of
  maximum fluorescence (dynamic Stokes shift)

• Data can be fit to multiple exponentials.
• Fourier transform gives spectral density!

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Pal and Zewail, Chem. Rev. (2004)
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An example

• ANS is
• chromophore

Pal, Peon, Zewail, PNAS (2002)
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Femtosecond laser spectroscopy Measurement of
the time-dependent spectral shift of a
chromophore in a solvated protein

• Increasing pH unfolds (denatures) protein and
  exposes chromophore to more solvent.
• Presence of protein reduces psec relaxation and
  adds 50 psec relaxation.
• Pal, Peon, Zewail, PNAS (2002)

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Measured spectral densities

• Three contributions of ohmic form
• Bulk water (solvent)
• as 1-10 ts 0.3-3 psec
• Water bound to the protein, esp. at surface
• ab 10-100 tb 10-100 psec
• Protein
• ap 100-1000 tp 1-100 nsec

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Spectral density for diverse range of
biomolecules solvents
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Classical molecular dynamics simulations
C(t) for Trp (green) and Trp-3 in monellin
(black) in aqueous solution at 300 K Nilsson and
Halle, PNAS (2005).
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Our continuum dielectric models for environment

• We have calculated J(w) for 5 models for
  environment
• Key feature is separation of time and distance
  scales Protein much larger
  than chromophore
• Relaxation time of Protein gtgt Bound water gtgt Bulk
  solvent
• J(w) is sum of Ohmic contributions which we can
  identify with 3 different environments, protein,
  bound water, and bulk water

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Key physics behind decoherence

• Most chromophores have a large difference between
  electric dipole moment of ground and excited
  states.
• Water is a very polar solvent (static dielectric
  constant ?s 80)
• Water molecules have a net electric dipole moment
• Dipole direction fluctuates due to thermal
  fluctuations (typical relaxation time at 300K is
  1 psec)
• Chromophore experiences fluctuating electric
  field
• Surrounding protein does not completely shield
  chromophore from solvent.

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What have we learned?

• Complete characterisation of system-environment
  interaction for biomolecular chromophores.
• These spectral densities can be used to make
  definitive statements about the importance of
  quantum effects in biomolecular processes.
• Due to their tuneable coupling to their
  environment biomolecular systems may be model
  systems to use to test ideas in quantum
  measurement theory.
• For chromophores the timescale of the
  collapse is less than 100 fsec.

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Criteria for quantum coherent transfer of
excitation energy between two chromophores
J. Gilmore RHM, Chem. Phys. Lett. (2006)
Location of excitation with time
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Realisation of spin-boson model for coupled
chromophores
What is the two level system?

• Excitation can be on either of two molecules
• Each two energy levels

If only one excitation is present, effectively a
two level system
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Realisation of spin-boson model for coupled
chromophores
What is ???the coupling?

• Excitations transferred by dipole-dipole
  interactions (Forster)
• Shine in blue, get out yellow!
• Basis of Fluorescent Resonant Energy Transfer
  (FRET) spectroscopy
• Used in photosynthesis to move excitations around

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Criteria for quantum coherent transfer of
excitation energy between two chromophores
J. Gilmore RHM, Chem. Phys. Lett. (2006)
Location of excitation with time
Coherent for alt1
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Questions

• How unusual is to have a physical system where
  the system-bath interaction is so well
  characterised?
• What experiment would best elucidate the
  collapse?

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A comparison Retinal vs. Green Fluorescent
Protein

• Green Fluorescent Protein
• Excited state 10000x longer
• Fluoresces with high quantum efficiency

• Bacteriorhodopsin
• Non-radiative decay in 200fs
• Specific conformational change

Very different quantum dynamics of Chromophore
determined by environment!
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Flouresence from differentamino acid residues
withinprotein

• Cohen et al, Science (2002)