1. Klaus’s work on the Spin-Boson Problem*
Tony Leggett Department of Physics University of Illinois at Urbana-Champaign
Klaus Schulten Memorial Symposium
The Beckman Institute, University of Illinois Wednesday, November 8, 2017
*K. Schulten and M. Tesch, Chemical Physics 158, 421 (1991)
Dong Xu and K. Schulten, Chemical Physics 182, 91 (1994) (“XS”)

2. Electron transfer ( 𝑃 𝑆 𝐻 𝐿 → 𝑃 𝑠 + 𝐻 𝐿 − ) in photosynthetic reaction center of Rhodopseudomonas Viridis
a naïve condensed-matter physicist’s cartoon:
𝑒 −
( )
 quinones etc.
𝑃 𝑠 (“special pair”)
𝐻 𝐿 (bacteriopheophytin)
Experimental fact:
backward rate at 𝑅𝑇~ 10n s −1
forward rate at 𝑅𝑇~ 3p s −1 
increases by factor 4 by 8K
Suggests: QM effects important
also: little dependence on redox energy

3. From QM point of view, special case of more general 2state problem
environment L R
𝜔 𝑜 v L R
redox energy ϵ
In present case, “environment” is vibrating nuclei of protein: nuclear coordinates coupled to tunnelling electron by Coulomb force.
In general case need to consider
effect of coupling to environment on barrier transmission
effect of coupling to environment on coherence between 𝐿 and 𝑅 .
but in present case Born-Oppenheimer approximation probably good,  can neglect effect (a)  spin-boson model

4. Generic spin-boson model:
𝐻 = 𝐻 sys + 𝐻 env + 𝐻 coup
  
system environment coupling
𝐻 sys = 1 2 Δ 𝜎 𝑥 𝜖 𝜎 𝑧
 “spin” 
“bare” tunneling matrix element
offset
𝜎 𝑧 =+1  ϵ
𝜎 𝑧 =−1 Δ
𝐻 env = 𝛼=1 𝑁 𝑝 𝛼 2 2 𝑚 𝛼 𝑚 𝛼 𝜔 𝛼 2 𝑥 𝛼 2
 “bosons” (SHO’s)
𝐻 coup = 𝜎 𝑧 𝛼=1 𝑁 𝑐 𝛼 𝑥 𝛼

5. In our case, multimode Marcus model:
Correspondence:
tunneling electron  spin
bare tunneling amplitude  Δ
nuclear coordinates  𝑥 𝛼 ≡ 𝑞 𝛼 − 1 2 𝑞 𝑜𝛼
coupling constant  𝑐 𝛼 ≡ 𝑚 𝛼 𝜔 𝑜𝛼 2 𝑞 𝑜𝛼
redox energy  ϵ
We would like to know transfer rate 𝑘 𝜖,𝑇 as function of redox energy ϵ and temperature T.

6. Crucial feature of spin-boson problem:
Provided we are interested only in dynamics of spin, complete information about effect of environment is encapsulated in
𝐽 𝜔 ≡ 𝜋 2 𝛼=1 𝑁 𝑐 𝛼 2 𝑚 𝛼 𝜔 𝛼 𝛿 𝜔− 𝜔 𝛼
 coupling spectral density
which can often be obtained from classical arguments (e.g. in superconducting devices, from experiment in classical regime). Schulten et al. obtain from classical MD simulation:
𝐽 𝜔 = 𝛼𝜔 1+ 𝜔 2 𝜏 2
dimensionless ( XS:𝜂)
For 𝜔𝜏≪1, dissipation is ohmic with α≫1, i.e. strongly overdamped.
Some estimated numbers for RPV (in secs)
ℏ/𝑘𝑇 ~25 f sec
τ (from simulation) ~100 f sec
ℏ/𝜎 (from simulation) ~3 f sec
en. fluctuation
hence 𝛼= 𝜎 2 𝜏/ℎ𝑘𝑇 ~25
Δ −1 𝑅𝑇 ~100 f sec
⟹ Δ 2 𝜏 −1 ~100 f sec
forward transfer rate 𝑘 exp −1 𝑅𝑇
so 𝑘 exp slowest rate in problem, but Δ~ 𝜏 −1 ~ cutoff for “ohmic” behavior however, 𝑘 exp ≪ 𝜏 −1 ↑
~10 psec

7. General treatment of spin-boson problem*
Formulation of problem: Given form of 𝐽 𝜔 (and Δ, ϵ, T), set initial condition 𝜎 𝑧 𝑡 =+1. Evolve according to 𝐻 𝑡𝑜𝑡 , calculate 𝜎 𝑧 𝑡 ≡𝑃 𝑡 , (and hence, if it is exponentially decaying, transfer rate 𝑘≡−𝑙𝑛𝑃 𝑡 /𝑡).
Step 1 (general). Derive exact formal expression for 𝑃 𝑡 in terms of 𝐽 𝜔 ,Δ,𝜖 and T, and represent in graphical form
(corr. to off- diagonal elements of 𝜌 )
(corr. to diagonal elements of 𝜌 )
Step 2. Under certain conditions, justify “noninteracting blip approximation” (NIBA), convert to much simpler form (in principle soluble by Laplace transform, if one can do the integrals)
Step 3. Obtain analytic expression for 𝑃 𝑡 and thus when appropriate for 𝑘 𝜖,𝑇 .
*A.J. Leggett, S. Chakravarty, A.T. Dorsey, Matthew P.A. Fisher, A. Garg, and W. Zwerger. Revs. Mod. Phys. 59, 1 (1987).

8. In the overdamped case (only) step 2 gives the “golden-rule” result
𝑃 𝑡 =𝑃  + 1−𝑃  exp−Γ𝑡
𝑃  ≡−𝑡𝑎𝑛ℎ 𝜖/ 𝑘 𝐵 𝑇
Γ Δ,𝜖,𝑇 = Δ ℏ 2 𝑂  𝑑𝑡 cos 𝜖𝑡/ℏ 𝑐𝑜𝑠 𝑄 1 𝑡 /𝜋ℏ exp− 𝑄 2 𝑡 /𝜋ℏ
where
𝑄 1 𝑡 ≡ 𝑂  𝑑𝜔 𝜔 2 𝐽 𝜔 sin 𝜔𝑡
𝑄 2 𝑡 ≡ 𝑂  𝑑𝜔 𝜔 2 𝐽 𝜔 1− cos 𝜔𝑡 coth 𝛽ℏ𝜔/2
In the limit 𝜏→ (pure ohmic dissipation with cut off 𝜔 𝑐 ≫ Δ,𝜖,𝑘 𝐵 𝑇) Γ Δ,𝜖,𝑇 ≡𝑘 𝜖,𝑇 can be evaluated analytically:
𝑘 𝜖,𝑇 = Δ 2 𝜔 𝑐 𝑓 𝜖,𝑇
where
for 𝜖=0, 𝑇≠0 𝑓 𝑇 =const. 𝑘 𝐵 𝑇/ℏ 𝜔 𝑐 2𝛼−1
for 𝑇=0, 𝜖 ≠0 𝑓 𝜖 =const. 𝜖/ℏ 𝜔 𝑐 2𝛼−1
for 𝑇,𝜖 both ≠0 messy formula involving Euler Γ-function

9. However, for the RPV tunneling problem the cut off frequency 𝜔 𝑐 ~ 𝜏 −1 is quite comparable to Δ,𝜖,𝑇, so we must keep the full form of 𝐽 𝜔 , namely
𝐽 𝜔 = 𝛼𝜔 1+ 𝜔 2 𝜏 2
XS show that for 𝑘 𝐵 𝑇≫ℏ/𝜏 the resulting expression for Γ, hence for 𝑘 𝜖,𝑇 can be evaluated analytically and the result expressed in the Marcus-like form (Garg et al., 1985)
𝑘 𝜖,𝑇 =const. 1 𝛿 exp − 𝜖− 𝜖 𝑚 2 /2 𝛿 2
where
𝜖≡ c-number well bias (≡ redox energy)
𝜖 𝑚 ≡ 𝜋 𝑂  𝐽 𝜔 𝜔
𝑑𝜔 (≡ solvation energy)
𝛿 2 ≡ ℏ 𝜋 𝑂  𝐽 𝜔
coth 𝛽ℏ𝜔/2 𝑑𝜔 (≡Δ 𝜖 2 , fluctuation energy)
quantum fluctuations

10. At lower T it is necessary to calculate 𝑘 𝜖,𝑇 numerically, inputting the specific values of α and τ for RPV. XS’s results:
Conclusions:
At RT, simple Marcus theory works well (not too surprising, since 𝑘 𝑇 𝑅 >ℏ/𝜏)
At lower T (but not 𝑇=0), generalized Marcus theory with 𝑇𝐹→𝑄𝐹 works well
At 𝑇~0, (or for more general problems, e.g. liquid solvation) need to evaluate GR expression numerically.
Golden Rule
(would “boson sampling” help?)