Abstract
In Sect. 1.1, we introduced the field of quantum transport theory and more specifically discussed the influence of quantum interference effects on transport phenomena. The present chapter, based on Walschaers et al. (2013, 2015), presents the first results of this dissertation, which fit into that framework.
Chaos isn’t a pit. Chaos is a ladder
Lord Petyr Baelish, played by Aidan Gillen in “Game of Thrones” (Sakharov 2013)
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Notes
- 1.
- 2.
We introduce the standard mathematics notation \(\lceil .\rceil \) for rounding up and \(\lfloor .\rfloor \) for rounding down. For example \(\lceil 1.1\rceil = 2\) and \(\lfloor 1.9\rfloor =1\)
- 3.
The factor two enhancement, as compared to the typical 1 / N, obtained for \(n=1\) is more closely related to coherent backscattering, since it is a manifestation of weak localisation.
- 4.
One retrieves the semicircle, but should incorporate finite size corrections for small N.
- 5.
The quantum recurrence theorem (Bocchieri and Loinger 1957) implies that for any state vector \(\psi \in \mathcal {H}\), any one-parameter group \(\{U_t \mid t \in \mathbb {R} \}\) and any \(\epsilon > 0\), there is a time \(t_0 > 0\) such that \(||(\mathbb {1} - U_{t_0})\left| \psi \right\rangle || < \epsilon \). However, note that this does not imply periodicity.
- 6.
Using the exchange operator which is later defined in (4.46), we can make this reasoning more rigorous by noting that \(e^{-it_0H}e^{i\theta }\left| \mathrm out\right\rangle = e^{-it_0H}e^{i\theta }J\left| \mathrm in\right\rangle \underset{[H,J]=0}{=} J e^{-it_0H}e^{i\theta }\left| \mathrm in\right\rangle \underset{(4.39)}{=} e^{2i\theta } J \left| \mathrm out\right\rangle \underset{(4.47)}{=} e^{2i\theta } \left| \mathrm in\right\rangle \).
- 7.
Which fixes a state’s parity.
- 8.
This is equivalent to sampling the Hamiltonian in the site basis, with
while explicitly fixing \(H_{i\, j}=H_{i\, N-j+1}=H_{N-i+1\, j}=H_{N-i+1\, N-j+1j}\).
- 9.
This symmetric behaviour on the time axis is quantified by \(\left|\left\langle e_i,e^{-i t H} \mathrm{in} \right\rangle \right|^2 = \left|\left\langle e_i,e^{-i t H} J^2 \mathrm{in} \right\rangle \right|^2 = \left|\left\langle J\, e_i,e^{-i t H} \mathrm{out} \right\rangle \right|^2 \approx \left|\left\langle J\, e_i,e^{-i (t + t_0) H} \mathrm{in} \right\rangle \right|^2\), where we explicitly use that there is a time \(t_0\) for which \(e^{- i t_0 H}\left| \mathrm in\right\rangle \approx e^{i \theta } \left| \mathrm out\right\rangle \). Moreover, an additional symmetry around \(t_0/2\) is implied, which follows from \(\left|\left\langle \mathrm{out},e^{-i t H} \mathrm{in} \right\rangle \right|^2 \approx \left|\left\langle e^{- i t_0 H} \mathrm{in},e^{-i t H} \mathrm{in} \right\rangle \right|^2 =\left|\left\langle e^{- i (t_0 - t) H} \mathrm{in},\mathrm{in} \right\rangle \right|^2 = \left|\overline{\left\langle \mathrm{in}, e^{- i (t_0 - t) H} \mathrm{in} \right\rangle }\right|^2 = \left|\left\langle \mathrm{in}, e^{- i (t_0 - t) H} \mathrm{in} \right\rangle \right|^2 \).
- 10.
Or, rather, the paradigmatic double-well potential.
- 11.
Throughout the remainder of this chapter, the terms “first passage time” and “transfer time” are therefore used interchangeably.
- 12.
We do notice that finite size effects slightly alter the prediction of Eq. (4.93), resulting for example in an actual \(\alpha '\approx 0.93\), obtained by numerics, instead of the analytically predicted \(\alpha '=0.95\) both for \(N=20\), \(\xi =2\) and \(\chi =0.0656234\).
- 13.
Due to the subtlety of the argument, the phrasing is delicate, hence the wording here is similar to that of Walschaers et al. (2015).
- 14.
We warn the reader that throughout the text \(\tau \) is used as a stochastic quantity, which may make definition (4.101) somewhat misleading.
- 15.
Given that a stochastic variable X is normally distributed, the distribution of \(\left|X\right|\) is called a half-normal (if \(\mathbb {E}(X)=0\)) or folded normal distribution (Leone et al. 1961). The terminology refers to the fact that the negative part of the probability distribution is literally folded to the positive side.
- 16.
All our attempts resulted in page-long expressions which were neither useful nor insightful.
- 17.
We use the property that \(x=e^{\log x}\), which implies that \(f(V')= \frac{1}{N}\log \left( e^{-{V'}^2} \left( \text {erfc}\left( V'\right) \right) ^{\frac{N}{2}-1} V'\right) \).
- 18.
Where \(\chi /\sqrt{N}\) is the typical (RMS) coupling between the input/output and the intermediate sites, and \(\xi /\sqrt{N}\) denotes the typical (RMS) coupling strength between the intermediate sites.
- 19.
This is also consistent with the physical idea behind the CAT mechanism: The coupling to a second chaotic (hence modelled by the GOE Bohigas et al. 1993) degree of freedom (here the randomly interacting intermediate sites, where the randomness comes from conformational changes in the macromolecular arrangement, in particular mimicking vibrational background degrees of freedom) enhances the tunnelling rate in a donor-acceptor system with vanishing direct coupling (Tomsovic and Ullmo 1994).
- 20.
Whether more efficient light harvesting is (or has ever been) evolutionarily beneficial (and to what extent) remains an open question. There is a chance that the answer to this question depends on the organism and even on its ecosystem.
- 21.
Due to the centrosymmetry we only consider even numbers of sites N. Moreover, \(N=2\) implies there are no intermediate sites and thus we obtain the benchmark system.
- 22.
Note that we use “\(\mathrm{cte}\)” to indicate an unspecified constant.
- 23.
Note that large densities of sites translate in many contributing energy levels in the perturbative series.
- 24.
Relevant network sizes in photosynthesis are of the order \(N\sim 10\) (Blankenship 2002).
- 25.
- 26.
Clock speed of 2.6 GHz, 8x256 KB of level 2 cache and 20 MB level 3 cache. All demanding computations, i.e. those that cannot be done on a normal laptop, were done on the bwGRiD.
- 27.
There is a much higher density of dominant doublet realisations in the centrosymmetric GOE than in the standard GOE.
- 28.
Common acronym to denote the community studying atomic, molecular, and optical physics.
- 29.
This provides a natural connection to the random matrix theory of many-particle systems.
- 30.
Whether the proposed mechanism can also be implemented in such systems is of course a very different question.
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Walschaers, M. (2018). Efficient Transport in Closed Systems. In: Statistical Benchmarks for Quantum Transport in Complex Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-93151-7_4
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