# Localisation determines the optimal noise rate for quantum transport

Alexandre R. Coates1, Brendon W. Lovett2 and Erik M. Gauger1 1 SUPA, Institute
of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS,
United Kingdom  2 SUPA, School of Physics and Astronomy, University of St
Andrews, St Andrews KY16 9SS, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Environmental noise plays a key role in determining the efficiency of
transport in quantum systems. However, disorder and localisation alter the
impact of such noise on energy transport. To provide a deeper understanding of
this relationship we perform a systematic study of the connection between
eigenstate localisation and the optimal dephasing rate in 1D chains. The
effects of energy gradients and disorder on chains of various lengths are
evaluated and we demonstrate how optimal transport efficiency is determined by
both size-independent, as well as size-dependent factors. By discussing how
size-dependent influences emerge from finite size effects we establish when
these effects are suppressed, and show that a simple power law captures the
interplay between size-dependent and size-independent responses. Moving beyond
phenomenological pure dephasing, we implement a finite temperature Bloch-
Redfield model that captures detailed balance. We show that the relationship
between localisation and optimal environmental coupling strength continues to
apply at intermediate and high temperature but breaks down in the low
temperature limit.
Keywords: Quantum Transport, Disorder, Localisation, Open Quantum Systems,
Noise-Assisted Quantum Transport

##  1 Introduction

Energy transport occurs in many contexts: from circuits and molecular
junctions to processes like photosynthesis [1, 2, 3, 4] and the electron
transport chain in biology [5]. This fundamental process has very different
features depending on the scale on which it acts and the specifics of the
system coupling to the environment [6, 7]. For over a decade, a lot of work
has exposed the mechanisms of Environmental Noise-Assisted Quantum Transport
(ENAQT) [8, 9, 10, 11, 12], a phenomenon describing how incoherent processes
from interactions with the environment around a system can improve energy
transport in quantum systems. This work was heavily motivated by the possible
connection between ENAQT and the efficiency of photosynthesis [1, 3, 8, 9, 10,
13, 14, 15], though recent work suggests the relationship between the two may
be more nuanced [16, 17, 18].

There are a number of different ways in which ENAQT can arise, as shown in
figure 1. These include line broadening which can help to overcome energetic
barriers; the breaking up of an ‘invariant subspace’ of the system Hamiltonian
that is inaccessible to extraction operators on a quantum system [10]; and
momentum rejuvenation which counteracts the tendency of a fraction of the
excitation to get stuck in only sluggishly propagating states [19]. Recent
studies of steady state populations have also shown that the occupation of
system sites becomes more uniform when transport efficiency is near-optimal
[11, 12, 20], this population uniformisation phenomenon is discussed in D.

Figure 1: Illustrations of the ENAQT mechanisms that are relevant in this
paper: dephasing induced line broadening (a), the invariant subspace (b) and
momentum rejuvenation (c). Dephasing (and other forms of decoherence) act to
broaden the linewidth of system states, making otherwise forbidden transitions
energetically possible, which enables faster energy transport in disordered
systems. The invariant subspace describes the eigenstates of a coupled system
that have zero overlap with the particular energy extraction site $\ket{i}$;
disorder and localisation increases the extent of this subspace, and
environmental noise is needed to access it from the extraction site. Momentum
rejuvenation is a finite size effect: it describes how high group velocity
components of a population leave a system first, producing a skewed velocity
distribution. Incoherent noise resets the distribution, effectively pumping
population from low to high group velocities.

In this paper, we perform a systematic study of how localising the eigenstates
of 1D chains modifies their transport efficiency. Figure 2 illustrates the
model we will consider here, which allows us to study three mechanisms that
limit the delocalisation of chain eigenstates: limiting the total length of
the chain, introducing static disorder, and applying a uniform energy
gradients. Varying static disorder induces Anderson localisation [21], while a
linear energy gradient produces Wannier-Stark localisation [22].

Figure 2: Schematic view of our system setup, showing a chain with ten sites
of different energies, the energy of each site is altered by some random
disorder $\zeta_{i}$ and a uniform and linear energy gradient $\eta$. Coupling
to an environment induces dephasing $\Gamma$ on each site. The measure of
transport efficiency that we use is the steady state current $I_{ss}$
extracted from the last site. After extraction the chain population is
reinjected back onto all sites equally. Our goal is to find $\Gamma_{optimal}$
where $I_{ss}$ is maximised for the given combination of $\eta$ and $\zeta$.

Previous studies on the effects of disorder on ENAQT have focused on how
disorder affects the extent of the invariant subspace [10, 23] as well as the
distribution of steady-state populations [20]. These studies have consistently
found that as static disorder increases, more dynamic disorder is needed to
improve transport efficiency[10, 20]. More static disorder means more pure
dephasing is needed to enable otherwise forbidden transitions, therefore the
optimal pure dephasing rate is generally positively correlated with static
disorder.

Momentum rejuvenation, unlike other ENAQT mechanisms is a finite size effect
[19]. High group velocity components of a propagating wave-packet explore and
quickly exit the finite sized system, leaving behind a skewed velocity
distribution which can be reset by environmental noise, repopulating the
depleted higher velocity states. A consequence of this mechanism is that
larger systems need longer before faster exciton components can escape,
therefore they need to be ‘reset’ less often, meaning the optimal noise rate
is reduced.

In this paper we aim to produce a deeper understanding of the relationship
between ENAQT and localisation, and we will also show that momentum
rejuvenation continues to apply in non-degenerate systems and in the steady
state. This allows us to compare the effect disorder has on size-dependent and
size-independent ENAQT mechanisms.

The focus of this work is on chains with short-range nearest-neighbour
coupling, as this model is widely studied and can be fully localised. Long-
range coupling has been observed in relevant experimental systems such as
molecular aggregates [24, 25, 26] or ion traps [27]. However, in general the
long-range interactions in 1D systems prevent full Anderson localisation [28,
29], and recent work has shown that homogeneous long-range coupling [30] or
coupling to cavities [31] can significantly alter 1D responses to disorder in
ways beyond the scope of this paper. Recent years have also seen broad
interest in the transient effects of dephasing on quantum diffusion, such as
stochastic resonance, and many-body localisation, especially focused on the
quasiperiodic Aubry-André model [32, 33, 34, 35, 36, 12, 37], as well as
quantum chaotic systems [38, 39, 40, 41]. We find no non-trivial transient
effects in our model (see F), so there remains open question of how the
findings presented here would apply to more complicated scenarios.

##  2 Theoretical Model

###  2.1 System Model

In this paper we model chains with the single excitation approximation,
defining the Hamiltonian as

$H=\sum_{i}\epsilon_{i}\ket{i}\bra{i}+J\sum_{i=1}^{N-1}\ket{i}\bra{i+1}+\text{H.c.},$
(1)

$\ket{i}$ represents a state with single excitation, on site $i$.
$\epsilon_{i}$ is the on-site energy for site $i$, H.c is the Hermitian
conjugate, and $J$ is the strength of the coupling between neighbouring sites.
For this work $\hbar=1$ and all quantities are given in terms of the coupling
strength $J$ so we can focus on capturing the influence of disorder and
gradients in a very general sense.

We consider chains of $N$ sites with site energies $\epsilon_{i}$ determined
by a combination of energy disorder and a gradient in average site energies.
As a convention we set the $\epsilon_{0}$ and $\epsilon_{N}$ to the highest
average energy and zero respectively, from this we can define
$\eta=\frac{\epsilon_{0}-\epsilon_{N}}{N\cdot J}$ which we use to define the
effective gradient applied to our system, scaled by system length and given in
terms of the coupling strength $J$. To each site energy we add a perturbation
$\zeta(\sigma)_{i}$ drawn from a Gaussian distribution, centred on zero with a
standard deviation $\sigma$; here, $\sigma$ denotes the disorder strength for
the system.

The three parameters of system size $N$, disorder strength $\sigma$ and
gradient $\eta$ all help define eigenstates and their localisations. The
disorder introduces energy gaps, constraining eigenstates through localisation
[21]. The gradient could be a result of the application of a field to the
system, and produces Wannier-Stark localisation [22, 42, 43, 44]. To measure
localisation we use the inverse participation ratio (IPR), which is a measure
of the number of sites over which each eigenstate $E_{\alpha}$ is delocalised.
The average IPR over all eigenstates is defined as

$\text{IPR}=\frac{1}{\sum_{i,\alpha}|\braket{i}{E_{\alpha}}|^{4}}.$ (2)

This single value represents how localised that system is, with greater
localisation implying not only a larger invariant subspace, but also a
decreased efficiency in coherent transport. The IPR captures the system-wide
impact of different gradients and disorder strengths on a system, making it a
natural measure to compare systems. The effects of gradients and random
disorder are illustrated in figure 3, the coloured areas in the left panel
show one standard deviation around the mean value at each point.

Figure 3: (Left) Average IPR against various disorder strengths $\sigma$ for
$N=40$, considered against four energy gradients $\eta$. Coloured areas show
$\pm$ one standard deviation, each point is averaged from 100 configurations
of disorder. An ordered ($\square$) and disordered ($\bigcirc$) point are
highlighted, and their eigenspectra shown in the centre and right panels,
respectively. (Centre) The eigenspectrum for a chain with
$\eta=0.1,\sigma=0J$, showing the slight localisation of eigenstates under a
uniform field. The size of each diamond is proportional to the probability of
observing each eigenstate on that site. (Right) The eigenspectrum for
$\eta=0.1,\sigma=0.3695J$, showing a mixture of field effects and disorder,
producing inconsistent eigenenergy spacing as well as slightly more localised
eigenstates.

###  2.2 Dynamics, Lindblad and Redfield Master Equations

We model each chain with a Lindblad master equation implemented with the QuTiP
package [45],

$\dot{\rho}=-i[H,\rho]+\Gamma\sum_{i=1}^{N}\mathcal{L}\left[A_{deph,i}\right]\rho+\gamma_{inj}\sum_{i=1}^{N}\mathcal{L}\left[A_{inj,i}\right]+\gamma_{trap}\mathcal{L}\left[A_{ext}\right]\rho,$
(3)

where $\mathcal{L}\left[A\right]\rho$ is the Lindbladian dissipator

$\mathcal{L}\left[A\right]\rho=\left(A\rho
A^{\dagger}-\frac{1}{2}\\{A^{\dagger}A,\rho\\}\right).$ (4)

$\Gamma$ sets the rate of (dephasing) noise in the system, for simplicity
assumed to be the same on each site, $\\{\cdot,\cdot\\}$ is the
anticommutator, and $A_{deph,i}$ are Lindblad operators describing the
environmental influence on each site $i$. For on-site dephasing in the single
excitation approximation, the operators for on-site energy noise take the form
$A_{deph,i}=2\ket{i}\bra{i}-\mathbb{I}$ [46, 47] . The extraction operator
projects population from the $N^{\text{th}}$ site (lowest end of chain) to an
external shelf state where it is trapped,
$A_{ext}={\sigma_{N}^{+}\sigma_{trap}^{-}}$. Similarly population is re-
injected from the trap back onto each site with the injection operators
$A_{inj,i}={\sigma_{trap}^{+}\sigma_{i}^{-}}$.

To treat these systems at finite temperatures we use the Bloch-Redfield master
equation. As we study disordered systems with very mixed energy splittings we
retain all non-secular terms to ensure it remains accurate [48]. The master
equation reads:

$\begin{split}\dot{\rho_{s}}&=-i[H,\rho_{s}]+\gamma_{inj}\sum_{i=1}^{N}\mathcal{L}\left[A_{inj,i}\right]+\gamma_{trap}\mathcal{L}\left[A_{ext}\right]\rho\\\
&+\Gamma\sum_{\omega}\sum_{m,n}S_{m,n}(\omega)\left(A_{n}(\omega)\rho_{s}A^{\dagger}_{m}(\omega)-\frac{1}{2}\\{A^{\dagger}_{m}(\omega)A_{n}(\omega),\rho_{s}\\}\right),\end{split}$
(5)

where the injection and extraction operators are the same as in equation 3,
$\rho_{s}$ is the system density matrix and the frequencies $\omega$ are the
eigenenergy splittings [49]. $A_{m}$ are system-environment interactions,
derived by transforming the relevant site basis operators
$A_{deph,i}=2\ket{i}\bra{i}-\mathbb{I}$ into the Hamiltonian eigenbasis [49]
and $S_{mn}(\omega)$ defines the noise-power spectrum associated with the
system-environment interaction. The noise-power spectrum function is

$S_{mn}(\omega)=\left(\mathcal{N}_{BE}(\omega,\beta)+\Theta(\omega)\right)\mathcal{J}(\omega),$
(6)

where $\mathcal{N}_{BE}(\omega)$ defines Bose-Einstein statistics at a given
phonon inverse temperature $\beta,\Theta(\omega)$ is the Heaviside function,
allowing phonon-assisted transitions from higher to lower eigenenergies
($\omega>0$) but not the reverse case, and $\mathcal{J}(\omega)$ is the
spectral density. We use a flat spectral density as assumed in equation 4,
such that $\mathcal{J}(\omega)=\mathcal{J}$, this is for a direct comparison
with the pure dephasing case. A Drude-Lorentz spectral density is considered
and presented in E.

###  2.3 Steady state setup and observables

As indicated by figure 2, we re-inject any extracted population back onto all
chain sites equally. By linearity, each injection site represents an initially
populated site in the dynamical approach, so this injection scheme is
equivalent to a mixed initial state. This choice ensures we capture the
general system response, minimising the influence from inversion symmetry
effects, while also ensuring that we can generically compare transport
properties across systems with different sizes without adding in extra
concerns about differing lengths between injection and extraction. For
completeness, we show in C that injection on a single site produces
qualitatively similar results.

We match the total injection to the extraction rate so that
$\gamma_{inj}=\frac{\gamma_{trap}}{N}$. Our focus on steady-state properties
is motivated by prior ENAQT studies which have shown that the steady state
approach is more natural for energy transport in photosynthetic systems [50,
51, 1].

The steady state $\rho_{ss}$ is found by calculating the zero eigenstate of
the system Liouvillian. In our work $\gamma_{trap}=3J$; changing this value
generally changes quantitative values but not the qualitative behaviour [20]
unless the rate is so high it begins to enter the Zeno regime [52]. The key
observable of transport efficiency is the steady state current $I_{ss}$, which
we aim to maximise. This is simply the product of the extraction rate and the
excited steady state population on the extraction site $N$,

$I_{ss}=\rho_{N,N}\gamma_{trap}.$ (7)

##  3 Results

In this section we show how random disorder, energy gradients and system size
affect ENAQT in the pure dephasing limit (section 3.1), demonstrating the
strikingly consistent relationship between IPR and $\Gamma_{optimal}$. We also
present a power law that fits the unbiased chain data, letting us separate the
influence of size-independent and size-dependent effects. We then go beyond
pure dephasing with the Bloch-Redfield master equation and show these effects
are still qualitatively robust at high to intermediate temperatures, but break
down in the lower temperature limit (section 3.2).

###  3.1 Pure Dephasing

Figure 4 shows how energy gradients and random disorder affect the optimal
dephasing for chains of 40 sites. In all cases we see that as the IPR
decreases linearly $\Gamma_{optimal}$ increases rapidly, once again confirming
positive correlation between ENAQT peak position and static disorder [10, 23,
20]. The main finding is that the results for each gradient largely overlap
once the chain is disordered enough, indicating that once the system
eigenstates are sufficiently localised the source of localisation does not
matter.

Figure 4: $\Gamma_{optimal}$ vs IPR for a variety of disordered $N$ = 40
chains, colour coded for each of the four different gradients $\eta$
considered. The inset shows how the curve points are generated: by varying the
dephasing rate $\Gamma$ until a peak current is found. We see that in the
majority of cases the trends overlap for each gradient, suggesting the IPR
matters more than specific energy landscape or gradient. Calculations are
repeated 100 times for each combination of gradient and disorder strength. The
inset curve is calculated for $\eta=0.1,\sigma=0.3695J$ as in figure 3.

Figure 4 shows that for sufficiently large IPRs (IPR $\geq 12$), the optimal
dephasing for no gradient ($\eta=0$) is lower than that for a weak gradient
($\eta=0.1$). With momentum rejuvenation we expect that the larger the system
is, the lower its $\Gamma_{optimal}$. As such, we infer that the presence of
nonzero gradients limits the maximum length momentum rejuvenation can work
over. So for $N=40$ the gradient $\eta=0.1$ is enough to slightly reduce the
impact of momentum rejuvenation as compared to when $\eta=0$. The result is a
higher $\Gamma_{optimal}$ for the weak gradient.

As discussed in Sec. 2, linear energy gradients localise eigenstates [22, 43]
and alter charge transport [53] differently from random disorder. Yet once the
chains are localised enough, momentum rejuvenation’s influence is negligible
and the optimal dephasing rate is determined only by the IPR, as can be seen
for $\eta=1,10$. Therefore gradient-induced localisation and disorder-induced
localisation only have different effects on ENAQT when the gradients are
strong enough to shorten the length scale over which momentum rejuvenation
acts, but weak enough to ensure it is still present.

The relative impact of momentum rejuvenation is then further affected by the
size of the system itself. As discussed above, gradients reduce the maximum
length over which momentum rejuvenation acts. This leads to differences in
$\Gamma_{optimal}$ if that length is less than the system length. By extension
we should then expect the differences between $\eta=0$ and $\eta=0.1$ to scale
with the number of chain sites $N$. We can observe this directly in figure 5
which demonstrates how the localising effects of energy gradients and random
disorder affect ENAQT for chains of different lengths.

Figure 5: $\Gamma_{optimal}$ vs IPR for 4 different chain lengths, calculated
with 100 realisations of disorder for each combination of gradients and
disorder strength. As $N$ increases the same $\Gamma_{optimal}$-IPR response
generally occurs, just stretched over a larger range of IPRs. The final panel
shows how the range of $\Gamma_{optimal}$ for each gradient varies with
length. For this final panel $N=50$ was also considered, using 25 different
realisations of disorder for each combination of disorder and energy gradient.
The upper limit of each bar shows the optimal dephasing for the most localised
chains, while the lower limit shows this for ordered chains where $\sigma=0$.
The momentum rejuvenation model predicts that as a system gets larger, the
optimal dephasing rate decreases, we observe this for the lower edge of the
$\eta=0$ bars and partially for $\eta=0.1$ before its effectiveness is reduced
at larger $N$ as discussed in section 3.1. For the stronger gradients the
lower range of $\Gamma_{optimal}$ does not significantly change with $N$,
confirming that finite size effects are effectively suppressed for these
chains, leaving only the effects of site to site detunings.

The upper limit of the $\Gamma_{optimal}$ bars in figure 5’s final panel shows
consistent scaling behaviour across length, that is independent of gradients.
This can be partially explained by regression to the mean as disorders are
sampled from a Gaussian distribution. Maximally localised systems have large
detunings between all sites, so the longer your system, the more detunings
there are to maximise. Therefore, the larger the system, the harder it is to
localise. Close inspection confirms this: the minimum IPR increases with chain
length, and by extension the highest $\Gamma_{optimal}$ decreases with chain
length.

We now focus our attention on the lower end of these $\Gamma_{optimal}$
ranges. First we note that the high gradient behaviour ($\eta=1,10$) has
consistent lower limits for all chain lengths considered, meaning ENAQT is
only determined by average site to site detunings, with little if any
sensitivity to size. Momentum rejuvenation suggests that the larger the
system, the lower its optimal noise rate, and we observe the zero gradient
data extends to lower and lower dephasing rates as $N$ increases, exactly as
predicted [19]. We note that the difference in $\Gamma_{optimal}$ between
$\eta=0$ and $\eta=0.1$ increases with $N$, indicating that the range of which
momentum rejuvenation acts has a finite length at $\eta=0.1$ and so becomes
less effective as system size increases. This can also be seen in how
$\Gamma_{optimal}$ changes with $N$. The lower range of $\Gamma_{optimal}$,
where $\eta=0.1$ initially decreases as expected with momentum rejuvenation,
then the trend reverses as increasing $N$ reduces the impact of momentum
rejuvenation.

So far we have described under what conditions the size dependent effects of
momentum rejuvenation can be observed, given the presence of energy gradients
and random disorder. We now focus on the $\eta=0$ limit we can directly
capture how static disorder alters the influence of finite size effects on
ENAQT. We consider the $N=10-40$ chains and fit the $\eta=0$ data with a power
law of the form

$\Gamma_{optimal}(\textnormal{IPR})\propto\text{IPR}^{\lambda+\kappa\cdot\textnormal{IPR}}.$
(8)

The exponent $\lambda$ captures the response across all IPR values,
corresponding to the influence of the invariant subspace and the need for line
broadening. Meanwhile, the exponent $\kappa$ captures a varying influence,
being negligible for very localised systems and most influential for systems
with large IPR, capturing the influence of finite size effects such as
momentum rejuvenation. We note that equation 8 is simply a phenomenological
fit that best captures the data produced by our results, the data is not well
fit by a single exponential and alternative functional forms likely require
additional fitting parameters. As we show in table 1 both $\lambda$ and
$\kappa$ scale monotonically with chain length as expected. Further details
and plots are presented in B.

N |  $A$ $(J)$ | $\lambda$ | $\kappa$ | SD $(\times 10^{-3})$
---|---|---|---|---
10 | 1.59 | -3.14 | 0.07 | 1.80
20 | 1.70 | -2.69 | 0.03 | 0.55
30 | 1.74 | -2.51 | 0.02 | 0.33
40 | 1.73 | -2.36 | 0.01 | 0.22
Table 1: Table of best fit values and standard deviation for each chain length
with $\eta=0$. $A$ captures the proportionality, $\lambda$ the size
independent response, and $\kappa$ the size-dependent response. As chains get
longer, the fits gets more accurate, and the parameters change monotonically
as the same behaviours stretch over a new range of IPRs.

###  3.2 Finite temperature Bloch Redfield model

As described in section 2.2, we can go from phenomenological pure dephasing
model—effectively an infinite temperature limit—to a microscopically-founded
finite-temperature approach with the use of the full, nonsecular Bloch-
Redfield master equation, equation 5. These calculations are done against a
flat spectral density for direct comparison with the pure dephasing results
above. We define the inverse temperature $\beta=\frac{1}{k_{B}T}$, and
consider 3 temperatures, $J\cdot\beta=10,1,0.1$ (low, medium and high
respectively), giving the results in figure 6111In $<$ 0.1% of cases we found
the steady state solver would fail, the optimisation procedure handled this by
moving to the next trial point and continuing..

Figure 6: $\Gamma_{optimal}$ vs IPR for $N=10$, considered at 3 inverse
temperatures, once again for 100 realisations of disorder at each combination
of gradient and disorder strengths. High and intermediate temperatures have
broadly the same monotonic form, we note that the cooler the system is, the
greater the $\Gamma_{optimal}$. These peaks are found using a bounded peak
finding function, with the range $0<\Gamma<50J$. In the finite temperature
limit we find some data points clustered at the edges of this range,
suggesting either monotonic $I_{ss}$ vs $\Gamma$ curves, or
$\Gamma_{optimal}\geq 50J$. We cut off all results within $10^{-3}J$ of either
limit and collect them in the offset sections above and below the central
axes. Each offset series is separated into points corresponding to different
temperatures and annotated with a percentage to illustrate what fraction of
data points corresponding to each temperature lie there.

Under these conditions we still recover the characteristic trend of a
monotonic relationship between $\Gamma_{optimal}$ and IPR, and we report
similar results for sufficiently wide non-flat spectra in figure 12. We note
that as temperatures lower, $\Gamma_{optimal}$ for a given IPR increases. As
temperatures decrease, the specific energy landscape of each chain becomes
more important [54], as it becomes harder to avoid trapping population in
energy minima. As a result, the range of $\Gamma_{optimal}$ associated with
any IPR gets broader continuously as temperatures get lower.

We therefore conclude that the general ENAQT response to disorder depends not
just on localisation, but also on avoiding the trapping of population in
energetic minima. So when transfer rates up and down in energy become
significantly different the chain population cannot explore all the system
sites, trapping population in energetic minima. In this limit the universal
response observed for pure dephasing breaks down, producing a regime which is
very sensitive to the specifics of the energy landscape. By the reverse
argument, if energy can move reasonably around a system then the monotonic
relationship between optimal environmental coupling and IPR is well-defined.

##  4 Conclusion

We have systematically shown how localisation and optimal ENAQT are related
for 1D chains, producing a universal trend strongly determined by the IPR. The
IPR in turn is determined by an interplay of energy gradients, random disorder
and system length. Comparing the range of $\Gamma_{optimal}$ for various
lengths of chain provided further insight into how strong gradients can
suppress the influence of finite size effects. Additionally we have found that
steady state current in unbiased, disorder systems can be described by a power
law containing size dependent and size independent contributions, illustrating
that finite size effects such as momentum rejuvenation still affect how ENAQT
acts on disordered systems.

Extending the model to include finite temperatures shows that the same
response holds for high to intermediate temperatures. By contrast, at lower
temperatures population can become trapped in local energetic minima. This
decouples transport efficiency from eigenstate localisation and the transport
becomes more sensitive to a chain’s specific energy landscape.

Throughout this paper we have shown that the localisation of a system’s
eigenstates is directly connected to what its optimal conditions for ENAQT
are. By considering this for a large range of possible conditions we have
developed new and broadly applicable insights into how localisation and finite
size effects alter ENAQT in 1D. More work is required to confirm if this
response is altered for higher dimensional systems where eigenstates may be
further delocalised. For example simple tight-binding honeycomb lattices as
found in graphene nano-ribbons can display quantum chaotic properties under
weak static fields [55], opening up a new class of system. The effects of
localisation could be further investigated, whether taking a more fine-grained
look at the unusual Wannier-Stark behaviour in A, or going to much larger
system sizes in order to limit the influence of finite size effects. Lastly,
quasiperiodic systems such as the Aubry-André model could be considered, where
transient effects such as stochastic resonance with anti-localised eigenstates
[32] may provide new insights into ENAQT beyond the steady state.

## Acknowledgements

We thank Scott Davidson, Dominic Rouse and Gerard Valentí-Rojas for the
helpful discussions. This work was supported by EPSRC Grant No. EP/L015110/1.
Computations were carried out using QuTiP [45], figures made in matplotlib
[56].

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##  Appendix A Gradient-only localisation vs IPR

In the limit where we apply a gradient but no disorder, we find a set of
behaviours quite different from that described in the main body of the paper,
we note these behaviours are only really visible on a log plot and for small
enough rates that it would be very hard to observe them experimentally.
Looking at figure 7 we see a consistent and strangely stepped behaviour where
$\Gamma_{optimal}$ increases irregularly as the IPR decreases. We speculate
this may be due to the eigenstates under Wannier-Stark localisation being very
consistent in spread and overlap. As a result of this ordering, changing the
IPR implies not only further localising eigenstates but also consistently
shifting the mutual participation any pair of eigenstates have on a common set
of sites. This is in stark contrast to systems with random static disorder
where there is no consistent mutual presence of eigenstates to disrupt.

Another potential explanation comes from prior experimental work on quantum
transport in biased, weakly coupled superlattices [57]. This work demonstrated
the existence of a single plateau in the current-bias relationship, and found
it could be physically explained by the influence of Wannier-Stark
localisation acting against field-assisted tunneling between adjacent wells.
More analysis is needed to determine if this could be connected to our work or
if the resemblance is merely superficial. The use of more local measures for
localisation, and studying larger systems would let one better distinguish the
effect these gradients have on eigenstates in the bulk as compared to the
edge.

Figure 7: $\Gamma_{optimal}$ vs IPR when there is no applied disorder, only
applied gradients. We see consistent stepped behaviour across all lengths as
the Wannier-Stark localisation increases.

##  Appendix B Curve fitting

Figure 8: $\Gamma_{optimal}$ vs IPR with power laws fitting each dataset. To
avoid the influence of gradients the data was only fit to subset of data with
no field gradient applied. Parameters shown, and listed in table 1.

We have explained that the curves in figure 4 show two different responses: a
steep disordered one largely determined by exponential localisation [21] and
the invariant subspace, and a flatter response characterised by the greater
presence of momentum rejuvenation and coherent transport. As we do not have an
analytic form for the curves in figures 4 and 5 we directly fit them, and find
that the change in response across IPR is modelled well by a power law of the
form

$\Gamma_{optimal,N}(\textnormal{IPR})=A\cdot\text{IPR}^{\lambda+\kappa\cdot\textnormal{IPR}},$
(9)

where the proportionality constant $A$ is determined by systematic factors
such as length, coupling strength, injection and extraction rates etc, while
$\lambda$ describes the constant response to localisation, and $\kappa$
captures the increasing presence of coherent effects as the IPR increases. As
such $\lambda$ captures the ever present exponential localisation from
disorder [21], $\kappa$ captures the varying presence of finite size effects
and momentum rejuvenation.

The best fits were found by using the logarithm of the data and equation 9.
The best fit parameters are listed in table 1, and the fit residuals shown in
figure 9. We find the fitting parameters scale monotonically with changes in
$N$ as expected. The error is calculated using the quadrature sum of the
covariance matrix diagonals generated by the fitting function.

Figure 9: Residuals due to fitting power laws to data from various disorder
strengths for N = 40.

##  Appendix C Injecting on single sites

Throughout this work we have used a pumping scheme that injects population
equally across all system sites. Here we show that response to localisation
remains qualitatively the same when injecting onto a single site, by
considering 2000 chains of length 20.

Figure 10: $\Gamma_{optimal}$ for 2000 length 20 chains under a range of
Wannier-Stark and Anderson Localisation, the pumping scheme is extracting out
the the $N^{th}$ site, and injecting on the $2^{nd}$ site. We see a very
similar response to pumping on all sites, but expect the optimal dephasing
rates to decrease the closer the injection site is to the extraction site.

##  Appendix D Population Uniformisation

There are multiple perspectives on ENAQT, and one prior, unified approach is
provided by population uniformisation [11], which has also been studied in
disordered systems [20]. The key measure used in this approach is the variance
of steady state populations across all chain sites, uniformity is highest when
variance is minimised. The population uniformisation theory predicts that this
occurs at, or near to peak steady state currents. In other words
$\Gamma_{min.var}\approx\Gamma_{optimal}$.

We tested this by generating a representative sample of length 10 chains, then
determined the optimal dephasing rates for steady state current and population
uniformisation. After considering 50 realisations of disorder at each point in
the parameter space and averaging, we indeed found that the two rates are
correlated, agreeing very closely in ordered systems, and less so in highly
disordered cases. As shown in figure 11, we find the average rates for peak
uniformisation are consistently greater than those for the peak current.
However, even in the divergent cases such as highly disordered systems, the
on-site populations are still relatively uniform when the steady state current
is maximised. We also note that the difference between these two rates is
mainly dependent on the IPR, and is insensitive to the gradient applied to the
system. As such we can say that while offering a different view, our results
are consistent with population uniformisation.

Figure 11: The averaged optimal dephasing rates with respect to the steady
state current and population uniformisation respectively. Each point presented
is the average of 50 realisations of disorder.

##  Appendix E Non-flat spectral densities

We mainly focus in this work on pure dephasing and non-peaked spectral
densities to show the generic influence of localisation on optimal ENAQT
conditions. Here we show the effects of a typical Drude-Lorentz spectral
density

$\mathcal{J}(\omega)=\lambda\cdot\frac{2}{\pi}\cdot\frac{\omega(\frac{1}{\tau})}{\omega^{2}+(\frac{1}{\tau})^{2}},$
(10)

where $\frac{1}{\tau}$ is the Lorentzian linewidth and $\lambda$ is the
coupling to the phonon bath.

We consider multiple linewidths and their effects on 2000 chains, generated
the same way as in the body of the text. We specifically use the linewidths
$\tau^{-1}=1,10J$ to see the effect of linewidths equivalent to average system
spacing, as shown in figure 12. We do not consider much narrower linewidths as
those would typically coincide with non-Markovian effects which are beyond the
scope of the model we use.

We find that for these Lorentzians we still recover the same qualitative
trends seen for flat spectral densities, though we note as $\tau^{-1}$
increases, the optimal phonon couplings increase as well. This is because as
$\tau^{-1}$ increases, $\mathcal{J}(\omega)$ begins to scale with
$\frac{\lambda}{\tau^{-1}}$. Thus for a system with an optimal set of
transition rates, a larger linewidth means larger phonon couplings $\lambda$
are optimal.

Figure 12: Optimal dephasing vs IPR with Drude-Lorentz spectral densities with
linewidths $\tau^{-1}=1J$ (left), and $10J$ (right). Note that as the
linewidth increases by an order of magnitude, so do the peak phonon couplings
$\lambda$. As a consequence, many more cases peak are above the upper limit of
our peak finding code for the larger linewidth.

##  Appendix F Transient effects

Given recent interest and investigations into transient effects in tight-
binding systems, we carried out dynamical calculations to check if effects
beyond the steady state were contained in our model. In all cases we found
that the dynamics converged onto the steady state behaviour, and remained
converged to long time. Analysis of the chain Liouvillians confirmed that in
all cases the steady state was uniquely defined, Hermitian and trace
preserving. The same held true for the Bloch-Redfield results.

Taking inspiration from a prior study on stochastic resonance [32] we
investigated the variance of a time-evolving localised initial state injected
onto the centre of a chain of 41 sites. This chain was closed, having no
extraction or pumping, and the variance was defined as

$variance=\sum_{n}n^{2}|\psi_{n}|^{2},$ (11)

where $n$ is the site index with respect to the central site, such that for an
odd length system $n\in[-\frac{N-1}{2},\frac{N-1}{2}]$, $\psi_{n}$ is the
normalised, initial state. We do not find any evidence of non-trivial
transient effects in our model, instead as shown in figure 13 we observe a set
of smooth continuous approaches to the steady state, with some over- or under-
damping depending on the dephasing rate.

Figure 13: Variance of a localised initial wavefunction injected onto the
centre of 3 representative length 41 chains with no injection or extraction.
All eventually converge onto the steady state (horizontal dashed line). All
quantities dimensionless.