arXiv:0807.4902v1  [quant-ph]  30 Jul 2008

Dephasing assisted transport: Quantum networks and biomolecules

M. B. Plenio1,2 and S. F. Huelga3

1 Institute for Mathematical Sciences, Imperial College London, London SW7 2PG, UK
2 QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK and
3 Quantum Physics Group, Department of Physics, Astronomy & Mathematics
University of Hertfordshire, Hatfield, Herts AL10 9AB, UK
(Dated: November 26, 2024)

Transport phenomena are fundamental in Physics. They allow for information and energy to be
exchanged between individual constituents of communication systems, networks or even biological
entities. Environmental noise will generally hinder the efficiency of the transport process. However,
and contrary to intuition, there are situations in classical systems where thermal fluctuations are ac-
tually instrumental in assisting transport phenomena. Here we show that, even at zero temperature,
transport of excitations across dissipative quantum networks can be enhanced by local dephasing
noise. We explain the underlying physical mechanisms behind this phenomenon, show that entangle-
ment does not play a supportive role and propose possible experimental demonstrations in quantum
optics. We argue that Nature may be routinely exploiting this effect and show that the transport
of excitations in light harvesting molecules does benefit from such noise assisted processes. These
results point towards the possibility for designing optimized structures for transport, for example
in artificial nano-structures, assisted by noise.

Introduction – Noise is an inevitable feature of any
physical system, be it natural or artificial.
Typically,
the presence of noise is associated with the deterioration
of performance for fundamental processes such as infor-
mation processing and storage, sensing or transport, in
systems ranging from proteins to computing devices.
However, the presence of noise does not always hinder
the efficiency of an information process and biological
systems provide a paradigm of efficient performance as-
sisted by a noisy environment [1]. A vivid illustration of
the counterintuitive role that noise may play is provided
by the phenomenon of stochastic resonance (SR)[2]. Here
thermal noise may enhance the response of the system to
a weak coherent signal, optimizing the response at an in-
termediate noise level [3]. Some experimental evidence
suggests that biological systems employ SR-like strate-
gies to enhance transport and sensing [4, 5]. Noise in the
form of thermal fluctuations may also lead to directed
transport in ratchets and play a helpful role in Brownian
motors [6, 7, 8]. It seems therefore natural to try and
draw analogies with complex classical networks so that
the physical mechanisms that underpin their functioning
when subject to noise can be perhaps mirrored and even-
tually used to optimize the performance of complex quan-
tum networks. Recently, tentative first steps towards the
exploration of the concept of SR in quantum many-body
systems [9, 10, 11] and quantum communication chan-
nels [12, 13, 14] have been undertaken while other studies
have focused into analyzing the persistence of coherence
effects in biological systems. In particular, detecting the
presence of quantum entanglement, has been the object
of considerable attention [16, 17].
It was noted, how-
ever, that even if found, it would be unclear whether
such entanglement has any functional importance or is
simply the unavoidable by-product of coherent quantum
dynamics in such systems [18].

Here we show that dephasing noise, which leads to

1

2

3

4

N+1

N

FIG. 1: Sites (blue spheres), modeled here as spin-1/2 parti-
cles or qubits, are interacting with each other (dashed line)
to form a network. The particles may suffer dissipative losses
as well as dephasing. The red arrow indicates an irreversible
transfer of excitations from the network to a sink that acts as
a receiver.

the destruction of quantum coherence and entanglement
as a result of phase randomization, may nevertheless
be an essential resource to enhance the transport of
excitations when combined with coherent dynamics.
Indeed, we show that a dissipative quantum network
subject to dephasing can exhibit an enhanced capacity
for transmission of classical information when seen as
a communication channel, even though its quantum
capacity and quantum coherence are diminished by
the presence of noise.
It is the constructive interplay
between dephasing noise and coherent dynamics, rather
than the presence of entanglement, that is responsible
for the improved transport of excitations.
Recently,
this enhancement of quantum transport due to the
interplay between coherence and the environment has
been demonstrated and quantified for chromophoric


2

complexes (see [19, 20, 21, 22] and Note Added).

In addition to the clarifying nature of these results, it is
intriguing to observe that Nature appears to exploit noise
assisted processes to maximize the system’s performance
and it will be worthwhile to explore how similar processes
may be useful for the design of improved transport in
nano-structures and perhaps even quantum information
processors.
The basic setting – We consider a network of N sites
that may support excitations which can be exchanged
between lattice sites by hopping (see Fig. 1). The Hamil-
tonian that describes this situation is then given by

H =

N
�

k=1
ℏωkσ+
k σ−
k +
�

k̸=l
ℏvk,l(σ−
k σ+
l + σ+
k σ−
l ),
(1)

where σ+
k (σ−
k ) are the raising and lowering operators for
site k, ℏωk is the local site excitation energies and vk,l
denotes the hopping rate of an excitation between the
sites k and l. It should be noted that the dynamics in
this system preserves the total excitation number in the
system. This is not an essential feature but makes the
system amenable to efficient numerical analysis. We will
assume that the system is susceptible simultaneously to
two distinct types of noise processes, a dissipative pro-
cess that reduces the number of excitations in the system
at rate Γk and a dephasing process that randomizes the
phase of local excitations at rate γk.
Initially we will assume that we can describe both pro-
cesses by using a Markovian master equation with local
dephasing and dissipation terms. It is important to note
however that the effects found here persist when tak-
ing account of the system-environment interaction in a
more detailed manner (see Methods). Dissipative pro-
cesses, which lead to energy loss, are then described by
the Lindblad super-operator

Ldiss(ρ) =

N
�

k=1
Γk[−{σ+
k σ−
k , ρ} + 2σ−
k ρσ+
k ],
(2)

while energy-conserving dephasing processes are de-
scribed by the operator

Ldeph(ρ) =

N
�

k=1
γk[−{σ+
k σ(−)
k
, ρ} + 2σ+
k σ−
k ρσ+
k σ−
k ]. (3)

Finally, in order to be able to measure the total transfer
of excitation, we designate an additional site, numbered
N + 1, which is populated by an irreversible decay pro-
cess from a chosen level k as described by the Lindblad
operator

Lsink(ρ) =
(4)
ΓN+1[−{σ+
k σ−
N+1σ+
N+1σ−
k , ρ} + 2σ+
N+1σ−
k ρσ+
k σ−
N+1].

The subindex ’sink’ emphasizes that no population can
escape of site N + 1. For definitiveness and simplicity,

the initial state of the network at t = 0 will be assumed
to be a single excitation in site 1 unless stated otherwise.
The key question that we will pose and answer is the
following: In a given time T , how much of the initial
population in site 1 will have been transferred to the sink
at site N + 1 and how is this transfer affected by the
presence of dephasing and dissipative noise.
In the remainder of this paper we will demonstrate
that, in certain settings, the presence of dephasing noise
can assist the transfer of population from site 1 to the
sink at site N + 1 considerably. It is an intriguing obser-
vation that this noise enhanced transfer does not occur
for all possible Hamiltonians of the type given by eq.(1)
and may depend also on properties of the noise such as
its spatial dependence. These noise rates can be opti-
mized numerically, and in very simple cases analytically,
to yield the strongest possible effect. One may suspect
that natural, biological systems, have actually made use
of such an optimization.
Linear chain – We begin with a brief analysis of the
uniform linear chain with only nearest neighbor inter-
actions so that in eq.
(1) the coupling strengths sat-
isfy vl,k = vk,l = vδl,k+1 for k = 1, . . . , N − 1 and
ωk = ω and Γk = Γ for k = 1, . . . , N. Extensive numer-
ical searches show that, for arbitrary choices of ΓN+1,
Γ and ω and arbitrary transmission times T and chains
of the length N = 2, . . . , 12, the optimal choice of de-
phasing noise rates vanish. We have used a directed ran-
dom walk algorithm with multiple initial states which
has never exceeded the values for the noise-free chain
and approached them to within at least 10−8. We were
able to derive formulae for the case T = ∞ and short
chains which demonstrate this behaviour analytically.
For N = 2, with ω1 = ω2 = ω and arbitrary v1,2, γi
and Γi, we find, with the abbreviation γ = γ1 + γ2 and
x = 2Γ3
1 + Γ1Γ3(3Γ1 + Γ3), that the population of the
sink is given by

psink =
Γ3v2
1,2

x + Γ1(Γ1 + Γ3)γ + (Γ3 + 2Γ1)v2
1,2
,
(5)

which is evidently maximized for γ = 0. One may also
obtain the analytical expressions for N = 3 and Γk = Γ
for k = 1, 2, 3 and demonstrate that the optimal dephas-
ing level is γ = 0 (see section on Methods).
This ap-
proach, though more tedious, may be taken to higher
values of N as well. Extensive numerical searches lend
further support to the observation that dephasing does
not improve excitation transfer for uniform chains but a
general proof has remained elusive.
So far, the findings are consistent with the expectation
that noise does not enhance the transport of excitations.
However, for non-uniform chains we encounter the dif-
ferent and perhaps surprising situation where noise can
significantly enhance the transfer rate of excitations.
As an illustrative example, we may keep the nearest
neighbor coupling uniform but allow for one site to have
a different site energy ω. If we chose N = 3, ω1 = ω3 = 1,
Γ1 = Γ2 = Γ3 = 1/100, v1,2 = v2,3 = 1/10, ΓN+1 = 1/5


3

0
0.5
1
1.5
2
0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

ω2

psink(γopt) − psink(γ = 0) 

FIG. 2: The optimal improvement of the transfer efficiency is
plotted versus the site frequency ω2 in a chain of length N = 3
and system parameters ω1 = ω3 = 1, Γ1 = Γ2 = Γ3 = 1/100,
v1,2 = v2,3 = 1/10, ΓN+1 = 1/5 and T = ∞. One observes
that dephasing only assists the transmission probability in
some frequency intervals.

and T = ∞ , then we obtain the results depicted in
Fig. 2. One observes that dephasing assists the trans-
mission only when site 2 is sufficiently detuned from the
neighboring sites. This example suggests a simple pic-
ture to explain the reason for the dephasing enhanced
population transfer through the chain. Site 2 is strongly
detuned from its neighboring sites and the coupling v to
its neighbors is comparatively weak, i.e. v ≪ δω with
δω = min[|ω2 − ω1|, |ω3 − ω2|].
Hence , the transport
rate is limited by a quantity of order v2/δω as it is a
second order process due to the lack of resonant modes
between neighboring sites. Introducing dephasing noise
leads to a broadening of the energy level at each site
k and a line-width proportional to the dephasing rate
γk.
Then, with increasing dephasing rate, the broad-
ened lines of neighboring sites begin to overlap and the
population transfer will be enhanced as resonant modes
are now available. Enhancing the dephasing rate further
will eventually lead to a weakening of the transfer as
the modes are distributed over a very large interval and
resonant modes have a small weight. Dissipation does
not lead to the same enhancement as, crucially, the gain
to the broadening of the line is overcompensated by the
irreversible loss of excitation.
This is corroborated by
numerical studies where increasing dissipation does not
assist the transport. The physical picture outlined above
is confirmed in Fig.
3.
We chose a chain of length 3
which suffers dephasing only in site 2 and uniform dis-
sipation with rates Γk = 1/100 along the chain while
ω1 = ω2/4 = ω3 = 1 and v1,2 = v2,3 = 1/10 (see fig.
3). The close relationship of this model to Raman tran-
sitions in quantum optics will be exploited to propose a
realizable experiment in a highly controlled environment
to verify these effects (see section on realizations).
In the examples above the improvement of excitation
transfer due to the dephasing is small. One can easily

0
2
4
6
8
10
0

0.005

0.01

0.015

γ2

psink(γ2) − psink(γ2=0)

FIG. 3: The difference between transfer efficiency and the
efficiency without dephasing is plotted versus the dephasing
rate γ2 in a chain of length N = 3 and ω1 = ω2/4 = ω3 = 1,
v1,2 = v2,3 = 1/10, γ1 = γ3 = 0, Γk = 1/100 for k = 1, . . . , N,
ΓN+1 = 1/5 and T = ∞. Initially increasing dephasing as-
sists the transfer of excitation while very strong dephasing
suppresses the transport.

show, however, that this improvement may be made ar-
bitrarily large in the sense that without noise the trans-
fer rate approaches zero while it approaches unity arbi-
trarily closely for optimal noise levels. As an example,
for N = 3, ω1 = ω3 = 1; ω2 = 100, v1,2 = v2,3 = v,
γ1 = γ3 = 0 and Γ1 = Γ2 = Γ3 = v2/f and Γ4 = 105v
we find for ∆p = psink(γ2,opt) − psink(γ2 = 0) that

lim
v→0 ∆p =
f 2γ2
2

f 2γ2
2 + 3fγ2((ω2 − 1)2 + γ2
2) + ((ω2 − 1)2 + γ2
2)2
(6)
This is maximized for γ2 = ω2 − 1 when it takes the
value ∆p = f 2/(f 2 + 6f(ω2 − 1) + 4(ω − 2 − 1)2). In the
limit f → ∞ this approaches 1, that is, without noise the
excitation transfer vanishes while with noise it achieves
unit efficiency! It should be noted that being a system of
fixed finite size, the effect may not be directly attributed
to Anderson localization [24] which, in addition does not
occur in systems attached to a sink, as is assumed here
[25].
Entanglement and coherence in the channel – We have
seen that the transport of excitations in the system may
be assisted considerably by local dephasing.
Now we
would like to discuss briefly the quantum coherence prop-
erties during transmission by studying the presence of
entanglement and the ability of the chain to transmit
quantum information. To this end, we consider how en-
tanglement is transported along the chain when it is used
to propagate one half of a maximally entangled state to
obtain an insight on how is the quantum capacity of this
channel affected by dephasing. To illustrate this, we con-
sider a chain of N = 4 sites. We chose the same param-
eters as in Fig. 2 and fix ω3 = 14. Comparison of the
entanglement between an uncoupled site and the various
sites in the chain for vanishing dephasing and the opti-
mal choice of the dephasing for excitation transfer show


4

that, while entanglement propagates through the system,
the amount of entanglement decreases with increasing
dephasing.
In fact, the dephasing rate that optimizes
the ability of the channel to transmit quantum informa-
tion vanishes, in contrast to the situation for excitation
transfer.
Therefore, although dephasing may enhance
the propagation of excitations, it also destroys quantum
coherence and in the present setting it leaves an overall
detrimental effect.
Complex networks and Light-harvesting molecules –
So far, we have demonstrated that in linear chains lo-
cal dephasing noise may enhance the transfer of excita-
tions. Going beyond this, we will now consider fully con-
nected networks and apply our observations to a model
that describes the transfer of excitons in the Fenna-
Matthews-Olson complex of Prosthecochloris aestuarii,
which is a pigment-protein complex that consists of seven
bacteriochlorophyll-a (BChla) molecules (see [20, 21, 22]
and Note Added for closely related work). This complex
is able to absorb light to create an exciton. This exci-
ton then propagates through the complex until it reaches
the reaction centre where its energy is then used to trig-
ger further processes that bind the energy in chemical
form [15, 23]. The Hamiltonian of this complex may be
approximated by eq.
(1), where the site energies and
coupling constants may be taken from table 2 and 4 of
[15]. We then find, in matrix form

H =












215 −104.1
5.1
−4.3
4.7 −15.1
−7.8
−104.1
220.0
32.6
7.1
5.4
8.3
0.8
5.1
32.6
0.0 −46.8
1.0
−8.1
5.1
−4.3
7.1 −46.8
125.0 −70.7 −14.7 −61.5
4.7
5.4
1.0 −70.7 450.0
89.7
−2.5
−15.1
8.3
−8.1 −14.7
89.7
330.0
32.7
−7.8
0.8
5.1 −61.5
−2.5
32.7
280.0












(7)
where
we
have
shifted
the
zero
of
energy
by
12230
(all
number
are
given
in
the
units
of
1.988865 · 10−23Nm
=
1.2414 10−4eV ) for all sites
corresponding to a wavelength of ∼= 800nm.
Recent
work [15] suggests that it is this site 3 that couples to
the reaction centre at site 8.
For this rate, somewhat
arbitrarily, we chose Γ3,8
=
10/1.88 corresponding
to about 1 ps−1 (value in the literature range from
0.25ps−1 [15] and 1 ps−1 [20] to 4 ps−1 [17]). Again, we
will assume the presence of both dissipative noise (loss
of excitons) and dephasing noise (due to the presence
of a phonon bath consisting of vibrational modes of the
molecule). The measured lifetime of excitons is of the
order of 1 ns which determines a dissipative decay rate
of 2Γk = 1/188 and that we assume to be the same for
each site [15].
If we neglect the presence of any form
of dephasing and we start with a single excitation on
site 1, then we observe that the excitation is transferred
to the reaction centre (site 8). For a time T = 5, we
find that the amount of excitation that is transferred
is psink = 0.551926.
Optimal dephasing rates that
maximize the transfer rate of the initial excitation in site
1 considerably improve on that. For T = 5 we find the

0
50
100
150
200
250
300
350
400
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

EN

 

 

0−1 No dephasing
0−2 No dephasing
0−3 No dephasing
0−4 No dephasing
0−1 Dephasing
0−2 Dephasing
0−3 Dephasing
0−4 Dephasing

FIG. 4: The time evolution of the entanglement between a
decoupled site and the sites in the chain of length N = 4
and system parameters ω1 = ω2 = ω4 = 10, ω = 14, v1,2 =
v2,3 = v3,4 = 1, Γk = 1/10 for k = 1, . . . , N and ΓN+1 =
1. The initial state is a maximally entangled state between
the decoupled site and the first site of the chain. Dephasing
destroys entanglement along the chain and has no beneficial
effect.

optimal
dephasing
rates
(γ1, γ2, γ3, γ4, γ5, γ6, γ7)
=
(469.34, 5.36, 99.13, 5.55, 114.86, 1.88, 291.08)
and
the much improved value psink
=
0.988526.
For
T
=
∞,
we
find
the
dephasing
free
transfer
probability of psink
=
0.81425 while for the op-
timal
dephasing
rates
(γ1, γ2, γ3, γ4, γ5, γ6, γ7)
=
(27.40, 26.84, 1.22, 87.12, 99.59, 232.76, 88.35)
we
find
psink = 0.99911. It should be noted that these dephasing
rates are comparable to the inter-site coupling rates
which suggests that a more accurate treatment will need
to go beyond master equations (see Methods for a brief
discussion).
We conclude that dephasing may lead to a very strong
enhancement of the transfer rate of excitations in a re-
alistic network. In fact, in models obtained from spec-
troscopic data measured on the FMO complex it is in-
deed observed that almost complete transport should
take place within time T = 5 [15]. It is remarkable that
such a rapid transfer cannot be explained from a purely
coherent dynamics and, as shown above, the underlying
reason for the speed up is the presence of dephasing which
may even be local.
Experimental Realizations – The FMO-complex pro-
vides a fascinating setting for the observation of dephas-
ing enhanced transport but it is also a very challeng-
ing environment to verify the effect precisely. Here we
present several physical systems in which the dephasing
enhanced excitation transfer may be observed and which
are at the same time highly controllable. Perhaps the
simplest such setting is found in atomic physics (see Fig.
5) where the behaviour of a chain of three sites may be
simulated using detuned Raman transitions in ions such
as Ca+, Sr+ or Ba+. The master equation of this system
simulates exactly that of a chain with a single excitation
as has been described throughout this paper.
Atomic


5

2

�
�

r

0

�S
�N �1
1

3

2

�
�

r

0

�S
�N �1
1

3

�
�

r

0

�S
�N �1
1

3

FIG. 5: A atomic system with Raman transitions provides a
transparent illustration of dephasing assisted transport. The
required level structure may be realized in Ca+, Sr+ or Ba+.
Each atomic level represents a site in the chain which may
be populated.
Starting with all the population in level 1,
one may then irradiate the system with classical laser fields
of Rabi-frequency Ω on the 1 ↔ 2 and the 3 ↔ 2 transition
[27]. Level 3 in turn is assumed to decay spontaneously into
an additional level |r⟩ that plays the role of the recipient.
Spontaneous decay of the chain as a whole is modelled by
spontaneous decay into level |0⟩ from which no population
can enter the levels |1⟩, |2⟩, |3⟩ and |r⟩ anymore. Dephasing
noise may now enter the system affecting level 2 for example
through magnetic field fluctuations.

populations may be measured with very high accuracy
using quantum jump detection [28, 29].
A variety of other natural implementations of dephas-
ing assisted excitation transport can be conceived and
will be studied in detail elsewhere. Firstly, the oscilla-
tions of ions in a linear ion trap transversal to the trap
axis realizes a harmonic chain [30] that allows for the
implementation of a variety of operations such as prepa-
ration of Fock states and is capable of supporting near-
est neighbor coupling between neighbouring ion oscilla-
tors [31] and allowing high efficiency readout by quantum
jump detection [28]. When restricting to the single exci-
tation space, the dynamics of the system is described by
master equations that become equivalent to those pre-
sented in this paper.
Furthermore, harmonic chains are also realized in cou-
pled arrays of cavities which have recently received con-
siderable attention in the context of quantum simulators
[32]. Ultra-cold atoms in optical lattices which have pre-
viously been used to study thermal assisted transport in
Brownian ratchets [33] presents another scenario in which
to study such dephasing assisted processes. Chains of su-
perconducting qubits or superconducting stripline cavi-
ties [34] may also provide a possible setting for the ob-
servation of the effects described above.
Conclusions and outlook – The results presented here
demonstrate that while dephasing noise destroys quan-
tum correlations, it may at the same time enhance the
transport of excitations. In fact, the efficient transport
observed in certain biological systems has been shown
to be incompatible with a fully coherent evolution while

it can be explained if the system is subject to local de-
phasing. Hence, in this context, the presence of quan-
tum coherence and therefore, entanglement in the sys-
tem, does not seem to be supporting excitation transfer.
This suggests that entanglement that may be present in
bio-molecules, though interesting, may not be a universal
functional resource.
Importantly, the results presented here suggest that it
may be possible to design and optimize the performance
of nano-fabricated transmission lines in naturally noisy
environments to achieve strongly enhanced transfer ef-
ficiencies employing the concept of noise assisted trans-
port.
Acknowledgements– We are grateful to Seth Lloyd
for helpful communications concerning [20, 21, 22], Neil
Oxtoby, Angel Rivas and Shashank Virmani for useful
comments on the manuscript and to Danny Segal for
advice on atomic physics.
This work was supported
by the EU via the Integrated Project QAP (‘Qubit
Applications’) and the STREP action CORNER and the
EPSRC through the QIP-IRC. MBP holds a Wolfson
Research Merit Award.

Note Added— While finalizing this work, we became
aware of independently obtained but closely related re-
sults presented in [20, 21, 22]. There it was showed that
quantum transport can be enhanced by an interplay be-
tween coherent dynamics and environment effects with
particular emphasis on excitonic energy transfer in light
harvesting complexes [20]. The role of the different phys-
ical processes that contribute to the energy transfer ef-
ficiency have been studied in [21] and the enhancement
of quantum transport due to a pure dephasing environ-
ment within the Haaken-Strobl model was demonstrated
in [22].


6

Methods –

Exact solutions for uniform chains – One may also ob-
tain the analytical expressions for a chain of length N = 3

described by eqs. (1) - (4) for the choice and Γk = Γ for
k = 1, 2, 3, 4 and demonstrate that the optimal dephasing
level is γ = 0. We find

psink =
(4Γ + γ1 + γ3)v2

36Γ5 + 6aΓ4 + 2Γ3(3γ2
1 + 3γ2
2 + 8b + 2γ2
3 + 32v2) + Γ2(2c + dv2) + Γv2(3γ2
1 + 7b + 4γ2
3 + 15v2) + 4(γ1 + γ3)v4

where a = (5γ1 + 5γ2 + 4γ3), b = γ1γ2 + γ1γ3 + γ2γ3,
c = γ1(γ2
2 + γ2
3) + γ2(γ2
1 + γ2
3) + γ3(γ2
1 + γ2
2) + 2γ1γ2γ3,
d = 32γ3 + 25γ2 + 29γ1. Then one first observes that
the optimal choice is γ2 = 0 as it only occurs in the
denominator with positive coefficients. In the remaining
expression one then substitutes γk = ˜γ2
k allowing also for
negative ˜γk. Then differentiation w.r.t these ˜γk shows
that the gradient only vanishes for ˜γ1 = ˜γ2 = 0.
Beyond Markovian master equations– So far we have
demonstrated the existence of dephasing enhanced exci-
tation transfer employing a master equation description.
The optimized dephasing rates that have been obtained,
in particular those in the context of the FMO complex,
can be comparable to the coherent interaction strengths
and may be similar to the spectral width of the bath re-
sponsible for the dephasing [15]. This may not be fully
compatible with the master equation approach employed
so far as its derivation relies on several assumptions in-
cluding the weak coupling hypothesis and the require-
ment for the bath to be Markovian [26]. The derivation is
further complicated for systems with several constituents
where the local coupling of its constituents is not com-
patible with non-local structure of the eigenmodes of the
systems. This is especially so when the coherent inter
sub-system coupling is of comparable strength to the sys-
tem environment coupling. The situation is made more
difficult due to spatial as well as temporal correlations in
the environmental noise (which is to be expected in par-
ticular for the FMO complex but also many other realisa-
tions of coupled chains in contact with an environment).
Bloch-Redfield equations and other effective description
are sometimes used but still represent approximations to
the correct dynamics [26] where the errors are often dif-
ficult to estimate precisely.
Therefore, we demonstrate briefly that dephasing as-
sisted transfer of excitation can also be observed when
one uses a microscopic model of an environment that
may, in addition, exhibit non-Markovian behaviour. To
this end we study the effect of an environment which is
modelled by brief interactions between two-level systems
and individual subsystem of the chain in which excita-

tion transport is taking place. The strength and nature
of the interactions can be chosen to implement dephasing
(elastic collisions) and dissipation (in-elastic collisions).
Non-markovian effects can be included in the model de-
pending on the spatial and temporal memory of the envi-
ronment particles. Interaction strengths are determined
for a single site system to obtain the dissipation rate Γ
and dephasing rate γ. This simplified model allows us
to study the effect of more realistic environments outside
the master equation picture and results are summarized
in Figure 6.
A more detailed simulation of excitation
transfer taking account of the full environment are be-
yond the scope of the present work and will be presented
elsewhere [35]

0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

psink

 

 

γ=0
γ=0.0064γopt
γ = 0.16γopt
γ = γopt

FIG. 6: Here we show how the transfer in the presence of
dephasing into a bath that is modelled by a collisional model
where local sites briefly interact with a single particle. The in-
teraction strength is chosen such that in an uncoupled systems
the sites suffer the optimal decoherence rates γopt as presented
in the previous section multiplied with factors 0, 0.0064, 0.16
and 1. The dynamics is similar to that observed for the mas-
ter equation approach and shows only minor deviations. In-
creased dephasing rates do improve the excitation transfer
also in this model.

[1] A. A. Faisal, L. P. J. Selen and D. M. Wolpert, Nature
Reviews on Neuroscience 9, 292 (2008).
[2] R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14,


7

L453 (1981).
[3] L. Gammaitoni, P. H¨anggi, P. Jung, and F. Marchesoni,
Rev. Mod. Phys. 70, 223 (1998).
[4] J. K Douglass, L. Wilkens, E. Pantazelou and F. Moss,
Nature 365, 337 (1993); K. Wiesenfeld and F. Moss, Na-
ture 373, 33 (1995).
[5] Y. H. Shang, A. Claridge-Chang, L. Sjulson, M. Pypaert
and G. Miesenbock, Cell 128, 601 (2007).
[6] P. Reimann, M. Grifoni, and P. H¨anggi, Phys. Rev. Lett.
79, 10 (1997)
[7] Special issue on Ratchets and Brownian Motors, edited
by H. Linke, Appl. Phys. A 75, 167 (2002)
[8] P. H¨anggi, F. Marchesoni and F. Nori, Ann. Phys.
(Berlin) 14, 51 (2005).
[9] L. Viola, E. M. Fortunato, S. Lloyd, C. H. Tseng, and
D. G. Cory, Phys. Rev. Lett. 84, 5466 (2000).
[10] M. B. Plenio and S. F. Huelga, Phys. Rev. Lett. 88,
197901 (2002).
[11] S. F. Huelga and M. B. Plenio, Phys. Rev. Lett. 98,
170601 (2007).
[12] J.-L. Ting, Phys. Rev. E 59, 2801 (1998)
[13] G. Bowen and S. Mancini, Phys. Lett. A 321, 1 (2004)
´ıbid 352, 272 (2006).
[14] C. Di Franco, M. Paternostro, D. I. Tsomokos and S. F.
Huelga, Phys. Rev. A 77, 062337 (2008).
[15] J. Adolphs and T. Renger, Biophys. J. 91, 2778 (2006)
[16] See G. S. Engel, T. R. Calhoun, E. L. Read EL, T. K.
Ahn, T. Mancal, Y. C. Cheng, R. E. Blankenship and
G. R. Fleming, Nature 446, 782 (2007) for recent exper-
imental results.
[17] A. Olaya-Castro, C. F. Lee, F. Fassioli-Olsen, and N. F.
Johnson, arXiv:0708.1159.
[18] H. J. Briegel and S. Popescu, arXiv:0806.4552.
[19] K. M. Gaab and C. J. Bardeen, J. Chem. Phys. 121,
7813 (2004).
[20] M. Mohseni, P. Rebentrost, S. Lloyd and A. Aspuru-

Guzik, arXiv:0805.2741.
[21] P. Rebentrost,
M.
Mohseni and
A. Aspuru-Guzik,
arXiv:0806.4725.
[22] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd, A.
Aspuru-Guzik, arXiv:0807.0929
[23] R. E. Fenna and B. W. Matthews, Nature 258, 573
(1975).
[24] P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[25] S. A. Gurvitz, Phys. Rev. Lett. 85, 812 (2000).
[26] H.-P. Breuer and F. Petruccione, The Theory of Open
Quantum Systems, Oxford University Press, 2002.
[27] M. O. Scully and S. Zubairy, Quantum Optics, Cam-
bridge University Press.
[28] M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101
(1998).
[29] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev.
Mod. Phys. 75, 281 (2003).
[30] M. B. Plenio, J. Hartley and J. Eisert, New J. Phys. 6,
36 (2004)
[31] A. Serafini, A. Retzker and M. B. Plenio,
E-print
arXiv:0708.0851 [quant-ph]
[32] M. J. Hartmann, F. G .S .L. Brand˜ao, M. B. Plenio,
Nature Phys. 2, 849 (2006); A. D. Greentree et al, Nature
Phys. 2, 856 (2006); D. G. Angelakis, M. F. Santos and
S. Bose, Phys. Rev. A 76, 031805(R) (2007).
[33] C. Mennerat-Robilliard, D. Lucas, S. Guibal, J. Tabosa,
C. Jurczak, J.-Y. Courtois, and G. Grynberg, Phys. Rev.
Lett. 82, 851 (1999).
[34] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R.
Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A.
Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M.
Girvin and R. J. Schoelkopf, Nature 449, 443 (2007);
A. Palacios-Laloy, F. Nguyen, F. Mallet, P. Bertet, D.
Vion and D. Esteve, arXiv:0712.0221.
[35] Work in progress.