text: 6 origin node m node m taxi i a m,m0 taxi i destination node m0 ni(t) = m ni (t) = 0 ni(t) = m ni (t) = a p p m,m0 (a) (b) FIG. 2. Speciﬁcation of the position of the taxi in the road network. Note that we update the states of all the agents in our simulation at every time stamp, and all the weights of the edges are given as an integer number indicating the number of the time stamps for a vehicle to travel from one node to another. taxis. When the taxi demand is low, taxi-booking is more eﬃcient for available taxis to reach waiting commuters, as compared to road-side pickups (with pseudo-random movement of empty taxis); with high taxi demand, the reverse is true, and we discuss this in more details in Sec. . The dynamics of the taxi system with taxi-bookiing is also quite diﬀerent from that with road-side hailing, as we will see in Sec. . More realistic simulations with a mixture of taxi-booking and road-side hailing will be studied elsewhere. D. A minimalist simulation approach While our simulation platform is capable of simulating complex taxi and commuter behaviours as well as diﬀerent routing/pickup/sharing algorithm, here we pursue a minimalist approach and employ the simplest possible simulation algorithms that still retain the essence of taxi dynamics. We are not aiming to quantitatively reproduce the features of the taxis in real cities. This requires a lot of empirical information (e.g. actual routing of the taxis, detailed spatiotemporal patterns of demand and commuter loading time, as well as the time dependence of the actual number of taxis running on the streets, etc.), even though given such information they can be easily implemented in our codes for more quantitative predictions. Instead, we aim to capture some of the universal trends of taxi dynamics in a qualitative way and shed light on the intricate interplay between supply, demand and road networks, from the commuters’, taxis drivers’ and policy-makers’ perspective. Following the principle of Occam’s razor, it is also useful to show that such universal trends can be captured by the minimalist model (with artiﬁcial road networks), as we compare numerical results in Sec. and Sec. . Unless otherwise stated, in our simulation each travelling party consists of only one commuter, and a taxi can take two travelling parties at most. Each taxi always moves towards its next destination via the shortest path as given by the Dijkstra’s algorithm, so that R always ﬁnds the shortest path between the original node and the ﬁnal node. o Once the taxi reaches its ﬁnal destination (after dropping oﬀ all the commuters in the taxi), instead of modelling its behaviour we make it either roam randomly or remain at the ﬁnal node until it is booked by another commuter. We can also make the empty taxis move towards some predeﬁned “hotspots”. We have done extensive (though not necessarily thorough) simuations, and it seems diﬀerent models only aﬀect the results in the following sections in minor and quantitative ways. The universal features of taxis and TRS discussed in this paper are not sensitive to this part of the taxi dynamics. Thus (cid:126)ti is empty if si (t) = 0, and R corresponds to a random walk. The algorithm e we use for assigning taxis to commuters is also simple: when a commuter calls a taxi from a certain node, we search through all the positions of available taxis at that time, and assign the taxi closest to the commuter (based on R , or o the shortest path) for the pick-up. Many route matching algorithms for TRS have been proposed in the literature[1, 2, 7, 23–28]. While this paper does not focus on proposing a new route matching algorithm or analysing the eﬃciency of various algorithms, we describe )0m ,m(¯A ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: origin

text: node

text: m

text: node

text: m

text: taxi

text: i

text: a

text: taxi

text: i

text: destination

text: node

text: ni(t)

text: m

text: ni

text: (t)

text: ni(t)

text: m

text: ni

text: (t)

text: a

text: p

text: p

text: (a)

text: (b)

text: FIG.

text: Speciﬁcation

text: of

text: the

text: position

text: of

text: the

text: taxi

text: in

text: the

text: road

text: network.

text: Note

text: that

text: we

text: update

text: the

text: states

text: of

text: all

text: the

text: agents

text: in

text: our

text: simulation

text: at

text: every

text: time

text: stamp,

text: and

text: all

text: the

text: weights

text: of

text: the

text: edges

text: are

text: given

text: as

text: an

text: integer

text: number

text: indicating

text: the

text: number

text: of

text: the

text: time

text: stamps

text: for

text: a

text: vehicle

text: to

text: travel

text: from

text: one

text: node

text: to

text: another.

text: taxis.

text: When

text: the

text: taxi

text: demand

text: is

text: low,

text: taxi-booking

text: is

text: more

text: eﬃcient

text: for

text: available

text: taxis

text: to

text: reach

text: waiting

text: commuters,

text: as

text: compared

text: to

text: road-side

text: pickups

text: (with

text: pseudo-random

text: movement

text: of

text: empty

text: taxis);

text: with

text: high

text: taxi

text: demand,

text: the

text: reverse

text: is

text: true,

text: and

text: we

text: discuss

text: this

text: in

text: more

text: details

text: in

text: Sec.

text: The

text: dynamics

text: of

text: the

text: taxi

text: system

text: with

text: taxi-bookiing

text: is

text: also

text: quite

text: diﬀerent

text: from

text: that

text: with

text: road-side

text: hailing,

text: as

text: we

text: will

text: see

text: in

text: Sec.

text: More

text: realistic

text: simulations

text: with

text: a

text: mixture

text: of

text: taxi-booking

text: and

text: road-side

text: hailing

text: will

text: be

text: studied

text: elsewhere.

text: D.

text: A

text: minimalist

text: simulation

text: approach

text: While

text: our

text: simulation

text: platform

text: is

text: capable

text: of

text: simulating

text: complex

text: taxi

text: and

text: commuter

text: behaviours

text: as

text: well

text: as

text: diﬀerent

text: routing/pickup/sharing

text: algorithm,

text: here

text: we

text: pursue

text: a

text: minimalist

text: approach

text: and

text: employ

text: the

text: simplest

text: possible

text: simulation

text: algorithms

text: that

text: still

text: retain

text: the

text: essence

text: of

text: taxi

text: dynamics.

text: We

text: are

text: not

text: aiming

text: to

text: quantitatively

text: reproduce

text: the

text: features

text: of

text: the

text: taxis

text: in

text: real

text: cities.

text: This

text: requires

text: a

text: lot

text: of

text: empirical

text: information

text: (e.g.

text: actual

text: routing

text: of

text: the

text: taxis,

text: detailed

text: spatiotemporal

text: patterns

text: of

text: demand

text: and

text: commuter

text: loading

text: time,

text: as

text: well

text: as

text: the

text: time

text: dependence

text: of

text: the

text: actual

text: number

text: of

text: taxis

text: running

text: on

text: the

text: streets,

text: etc.),

text: even

text: though

text: given

text: such

text: information

text: they

text: can

text: be

text: easily

text: implemented

text: in

text: our

text: codes

text: for

text: more

text: quantitative

text: predictions.

text: Instead,

text: we

text: aim

text: to

text: capture

text: some

text: of

text: the

text: universal

text: trends

text: of

text: taxi

text: dynamics

text: in

text: a

text: qualitative

text: way

text: and

text: shed

text: light

text: on

text: the

text: intricate

text: interplay

text: between

text: supply,

text: demand

text: and

text: road

text: networks,

text: from

text: the

text: commuters’,

text: taxis

text: drivers’

text: and

text: policy-makers’

text: perspective.

text: Following

text: the

text: principle

text: of

text: Occam’s

text: razor,

text: it

text: is

text: also

text: useful

text: to

text: show

text: that

text: such

text: universal

text: trends

text: can

text: be

text: captured

text: by

text: the

text: minimalist

text: model

text: (with

text: artiﬁcial

text: road

text: networks),

text: as

text: we

text: compare

text: numerical

text: results

text: in

text: Sec.

text: and

text: Sec.

text: Unless

text: otherwise

text: stated,

text: in

text: our

text: simulation

text: each

text: travelling

text: party

text: consists

text: of

text: only

text: one

text: commuter,

text: and

text: a

text: taxi

text: can

text: take

text: two

text: travelling

text: parties

text: at

text: most.

text: Each

text: taxi

text: always

text: moves

text: towards

text: its

text: next

text: destination

text: via

text: the

text: shortest

text: path

text: as

text: given

text: by

text: the

text: Dijkstra’s

text: algorithm,

text: so

text: that

text: R

text: always

text: ﬁnds

text: the

text: shortest

text: path

text: between

text: the

text: original

text: node

text: and

text: the

text: ﬁnal

text: node.

text: o

text: Once

text: the

text: taxi

text: reaches

text: its

text: ﬁnal

text: destination

text: (after

text: dropping

text: oﬀ

text: all

text: the

text: commuters

text: in

text: the

text: taxi),

text: instead

text: of

text: modelling

text: its

text: behaviour

text: we

text: make

text: it

text: either

text: roam

text: randomly

text: or

text: remain

text: at

text: the

text: ﬁnal

text: node

text: until

text: it

text: is

text: booked

text: by

text: another

text: commuter.

text: We

text: can

text: also

text: make

text: the

text: empty

text: taxis

text: move

text: towards

text: some

text: predeﬁned

text: “hotspots”.

text: We

text: have

text: done

text: extensive

text: (though

text: not

text: necessarily

text: thorough)

text: simuations,

text: and

text: it

text: seems

text: diﬀerent

text: models

text: only

text: aﬀect

text: the

text: results

text: in

text: the

text: following

text: sections

text: in

text: minor

text: and

text: quantitative

text: ways.

text: The

text: universal

text: features

text: of

text: taxis

text: and

text: TRS

text: discussed

text: in

text: this

text: paper

text: are

text: not

text: sensitive

text: to

text: this

text: part

text: of

text: the

text: taxi

text: dynamics.

text: Thus

text: is

text: empty

text: if

text: si

text: (t)

text: and

text: R

text: corresponds

text: to

text: a

text: random

text: walk.

text: The

text: algorithm

text: e

text: we

text: use

text: for

text: assigning

text: taxis

text: to

text: commuters

text: is

text: also

text: simple:

text: when

text: a

text: commuter

text: calls

text: a

text: taxi

text: from

text: a

text: certain

text: node,

text: we

text: search

text: through

text: all

text: the

text: positions

text: of

text: available

text: taxis

text: at

text: that

text: time,

text: and

text: assign

text: the

text: taxi

text: closest

text: to

text: the

text: commuter

text: (based

text: on

text: R

text: or

text: o

text: the

text: shortest

text: path)

text: for

text: the

text: pick-up.

text: Many

text: route

text: matching

text: algorithms

text: for

text: TRS

text: have

text: been

text: proposed

text: in

text: the

text: While

text: this

text: paper

text: does

text: not

text: focus

text: on

text: proposing

text: a

text: new

text: route

text: matching

text: algorithm

text: or

text: analysing

text: the

text: eﬃciency

text: of

text: various

text: algorithms,

text: we

text: describe

text: ,m(¯A

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Let n be a nonnegative integer, and let (cid:96) denote the side length of the square at the n n-th iteration. Then, (cid:96) = 2(cid:96) + (cid:96) . (1) n n+1 n+2 The scaling factor constraint implies (cid:96) (cid:96) n+1 n+2 k = = . (2) (cid:96) (cid:96) n n+1 √ − Combining formulas (1) and (2), we obtain an irrational ratio k = 2 1. This ratio suggests that the Jerusalem square cannot be built from a simple integer grid [1, 2]. × However, a naive method is to consider a 5 5 square, and then remove a cross which consists of ﬁve unit squares as shown in Figure 2. Figure 2: An integer grid approximation to the ﬁrst iteration of the Jerusalem square. We shall see in the next section that it is in fact a nice approximation of the actual fractal for the corresponding iteration. Indeed, the present paper is motivated by formulas (1), (2) and the observation of Figure 2. On the one hand, notice the similarities between the recurrence relation (1) and the deﬁnition of the Pell numbers A000129, and on the other hand, notice that the side lengths of the squares (of rank +1 and +2, as well as the original) in Figure 2 are exactly 1, 2 and 5, some of the ﬁrst few terms of the Pell numbers. 2 Pell numbers in action Firstly, recall that the Pell numbers are deﬁned as the sequence of integers ≥ p = 0, p = 1, and p = 2p − + p − for all n 2. (3) 0 1 n n 1 n 2 The ﬁrst few Pell numbers are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, . . . (sequence A000129 in the OEIS [5]). √ p n It is well-known that the integer ratio rapidly approach 1 + 2 [4, p. 138]. p − n 1 2 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Let

label: paragraph

text: n

text: be

text: a

text: nonnegative

text: integer,

text: and

text: let

text: denote

text: the

text: side

text: length

text: of

text: the

text: square

text: at

text: the

text: n

text: n-th

text: iteration.

text: Then,

label: equation

text: n

text: The

text: scaling

text: factor

text: constraint

text: implies

text: k

text: n

text: Combining

text: formulas

text: and

text: we

text: obtain

text: an

text: irrational

text: ratio

text: k

text: This

text: ratio

text: suggests

text: that

text: the

text: Jerusalem

text: square

text: cannot

text: be

text: built

text: from

text: a

text: simple

text: integer

text: grid

text: However,

text: a

text: naive

text: method

text: is

text: to

text: consider

text: a

text: square,

text: and

text: then

text: remove

text: a

text: cross

text: which

text: consists

text: of

text: ﬁve

text: unit

text: squares

text: as

text: shown

text: in

text: Figure

text: Figure

text: An

text: integer

text: grid

text: approximation

text: to

text: the

text: ﬁrst

text: iteration

text: of

text: the

text: Jerusalem

text: square.

text: We

text: shall

text: see

text: in

text: the

text: next

text: section

text: that

text: it

text: is

text: in

text: fact

text: a

text: nice

text: approximation

text: of

text: the

text: actual

text: fractal

text: for

text: the

text: corresponding

text: iteration.

text: Indeed,

text: the

text: present

text: paper

text: is

text: motivated

text: by

text: formulas

text: and

text: the

text: observation

text: of

text: Figure

text: On

text: the

text: one

text: hand,

text: notice

text: the

text: similarities

text: between

text: the

text: recurrence

text: relation

text: and

text: the

text: deﬁnition

text: of

text: the

text: Pell

text: numbers

text: and

text: on

text: the

text: other

text: hand,

text: notice

text: that

text: the

text: side

text: lengths

text: of

text: the

text: squares

text: (of

text: rank

text: and

text: as

text: well

text: as

text: the

text: original)

text: in

text: Figure

text: are

text: exactly

text: and

text: some

text: of

text: the

text: ﬁrst

text: few

text: terms

text: of

text: the

text: Pell

text: numbers.

text: Pell

text: numbers

text: in

text: action

text: Firstly,

text: recall

text: that

text: the

text: Pell

text: numbers

text: are

text: deﬁned

text: as

text: the

text: sequence

text: of

text: integers

text: p

text: p

text: and

text: p

text: p

text: for

text: all

text: n

text: n

text: n

text: n

text: The

text: ﬁrst

text: few

text: Pell

text: numbers

text: are

text: (sequence

text: in

text: the

text: OEIS

text: p

text: n

text: It

text: is

text: well-known

text: that

text: the

text: integer

text: ratio

text: rapidly

text: approach

text: p.

text: p

text: n

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: u0 x0 u0 h k/2 D x0 witc u0 x0 s u1 x1 u1 x1 u1 k/2 D x1 u2 x2 XP SK u3 x3 u4 x4 u0 x0 u0 A x0 u5 x5 u1 x1 u1 A x1 k u6 x6 uu32 xx32 u...k A... x...k u7 x7 P8 P4 IK A Fig. 1. The data-ﬂow graph of an 8-bit polar encoder. Fig. 3. The symbols and corresponding hardware modules in the formula. be easily denoted by a general formula F(N, M). Then we achieves: x0 = u0 + u1 (in GF(2)) and x1 = u1. The number show how to use an algorithm to derive a speciﬁc formula of inputs of XP is ﬁxed and equals to 2 in our design. fN,M from F(N, M) based on the values of N and M. Symbol SK (K is a power of 2, K > 1) represents a switch Finally, a compiler is employed to translate f into RTL with k/2 delay elements (denoted by D) on each side. A N,M description. The hardware generation system is illustrated in log K-bit counter is needed to control the switch: the value 2 Fig. 2. 0 of the most signiﬁcant bit of the counter infers direct data transfer, and the value 1 infers cross data transfer. The number of inputs of S is ﬁxed and equals to 2. K Symbol P (N is a power of 2, N > 2) denotes the General Framework N permutation on an N-dimensional vector. The detail function Code Length n of P is illustrated in Algorithm 1. Intuitively, P is the N N duplication of P . For example, P could be viewed as N/2 8 Parallelism m Formula: F (N,M) partial overlap of two P modules with red wires and black 4 wires respectively. ⊗ Algorithm Symbol (I A) (K is a power of 2, K > 0) is a Kronecker K Formula: fn,m product representing K parallel instances of module A, where A is an abstract module and A could be replaced by XP, S Compiler ⊗ K or P . Note that when K = 1, (I A) equals to A. Suppose N K ⊗ RTL verilog tha×t A has X inputs, the number of inputs of (IK A) equals K X. The general formula F(N, M) is composed of symbols Fig. 2. The hardware generation system for polar encoder. mentioned above, except that the W in Eq. (2) is a variable module. When deriving f from F(N, M), symbol W N,M A. From General Framework to Formula should be replaced by P or S according to its subscript. In N K Algorithm 2, as the code length and the level of parallelism Consider that the general framework is expected to imple- ment polar encoder with arbitrary code length and arbitrary are given, all the subscripts of each symbol in F(N, M) will ⊗ ⊗ level of parallelism, the framework should be scalable, i.e., be ﬁgure out. Then the module (I W) is replaced by (I P) ⊗ the number of stages and the number of hardware modules in or (I S) based on the value of the subscript of W. Finally, each stage should change with the values of N and M. Such the formula fN,M is determined. B. Compiler a scalable framework could represented by formula F(N, M) shown in Eq. (2). Here the parameters N and M are powers of We have built a compiler in Python that takes f as input (cid:54) (cid:54) N,M 2, and 4 M N/2. Before we go into details of F(N, M), and automatically connects all the basic modules in f in N,M we introduce all the symbols that might be used in F(N, M) left-to-right order. Speciﬁcally, as we input N and M into and f , as well as the symbols’ corresponding hardware F(N, M), the f will be determined and transformed into N,M N,M modules. Note that the ﬁnal hardware implementation of f the register-transfer level (RTL) Verilog by the compiler. The N,M is the serial connection of the individual modules of different detail of the compiler is beyond the scope of this paper; we symbols. Fig. 3 illustrates all the exemplary modules, as well only provide a brief introduction here. as symbols, that might be used in our design, all of which take There are totally three types of basic modules in the formula − − uN 1 as input and xN 1 as output. f : the XOR-and-PASS module XP, the switch module S , 0 0 N,M K Symbol XP represents an XOR-and-PASS module that and the permutation module P . There are two ways to expand N ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: h

text: D

text: witc

text: s

text: D

text: XP

text: SK

text: A

text: A

text: k

text: u...k

text: A...

text: x...k

text: IK

text: A

text: Fig.

text: The

text: data-ﬂow

text: graph

text: of

text: an

text: polar

text: encoder.

text: Fig.

text: The

text: symbols

text: and

text: corresponding

text: hardware

text: modules

text: in

text: the

text: formula.

text: be

text: easily

text: denoted

text: by

text: a

text: general

text: formula

text: F(N,

text: M).

text: Then

text: we

text: achieves:

text: (in

text: and

text: The

text: number

text: show

text: how

text: to

text: use

text: an

text: algorithm

text: to

text: derive

text: a

text: speciﬁc

text: formula

text: of

text: inputs

text: of

text: XP

text: is

text: ﬁxed

text: and

text: equals

text: to

text: in

text: our

text: design.

text: fN,M

text: from

text: F(N,

text: M)

text: based

text: on

text: the

text: values

text: of

text: N

text: and

text: M.

text: Symbol

text: SK

text: (K

text: is

text: a

text: power

text: of

text: K

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text: a

text: switch

text: Finally,

text: a

text: compiler

text: is

text: employed

text: to

text: translate

text: f

text: into

text: RTL

text: with

text: delay

text: elements

text: (denoted

text: by

text: D)

text: on

text: each

text: side.

text: A

text: N,M

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text: The

text: hardware

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text: illustrated

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text: log

text: K-bit

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text: needed

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text: the

text: switch:

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text: value

text: Fig.

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text: 6 Zeeman eﬀects in monolayer WS and MoS to 65 Tesla, Spin-Polarized Electronic Band Structure in Monolayer 2 2 Nat. Commun. 7, 10643 (2016). Transition Metal Dichalcogenides by Optical Spectroscopy, [31] D. W. Kidd, D. K. Zhang, and K. Varga, Binding energies Nano Lett. 17, 740 (2017). and structures of two-dimensional excitonic complexes in [47] A. M. Jones, H. Yu, J. Schaibley, J. Yan, D. G. Man- transition metal dichalcogenides, Phys. Rev. B 93, 125423 drus, T. Taniguchi, K. Watanabe, H. Dery, W. Yao, and (2016). X. Xu, Excitonic Luminescence Upconversion in a Two- [32] R. Ya. Kezerashvili and S. M.Tsiklauri, Trion and Biexci- Dimensional Semiconductor, Nat. Phys. 12, 323 (2016). ton in Monolayer Transition Metal Dichalcogenides, Few- [48] J. S. Ross, S. Wu, H. Yu, N. J. Ghimire, A. M. Jones, G. Body Syst. 58,18 (2017). Aivazian, J. Yan, D. G. Mandrus, D. Xiao, W. Yao, and X. [33] E. Mostaani, M. Szyniszewski, C. H. Price, R. Maezono, Xu, Electrical control of neutral and charged excitons in a M. Danovich, R. J. Hunt, N. D. Drummond, and V. I. monolayer semiconductor, Nat. Commun. 4, 1474 (2013). Fal’ko, Diﬀusion quantum Monte Carlo study of excitonic [49] G. D. Shepard, J. V. Ardelean, D. A. Rhodes, X.-Y. Zhu, complexes in two-dimensional transition-metal dichalco- J. C. Hone, and S. Strauf, Trion Species-Resolved Quan- genides, Phys. Rev. B 96, 075431 (2017). tum Beats in MoSe , arXiv:1709.03233. 2 [34] M. Szyniszewski, E. Mostaani, N. D. Drummond, and [50] D. Van Tuan, B. Scharf, I. Zˇuti´c, and H. Dery, Marry- V. I. Fal’Ko, Binding energies of trions and biexcitons in ing excitons and plasmons in monolayer transition-metal two-dimensional semiconductors from diﬀusion quantum dichalcogenides, Phys. Rev. X. 7, 041040 (2017). Monte Carlo calculations, Phys. Rev. B 95, 716 (2017). [51] G. Plechinger, P. Nagler, A. Arora, R. Schmidt, A. [35] E. Courtade, M. Semina, M. Manca, M. M. Glazov, C. Chernikov, A. Granados del A´guila, P. C.M. Christianen, Robert, F. Cadiz, G. Wang, T. Taniguchi, K. Watanabe, R. Bratschitsch, C. Schu¨ller, and T. Korn, Trion Fine M. Pierre, W. Escoﬃer, E. L. Ivchenko, P. Renucci, X. Structure and Coupled Spin-Valley Dynamics in Mono- Marie, T. Amand, and B. Urbaszek, Charged excitons in layer Tungsten Disulﬁde, Nat. Commun. 7, 12715 (2016). monolayer WSe2: experiment and theory, Phys. Rev. B [52] The spin-splitting in the valence band, ∆v, is compara- 96, 085302 (2017). ble to ∆ in molybdenum-based MLs, while being much 12 [36] A. V. Stier, N. P. Wilson, K. A. Velizhanin, J. Kono, larger than ∆ in tungsten-based MLs. Therefore, it is 12 X. Xu, and S. A. Crooker, Magneto-Optics of Ex- possible that the strong absorption of excitons from the citon Rydberg States in a Monolayer Semiconductor, lower spin-split valley in the valence band (type B exci- arXiv:1709.00123. tons) masks the weaker absorption of the 2s exciton state [37] The contribution from the nonzero charge of the inﬁnite from the top spin-split valley in molybdenum-based MLs. cationic and anionic sheets to the electric-ﬁeld vanishes [53] A. Korma´nyos, G. Burkard, M. Gmitra, J. Fabian, V. · in the mid-plane by symmetry. Thus, we only consider Zo´lyomi, N. D. Drummond, and V. Fal’ko, k p theory how the polarizabilities of these sheets aﬀect the Coulomb for two-dimensional transition metal dichalcogenide semi- interaction between itinerant charges in the mid-plane. conductors, 2D Mater. 2, 022001 (2015). [38] A. V. Stier, N. P. Wilson, G. Clark, X. Xu, and S. A. [54] H. Dery and Y. Song, Polarization analysis of excitons Crooker, Probing the Inﬂuence of Dielectric Environment in monolayer and bilayer transition-metal dichalcogenides, on Excitons in Monolayer WSe2: Insight from High Mag- Phys. Rev. B 92, 125431 (2015). netic Fields, Nano Lett. 16, 7054 (2016). [55] The spin-splitting in the conduction band is not related [39] Y. Suzuki and K. Varga, Stochastic Variational Approach to the binding energy of trions [35]. to Quantum Mechanical Few-Body Problems, Springer- [56] The polaron parameters in bulk CdS and ZnSe semicon- (cid:39) (cid:39) Verlag (1998). ductors are α 0.53 and α 0.43, respectively [59– ∗ (cid:39) [40] K. Varga and Y. Suzuki, Solution of few-body problems 61]. The correction to the eﬀective mass is then m ∗ (cid:39) with the stochastic variational method I. Central forces m(1 + α/6) in 3D systems and about m m(1 + πα/8) with zero orbital momentum, Comp. Phys. Comm. 106, in 2D [62]. These values lead to about 10% increase in the 157 (1997). eﬀective mass, similar to what we use in this work. [41] K. Varga and Y. Suzuki, Precise solution of few-body [57] The Fro¨hlich interaction is stronger in MoSe [63], mean- 2 problems with the stochastic variational method on a cor- ing that larger polarizability parameters, (cid:96)± = 2πχ±, related Gaussian basis, Phys. Rev. C 52, 2885 (1995). should be used in this material, leading to reduced binding [42] K. Varga, Solution of few-body problems with the energies. On the other hand, the stronger Fro¨hlich inter- stochastic variational method II: Two-dimensional sys- action also means that we should increase the eﬀective tems, Comp. Phys. Comm. 179, 591 (2008). masses by a larger amount due to a stronger polaron ef- [43] A. Branny, G. Wang, S. Kumar, C. Robert, B. Lassagne, fect, leading to increased binding energies. Since the two X. Marie, B. D. Gerardot, and B. Urbaszek, Discrete quan- eﬀects oﬀset each other, we choose to use similar values of tum dot like emitters in monolayer MoSe2: Spatial map- (cid:96)± in WSe2 and MoSe2 in order to reduce the dependence ping, magneto-optics, and charge tuning, Appl. Phys. Lett. of the results on ﬁtting parameters. 108, 142101 (2016). [58] The in-plane and out-of-plane dielectric constants h-BN [44] S. Borghardt, J.-S. Tu, F. Winkler, J. Schubert, W. Zan- were originally measured by Geick et al. who found (cid:15)(cid:107) = der, K. Leosson, and B. E. Kardynal, Engineering of opti- 6.9 and (cid:15)⊥ = 5.06 [64]. We have used their average value in cal and electronic band gaps in transition metal dichalco- the main paper, (cid:15) = 6. Note that the results of Geick h-BN genide monolayers through external dielectric screening, et al. are inconsistent with recent experiments that report ∼ − Phys. Rev. Materials 1, 054001 (2017). (cid:15) 2 4 [65]. However, we could not ﬁnd reasonable h-BN (cid:46) [45] K. 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text: Zeeman

label: reference

text: eﬀects

text: in

text: monolayer

text: WS

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text: MoS

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text: Tesla,

text: Spin-Polarized

text: Electronic

text: Band

text: Structure

text: in

text: Monolayer

text: Nat.

text: Commun.

text: Transition

text: Metal

text: Dichalcogenides

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text: excitonic

text: complexes

text: in

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text: drus,

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text: Dery,

text: W.

text: Yao,

text: and

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text: Xu,

text: Excitonic

text: Luminescence

text: Upconversion

text: in

text: a

text: Two-

text: R.

text: Ya.

text: Kezerashvili

text: and

text: S.

text: M.Tsiklauri,

text: Trion

text: and

text: Biexci-

text: Dimensional

text: Semiconductor,

text: Nat.

text: Phys.

text: ton

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text: Monolayer

text: Transition

text: Metal

text: Dichalcogenides,

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text: W.

text: Yao,

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text: E.

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text: M.

text: Szyniszewski,

text: C.

text: H.

text: Price,

text: R.

text: Maezono,

text: Xu,

text: Electrical

text: control

text: of

text: neutral

text: and

text: charged

text: excitons

text: in

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text: M.

text: Danovich,

text: R.

text: J.

text: Hunt,

text: N.

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text: Drummond,

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text: theory,

text: Phys.

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text: The

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text: the

text: valence

text: band,

text: ∆v,

text: is

text: compara-

text: ble

text: to

text: in

text: molybdenum-based

text: MLs,

text: while

text: being

text: much

text: A.

text: V.

text: Stier,

text: N.

text: P.

text: Wilson,

text: K.

text: A.

text: Velizhanin,

text: J.

text: Kono,

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text: than

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text: tungsten-based

text: MLs.

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text: it

text: is

text: X.

text: Xu,

text: and

text: S.

text: A.

text: Crooker,

text: Magneto-Optics

text: of

text: Ex-

text: possible

text: that

text: the

text: strong

text: absorption

text: of

text: excitons

text: from

text: the

text: citon

text: Rydberg

text: States

text: in

text: a

text: Monolayer

text: Semiconductor,

text: lower

text: spin-split

text: valley

text: in

text: the

text: valence

text: band

text: (type

text: B

text: exci-

text: tons)

text: masks

text: the

text: weaker

text: absorption

text: of

text: the

text: exciton

text: state

text: The

text: contribution

text: from

text: the

text: nonzero

text: charge

text: of

text: the

text: inﬁnite

text: from

text: the

text: top

text: spin-split

text: valley

text: in

text: molybdenum-based

text: MLs.

text: cationic

text: and

text: anionic

text: sheets

text: to

text: the

text: electric-ﬁeld

text: vanishes

text: A.

text: Korma´nyos,

text: G.

text: Burkard,

text: M.

text: Gmitra,

text: J.

text: Fabian,

text: V.

text: in

text: the

text: mid-plane

text: by

text: symmetry.

text: Thus,

text: we

text: only

text: consider

text: Zo´lyomi,

text: N.

text: D.

text: Drummond,

text: and

text: V.

text: Fal’ko,

text: k

text: p

text: theory

text: how

text: the

text: polarizabilities

text: of

text: these

text: sheets

text: aﬀect

text: the

text: Coulomb

text: for

text: two-dimensional

text: transition

text: metal

text: dichalcogenide

text: semi-

text: interaction

text: between

text: itinerant

text: charges

text: in

text: the

text: mid-plane.

text: conductors,

text: Mater.

text: A.

text: V.

text: Stier,

text: N.

text: P.

text: Wilson,

text: G.

text: Clark,

text: X.

text: Xu,

text: and

text: S.

text: A.

text: H.

text: Dery

text: and

text: Y.

text: Song,

text: Polarization

text: analysis

text: of

text: excitons

text: Crooker,

text: Probing

text: the

text: Inﬂuence

text: of

text: Dielectric

text: Environment

text: in

text: monolayer

text: and

text: bilayer

text: transition-metal

text: dichalcogenides,

text: on

text: Excitons

text: in

text: Monolayer

text: Insight

text: from

text: High

text: Mag-

text: Phys.

text: Rev.

text: B

text: netic

text: Fields,

text: Nano

text: Lett.

text: The

text: spin-splitting

text: in

text: the

text: conduction

text: band

text: is

text: not

text: related

text: Y.

text: Suzuki

text: and

text: K.

text: Varga,

text: Stochastic

text: Variational

text: Approach

text: to

text: the

text: binding

text: energy

text: of

text: trions

text: to

text: Quantum

text: Mechanical

text: Few-Body

text: Problems,

text: Springer-

text: The

text: polaron

text: parameters

text: in

text: bulk

text: CdS

text: and

text: ZnSe

text: semicon-

text: Verlag

text: ductors

text: are

text: α

text: and

text: α

text: respectively

text: K.

text: Varga

text: and

text: Y.

text: Suzuki,

text: Solution

text: of

text: few-body

text: problems

text: The

text: correction

text: to

text: the

text: eﬀective

text: mass

text: is

text: then

text: m

text: with

text: the

text: stochastic

text: variational

text: method

text: I.

text: Central

text: forces

text: in

text: systems

text: and

text: about

text: m

text: with

text: zero

text: orbital

text: momentum,

text: Comp.

text: Phys.

text: Comm.

text: in

text: These

text: values

text: lead

text: to

text: about

text: increase

text: in

text: the

text: eﬀective

text: mass,

text: similar

text: to

text: what

text: we

text: use

text: in

text: this

text: work.

text: K.

text: Varga

text: and

text: Y.

text: Suzuki,

text: Precise

text: solution

text: of

text: few-body

text: The

text: Fro¨hlich

text: interaction

text: is

text: stronger

text: in

text: MoSe

text: mean-

text: problems

text: with

text: the

text: stochastic

text: variational

text: method

text: on

text: a

text: cor-

text: ing

text: that

text: larger

text: polarizability

text: parameters,

text: 2πχ±,

text: related

text: Gaussian

text: basis,

text: Phys.

text: Rev.

text: C

text: should

text: be

text: used

text: in

text: this

text: material,

text: leading

text: to

text: reduced

text: binding

text: K.

text: Varga,

text: Solution

text: of

text: few-body

text: problems

text: with

text: the

text: energies.

text: On

text: the

text: other

text: hand,

text: the

text: stronger

text: Fro¨hlich

text: inter-

text: stochastic

text: variational

text: method

text: II:

text: Two-dimensional

text: sys-

text: action

text: also

text: means

text: that

text: we

text: should

text: increase

text: the

text: eﬀective

text: tems,

text: Comp.

text: Phys.

text: Comm.

text: masses

text: by

text: a

text: larger

text: amount

text: due

text: to

text: a

text: stronger

text: polaron

text: ef-

text: A.

text: Branny,

text: G.

text: Wang,

text: S.

text: Kumar,

text: C.

text: Robert,

text: B.

text: Lassagne,

text: fect,

text: leading

text: to

text: increased

text: binding

text: energies.

text: Since

text: the

text: two

text: X.

text: Marie,

text: B.

text: D.

text: Gerardot,

text: and

text: B.

text: Urbaszek,

text: Discrete

text: quan-

text: eﬀects

text: oﬀset

text: each

text: other,

text: we

text: choose

text: to

text: use

text: similar

text: values

text: of

text: tum

text: dot

text: like

text: emitters

text: in

text: monolayer

text: Spatial

text: map-

text: in

text: and

text: in

text: order

text: to

text: reduce

text: the

text: dependence

text: ping,

text: magneto-optics,

text: and

text: charge

text: tuning,

text: Appl.

text: Phys.

text: Lett.

text: of

text: the

text: results

text: on

text: ﬁtting

text: parameters.

text: The

text: in-plane

text: and

text: out-of-plane

text: dielectric

text: constants

text: h-BN

text: S.

text: Borghardt,

text: J.-S.

text: Tu,

text: F.

text: Winkler,

text: J.

text: Schubert,

text: W.

text: Zan-

text: were

text: originally

text: measured

text: by

text: Geick

text: et

text: al.

text: who

text: found

text: der,

text: K.

text: Leosson,

text: and

text: B.

text: E.

text: Kardynal,

text: Engineering

text: of

text: opti-

text: and

text: We

text: have

text: used

text: their

text: average

text: value

text: in

text: cal

text: and

text: electronic

text: band

text: gaps

text: in

text: transition

text: metal

text: dichalco-

text: the

text: main

text: paper,

text: Note

text: that

text: the

text: results

text: of

text: Geick

text: h-BN

text: genide

text: monolayers

text: through

text: external

text: dielectric

text: screening,

text: et

text: al.

text: are

text: inconsistent

text: with

text: recent

text: experiments

text: that

text: report

text: Phys.

text: Rev.

text: Materials

text: However,

text: we

text: could

text: not

text: ﬁnd

text: reasonable

text: h-BN

text: K.

text: He,

text: N.

text: Kumar,

text: L.

text: Zhao,

text: Z.

text: Wang,

text: K.

text: F.

text: Mak,

text: H.

text: Zhao,

text: parameters

text: for

text: when

text: using

text: and

text: h-BN

text: and

text: J.

text: Shan,

text: Tightly

text: Bound

text: Excitons

text: in

text: Monolayer

text: That

text: is,

text: these

text: parameters

text: cannot

text: yield

text: meV

text: Phys.

text: Rev.

text: Lett.

text: for

text: ML-WSe

text: encapsulated

text: in

text: h-BN

text: and

text: meV

text: Z.

text: Wang,

text: L.

text: Zhao,

text: K.

text: F.

text: Mak,

text: and

text: J.

text: Shan,

text: Probing

text: the


text: 4 − − f(r) = 2m/r 2m2/r2 + 3m3/2r3 m4/r4 (4) g(r) = 2m/r + 3m2/2r2 + m3/2r3 + m4/16r4. (5) Now we study the apparent horizon for this metric. In the exact model the apparent horizon is located in R = 2m = 4r. In Isotropic coordinates to fourth order, we obtain the expression for the apparent horizon as 1 4M 11M2 − 165M4 θ = 2( + + + O (5)). (6) r r2 2r3 16r5 We see that the expansion vanishes at the fourth order leading to the apparent horizon. This perturbative expansion is equivalent to expansion in terms of the gravitational potential x = 2m < 1. It R∼ can be seen that the apparent horizon can form in the regime that the perturbation holds i.e x 0.5. B. Dynamical Black hole formation The spherically symmetric exact solution of Einstein equations with dust is given by the LTB metric. This metric is given by − (∂ R(t, r))2 ds2 = dt2 + r dr2 + R(t, r)2dΩ2 (7) 1 + 2f(r) where f(r) is the energy for each spherical shell with radius r. The Einstein equations for this metric lead to R˙ (t, r) = 2M(r)/R + 2f(r) (8) ∂ M(r) ρ = r , (9) R(t, r)2∂ R(t, r) r where dot means derivative with respect to time. We can have three classes of solutions depending on the curvature function, f(r). M(r) is the Misner-Sharp mass that can be applied to the cosmological structure mass proﬁles [4]. In this metric the surface where R(t, r) = 2M(r) is the apparent horizon. One can show that the metric is also nonlinear at the time of the apparent horizon formation. First we transform the metric to a new coordinate (t, R) given by(cid:113) − ds2 = −(1 2RM )dt2 + dR2 − 2 2RM + 2f dRdt + R2dΩ2. (10) 1 + 2f 1 + 2f 1 + 2f Here, we used the Einstein equation (8) to simplify the metric. If we interpret Φ = M as the R gravitational potential, in the regime that the gravitational potential tends to zero and the spacetime becomes ﬂat, f(r) = 0, this metric asymptotes to the Minkowski metric at inﬁnity in these coordinates. We see that the apparent horizon forms at g = 0 which gives R(t, r) = 2M(r). It is clear from the metric tt that as long as the gravitational potential of dust is in the linear regime 2Φ < 1, the apparent horizon can not form. Thus, the gravitational potential which is deﬁned by the Misner-Sharp mass is a good criterion ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: f(r)

text: g(r)

text: Now

text: we

text: study

text: the

text: apparent

text: horizon

text: for

text: this

text: metric.

text: In

text: the

text: exact

text: model

text: the

text: apparent

text: horizon

text: is

text: located

text: in

text: R

text: In

text: Isotropic

text: coordinates

text: to

text: fourth

text: order,

text: we

text: obtain

text: the

text: expression

text: for

text: the

text: apparent

text: horizon

text: as

text: θ

text: O

text: r

text: We

text: see

text: that

text: the

text: expansion

text: vanishes

text: at

text: the

text: fourth

text: order

text: leading

text: to

text: the

text: apparent

text: horizon.

text: This

text: perturbative

text: expansion

text: is

text: equivalent

text: to

text: expansion

text: in

text: terms

text: of

text: the

text: gravitational

text: potential

text: x

text: It

text: R∼

text: can

text: be

text: seen

text: that

text: the

text: apparent

text: horizon

text: can

text: form

text: in

text: the

text: regime

text: that

text: the

text: perturbation

text: holds

text: i.e

text: x

text: B.

text: Dynamical

text: Black

text: hole

text: formation

text: The

text: spherically

text: symmetric

text: exact

text: solution

text: of

text: Einstein

text: equations

text: with

text: dust

text: is

text: given

text: by

text: the

text: LTB

text: metric.

text: This

text: metric

text: is

text: given

text: by

text: R(t,

text: r

text: R(t,

text: where

text: f(r)

text: is

text: the

text: energy

text: for

text: each

text: spherical

text: shell

text: with

text: radius

text: r.

text: The

text: Einstein

text: equations

text: for

text: this

text: metric

text: lead

text: to

text: R˙

text: (t,

text: r)

text: M(r)

text: ρ

text: r

text: R(t,

text: R(t,

text: r)

text: r

text: where

text: dot

text: means

text: derivative

text: with

text: respect

text: to

text: time.

text: We

text: can

text: have

text: three

text: classes

text: of

text: solutions

text: depending

text: on

text: the

text: curvature

text: function,

text: f(r).

text: M(r)

text: is

text: the

text: Misner-Sharp

text: mass

text: that

text: can

text: be

text: applied

text: to

text: the

text: cosmological

text: structure

text: mass

text: proﬁles

text: In

text: this

text: metric

text: the

text: surface

text: where

text: R(t,

text: r)

text: is

text: the

text: apparent

text: horizon.

text: One

text: can

text: show

text: that

text: the

text: metric

text: is

text: also

text: nonlinear

text: at

text: the

text: time

text: of

text: the

text: apparent

text: horizon

text: formation.

text: First

text: we

text: transform

text: the

text: metric

text: to

text: a

text: new

text: coordinate

text: (t,

text: R)

text: given

text: dRdt

text: Here,

text: we

text: used

text: the

text: Einstein

text: equation

text: to

text: simplify

text: the

text: metric.

text: If

text: we

text: interpret

text: Φ

text: M

text: as

text: the

text: R

text: gravitational

text: potential,

text: in

text: the

text: regime

text: that

text: the

text: gravitational

text: potential

text: tends

text: to

text: zero

text: and

text: the

text: spacetime

text: becomes

text: ﬂat,

text: f(r)

text: this

text: metric

text: asymptotes

text: to

text: the

text: Minkowski

text: metric

text: at

text: inﬁnity

text: in

text: these

text: coordinates.

text: We

text: see

text: that

text: the

text: apparent

text: horizon

text: forms

text: at

text: g

text: which

text: gives

text: R(t,

text: r)

text: It

text: is

text: clear

text: from

text: the

text: metric

text: tt

text: that

text: as

text: long

text: as

text: the

text: gravitational

text: potential

text: of

text: dust

text: is

text: in

text: the

text: linear

text: regime

text: 2Φ

text: the

text: apparent

text: horizon

text: can

text: not

text: form.

text: Thus,

text: the

text: gravitational

text: potential

text: which

text: is

text: deﬁned

text: by

text: the

text: Misner-Sharp

text: mass

text: is

text: a

text: good

text: criterion

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 interactions and EET eﬃciency are related to each other Ini�al [7–19]. Excita�on The standard computational treatment to study the eﬀects of an environment on an open quantum system, which has been used in the study of pigment-protein 11 molecular aggregates in Refs. [8, 9, 13–16], has been to employ Markovian master equations to model the dy- namics. The Markovian approximation is to assume that the environment is in equilibrium during the time evo- lution of the system. For some processes, such as EET in photosynthetic complexes where the strength of the 22 system-environment interaction is of the same order as Pigment the intra-system interactions, Markovian master equa- 33 tions are not suﬃcient to capture relevant environmental eﬀects. In order to accurately take those eﬀects into ac- count, the dynamics has to be modeled by master equa- tions within the non-Markovian regime. As a result, there is a possibility for information to ﬂow back-and- Protein forth between the system and the environment during Reac�on the time evolution of the system. Hence, environmental- scaffold Centre induced revivals of quantities such as coherence and cor- relations within the system may occur. In the case of photosynthetic complexes, the need for FIG. 1: Schematic illustration of EET in a trimer non-Markovian master equations to capture the behav- pigment-protein molecular aggregate. ior of the system-environment interactions in a realistic manner has resulted in developments of new theoretical methods. Especially the hierarchical equations of mo- fold when the HEOM method is used to model the EET tion (HEOM) method, originally derived in Refs. [20– dynamics. Within a range of possible environmental pa- 22] and then extended in Refs. [23, 24], has served as the rameters, we seek for an optimum in the eﬃciency of benchmarking method to model EET in photosynthetic the EET. We investigate a number of pigment conﬁgu- complexes. rations (deﬁned by the pigments excitation energies and Non-Markovian dynamics has been found to be a re- inter-pigment couplings) in order to evaluate how opti- source in some processes. Examples include entangle- mal environmental parameters may diﬀer depending on ment generation [25–29], information transfer in a noisy the system under consideration. We further determine channel [30–32] and quantum communication [33, 34]. In whether the dynamics imposed by the optimal environ- Ref. [35] it is shown that non-Markovianity can function mental parameters (with respect to EET eﬃciency) is as a resource for prolonging the duration of coherence in within the non-Markovian regime or not. To explore the a dimer pigment-protein model system. Non-Markovian interplay between coherence and environmental eﬀects, eﬀects in pigment-protein molecular aggregates have also as well as the possible connection between coherence and been studied in Refs. [36–38]. non-Markovianity, we quantify the amount of coherence As in the case of any potential quantum resource, it in our systems by the l -norm of coherence [42]. 1 is crucial to be able to quantify non-Markovian eﬀects This study attempts to reveal design principles on how in a system in order to be able to distinguish diﬀerent to create environmental conditions for eﬃcient EET in environments in terms of their ability to function as a pigment-protein molecular aggregates. Further, we aim resource. Indeed, a number of diﬀerent measures of non- to evaluate the possible role of non-Markovianity for eﬃ- Markovianity has been developed and proposed [39–41]. cient EET in such aggregates. The results can hopefully One that has been frequently used in studies of non- be useful for gaining insights on how to design artiﬁcial Markovianity is the BLP-measure [39]. It is based on molecular aggregates for light harvesting. the observation that the distinguishability of two initial II. SYSTEMS states can never increase under a Markovian evolution. The distinguishability is quantiﬁed by the trace distance The systems considered in this study are networks of and an increase in the trace distance in a certain time N (N = 2, 3) sites where each site represents a pigment. interval is interpreted as a backﬂow of information from These two types of systems can capture both pure quan- the environment to the system. tum tunneling between pigments (dimer, i.e., N = 2) In this study, we investigate numerically how the pa- and interference between diﬀerent pathways (trimer, i.e., rameters describing the environment and its coupling to N = 3), which are two important mechanisms in EET. the system inﬂuence EET eﬃciency between end sites in A general trimer system (N = 3) is illustrated in Fig. a network of N (N = 2, 3) pigments in a protein scaf- 1 along with the EET direction and possible EET path- ##LTFigure## ##LTFigure##

text: interactions

text: and

text: EET

text: eﬃciency

text: are

text: related

text: to

text: each

text: other

text: Ini�al

text: Excita�on

text: The

text: standard

text: computational

text: treatment

text: to

text: study

text: the

text: eﬀects

text: of

text: an

text: environment

text: on

text: an

text: open

text: quantum

text: system,

text: which

text: has

text: been

text: used

text: in

text: the

text: study

text: of

text: pigment-protein

text: molecular

text: aggregates

text: in

text: Refs.

text: has

text: been

text: to

text: employ

text: Markovian

text: master

text: equations

text: to

text: model

text: the

text: dy-

text: namics.

text: The

text: Markovian

text: approximation

text: is

text: to

text: assume

text: that

text: the

text: environment

text: is

text: in

text: equilibrium

text: during

text: the

text: time

text: evo-

text: lution

text: of

text: the

text: system.

text: For

text: some

text: processes,

text: such

text: as

text: EET

text: in

text: photosynthetic

text: complexes

text: where

text: the

text: strength

text: of

text: the

text: system-environment

text: interaction

text: is

text: of

text: the

text: same

text: order

text: as

text: Pigment

text: the

text: intra-system

text: interactions,

text: Markovian

text: master

text: equa-

text: tions

text: are

text: not

text: suﬃcient

text: to

text: capture

text: relevant

text: environmental

text: eﬀects.

text: In

text: order

text: to

text: accurately

text: take

text: those

text: eﬀects

text: into

text: ac-

text: count,

text: the

text: dynamics

text: has

text: to

text: be

text: modeled

text: by

text: master

text: equa-

text: tions

text: within

text: the

text: non-Markovian

text: regime.

text: As

text: a

text: result,

text: there

text: is

text: a

text: possibility

text: for

text: information

text: to

text: ﬂow

text: back-and-

text: Protein

text: forth

text: between

text: the

text: system

text: and

text: the

text: environment

text: during

text: Reac�on

text: the

text: time

text: evolution

text: of

text: the

text: system.

text: Hence,

text: environmental-

text: scaffold

text: Centre

text: induced

text: revivals

text: of

text: quantities

text: such

text: as

text: coherence

text: and

text: cor-

text: relations

text: within

text: the

text: system

text: may

text: occur.

text: In

text: the

text: case

text: of

text: photosynthetic

text: complexes,

text: the

text: need

text: for

text: FIG.

text: Schematic

text: illustration

text: of

text: EET

text: in

text: a

text: trimer

text: non-Markovian

text: master

text: equations

text: to

text: capture

text: the

text: behav-

text: pigment-protein

text: molecular

text: aggregate.

text: ior

text: of

text: the

text: system-environment

text: interactions

text: in

text: a

text: realistic

text: manner

text: has

text: resulted

text: in

text: developments

text: of

text: new

text: theoretical

text: methods.

text: Especially

text: the

text: hierarchical

text: equations

text: of

text: mo-

text: fold

text: when

text: the

text: HEOM

text: method

text: is

text: used

text: to

text: model

text: the

text: EET

text: tion

text: (HEOM)

text: method,

text: originally

text: derived

text: in

text: Refs.

text: dynamics.

text: Within

text: a

text: range

text: of

text: possible

text: environmental

text: pa-

text: and

text: then

text: extended

text: in

text: Refs.

text: has

text: served

text: as

text: the

text: rameters,

text: we

text: seek

text: for

text: an

text: optimum

text: in

text: the

text: eﬃciency

text: of

text: benchmarking

text: method

text: to

text: model

text: EET

text: in

text: photosynthetic

text: the

text: EET.

text: We

text: investigate

text: a

text: number

text: of

text: pigment

text: conﬁgu-

text: complexes.

text: rations

text: (deﬁned

text: by

text: the

text: pigments

text: excitation

text: energies

text: and

text: Non-Markovian

text: dynamics

text: has

text: been

text: found

text: to

text: be

text: a

text: re-

text: inter-pigment

text: couplings)

text: in

text: order

text: to

text: evaluate

text: how

text: opti-

text: source

text: in

text: some

text: processes.

text: Examples

text: include

text: entangle-

text: mal

text: environmental

text: parameters

text: may

text: diﬀer

text: depending

text: on

text: ment

text: generation

text: information

text: transfer

text: in

text: a

text: noisy

text: the

text: system

text: under

text: consideration.

text: We

text: further

text: determine

text: channel

text: and

text: quantum

text: communication

text: In

text: whether

text: the

text: dynamics

text: imposed

text: by

text: the

text: optimal

text: environ-

text: Ref.

text: it

text: is

text: shown

text: that

text: non-Markovianity

text: can

text: function

text: mental

text: parameters

text: (with

text: respect

text: to

text: EET

text: eﬃciency)

text: is

text: as

text: a

text: resource

text: for

text: prolonging

text: the

text: duration

text: of

text: coherence

text: in

text: within

text: the

text: non-Markovian

text: regime

text: or

text: not.

text: To

text: explore

text: the

text: a

text: dimer

text: pigment-protein

text: model

text: system.

text: Non-Markovian

text: interplay

text: between

text: coherence

text: and

text: environmental

text: eﬀects,

text: eﬀects

text: in

text: pigment-protein

text: molecular

text: aggregates

text: have

text: also

text: as

text: well

text: as

text: the

text: possible

text: connection

text: between

text: coherence

text: and

text: been

text: studied

text: in

text: Refs.

text: non-Markovianity,

text: we

text: quantify

text: the

text: amount

text: of

text: coherence

text: As

text: in

text: the

text: case

text: of

text: any

text: potential

text: quantum

text: resource,

text: it

text: in

text: our

text: systems

text: by

text: the

text: l

text: -norm

text: of

text: coherence

text: is

text: crucial

text: to

text: be

text: able

text: to

text: quantify

text: non-Markovian

text: eﬀects

text: This

text: study

text: attempts

text: to

text: reveal

text: design

text: principles

text: on

text: how

text: in

text: a

text: system

text: in

text: order

text: to

text: be

text: able

text: to

text: distinguish

text: diﬀerent

text: to

text: create

text: environmental

text: conditions

text: for

text: eﬃcient

text: EET

text: in

text: environments

text: in

text: terms

text: of

text: their

text: ability

text: to

text: function

text: as

text: a

text: pigment-protein

text: molecular

text: aggregates.

text: Further,

text: we

text: aim

text: resource.

text: Indeed,

text: a

text: number

text: of

text: diﬀerent

text: measures

text: of

text: non-

text: to

text: evaluate

text: the

text: possible

text: role

text: of

text: non-Markovianity

text: for

text: eﬃ-

text: Markovianity

text: has

text: been

text: developed

text: and

text: proposed

text: cient

text: EET

text: in

text: such

text: aggregates.

text: The

text: results

text: can

text: hopefully

text: One

text: that

text: has

text: been

text: frequently

text: used

text: in

text: studies

text: of

text: non-

text: be

text: useful

text: for

text: gaining

text: insights

text: on

text: how

text: to

text: design

text: artiﬁcial

text: Markovianity

text: is

text: the

text: BLP-measure

text: It

text: is

text: based

text: on

text: molecular

text: aggregates

text: for

text: light

text: harvesting.

text: the

text: observation

text: that

text: the

text: distinguishability

text: of

text: two

text: initial

text: II.

text: SYSTEMS

text: states

text: can

text: never

text: increase

text: under

text: a

text: Markovian

text: evolution.

text: The

text: distinguishability

text: is

text: quantiﬁed

text: by

text: the

text: trace

text: distance

text: The

text: systems

text: considered

text: in

text: this

text: study

text: are

text: networks

text: of

text: and

text: an

text: increase

text: in

text: the

text: trace

text: distance

text: in

text: a

text: certain

text: time

text: N

text: (N

text: sites

text: where

text: each

text: site

text: represents

text: a

text: pigment.

text: interval

text: is

text: interpreted

text: as

text: a

text: backﬂow

text: of

text: information

text: from

text: These

text: two

text: types

text: of

text: systems

text: can

text: capture

text: both

text: pure

text: quan-

text: the

text: environment

text: to

text: the

text: system.

text: tum

text: tunneling

text: between

text: pigments

text: (dimer,

text: i.e.,

text: N

text: In

text: this

text: study,

text: we

text: investigate

text: numerically

text: how

text: the

text: pa-

text: and

text: interference

text: between

text: diﬀerent

text: pathways

text: (trimer,

text: i.e.,

text: rameters

text: describing

text: the

text: environment

text: and

text: its

text: coupling

text: to

text: N

text: which

text: are

text: two

text: important

text: mechanisms

text: in

text: EET.

text: the

text: system

text: inﬂuence

text: EET

text: eﬃciency

text: between

text: end

text: sites

text: in

text: A

text: general

text: trimer

text: system

text: (N

text: is

text: illustrated

text: in

text: Fig.

text: a

text: network

text: of

text: N

text: (N

text: pigments

text: in

text: a

text: protein

text: scaf-

text: along

text: with

text: the

text: EET

text: direction

text: and

text: possible

text: EET

text: path-

text: ##LTFigure##

text: ##LTFigure##


text: 11 Previous studies on environment-assisted quantum the most eﬃcient EET process might be a combination transport (ENAQT) in pigment-protein molecular aggre- of coherent tunneling between sites and a downhill fun- gates [7–19] have, among others, proposed the following neling from site to site. We believe that the EET in the mec•hanisms for an enhanced EET eﬃciency: system with Hamiltonian Hˆ (3) is an example of such a The environment can suppress pathways not lead- combined process. FMO ing to an enhanced probability of ﬁnding the exci- Previous studies on non-Markovian eﬀects in pigment- tation on the end site. protein molecular aggregates have mainly connected the • occurrence of non-Markovianity to a prolonged duration Strongly coupled vibrational modes in the environ- of coherence [35, 36, 45]. In [37] it is noted though that in ment can induce resonant energy transfer between a FMO complex monomer, maximal non-Markovianity is two sites/excitons that in the static picture are non- within the same environmental regime as optimal EET resonant. eﬃciency according to Ref. [14]. The authors suggest that the role of non-Markovianity could be to preserve In particular it has been noted that for EET that is very coherence, which could be of importance for the ﬁrst ineﬃcient in the coherent regime, as the EET from site 1 to site 3 in the FMO-complex monomer, the eﬃciency EET route (site 1, 2 and 3) since there is an energy can be increased considerably by adding environmental barrier between site 1 and 2 (which might require quan- tum tunneling). Even though the important features of eﬀects. In this study, we do not aim to explain the ex- the system Hamiltonian parameter space for these three act mechanisms for ENAQT, but rather try to pin down sites coincide with our trimer system with Hamiltonian some simple design principles (in terms of environmen- tal properties) for how to achieve eﬃcient EET in model Hˆ (3) (with only the exact parameter values diﬀering), FMO pigment-protein molecular aggregates. the dynamics will diﬀer a bit due to the omission of the Our study indicates that the information about opti- other four pigments in the FMO-complex monomer in our mal environmental conditions for eﬃcient EET is in the calculations. It should be noted that even though non- structure of the closed system Hamiltonian and in par- Markovianity is quantiﬁed by the same measure as in ticular, the relationship between the site energies. In the present study, the maximization over initial states is general, a strategy to predict the optimal environment ignored. This might aﬀect the conclusion about the con- for a system is to compare F to F of site N. If nection between non-Markovianity and EET eﬃciency. CS,max eq F > F , the most beneﬁcial role of the environ- Indeed our study does not indicate a general connection CS,max eq ment to promote eﬃcient EET is to preserve coherent between EET eﬃciency and non-Markovianity. EET dynamics (i.e., the system-environment coupling, λ, The parameter that seemingly aﬀects the amount of should be small). If instead F > F , most eﬃcient non-Markovianity - and hence, the backﬂow of informa- eq CS,max EET is achieved when the the system is driven towards tion from the environment to the system - the most, is equilibrium as fast as possible (i.e., the environmental the environmental time scale (τ). The systems where time scale, τ, should be small). In this study, eﬃcient FOS,max is achieved in the coherent regime are not par- EET hence occurs in either of the two regimes “coher- ticularly sensitive to a potential backﬂow of information ent dynamics” (as close to the closed system dynamics as as accompanied with large values of τ, but they do not possible) or “equilibrating” (as close to the equilibrium seem to beneﬁt from it either. population as possible). Another general feature worth noticing is that the pat- In terms of the system Hamiltonian parameter space, tern of the trace distance is very similar to the pattern of our results indicate that for systems where site N is coherence in the exciton basis when λ is small (see Figs. higher in energy than site 1, the EET beneﬁts from quan- 6 and 7). This result suggests that a temporal increase of tum tunneling between the sites and hence requires a dy- coherence in the exciton basis indeed may be interpreted namics within the coherent regime. The same is true as a backﬂow of information from the environment to the for the system Hamiltonians where E = E . In such system and could possibly detect non-Markovian dynam- N 1 a case, the most important design principle is to create ics. an environment that allows for coherent EET dynamics Lastly, our study shows the necessity of using the BLP- to persist, i.e., the system should be weakly coupled to measure the way that it is intended to since a predeﬁned the environment. The environmental time scale is less choice of an initial state pair may give an arbitrary result. important. If instead E > E , the system is favor- 1 N ing a fast return to equilibrium where E obviously is N more populated than E . Here, the environmental time ACKNOWLEDGMENTS 1 scale should be designed to be as small as possible and the coupling strength between the system and the envi- The computations were performed on resources pro- ronment is of less importance. Still, depending on the vided by the Swedish National Infrastructure for Com- energy landscape of the other sites in the network as well puting (SNIC) at Uppsala Multidisciplinary Center for as their intersite couplings, the time scale of equilibration Advanced Computational Science (UPPMAX) under might be beyond the given time frame. Hence, in reality Project snic2017-7-140.

text: Previous

text: studies

text: on

text: environment-assisted

text: quantum

text: the

text: most

text: eﬃcient

text: EET

text: process

text: might

text: be

text: a

text: combination

text: transport

text: (ENAQT)

text: in

text: pigment-protein

text: molecular

text: aggre-

text: of

text: coherent

text: tunneling

text: between

text: sites

text: and

text: a

text: downhill

text: fun-

text: gates

text: have,

text: among

text: others,

text: proposed

text: the

text: following

text: neling

text: from

text: site

text: to

text: site.

text: We

text: believe

text: that

text: the

text: EET

text: in

text: the

text: mec•hanisms

label: list

text: for

text: an

text: enhanced

text: EET

text: eﬃciency:

text: system

text: with

text: Hamiltonian

text: Hˆ

text: is

text: an

text: example

text: of

text: such

text: a

text: The

text: environment

text: can

text: suppress

text: pathways

text: not

text: lead-

text: combined

text: process.

text: FMO

text: ing

text: to

text: an

text: enhanced

text: probability

text: of

text: ﬁnding

text: the

text: exci-

text: Previous

text: studies

text: on

text: non-Markovian

text: eﬀects

text: in

text: pigment-

text: tation

text: on

text: the

text: end

text: site.

text: protein

text: molecular

text: aggregates

text: have

text: mainly

text: connected

text: the

text: occurrence

text: of

text: non-Markovianity

text: to

text: a

text: prolonged

text: duration

text: Strongly

text: coupled

text: vibrational

text: modes

text: in

text: the

text: environ-

text: of

text: coherence

text: In

text: it

text: is

text: noted

text: though

text: that

text: in

text: ment

text: can

text: induce

text: resonant

text: energy

text: transfer

text: between

text: a

text: FMO

text: complex

text: monomer,

text: maximal

text: non-Markovianity

text: is

text: two

text: sites/excitons

text: that

text: in

text: the

text: static

text: picture

text: are

text: non-

text: within

text: the

text: same

text: environmental

text: regime

text: as

text: optimal

text: EET

text: resonant.

text: eﬃciency

text: according

text: to

text: Ref.

text: The

text: authors

text: suggest

text: that

text: the

text: role

text: of

text: non-Markovianity

text: could

text: be

text: to

text: preserve

text: In

text: particular

text: it

text: has

text: been

text: noted

text: that

text: for

text: EET

text: that

text: is

text: very

text: coherence,

text: which

text: could

text: be

text: of

text: importance

text: for

text: the

text: ﬁrst

text: ineﬃcient

text: in

text: the

text: coherent

text: regime,

text: as

text: the

text: EET

text: from

text: site

text: to

text: site

text: in

text: the

text: FMO-complex

text: monomer,

text: the

text: eﬃciency

text: EET

text: route

text: (site

text: and

text: since

text: there

text: is

text: an

text: energy

text: can

text: be

text: increased

text: considerably

text: by

text: adding

text: environmental

text: barrier

text: between

text: site

text: and

text: (which

text: might

text: require

text: quan-

text: tum

text: tunneling).

text: Even

text: though

text: the

text: important

text: features

text: of

text: eﬀects.

text: In

text: this

text: study,

text: we

text: do

text: not

text: aim

text: to

text: explain

text: the

text: ex-

text: the

text: system

text: Hamiltonian

text: parameter

text: space

text: for

text: these

text: three

text: act

text: mechanisms

text: for

text: ENAQT,

text: but

text: rather

text: try

text: to

text: pin

text: down

text: sites

text: coincide

text: with

text: our

text: trimer

text: system

text: with

text: Hamiltonian

text: some

text: simple

text: design

text: principles

text: (in

text: terms

text: of

text: environmen-

text: tal

text: properties)

text: for

text: how

text: to

text: achieve

text: eﬃcient

text: EET

text: in

text: model

text: Hˆ

text: (with

text: only

text: the

text: exact

text: parameter

text: values

text: diﬀering),

text: FMO

text: pigment-protein

text: molecular

text: aggregates.

text: the

text: dynamics

text: will

text: diﬀer

text: a

text: bit

text: due

text: to

text: the

text: omission

text: of

text: the

text: Our

text: study

text: indicates

text: that

text: the

text: information

text: about

text: opti-

text: other

text: four

text: pigments

text: in

text: the

text: FMO-complex

text: monomer

text: in

text: our

text: mal

text: environmental

text: conditions

text: for

text: eﬃcient

text: EET

text: is

text: in

text: the

text: calculations.

text: It

text: should

text: be

text: noted

text: that

text: even

text: though

text: non-

text: structure

text: of

text: the

text: closed

text: system

text: Hamiltonian

text: and

text: in

text: par-

text: Markovianity

text: is

text: quantiﬁed

text: by

text: the

text: same

text: measure

text: as

text: in

text: ticular,

text: the

text: relationship

text: between

text: the

text: site

text: energies.

text: In

text: the

text: present

text: study,

text: the

text: maximization

text: over

text: initial

text: states

text: is

text: general,

text: a

text: strategy

text: to

text: predict

text: the

text: optimal

text: environment

text: ignored.

text: This

text: might

text: aﬀect

text: the

text: conclusion

text: about

text: the

text: con-

text: for

text: a

text: system

text: is

text: to

text: compare

text: F

text: to

text: F

text: of

text: site

text: N.

text: If

text: nection

text: between

text: non-Markovianity

text: and

text: EET

text: eﬃciency.

text: CS,max

text: eq

text: F

text: F

text: the

text: most

text: beneﬁcial

text: role

text: of

text: the

text: environ-

text: Indeed

text: our

text: study

text: does

text: not

text: indicate

text: a

text: general

text: connection

text: CS,max

text: eq

text: ment

text: to

text: promote

text: eﬃcient

text: EET

text: is

text: to

text: preserve

text: coherent

text: between

text: EET

text: eﬃciency

text: and

text: non-Markovianity.

text: EET

text: dynamics

text: (i.e.,

text: the

text: system-environment

text: coupling,

text: λ,

text: The

text: parameter

text: that

text: seemingly

text: aﬀects

text: the

text: amount

text: of

text: should

text: be

text: small).

text: If

text: instead

text: F

text: F

text: most

text: eﬃcient

text: non-Markovianity

text: and

text: hence,

text: the

text: backﬂow

text: of

text: informa-

text: eq

text: CS,max

text: EET

text: is

text: achieved

text: when

text: the

text: the

text: system

text: is

text: driven

text: towards

text: tion

text: from

text: the

text: environment

text: to

text: the

text: system

text: the

text: most,

text: is

text: equilibrium

text: as

text: fast

text: as

text: possible

text: (i.e.,

text: the

text: environmental

text: the

text: environmental

text: time

text: scale

text: (τ).

text: The

text: systems

text: where

text: time

text: scale,

text: τ,

text: should

text: be

text: small).

text: In

text: this

text: study,

text: eﬃcient

text: FOS,max

text: is

text: achieved

text: in

text: the

text: coherent

text: regime

text: are

text: not

text: par-

text: EET

text: hence

text: occurs

text: in

text: either

text: of

text: the

text: two

text: regimes

text: “coher-

text: ticularly

text: sensitive

text: to

text: a

text: potential

text: backﬂow

text: of

text: information

text: ent

text: dynamics”

text: (as

text: close

text: to

text: the

text: closed

text: system

text: dynamics

text: as

text: as

text: accompanied

text: with

text: large

text: values

text: of

text: τ,

text: but

text: they

text: do

text: not

text: possible)

text: or

text: “equilibrating”

text: (as

text: close

text: to

text: the

text: equilibrium

text: seem

text: to

text: beneﬁt

text: from

text: it

text: either.

text: population

text: as

text: possible).

text: Another

text: general

text: feature

text: worth

text: noticing

text: is

text: that

text: the

text: pat-

text: In

text: terms

text: of

text: the

text: system

text: Hamiltonian

text: parameter

text: space,

text: tern

text: of

text: the

text: trace

text: distance

text: is

text: very

text: similar

text: to

text: the

text: pattern

text: of

text: our

text: results

text: indicate

text: that

text: for

text: systems

text: where

text: site

text: N

text: is

text: coherence

text: in

text: the

text: exciton

text: basis

text: when

text: λ

text: is

text: small

text: (see

text: Figs.

text: higher

text: in

text: energy

text: than

text: site

text: the

text: EET

text: beneﬁts

text: from

text: quan-

text: and

text: This

text: result

text: suggests

text: that

text: a

text: temporal

text: increase

text: of

text: tum

text: tunneling

text: between

text: the

text: sites

text: and

text: hence

text: requires

text: a

text: dy-

text: coherence

text: in

text: the

text: exciton

text: basis

text: indeed

text: may

text: be

text: interpreted

text: namics

text: within

text: the

text: coherent

text: regime.

text: The

text: same

text: is

text: true

text: as

text: a

text: backﬂow

text: of

text: information

text: from

text: the

text: environment

text: to

text: the

text: for

text: the

text: system

text: Hamiltonians

text: where

text: E

text: E

text: In

text: such

text: system

text: and

text: could

text: possibly

text: detect

text: non-Markovian

text: dynam-

text: N

text: a

text: case,

text: the

text: most

text: important

text: design

text: principle

text: is

text: to

text: create

text: ics.

text: an

text: environment

text: that

text: allows

text: for

text: coherent

text: EET

text: dynamics

text: Lastly,

text: our

text: study

text: shows

text: the

text: necessity

text: of

text: using

text: the

text: BLP-

text: to

text: persist,

text: i.e.,

text: the

text: system

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text: predeﬁned

text: the

text: environment.

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text: environmental

text: time

text: scale

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text: choice

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text: initial

text: state

text: pair

text: may

text: give

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text: If

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text: favor-

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text: obviously

text: is

text: N

text: more

text: populated

text: than

text: E

text: Here,

text: the

text: environmental

text: time

text: ACKNOWLEDGMENTS

text: scale

text: should

text: be

text: designed

text: to

text: be

text: as

text: small

text: as

text: possible

text: and

text: the

text: coupling

text: strength

text: between

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text: system

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text: envi-

text: The

text: computations

text: were

text: performed

text: on

text: resources

text: pro-

text: ronment

text: is

text: of

text: less

text: importance.

text: Still,

text: depending

text: on

text: the

text: vided

text: by

text: the

text: Swedish

text: National

text: Infrastructure

text: for

text: Com-

text: energy

text: landscape

text: of

text: the

text: other

text: sites

text: in

text: the

text: network

text: as

text: well

text: puting

text: (SNIC)

text: at

text: Uppsala

text: Multidisciplinary

text: Center

text: for

text: as

text: their

text: intersite

text: couplings,

text: the

text: time

text: scale

text: of

text: equilibration

text: Advanced

text: Computational

text: Science

text: (UPPMAX)

text: under

text: might

text: be

text: beyond

text: the

text: given

text: time

text: frame.

text: Hence,

text: in

text: reality

text: Project


text: The Cyclic Douglas-Rachford Algorithm with r-sets-Douglas-Rachford Operators Yair Censor1 and Aviv Gibali2 1Department of Mathematics, University of Haifa { } Mt. Carmel, Haifa 3498838, Israel. yair@math.haifa.ac.il 2Department of Mathematics, ORT Braude College { } Karmiel 2161002, Israel. avivg@braude.ac.il January 1, 2018. Abstract The Douglas-Rachford (DR) algorithm is an iterative procedure that uses sequential reﬂections onto convex sets and which has become popular for convex feasibility problems. In this paper we propose a structural generalization that allows to use r-sets-DR operators in a cyclic fashion. We prove convergence and illustrate the advantage of such operators with r > 2 over the classical 2-sets-DR operators in a cyclic algorithm. Keywords: Douglas–Rachford; reﬂections; feasibility problems; r-sets-Douglas-Rachford operator. 2010 Mathematics Subject Classiﬁcation: 65K05; 90C25. 1 Introduction We consider the convex feasibility problem (CFP) in a real Hilbert space H · · · − ⊆ H . For i = 0, 1, , m 1, let C be nonempty, closed and convex sets. i The CFP is to 1

text: The

label: title

text: Cyclic

text: Douglas-Rachford

text: Algorithm

text: with

text: r-sets-Douglas-Rachford

text: Operators

text: Yair

label: author

text: and

text: Aviv

text: of

text: Mathematics,

text: University

text: of

text: Haifa

text: Mt.

text: Carmel,

text: Haifa

text: Israel.

text: yair@math.haifa.ac.il

text: of

text: Mathematics,

text: ORT

text: Braude

text: College

text: Karmiel

text: Israel.

text: avivg@braude.ac.il

text: January

label: date

text: Abstract

text: The

label: abstract

text: Douglas-Rachford

text: (DR)

text: algorithm

text: is

text: an

text: iterative

text: procedure

text: that

text: uses

text: sequential

text: reﬂections

text: onto

text: convex

text: sets

text: and

text: which

text: has

text: become

text: popular

text: for

text: convex

text: feasibility

text: problems.

text: In

text: this

text: paper

text: we

text: propose

text: a

text: structural

text: generalization

text: that

text: allows

text: to

text: use

text: r-sets-DR

text: operators

text: in

text: a

text: cyclic

text: fashion.

text: We

text: prove

text: convergence

text: and

text: illustrate

text: the

text: advantage

text: of

text: such

text: operators

text: with

text: r

text: over

text: the

text: classical

text: operators

text: in

text: a

text: cyclic

text: algorithm.

text: Keywords:

text: Douglas–Rachford;

text: reﬂections;

text: feasibility

text: problems;

text: r-sets-Douglas-Rachford

text: operator.

text: Mathematics

text: Subject

text: Classiﬁcation:

text: Introduction

text: We

text: consider

text: the

text: convex

text: feasibility

text: problem

text: (CFP)

text: in

text: a

text: real

text: Hilbert

text: space

text: H

text: H

text: For

text: i

text: m

text: let

text: C

text: be

text: nonempty,

text: closed

text: and

text: convex

text: sets.

text: i

text: The

text: CFP

text: is

text: to


text: Acknowledgments. We thank Prof. Touﬁk Mansour for help in for- mulating our ideas, Yehuda Zur for the meticulous work with Matlab and Raﬁq Mansour for enlightening comments about the manuscript. The ﬁrst author’s work was supported by research grant no. 2013003 of the United States-Israel Binational Science Foundation (BSF). The second author’s work was supported by the EU FP7 IRSES program STREVCOMS, grant no. PIRSES-GA-2013-612669. References [1] F.J. Arago´n Artacho, J.M. Borwein and M.K. Tam, Recent results on Douglas-Rachford methods for combinatorial optimization problems, Journal of Optimizaion Theory and Applications 163 (2014), 1–30. [2] F.J. Arago´n Artacho and R. Campoy, A new projection method for ﬁnding the closest point in the intersection of convex sets, to appear in Computational Optimization and Applications (2017). https://doi.org/10.1007/s10589-017-9942-5 (34pp). [3] J.-B. Baillon, R.E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston Journal of Mathematics 4 (1978), 1–9. [4] H.H. Bauschke and J.M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review 38 (1996), 367–426. [5] H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Op- erator Theory in Hilbert Spaces, Second Edition, Springer International Publishing AG, 2017. [6] H.H. Bauschke and W.M. Moursi, On the Douglas-Rachford algorithm, Mathematical Programming, Series A 164 (2017), 263–284. [7] J.M. Borwein and M.K. Tam, A cyclic Douglas-Rachford iteration scheme, Journal of Optimizaion Theory and Applications 160 (2014), 1–29. [8] R.E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston Journal of Mathematics 3 (1977), 459–470. 13

text: Acknowledgments.

text: We

text: thank

text: Prof.

text: Touﬁk

text: Mansour

text: for

text: help

text: in

text: for-

text: mulating

text: our

text: ideas,

text: Yehuda

text: Zur

text: for

text: the

text: meticulous

text: work

text: with

text: Matlab

text: and

text: Raﬁq

text: Mansour

text: for

text: enlightening

text: comments

text: about

text: the

text: manuscript.

text: The

text: ﬁrst

text: author’s

text: work

text: was

text: supported

text: by

text: research

text: grant

text: no.

text: of

text: the

text: United

text: States-Israel

text: Binational

text: Science

text: Foundation

text: (BSF).

text: The

text: second

text: author’s

text: work

text: was

text: supported

text: by

text: the

text: EU

text: IRSES

text: program

text: STREVCOMS,

text: grant

text: no.

text: References

text: F.J.

text: Arago´n

text: Artacho,

text: J.M.

text: Borwein

text: and

text: M.K.

text: Tam,

text: Recent

text: results

text: on

text: Douglas-Rachford

text: methods

text: for

text: combinatorial

text: optimization

text: problems,

text: Journal

text: of

text: Optimizaion

text: Theory

text: and

text: Applications

text: F.J.

text: Arago´n

text: Artacho

text: and

text: R.

text: Campoy,

text: A

text: new

text: projection

text: method

text: for

text: ﬁnding

text: the

text: closest

text: point

text: in

text: the

text: intersection

text: of

text: convex

text: sets,

text: to

text: appear

text: in

text: Computational

text: Optimization

text: and

text: Applications

text: J.-B.

text: Baillon,

text: R.E.

text: Bruck

text: and

text: S.

text: Reich,

text: On

text: the

text: asymptotic

text: behavior

text: of

text: nonexpansive

text: mappings

text: and

text: semigroups

text: in

text: Banach

text: spaces,

text: Houston

text: Journal

text: of

text: Mathematics

text: H.H.

text: Bauschke

text: and

text: J.M.

text: Borwein,

text: On

text: projection

text: algorithms

text: for

text: solving

text: convex

text: feasibility

text: problems,

text: SIAM

text: Review

text: H.H.

text: Bauschke

text: and

text: P.L.

text: Combettes,

text: Convex

text: Analysis

text: and

text: Monotone

text: Op-

text: erator

text: Theory

text: in

text: Hilbert

text: Spaces,

text: Second

text: Edition,

text: Springer

text: International

text: Publishing

text: AG,

text: H.H.

text: Bauschke

text: and

text: W.M.

text: Moursi,

text: On

text: the

text: Douglas-Rachford

text: algorithm,

text: Mathematical

text: Programming,

text: Series

text: A

text: J.M.

text: Borwein

text: and

text: M.K.

text: Tam,

text: A

text: cyclic

text: Douglas-Rachford

text: iteration

text: scheme,

text: Journal

text: of

text: Optimizaion

text: Theory

text: and

text: Applications

text: R.E.

text: Bruck

text: and

text: S.

text: Reich,

text: Nonexpansive

text: projections

text: and

text: resolvents

text: of

text: accretive

text: operators

text: in

text: Banach

text: spaces,

text: Houston

text: Journal

text: of

text: Mathematics