text: of treatments across blocks. They further show that, for experiments with non-negligible eﬀect sizes, the Poisson GLMM-based optimal designs may be substantially more eﬃcient than optimal designs from the classical setting assuming additivity. With these methods in hand, experimenters will be enabled in correctly answering their research questions with minimum eﬀort, subjects, or other resources. 2 Models In this section we introduce generalised linear mixed models as an extension of both generalised linear models and of linear mixed models. As we progress to derive expressions for design optimality criteria, we observe that features of eﬃcient designs for both generalised linear models and of linear mixed models are present in eﬃcient designs for generalised linear mixed models. 2.1 The generalised linear model Consider an experiment in which t treatments are arranged in a completely randomised design comprising n × experimental units, where n is a multiple of t. We deﬁne a linear model for an n 1 vector of observations as y = Xβ + e, (1) × where β = (α, τT)T is a p 1 vector of parameters containing the ﬁxed eﬀects α, denoting the overall mean × of the observations, and τ = (τ , . . . , τ )T, denoting the t treatment eﬀect parameters. The n p treatment 1 t design matrix, X, characterises the allocation of treatments to experimental units and, therefore, the ﬁxed × eﬀect parameters associated with each observation in y. The n 1 vector of residual errors, e, is assumed to be independently and identically distributed normal with constant variance. If the response variable does not give rise to this error distribution, then an alternative model needs to be considered. One such alternative is the generalised linear model (GLM). GLMs are used when the responses in y are assumed to arise from a distribution belonging to the exponential family of distributions. Such distributions include, for example, the binomial distribution for binary responses and the Poisson distribution for count responses. In a GLM· the ve·ctor of mean responses, µ, and the linear (cid:48) − predictors, Xβ, are related by a canonical link function, g( ) = b ( ) 1. This gives rise to the model form η = g(µ) = Xβ. (2) (cid:48) (cid:48)(cid:48) It follows that E(y) = µ = b (θ) and Var(y) = b (θ)a(φ), where b(θ) and a(φ) denote functions of the natural parameter, θ, and the dispersion parameter, φ, respectively, for the independent observations in y. (cid:98) 2.1.1 Fisher information matrix for GLMs (cid:98) The maximum likelihood estimator, β, of the ﬁxed model eﬀects, β, in a GLM is asymptotically normally distributed. The covariance ma(cid:40)trix of β is (cid:41)the inv(cid:40)erse of th(cid:104)e(cid:0)Fisher informa(cid:1)t(cid:105)io(cid:41)n matrix which is derived from the log-likelihood function of the GLM, i.e. − − ∂2(cid:96)(θ) − ∂2(cid:96) θ g 1(Xβ; y, φ) M(ξ, β) = E = E = XTWX, ∂β2 ∂β2 − where W = (DV D) 1, V = diag[Var(y )] and D = diag[∂η /∂µ ]. Hence, the information matrix depends (cid:48) i i i on the link function, since ∂η /∂µ = g (µ ), the design, ξ, through the design matrix X and the parameters i i i in β. 2.1.2 Poisson GLM Responses in many modern experiments are drawn from distributions such as the one- and two-parameter exponential families, e.g., RNA sequence counts from a negative binomial distribution. Here we focus exclusively on experiments in which responses from ∼the ith treatment group are counts independe·ntly draw·n from the one-parameter Poisson distribution, i.e. y Poisson(λ ), with canonical link function g( ) = log( ). i i 2.2 The linear mixed model Consider now an experiment in which t treatments are arranged in a generalised block design with b blocking × factors. We deﬁne the linear mixed model (LMM) for an n 1 vector of observations, y, using the general matrix notation 3 ##LTLine## ##LTLine##

text: of

text: treatments

text: across

text: blocks.

text: They

text: further

text: show

text: that,

text: for

text: experiments

text: with

text: non-negligible

text: eﬀect

text: sizes,

text: the

text: Poisson

text: GLMM-based

text: optimal

text: designs

text: may

text: be

text: substantially

text: more

text: eﬃcient

text: than

text: optimal

text: designs

text: from

text: the

text: classical

text: setting

text: assuming

text: additivity.

text: With

text: these

text: methods

text: in

text: hand,

text: experimenters

text: will

text: be

text: enabled

text: in

text: correctly

text: answering

text: their

text: research

text: questions

text: with

text: minimum

text: eﬀort,

text: subjects,

text: or

text: other

text: resources.

label: section

text: Models

text: In

text: this

text: section

text: we

text: introduce

text: generalised

text: linear

text: mixed

text: models

text: as

text: an

text: extension

text: of

text: both

text: generalised

text: linear

text: models

text: and

text: of

text: linear

text: mixed

text: models.

text: As

text: we

text: progress

text: to

text: derive

text: expressions

text: for

text: design

text: optimality

text: criteria,

text: we

text: observe

text: that

text: features

text: of

text: eﬃcient

text: designs

text: for

text: both

text: generalised

text: linear

text: models

text: and

text: of

text: linear

text: mixed

text: models

text: are

text: present

text: in

text: eﬃcient

text: designs

text: for

text: generalised

text: linear

text: mixed

text: models.

text: The

text: generalised

text: linear

text: model

text: Consider

text: an

text: experiment

text: in

text: which

text: t

text: treatments

text: are

text: arranged

text: in

text: a

text: completely

text: randomised

text: design

text: comprising

text: n

text: experimental

text: units,

text: where

text: n

text: is

text: a

text: multiple

text: of

text: t.

text: We

text: deﬁne

text: a

text: linear

text: model

text: for

text: an

text: n

text: vector

text: of

text: observations

text: as

text: y

text: Xβ

text: e,

text: where

text: β

text: (α,

text: τT)T

text: is

text: a

text: p

text: vector

text: of

text: parameters

text: containing

text: the

text: ﬁxed

text: eﬀects

text: α,

text: denoting

text: the

text: overall

text: mean

text: of

text: the

text: observations,

text: and

text: τ

text: (τ

text: τ

text: )T,

text: denoting

text: the

text: t

text: treatment

text: eﬀect

text: parameters.

text: The

text: n

text: p

text: treatment

text: t

text: design

text: matrix,

text: X,

text: characterises

text: the

text: allocation

text: of

text: treatments

text: to

text: experimental

text: units

text: and,

text: therefore,

text: the

text: ﬁxed

text: eﬀect

text: parameters

text: associated

text: with

text: each

text: observation

text: in

text: y.

text: The

text: n

text: vector

text: of

text: residual

text: errors,

text: e,

text: is

text: assumed

text: to

text: be

text: independently

text: and

text: identically

text: distributed

text: normal

text: with

text: constant

text: variance.

text: If

text: the

text: response

text: variable

text: does

text: not

text: give

text: rise

text: to

text: this

text: error

text: distribution,

text: then

text: an

text: alternative

text: model

text: needs

text: to

text: be

text: considered.

text: One

text: such

text: alternative

text: is

text: the

text: generalised

text: linear

text: model

text: (GLM).

text: GLMs

text: are

text: used

text: when

text: the

text: responses

text: in

text: y

text: are

text: assumed

text: to

text: arise

text: from

text: a

text: distribution

text: belonging

text: to

text: the

text: exponential

text: family

text: of

text: distributions.

text: Such

text: distributions

text: include,

text: for

text: example,

text: the

text: binomial

text: distribution

text: for

text: binary

text: responses

text: and

text: the

text: Poisson

text: distribution

text: for

text: count

text: responses.

text: In

text: a

text: GLM·

text: the

text: ve·ctor

text: of

text: mean

text: responses,

text: µ,

text: and

text: the

text: linear

text: predictors,

text: Xβ,

text: are

text: related

text: by

text: a

text: canonical

text: link

text: function,

text: g(

text: b

text: This

text: gives

text: rise

text: to

text: the

text: model

text: form

text: η

text: g(µ)

text: Xβ.

text: It

text: follows

text: that

text: E(y)

text: µ

text: b

text: (θ)

text: and

text: Var(y)

text: b

text: (θ)a(φ),

text: where

text: b(θ)

text: and

text: a(φ)

text: denote

text: functions

text: of

text: the

text: natural

text: parameter,

text: θ,

text: and

text: the

text: dispersion

text: parameter,

text: φ,

text: respectively,

text: for

text: the

text: independent

text: observations

text: in

text: y.

text: Fisher

text: information

text: matrix

text: for

text: GLMs

text: The

text: maximum

text: likelihood

text: estimator,

text: β,

text: of

text: the

text: ﬁxed

text: model

text: eﬀects,

text: β,

text: in

text: a

text: GLM

text: is

text: asymptotically

text: normally

text: distributed.

text: The

text: covariance

text: of

text: β

text: is

text: of

text: matrix

text: which

text: is

text: derived

text: from

text: the

text: log-likelihood

text: function

text: of

text: the

text: GLM,

text: i.e.

text: θ

text: g

text: y,

text: φ)

text: M(ξ,

text: β)

text: E

text: E

text: XTWX,

text: ∂β2

text: ∂β2

text: where

text: W

text: (DV

text: D)

text: V

text: diag[Var(y

text: and

text: D

text: diag[∂η

text: /∂µ

text: Hence,

text: the

text: information

text: matrix

text: depends

text: i

text: i

text: i

text: on

text: the

text: link

text: function,

text: since

text: ∂η

text: /∂µ

text: g

text: (µ

text: the

text: design,

text: ξ,

text: through

text: the

text: design

text: matrix

text: X

text: and

text: the

text: parameters

text: i

text: i

text: i

text: in

text: β.

text: Poisson

text: GLM

text: Responses

text: in

text: many

text: modern

text: experiments

text: are

text: drawn

text: from

text: distributions

text: such

text: as

text: the

text: one-

text: and

text: two-parameter

text: exponential

text: families,

text: e.g.,

text: RNA

text: sequence

text: counts

text: from

text: a

text: negative

text: binomial

text: distribution.

text: Here

text: we

text: focus

text: exclusively

text: on

text: experiments

text: in

text: which

text: responses

text: from

text: ∼the

text: ith

text: treatment

text: group

text: are

text: counts

text: independe·ntly

text: draw·n

text: from

text: the

text: one-parameter

text: Poisson

text: distribution,

text: i.e.

text: y

text: Poisson(λ

text: with

text: canonical

text: link

text: function

text: g(

text: log(

text: i

text: i

text: The

text: linear

text: mixed

text: model

text: Consider

text: now

text: an

text: experiment

text: in

text: which

text: t

text: treatments

text: are

text: arranged

text: in

text: a

text: generalised

text: block

text: design

text: with

text: b

text: blocking

text: factors.

text: We

text: deﬁne

text: the

text: linear

text: mixed

text: model

text: (LMM)

text: for

text: an

text: n

text: vector

text: of

text: observations,

text: y,

text: using

text: the

text: general

text: matrix

text: notation

text: ##LTLine##

text: ##LTLine##


text: A full derivation is provided in the Supplementary Material. The following example illustrates the structure of these objective functions. Example 1 Suppose that we wish to ﬁnd the optimal arrangement of t = 3 treatments in b = 2 blocks of size k = 3, and will observe a count response that we wish to model by the | ∼ Poisson GLMM η = α + τ + u + e , where y u , e Poisson(exp(η )) for i = 1, 2, 3, ij R(i,j) i ij ij i ij ij and j = 1, 2. ⊗ The components of the Fisher information matrix, deﬁned in (10), are Z = (I j , I ), W = (cid:20) (cid:21)2 3 6 diag(λ ), and G = diag(σ2jT, σ2jT). It follows that R(i,j) u 2 6  ⊗ | ⊗ | (1/σ2)I 0  M = [I j I ]Tdiag(λ )[I j I ] + u 2 22 2 3 6 R(i) 2 3 6 0 (1/σ2)I6     λ · 0 λ λ λ 0 0 0   R(1, ) R(1,1) R(1,2) R(1,3)   0 λ · 0 0 0 λ λ λ   R(2, ) R(2,1) R(2,2) R(2,3)   λ 0 λ 0 0 0 0 0   R(1,1) R(1,1)   λ 0 0 λ 0 0 0 0  =  R(1,2) R(1,2)   λ 0 0 0 λ 0 0 0  R(1,3) R(1,3) 0 λ 0 0 0 λ 0 0 R(2,1) R(2,1) (cid:20) (cid:21) 0 λ 0 0 0 0 λ 0 R(2,2) R(2,2) 0 λ 0 0 0 0 0 λ R(2,3) R(2,3) (1/σ2)I 0 + u 2 0 (1/σ2)I 6 (cid:80) (cid:34) (cid:35) where λ · = k λ and R(i, ) j=1 R(i,j) σ2 σ2 σ2 (cid:96) = b− , b− , b− . i σ2 + λ 1 σ2 + λ 1 σ2 + λ 1 R(i,1) R(i,2) R(i,3) The ith sub-matrix of the marginal information matrix, corresponding to block i in the design, is given by (cid:32) (cid:33) Mβmarg(ξ, β, σ, σb)i (cid:16) (cid:17) 1 − (cid:96) (cid:96)T = diag − 1 1  σ2 + λ 1 σ2 1 + ((cid:96)1/2)T(cid:96)1/2  R(1,j) b 1  1    1− 0 0  σ2+λ 1 R(1,1) = 0 σ2+λ1−1 0 R(1,2) 0 0 1− σ2+λ 1 R(1,3) − 1 ×   σ2 σ2 σ2 1 + −b + −b + −b  σ2+λ 1 σ2+λ 1 σ2+λ 1   R(1,1) R(1,2) R(1,3)   σ2 σ2 σ2   −b − b − − b −   (σ2+λ 1 )2 (σ2+λ 1 )(σ2+λ 1 ) (σ2+λ 1 )(σ2+λ 1 )  R(1,1) R(1,1) R(1,2) R(1,1) R(1,3) σ2 σ2 σ2 − b − −b − b − , (σ2+λ 1 )(σ2+λ 1 ) (σ2+λ 1 )2 (σ2+λ 1 )(σ2+λ 1 ) R(1,2) R(1,1) R(1,2) R(1,2) R(1,3) σ2 σ2 σ2 − b − − b − −b (σ2+λ 1 )(σ2+λ 1 ) (σ2+λ 1 )(σ2+λ 1 ) (σ2+λ 1 )2 R(1,3) R(1,1) R(1,3) R(1,2) R(1,3) with the corresponding structure of the second block taking a similar form. To investigate the optimal designs that are produced, Table 1 gives the D –optimal and C– A optimal designs for a variety of treatment means and and values of σ2, with σ2 = 0.25. We b observe that as the size of the block variance increases relative to the treatment means, the optimal design becomes more balanced. For small block variances the eﬀect of the diﬀerent treatment variances becomes more dominant in determining the optimal design. [Table 1 about here.] 7 ##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##

text: A

label: paragraph

text: full

text: derivation

text: is

text: provided

text: in

text: the

text: Supplementary

text: Material.

text: The

text: following

text: example

text: illustrates

text: the

text: structure

text: of

text: these

text: objective

text: functions.

text: Example

text: Suppose

text: that

text: we

text: wish

text: to

text: ﬁnd

text: the

text: optimal

text: arrangement

text: of

text: t

text: treatments

text: in

text: b

text: blocks

text: of

text: size

text: k

text: and

text: will

text: observe

text: a

text: count

text: response

text: that

text: we

text: wish

text: to

text: model

text: by

text: the

text: Poisson

text: GLMM

text: η

text: α

text: τ

text: u

text: e

text: where

text: y

text: u

text: e

text: Poisson(exp(η

text: for

text: i

text: ij

text: R(i,j)

text: i

text: ij

text: ij

text: i

text: ij

text: ij

text: and

text: j

text: The

text: components

text: of

text: the

text: Fisher

text: information

text: matrix,

text: deﬁned

text: in

text: are

text: Z

text: (I

text: j

text: I

text: W

text: diag(λ

text: and

text: G

text: It

text: follows

text: that

text: R(i,j)

text: u

text: M

text: [I

text: j

text: I

text: ]Tdiag(λ

text: )[I

text: j

text: I

text: u

text: 2

text: R(i)

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: λ

text: u

text: where

text: λ

text: k

text: λ

text: and

text: R(i,

text: R(i,j)

text: σ2

text: σ2

text: σ2

text: b−

text: b−

text: b−

text: i

text: σ2

text: λ

text: σ2

text: λ

text: σ2

text: λ

text: The

text: ith

text: sub-matrix

text: of

text: the

text: marginal

text: information

text: matrix,

text: corresponding

text: to

text: block

text: i

text: in

text: the

text: design,

text: is

text: given

text: by

text: Mβmarg(ξ,

text: β,

text: σ,

text: σb)i

text: diag

text: σ2

text: λ

text: σ2

text: b

text: 1−

text: σ2+λ

text: 1−

text: σ2+λ

text: σ2

text: σ2

text: σ2

text: −b

text: −b

text: −b

text: σ2+λ

text: σ2+λ

text: σ2+λ

text: σ2

text: σ2

text: σ2

text: −b

text: b

text: b

text: σ2

text: σ2

text: σ2

text: b

text: −b

text: b

text: σ2

text: σ2

text: σ2

text: b

text: b

text: −b

text: with

text: the

text: corresponding

text: structure

text: of

text: the

text: second

text: block

text: taking

text: a

text: similar

text: form.

text: To

text: investigate

text: the

text: optimal

text: designs

text: that

text: are

text: produced,

text: Table

text: gives

text: the

text: D

text: –optimal

text: and

text: C–

text: A

text: optimal

text: designs

text: for

text: a

text: variety

text: of

text: treatment

text: means

text: and

text: and

text: values

text: of

text: with

text: σ2

text: We

text: b

text: observe

text: that

text: as

text: the

text: size

text: of

text: the

text: block

text: variance

text: increases

text: relative

text: to

text: the

text: treatment

text: means,

text: the

text: optimal

text: design

text: becomes

text: more

text: balanced.

text: For

text: small

text: block

text: variances

text: the

text: eﬀect

text: of

text: the

text: diﬀerent

text: treatment

text: variances

text: becomes

text: more

text: dominant

text: in

text: determining

text: the

text: optimal

text: design.

text: [Table

text: about

text: here.]

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 4 Locally optimal block designs using simulated annealing Generating D –optimal or C–optimal block designs requires an iterative search algorithm for which we have A elected to use simulated annealing (SA) (Kirkpatrick et al., 1983). Since the design criterion to be minimised for both D – and C–optimality is the generalised variance of Bβ, which for count data is functionally A dependent on the treatment group means, we constrain the SA algorithm to search for locally optimal designs, i.e. designs that are optimal for a set of point priors for the expected treatment counts, λ , h h = 1, . . . , t, and the variance components between blocks, σ2, and residuals, σ2. u Three key inputs are required by the SA algorithm: a starting design, an objective function and candidate generator procedure, which we now discuss. The SA algorithm is initialised with a starting design, D , generated by randomly assigning treatments to 0 blocks with objective function value O(D ). At each iteration a new design, D , is generated by random 0 i exchanges of treatments in randomly selected experimental units in design D − at the previous iteration, i 1 where D − = D at the ﬁrst iteration. Since the SA algorithm searches for candidate designs which minimise i 1 0 the objective function, D always replaces D − if O(D ) < O(D − ), and has a small probability of replacing i i 1 i i 1 D − even it is a slightly worse design. The acceptance probability of a worse design depends on the so-called i 1 temperature of the algorithm, which is initially set high to enable the algorithm to escape local optima in early iterations. As the iterations continue the temperature gradually cools, and with it the probability of accepting worse designs. In this way, the algorithm converges to the global optimum within the constrained set of competing designs. As discussed in section , we consider two objective functions based on the D –optimality and C–optimality A criteria, both deﬁned in Atkinson et al. (2007). The goal is search the space of candidate designs that minimise these functions. In contrast to block designs from the classical setting, where optimal eﬃciency is achieved by allocating treatments as equally as possible among blocks, optimal designs based on responses drawn from Poisson distributions have treatment replication inversely proportional to their treatment means. Our selection of a starting design, therefore, makes no assumption of equal replication or balance. Consequently, a candidate design generating procedure which makes random exchanges of treatments between blocks is unsatisfactory. Instead, we propose starting with a random design and then substitute the treatment assigned to a randomly chosen experimental unit in the design with a randomly chosen treatment from the treatment set. Our preliminary testing of this strategy showed that, for some sets of design parameters (i.e. number of treatments, blocks and block size) and point priors, the SA algorithm converged very slowly and sometimes would get caught in local minima, even for reasonably high initial temperatures. To overcome these limita- (cid:62) tions, our candidate design generating procedure includes an option for m 1 substitutions to be made at { ≥ · · · ≥ · · · } each iteration. A vector of probabilities P = (p , . . . , p ) : p p and p + + p = 1 , where 1 m 1 m 1 m p denotes the probability that m experimental units will have treatment substitutions at a given iteration. m The SA algorithm for performing the optimisation described above is implemented in the designGLMM R package which is available from the Comprehensive R Archive Network. We now consider two examples where we ﬁnd optimal block designs for experiments in which responses are drawn from a Poisson distribution using our SA algorithm. 5 Examples 5.1 Diﬀerential striatal gene expression between two strains of mouse We consider a comparative experiment to assess the level of diﬀerential striatal gene expression between two mice strains using the Illumina GAIIx next-generation sequencing (NGS) platform (Bottomly et al., 2011). cDNA, copied from ampliﬁed RNA isolated from cells in the striatum of twenty-one mice – ten from the C57BL/6J strain (strain 1) and eleven from the DBA/2J strain (strain 2) – was loaded into individual lanes (plots) of three ﬂow cells (blocks) for sequencing. The design used by Bottomly et al. (2011) comprised two ﬂow cells with three replicates of strain 1 and four replicates of strain 2, and a third ﬂow cell with four replicates of strain 1 and three replicates of strain 2. The question that we wish to answer is whether this design, or a diﬀerent design with the same number of samples, is optimal for the estimation of strain eﬀects. Table 2 presents counts from four of the 36536 identiﬁed genes (labelled A = ENSMUSG00000046994, B = ENSMUSG00000039967, C = ENSMUSG00000050141 and D = ENSMUSG00000033826) in this experiment, selected to represent the diﬀerent eﬀect sizes observed across the entire data set (see ReCount resource (Frazee et al., 2011)). A per-gene GLMM of the form presented in (9) was ﬁtted to the count data yielding the parameter estimates shown in Table 3. We now use the eﬀect sizes obtained from these estimates as point priors in searching for optimal designs for the estimation of the strain eﬀect. Table 3 shows that the size of the strain eﬀect in genes A and B are quite small, with the relative abundances 8

text: Locally

text: optimal

text: block

text: designs

text: using

text: simulated

text: annealing

text: Generating

text: D

text: –optimal

text: or

text: C–optimal

text: block

text: designs

text: requires

text: an

text: iterative

text: search

text: algorithm

text: for

text: which

text: we

text: have

text: A

text: elected

text: to

text: use

text: simulated

text: annealing

text: (SA)

text: (Kirkpatrick

text: et

text: al.,

text: Since

text: the

text: design

text: criterion

text: to

text: be

text: minimised

text: for

text: both

text: D

text: and

text: C–optimality

text: is

text: the

text: generalised

text: variance

text: of

text: Bβ,

text: which

text: for

text: count

text: data

text: is

text: functionally

text: A

text: dependent

text: on

text: the

text: treatment

text: group

text: means,

text: we

text: constrain

text: the

text: SA

text: algorithm

text: to

text: search

text: for

text: locally

text: optimal

text: designs,

text: i.e.

text: designs

text: that

text: are

text: optimal

text: for

text: a

text: set

text: of

text: point

text: priors

text: for

text: the

text: expected

text: treatment

text: counts,

text: λ

text: h

text: h

text: t,

text: and

text: the

text: variance

text: components

text: between

text: blocks,

text: and

text: residuals,

text: u

text: Three

text: key

text: inputs

text: are

text: required

text: by

text: the

text: SA

text: algorithm:

text: a

text: starting

text: design,

text: an

text: objective

text: function

text: and

text: candidate

text: generator

text: procedure,

text: which

text: we

text: now

text: discuss.

text: The

text: SA

text: algorithm

text: is

text: initialised

text: with

text: a

text: starting

text: design,

text: D

text: generated

text: by

text: randomly

text: assigning

text: treatments

text: to

text: blocks

text: with

text: objective

text: function

text: value

text: O(D

text: At

text: each

text: iteration

text: a

text: new

text: design,

text: D

text: is

text: generated

text: by

text: random

text: i

text: exchanges

text: of

text: treatments

text: in

text: randomly

text: selected

text: experimental

text: units

text: in

text: design

text: D

text: at

text: the

text: previous

text: iteration,

text: i

text: where

text: D

text: D

text: at

text: the

text: ﬁrst

text: iteration.

text: Since

text: the

text: SA

text: algorithm

text: searches

text: for

text: candidate

text: designs

text: which

text: minimise

text: i

text: the

text: objective

text: function,

text: D

text: always

text: replaces

text: D

text: if

text: O(D

text: O(D

text: and

text: has

text: a

text: small

text: probability

text: of

text: replacing

text: i

text: i

text: i

text: i

text: D

text: even

text: it

text: is

text: a

text: slightly

text: worse

text: design.

text: The

text: acceptance

text: probability

text: of

text: a

text: worse

text: design

text: depends

text: on

text: the

text: so-called

text: i

text: temperature

text: of

text: the

text: algorithm,

text: which

text: is

text: initially

text: set

text: high

text: to

text: enable

text: the

text: algorithm

text: to

text: escape

text: local

text: optima

text: in

text: early

text: iterations.

text: As

text: the

text: iterations

text: continue

text: the

text: temperature

text: gradually

text: cools,

text: and

text: with

text: it

text: the

text: probability

text: of

text: accepting

text: worse

text: designs.

text: In

text: this

text: way,

text: the

text: algorithm

text: converges

text: to

text: the

text: global

text: optimum

text: within

text: the

text: constrained

text: set

text: of

text: competing

text: designs.

text: As

text: discussed

text: in

text: section

text: we

text: consider

text: two

text: objective

text: functions

text: based

text: on

text: the

text: D

text: –optimality

text: and

text: C–optimality

text: A

text: criteria,

text: both

text: deﬁned

text: in

text: Atkinson

text: et

text: al.

text: The

text: goal

text: is

text: search

text: the

text: space

text: of

text: candidate

text: designs

text: that

text: minimise

text: these

text: functions.

text: In

text: contrast

text: to

text: block

text: designs

text: from

text: the

text: classical

text: setting,

text: where

text: optimal

text: eﬃciency

text: is

text: achieved

text: by

text: allocating

text: treatments

text: as

text: equally

text: as

text: possible

text: among

text: blocks,

text: optimal

text: designs

text: based

text: on

text: responses

text: drawn

text: from

text: Poisson

text: distributions

text: have

text: treatment

text: replication

text: inversely

text: proportional

text: to

text: their

text: treatment

text: means.

text: Our

text: selection

text: of

text: a

text: starting

text: design,

text: therefore,

text: makes

text: no

text: assumption

text: of

text: equal

text: replication

text: or

text: balance.

text: Consequently,

text: a

text: candidate

text: design

text: generating

text: procedure

text: which

text: makes

text: random

text: exchanges

text: of

text: treatments

text: between

text: blocks

text: is

text: unsatisfactory.

text: Instead,

text: we

text: propose

text: starting

text: with

text: a

text: random

text: design

text: and

text: then

text: substitute

text: the

text: treatment

text: assigned

text: to

text: a

text: randomly

text: chosen

text: experimental

text: unit

text: in

text: the

text: design

text: with

text: a

text: randomly

text: chosen

text: treatment

text: from

text: the

text: treatment

text: set.

text: Our

text: preliminary

text: testing

text: of

text: this

text: strategy

text: showed

text: that,

text: for

text: some

text: sets

text: of

text: design

text: parameters

text: (i.e.

text: number

text: of

text: treatments,

text: blocks

text: and

text: block

text: size)

text: and

text: point

text: priors,

text: the

text: SA

text: algorithm

text: converged

text: very

text: slowly

text: and

text: sometimes

text: would

text: get

text: caught

text: in

text: local

text: minima,

text: even

text: for

text: reasonably

text: high

text: initial

text: temperatures.

text: To

text: overcome

text: these

text: limita-

text: tions,

text: our

text: candidate

text: design

text: generating

text: procedure

text: includes

text: an

text: option

text: for

text: m

text: substitutions

text: to

text: be

text: made

text: at

text: each

text: iteration.

text: A

text: vector

text: of

text: probabilities

text: P

text: (p

text: p

text: p

text: p

text: and

text: p

text: p

text: where

text: m

text: m

text: m

text: p

text: denotes

text: the

text: probability

text: that

text: m

text: experimental

text: units

text: will

text: have

text: treatment

text: substitutions

text: at

text: a

text: given

text: iteration.

text: m

text: The

text: SA

text: algorithm

text: for

text: performing

text: the

text: optimisation

text: described

text: above

text: is

text: implemented

text: in

text: the

text: designGLMM

text: R

text: package

text: which

text: is

text: available

text: from

text: the

text: Comprehensive

text: R

text: Archive

text: Network.

text: We

text: now

text: consider

text: two

text: examples

text: where

text: we

text: ﬁnd

text: optimal

text: block

text: designs

text: for

text: experiments

text: in

text: which

text: responses

text: are

text: drawn

text: from

text: a

text: Poisson

text: distribution

text: using

text: our

text: SA

text: algorithm.

text: Examples

text: Diﬀerential

text: striatal

text: gene

text: expression

text: between

text: two

text: strains

text: of

text: mouse

text: We

text: consider

text: a

text: comparative

text: experiment

text: to

text: assess

text: the

text: level

text: of

text: diﬀerential

text: striatal

text: gene

text: expression

text: between

text: two

text: mice

text: strains

text: using

text: the

text: Illumina

text: GAIIx

text: next-generation

text: sequencing

text: (NGS)

text: platform

text: (Bottomly

text: et

text: al.,

text: cDNA,

text: copied

text: from

text: ampliﬁed

text: RNA

text: isolated

text: from

text: cells

text: in

text: the

text: striatum

text: of

text: twenty-one

text: mice

text: ten

text: from

text: the

text: strain

text: (strain

text: and

text: eleven

text: from

text: the

text: strain

text: (strain

text: was

text: loaded

text: into

text: individual

text: lanes

text: (plots)

text: of

text: three

text: ﬂow

text: cells

text: (blocks)

text: for

text: sequencing.

text: The

text: design

text: used

text: by

text: Bottomly

text: et

text: al.

text: comprised

text: two

text: ﬂow

text: cells

text: with

text: three

text: replicates

text: of

text: strain

text: and

text: four

text: replicates

text: of

text: strain

text: and

text: a

text: third

text: ﬂow

text: cell

text: with

text: four

text: replicates

text: of

text: strain

text: and

text: three

text: replicates

text: of

text: strain

text: The

text: question

text: that

text: we

text: wish

text: to

text: answer

text: is

text: whether

text: this

text: design,

text: or

text: a

text: diﬀerent

text: design

text: with

text: the

text: same

text: number

text: of

text: samples,

text: is

text: optimal

text: for

text: the

text: estimation

text: of

text: strain

text: eﬀects.

text: Table

text: presents

text: counts

text: from

text: four

text: of

text: the

text: identiﬁed

text: genes

text: (labelled

text: A

text: B

text: C

text: and

text: D

text: in

text: this

text: experiment,

text: selected

text: to

text: represent

text: the

text: diﬀerent

text: eﬀect

text: sizes

text: observed

text: across

text: the

text: entire

text: data

text: set

text: (see

text: ReCount

text: resource

text: (Frazee

text: et

text: al.,

text: A

text: per-gene

text: GLMM

text: of

text: the

text: form

text: presented

text: in

text: was

text: ﬁtted

text: to

text: the

text: count

text: data

text: yielding

text: the

text: parameter

text: estimates

text: shown

text: in

text: Table

text: We

text: now

text: use

text: the

text: eﬀect

text: sizes

text: obtained

text: from

text: these

text: estimates

text: as

text: point

text: priors

text: in

text: searching

text: for

text: optimal

text: designs

text: for

text: the

text: estimation

text: of

text: the

text: strain

text: eﬀect.

text: Table

text: shows

text: that

text: the

text: size

text: of

text: the

text: strain

text: eﬀect

text: in

text: genes

text: A

text: and

text: B

text: are

text: quite

text: small,

text: with

text: the

text: relative

text: abundances


text: 6 Discussion In this paper, we ﬁnd optimal designs for Poisson regression with a single unstructured treatment factor and a single variable that creates blocks of equal size. The methods discussed here are implemented in the R statistical software package designGLMM, which is available on the Comprehensive R Archive Network (cran.r-project.org) under a GPL3 license. We observe that for experiments where the treatment means are suﬃciently diﬀerent, and the block eﬀect is not dominant, the optimal designs diﬀer from those for linear models. This is because in a non–linear setting the Fisher information matrix, and hence the optimal designs, depend on the values of the model parameters. The optimal designs for generalised linear mixed models depend on the functional form of the model and the values of the model parameters. In this paper, we have used a Poisson–Lognormal model, as discussed in Stroup (2012) and Nettleton (2014). Many other model conﬁgurations are available for modelling count data, most notably the negative binomial model (Lawless (1987)). Hilbe (2011) presents yet other possibil- ities, including alternate mean–variance relationships and hurdle models. Use of these alternate modelling approaches may yield optimal designs for the estimation of treatment eﬀects which diﬀer from those based on the Poisson GLMM. The objective functions considered in this paper assume that the treatment eﬀects are of primary interest. Speciﬁcally, a set of linear combinations of treatment eﬀects are to be estimated with as small variance as possible, while being distinguishable from block eﬀects. In some experiments, researchers are also interested in estimating the block eﬀects eﬃciently, which may give rise to diﬀerent optimal designs. In Example 1 of Section , we considered the optimal design of a NGS experiment in which seven samples were placed onto individual lanes of three diﬀerent ﬂow cells. In generating optimal designs for this experiment, we considered only the variability between chips, and not between lanes. Auer and Doerge (2010) suggest that variation between lanes should also be a consideration. Furthermore, some NGS experiments use a process called barcoding to place multiple samples onto a single lane. This would suggest that more complex design structures, such as row–column designs, may be appropriate. Additional complications arise from the design of NGS experiments. For instance, Auer and Doerge (2010, Eq.3) use an oﬀset term, such as log(c ), to normalise the number of reads per lane, and is common practise ij in the modelling of NGS data (see, for example Mortazavi et al. 2008). We are currently looking at how we can address some of these issues in the optimal design of NGS exper- iments. Other areas which require further investigation include incorporating prior distributions for each of the model parameters to develop Bayesian optimal designs, and the investigation of alternative search algorithms that may be more eﬃcient than simulated annealing in ﬁnding optimal designs. Supplementary Materials Web Appendix A, referenced in Section , is available with this paper at the Biometrics website on Wiley Online Library. References Atkinson, A., Donev, A., and Tobias, R. (2007). Optimum experimental designs, with SAS. Oxford, UK: Oxford Univiversity Press. Auer, P. L. and Doerge, R. W. (2010). Statistical design and analysis of RNA sequencing data. Genetics 185, 405–416. Bottomly, D., Walter, N., Hunter, J., Darakjian, P., Kawane, S., Buck, K., Searles, R., Mooney, M., McWeeney, S., and Hitzemann, R. (2011). Evaluating gene expression in C57BL/6J and DBA/2J mouse striatum using RNA-seq and microarrays. PloS One 6, e17820. Cox, D. (1988). A note on design when response has an exponential family distribution. Biometrika 75, 161–164. Craig, D. W., Pearson, J. V., Szelinger, S., Sekar, A., Redman, M., Corneveaux, J. J., Pawlowski, T. L., Laub, T., Nunn, G., Stephan, D. A., et al. (2008). Identiﬁcation of genetic variants using bar-coded multiplexed sequencing. Nature Methods 5, 887–893. Fisher, R. A. (1926). The arrangement of ﬁeld experiments. Journal of the Ministry of Agriculture of Great Britain 33, 503–513. Frazee, A., Langmead, B., and Leek, J. (2011). Recount: a multi-experiment resource of analysis-ready RNA-seq gene count datasets. BMC Bioinformatics 12, 449. 10

text: Discussion

text: In

text: this

text: paper,

text: we

text: ﬁnd

text: optimal

text: designs

text: for

text: Poisson

text: regression

text: with

text: a

text: single

text: unstructured

text: treatment

text: factor

text: and

text: a

text: single

text: variable

text: that

text: creates

text: blocks

text: of

text: equal

text: size.

text: The

text: methods

text: discussed

text: here

text: are

text: implemented

text: in

text: the

text: R

text: statistical

text: software

text: package

text: designGLMM,

text: which

text: is

text: available

text: on

text: the

text: Comprehensive

text: R

text: Archive

text: Network

text: (cran.r-project.org)

text: under

text: a

text: license.

text: We

text: observe

text: that

text: for

text: experiments

text: where

text: the

text: treatment

text: means

text: are

text: suﬃciently

text: diﬀerent,

text: and

text: the

text: block

text: eﬀect

text: is

text: not

text: dominant,

text: the

text: optimal

text: designs

text: diﬀer

text: from

text: those

text: for

text: linear

text: models.

text: This

text: is

text: because

text: in

text: a

text: non–linear

text: setting

text: the

text: Fisher

text: information

text: matrix,

text: and

text: hence

text: the

text: optimal

text: designs,

text: depend

text: on

text: the

text: values

text: of

text: the

text: model

text: parameters.

text: The

text: optimal

text: designs

text: for

text: generalised

text: linear

text: mixed

text: models

text: depend

text: on

text: the

text: functional

text: form

text: of

text: the

text: model

text: and

text: the

text: values

text: of

text: the

text: model

text: parameters.

text: In

text: this

text: paper,

text: we

text: have

text: used

text: a

text: Poisson–Lognormal

text: model,

text: as

text: discussed

text: in

text: Stroup

text: and

text: Nettleton

text: Many

text: other

text: model

text: conﬁgurations

text: are

text: available

text: for

text: modelling

text: count

text: data,

text: most

text: notably

text: the

text: negative

text: binomial

text: model

text: (Lawless

text: Hilbe

text: presents

text: yet

text: other

text: possibil-

text: ities,

text: including

text: alternate

text: mean–variance

text: relationships

text: and

text: hurdle

text: models.

text: Use

text: of

text: these

text: alternate

text: modelling

text: approaches

text: may

text: yield

text: optimal

text: designs

text: for

text: the

text: estimation

text: of

text: treatment

text: eﬀects

text: which

text: diﬀer

text: from

text: those

text: based

text: on

text: the

text: Poisson

text: GLMM.

text: The

text: objective

text: functions

text: considered

text: in

text: this

text: paper

text: assume

text: that

text: the

text: treatment

text: eﬀects

text: are

text: of

text: primary

text: interest.

text: Speciﬁcally,

text: a

text: set

text: of

text: linear

text: combinations

text: of

text: treatment

text: eﬀects

text: are

text: to

text: be

text: estimated

text: with

text: as

text: small

text: variance

text: as

text: possible,

text: while

text: being

text: distinguishable

text: from

text: block

text: eﬀects.

text: In

text: some

text: experiments,

text: researchers

text: are

text: also

text: interested

text: in

text: estimating

text: the

text: block

text: eﬀects

text: eﬃciently,

text: which

text: may

text: give

text: rise

text: to

text: diﬀerent

text: optimal

text: designs.

text: In

text: Example

text: of

text: Section

text: we

text: considered

text: the

text: optimal

text: design

text: of

text: a

text: NGS

text: experiment

text: in

text: which

text: seven

text: samples

text: were

text: placed

text: onto

text: individual

text: lanes

text: of

text: three

text: diﬀerent

text: ﬂow

text: cells.

text: In

text: generating

text: optimal

text: designs

text: for

text: this

text: experiment,

text: we

text: considered

text: only

text: the

text: variability

text: between

text: chips,

text: and

text: not

text: between

text: lanes.

text: Auer

text: and

text: Doerge

text: suggest

text: that

text: variation

text: between

text: lanes

text: should

text: also

text: be

text: a

text: consideration.

text: Furthermore,

text: some

text: NGS

text: experiments

text: use

text: a

text: process

text: called

text: barcoding

text: to

text: place

text: multiple

text: samples

text: onto

text: a

text: single

text: lane.

text: This

text: would

text: suggest

text: that

text: more

text: complex

text: design

text: structures,

text: such

text: as

text: row–column

text: designs,

text: may

text: be

text: appropriate.

text: Additional

text: complications

text: arise

text: from

text: the

text: design

text: of

text: NGS

text: experiments.

text: For

text: instance,

text: Auer

text: and

text: Doerge

text: use

text: an

text: oﬀset

text: term,

text: such

text: as

text: log(c

text: to

text: normalise

text: the

text: number

text: of

text: reads

text: per

text: lane,

text: and

text: is

text: common

text: practise

text: ij

text: in

text: the

text: modelling

text: of

text: NGS

text: data

text: (see,

text: for

text: example

text: Mortazavi

text: et

text: al.

text: We

text: are

text: currently

text: looking

text: at

text: how

text: we

text: can

text: address

text: some

text: of

text: these

text: issues

text: in

text: the

text: optimal

text: design

text: of

text: NGS

text: exper-

text: iments.

text: Other

text: areas

text: which

text: require

text: further

text: investigation

text: include

text: incorporating

text: prior

text: distributions

text: for

text: each

text: of

text: the

text: model

text: parameters

text: to

text: develop

text: Bayesian

text: optimal

text: designs,

text: and

text: the

text: investigation

text: of

text: alternative

text: search

text: algorithms

text: that

text: may

text: be

text: more

text: eﬃcient

text: than

text: simulated

text: annealing

text: in

text: ﬁnding

text: optimal

text: designs.

text: Supplementary

text: Materials

text: Web

text: Appendix

text: A,

text: referenced

text: in

text: Section

text: is

text: available

text: with

text: this

text: paper

text: at

text: the

text: Biometrics

text: website

text: on

text: Wiley

text: Online

text: Library.

text: References

text: Atkinson,

text: A.,

text: Donev,

text: A.,

text: and

text: Tobias,

text: R.

text: Optimum

text: experimental

text: designs,

text: with

text: SAS.

text: Oxford,

text: UK:

text: Oxford

text: Univiversity

text: Press.

text: Auer,

text: P.

text: L.

text: and

text: Doerge,

text: R.

text: W.

text: Statistical

text: design

text: and

text: analysis

text: of

text: RNA

text: sequencing

text: data.

text: Genetics

text: Bottomly,

text: D.,

text: Walter,

text: N.,

text: Hunter,

text: J.,

text: Darakjian,

text: P.,

text: Kawane,

text: S.,

text: Buck,

text: K.,

text: Searles,

text: R.,

text: Mooney,

text: M.,

text: McWeeney,

text: S.,

text: and

text: Hitzemann,

text: R.

text: Evaluating

text: gene

text: expression

text: in

text: and

text: mouse

text: striatum

text: using

text: RNA-seq

text: and

text: microarrays.

text: PloS

text: One

text: Cox,

text: D.

text: A

text: note

text: on

text: design

text: when

text: response

text: has

text: an

text: exponential

text: family

text: distribution.

text: Biometrika

text: Craig,

text: D.

text: W.,

text: Pearson,

text: J.

text: V.,

text: Szelinger,

text: S.,

text: Sekar,

text: A.,

text: Redman,

text: M.,

text: Corneveaux,

text: J.

text: J.,

text: Pawlowski,

text: T.

text: L.,

text: Laub,

text: T.,

text: Nunn,

text: G.,

text: Stephan,

text: D.

text: A.,

text: et

text: al.

text: Identiﬁcation

text: of

text: genetic

text: variants

text: using

text: bar-coded

text: multiplexed

text: sequencing.

text: Nature

text: Methods

text: Fisher,

text: R.

text: A.

text: The

text: arrangement

text: of

text: ﬁeld

text: experiments.

text: Journal

text: of

text: the

text: Ministry

text: of

text: Agriculture

text: of

text: Great

text: Britain

text: Frazee,

text: A.,

text: Langmead,

text: B.,

text: and

text: Leek,

text: J.

text: Recount:

text: a

text: multi-experiment

text: resource

text: of

text: analysis-ready

text: RNA-seq

text: gene

text: count

text: datasets.

text: BMC

text: Bioinformatics


text: Unitarity limit !$ BBllaacckk--ddiisskk lliimmit !"#$ b2 Figure 2: Schematic impact-parameter dependence of the amplitude f(s, b) in the two cases of absorptive (solid line) and reﬂective (dashed line) forms of unitarization. √ not very signiﬁcant at s = 7 TeV, but we expect its increase at higher ener- gies. That means that the difﬁculties in the simultaneous description of the total cross-section and the slope parameter in absorptive (eikonal) models would become more noticeable with the collision energy growth. Indeed, widening distribution over impact parameter space leads to the narrowing of the corresponding distribution over transferred momentum. Thus, the two discussed options—saturation of the black disk limit or satu- ration of the unitarity limit—g√enerate different mechanisms of the total cross– section growth at the energies s > 7 TeV. Namely, saturation of the black disk limit assumes the growth of the total cross–section occurs due to increasing impact parameter values only, while the growth of total cross-section can, in fact, be due to both factors — increase of impact parameter values combined with an increase of the elastic scattering amplitude f(s, b)√with energy at ﬁxed b. Therefore, the measurements of the elastic scattering at s = 13 TeV (especially in the region − of large t) could be quite decisive regarding this aspect of asymptotic picture and would help to determine which scattering picture —absorptive or reﬂective — should be expected at the asymptotical energies. 6 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: Unitarity

font: default

text: limit

text: BBllaacckk--ddiisskk

text: lliimmit

text: Figure

text: Schematic

label: caption

text: impact-parameter

text: dependence

text: of

text: the

text: amplitude

text: f(s,

text: b)

text: in

text: the

text: two

text: cases

text: of

text: absorptive

text: (solid

text: line)

text: and

text: reﬂective

text: (dashed

text: line)

text: forms

text: of

text: unitarization.

label: list

text: not

text: very

text: signiﬁcant

text: at

text: s

text: TeV,

text: but

text: we

text: expect

text: its

text: increase

text: at

text: higher

text: ener-

text: gies.

text: That

text: means

text: that

text: the

text: difﬁculties

text: in

text: the

text: simultaneous

text: description

text: of

text: the

text: total

text: cross-section

text: and

text: the

text: slope

text: parameter

text: in

text: absorptive

text: (eikonal)

text: models

text: would

text: become

text: more

text: noticeable

text: with

text: the

text: collision

text: energy

text: growth.

text: Indeed,

text: widening

text: distribution

text: over

text: impact

text: parameter

text: space

text: leads

text: to

text: the

text: narrowing

text: of

text: the

text: corresponding

text: distribution

text: over

text: transferred

text: momentum.

text: Thus,

text: the

text: two

text: discussed

text: options—saturation

text: of

text: the

text: black

text: disk

text: limit

text: or

text: satu-

text: ration

text: of

text: the

text: unitarity

text: limit—g√enerate

text: different

text: mechanisms

text: of

text: the

text: total

text: cross–

text: section

text: growth

text: at

text: the

text: energies

text: s

text: TeV.

text: Namely,

text: saturation

text: of

text: the

text: black

text: disk

text: limit

text: assumes

text: the

text: growth

text: of

text: the

text: total

text: cross–section

text: occurs

text: due

text: to

text: increasing

text: impact

text: parameter

text: values

text: only,

text: while

text: the

text: growth

text: of

text: total

text: cross-section

text: can,

text: in

text: fact,

text: be

text: due

text: to

text: both

text: factors

text: increase

text: of

text: impact

text: parameter

text: values

text: combined

text: with

text: an

text: increase

text: of

text: the

text: elastic

text: scattering

text: amplitude

text: f(s,

text: b)√with

text: energy

text: at

text: ﬁxed

text: b.

text: Therefore,

text: the

text: measurements

text: of

text: the

text: elastic

text: scattering

text: at

text: s

text: TeV

text: (especially

text: in

text: the

text: region

text: of

text: large

text: t)

text: could

text: be

text: quite

text: decisive

text: regarding

text: this

text: aspect

text: of

text: asymptotic

text: picture

text: and

text: would

text: help

text: to

text: determine

text: which

text: scattering

text: picture

text: —absorptive

text: or

text: reﬂective

text: should

text: be

text: expected

text: at

text: the

text: asymptotical

text: energies.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 1 (M + hM )z˜µ+1 = hN z˜µ + hN zν + h(N + N )yk A 3 n+1 3 n+1 2 n+1 1 h A n+1 − +M y N y + hf , A n A n n+1 z˜0 (t ) = zνk , µ = 0, 1, . . . , µ , ﬁrst inner iteration n+1 n n+1 νk z0 (t ) = yk , ν = 0, 1, . . . , ν , second inner iteration n+1 n n+1 k yk+1 = z˜µνk , k = 0, 1, . . . , K, outer iteration n+1 n+1 z˜0 (t ) = y , initialization n+1 n n n = 0, 1, 2, . . . , J, . Three-Stage WR algorithm (serial) 2 is the given as: Given the initial vector y = y(0) , 0 z0 (0) = y , n+1 0 for n = 0, 1, . . . , J do for k = 0, 1, . . . , K do for ν = 0, 1, . . . , ν do k for µ = 0, 1, . . . , µ do νk (M + hM )z˜µ+1 = hN z˜µ + hN zν + h(N + A 3 n+1 − 3 n+1 2 n+1 1 1 N )yk + M y N y + hf , h A n+1 A n A n n+1 end zν+1 = z˜µν , n+1 n+1 z˜0 = zν+1, n+1 n+1 end yk+1 = zνk , n+1 n+1 z0 = yk+1, n+1 n+1 end y = zνK , n+1 n+1 z0 (t ) = y , n+1 n+1 n+1 end Algorithm 2: Three-Stage WR algorithm (serial) where we apply the algorithm with p = 50, q = 6, J = 20, h = 0.1. Further we have ∆x = 1.0 and D = 1.0. ∆x2 We have 2 possible stopping criteria: (a) Error bound: We have an stopping error norm : || − || ≤ − for the outer iteration with yk+1 yk 10 3, || n+1− n+1|| ≤ − and for the inner iteration zν+1 zν 10 3 n+1 || n+1 − || ≤ − and for the second inner iteration z˜µ+1 z˜µ 10 3 n+1 n+1 (b) Fix number of outer-iterative steps: K = 5 and inner iterative steps ν = 2, µ = 2. k νk ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: (M

text: hM

text: hN

text: z˜µ

text: hN

text: zν

text: h(N

text: N

text: )yk

text: A

text: h

text: A

text: +M

text: y

text: N

text: y

text: hf

text: A

text: n

text: A

text: n

text: (t

text: zνk

text: µ

text: µ

text: ﬁrst

text: inner

text: iteration

text: n

text: νk

text: (t

text: yk

text: ν

text: ν

text: second

text: inner

text: iteration

text: n

text: k

text: z˜µνk

text: k

text: K,

text: outer

text: iteration

text: (t

text: y

text: initialization

text: n

text: n

text: n

text: J,

text: Three-Stage

text: WR

text: algorithm

text: (serial)

text: is

text: the

text: given

text: as:

text: Given

text: the

text: initial

text: vector

text: y

text: y

text: for

text: n

text: J

text: do

text: for

text: k

text: K

text: do

text: for

text: ν

text: ν

text: do

text: k

text: for

text: µ

text: µ

text: do

text: νk

text: (M

text: hM

text: hN

text: z˜µ

text: hN

text: zν

text: h(N

text: A

text: N

text: )yk

text: M

text: y

text: N

text: y

text: hf

text: h

text: A

text: A

text: n

text: A

text: n

text: end

text: z˜µν

text: end

text: zνk

text: end

text: y

text: zνK

text: (t

text: y

text: end

text: Algorithm

text: Three-Stage

text: WR

text: algorithm

text: (serial)

text: where

text: we

text: apply

text: the

text: algorithm

text: with

text: p

text: q

text: J

text: h

text: Further

text: we

text: have

text: ∆x

text: and

text: D

text: We

text: have

text: possible

text: stopping

text: criteria:

text: (a)

text: Error

text: bound:

text: We

text: have

text: an

text: stopping

text: error

text: norm

text: for

text: the

text: outer

text: iteration

text: with

text: yk

text: and

text: for

text: the

text: inner

text: iteration

text: zν

text: and

text: for

text: the

text: second

text: inner

text: iteration

text: z˜µ

text: (b)

text: Fix

text: number

text: of

text: outer-iterative

text: steps:

text: K

text: and

text: inner

text: iterative

text: steps

text: ν

text: µ

text: k

text: νk

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: We note, in this paper, that the Lindblad operators are time-independent and the de- ≥ phasing rates are positive, i.e. γ 0, therefore the dynamics for the second conﬁguration, j as an open system, can be described by time-independent Markovian master equation in the following Lindblad form [22] dρ − = i[H, ρ] + L(ρ), (11) dt L L − where L(ρ) = (ρ) + (ρ). Remember that, the term i[H, ρ] is responsible for deph sink the transfer of excitation to the reaction center in the ﬁrst conﬁguration, while in the second L conﬁguration this task is performed by the term (ρ). When all of the sites are ﬁxed, deph both of the systems are initially prepared with a single excitation localized at the site 1, i.e. | (cid:105)(cid:104) | ρ(0) = 1 1 , and it is assumed that, γ = γ = γ and Γ = 2γ. For the second system, the 2 3 maximal eﬃciency of incoherent transport is occurred for the optimal value of dephasing rate γ = γ = 1.05. Fig. 3, shows the population of the sink versus time for both systems. It is opt clear that the optimal eﬃciency of coherent transport is more than the optimal eﬃciency of incoherent transport. This observations is in complete agreement with results of Refs [18]. 3 Eﬀects of harmonic deformations We consider the harmonic deformations as mechanical oscillations of the vertices around their respective equilibrium positions which change their relative distance and therefore, modulate the distance-dependent dipolar coupling. Time-dependent distance between the sites i and j is as follows − − − d (t) = d [u (t) u (t)] = d (1 2a sin(ω t + φ )) (12) i,j 0 i j 0 i,j 0 i,j where d is the equilibrium distance between two connected sites, u is the displacement 0 i of the ith site from its equilibrium position and a is the individual relative amplitude of i,j oscillations of sites ith and jth when they move with opposite phase around their equilibrium positions. The dipole-dipole coupling between the sites i and j has the form [17] as ˜ J J J (t) = 0 = − 0 , (13) i,j [d (t)]3 (1 2a sin(ω t + φ ))3 i,j i,j 0 i,j where J˜ contains the dipole moments and physical constants, and we deﬁne J = J˜ /d3, 0 0 0 0 which has the units of energy. Henceforth, all energies, time scales and rates will be expressed in units of J . Therefore, by existence some time-dependent coupling strength such as J (t), 0 i,j the equation (1) represents a time-dependent Hamiltonian. ≡ We consider deformations for which the conditions, J (t) = J (t) ζ (t) and J (t) = ± ≡ 1,2 1,3 1 2,4 J (t) ζ (t), are satisﬁed for all of times (+ is devoted for the coherent evolution of the 3,4 2 ﬁrst conﬁguration and - for the incoherent evolution of the second system). Therefore, the deformed time-dependent Hamiltonian for the coherent evolution becomes as 5 ##LTLine## ##LTLine## ##LTLine##

text: We

text: note,

text: in

text: this

text: paper,

text: that

text: the

text: Lindblad

text: operators

text: are

text: time-independent

text: and

text: the

text: de-

text: phasing

text: rates

text: are

text: positive,

text: i.e.

text: γ

text: therefore

text: the

text: dynamics

text: for

text: the

text: second

text: conﬁguration,

text: j

text: as

text: an

text: open

text: system,

text: can

text: be

text: described

text: by

text: time-independent

text: Markovian

text: master

text: equation

text: in

text: the

text: following

text: Lindblad

text: form

text: dρ

text: i[H,

text: ρ]

text: L(ρ),

text: dt

text: L

text: L

text: where

text: L(ρ)

text: (ρ)

text: (ρ).

text: Remember

text: that,

text: the

text: term

text: i[H,

text: ρ]

text: is

text: responsible

text: for

text: deph

text: sink

text: the

text: transfer

text: of

text: excitation

text: to

text: the

text: reaction

text: center

text: in

text: the

text: ﬁrst

text: conﬁguration,

text: while

text: in

text: the

text: second

text: L

text: conﬁguration

text: this

text: task

text: is

text: performed

text: by

text: the

text: term

text: (ρ).

text: When

text: all

text: of

text: the

text: sites

text: are

text: ﬁxed,

text: deph

text: both

text: of

text: the

text: systems

text: are

text: initially

text: prepared

text: with

text: a

text: single

text: excitation

text: localized

text: at

text: the

text: site

text: i.e.

text: and

text: it

text: is

text: assumed

text: that,

text: γ

text: γ

text: γ

text: and

text: Γ

text: For

text: the

text: second

text: system,

text: the

text: maximal

text: eﬃciency

text: of

text: incoherent

text: transport

text: is

text: occurred

text: for

text: the

text: optimal

text: value

text: of

text: dephasing

text: rate

text: γ

text: γ

text: Fig.

text: shows

text: the

text: population

text: of

text: the

text: sink

text: versus

text: time

text: for

text: both

text: systems.

text: It

text: is

text: opt

text: clear

text: that

text: the

text: optimal

text: eﬃciency

text: of

text: coherent

text: transport

text: is

text: more

text: than

text: the

text: optimal

text: eﬃciency

text: of

text: incoherent

text: transport.

text: This

text: observations

text: is

text: in

text: complete

text: agreement

text: with

text: results

text: of

text: Refs

text: Eﬀects

text: of

text: harmonic

text: deformations

text: We

text: consider

text: the

text: harmonic

text: deformations

text: as

text: mechanical

text: oscillations

text: of

text: the

text: vertices

text: around

text: their

text: respective

text: equilibrium

text: positions

text: which

text: change

text: their

text: relative

text: distance

text: and

text: therefore,

text: modulate

text: the

text: distance-dependent

text: dipolar

text: coupling.

text: Time-dependent

text: distance

text: between

text: the

text: sites

text: i

text: and

text: j

text: is

text: as

text: follows

text: d

text: (t)

text: d

text: [u

text: (t)

text: u

text: (t)]

text: d

text: sin(ω

text: t

text: φ

text: i,j

text: i

text: j

text: i,j

text: i,j

text: where

text: d

text: is

text: the

text: equilibrium

text: distance

text: between

text: two

text: connected

text: sites,

text: u

text: is

text: the

text: displacement

text: i

text: of

text: the

text: ith

text: site

text: from

text: its

text: equilibrium

text: position

text: and

text: a

text: is

text: the

text: individual

text: relative

text: amplitude

text: of

text: i,j

text: oscillations

text: of

text: sites

text: ith

text: and

text: jth

text: when

text: they

text: move

text: with

text: opposite

text: phase

text: around

text: their

text: equilibrium

text: positions.

text: The

text: dipole-dipole

text: coupling

text: between

text: the

text: sites

text: i

text: and

text: j

text: has

text: the

text: form

text: as

text: J

text: J

text: J

text: (t)

text: i,j

text: [d

text: sin(ω

text: t

text: φ

text: i,j

text: i,j

text: i,j

text: where

text: J˜

text: contains

text: the

text: dipole

text: moments

text: and

text: physical

text: constants,

text: and

text: we

text: deﬁne

text: J

text: J˜

text: which

text: has

text: the

text: units

text: of

text: energy.

text: Henceforth,

text: all

text: energies,

text: time

text: scales

text: and

text: rates

text: will

text: be

text: expressed

text: in

text: units

text: of

text: J

text: Therefore,

text: by

text: existence

text: some

text: time-dependent

text: coupling

text: strength

text: such

text: as

text: J

text: (t),

text: i,j

text: the

text: equation

text: represents

text: a

text: time-dependent

text: Hamiltonian.

text: We

text: consider

text: deformations

text: for

text: which

text: the

text: conditions,

text: J

text: (t)

text: J

text: (t)

text: ζ

text: (t)

text: and

text: J

text: (t)

text: J

text: (t)

text: ζ

text: (t),

text: are

text: satisﬁed

text: for

text: all

text: of

text: times

text: is

text: devoted

text: for

text: the

text: coherent

text: evolution

text: of

text: the

text: ﬁrst

text: conﬁguration

text: and

text: for

text: the

text: incoherent

text: evolution

text: of

text: the

text: second

text: system).

text: Therefore,

text: the

text: deformed

text: time-dependent

text: Hamiltonian

text: for

text: the

text: coherent

text: evolution

text: becomes

text: as

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: References [1] M. B. Plenio, J. Hartley, and J. Eisert, New J. Phys. 6, 36 (2004). [2] P. J. Pemberton-Ross, and Alastair Kay, Phys. Rev. Leet. 106 (2011) 020503. [3] V. karimipour, M. Sarmadi Rad, and M. Asoudeh, Phys. Rev. A 85 (2012) 010302(R). [4] N. Behzadi, S. Kazemi Rudsary, B. Ahansaz Salmasi, Eur. Phys J. D——(2013). [5] M. Mohseni, P. Rebentrost, S. Lloyd and A. Aspuru-Guzik, J. Chem. Phys. 129, 174106 (2008). [6] R.J. Sension, Nature 446, 740 (2007). [7] M. B. Plenio and S. F. Huelga. New J. Phys. 10, 113019 (2008). [8] F. Caruso, A. W. Chin, A. Datta, S. F. Huelga, and M. B. Plenio, J. Chem. Phys. 131, 105106 (2009). [9] A. W. Chin, A. Datta, F. Caruso, S. F. Huelga, and M. B. Plenio, New J. Phys. 12, 065002 (2010). [10] P. Rebentrost, M. Mohseni, I. Kassal, S. Lloyd and A. Aspuru-Guzik, New J. Phys. 11, 0330 (2009). [11] I. Kassal and A. Aspuru-Guzik, New J. Phys. 14, 053041 (2012). [12] I. Sinayskiy, A. Marais, F. Petruccione and A. Ekert, Phys. ev. Lett. 108, 020602 (2012). [13] J. Lim, M. Tame, K. H. Yee, J-S. Lee and J. Lee, New J. Phys. 16, 053018 (2014). [14] E. J. OReilly and A. Olaya-Castro, Nature Communications,——-, (2014). [15] E. K. Irish, R. Gomez-Bombarelli and B. W. Lovett, Phys. Rev. A 90, 012510 (2014). [16] F. L. Semi, K. Furuya and G. J. Milburn, New J. Phys. 12, 083033 (2010). [17] A. Asadian, M. Tiersch, G. G. Guerreschi, J. Cai, S. Popescu and H. J. Briegel, New J. Phys. 12, 075019 (2010). [18] F. Caruso, New J. Phys. 16 (2014) 055015 [19] B. Cui, X. X. Yi and C. H. Oh, J. Phys. B: At. Mol. Opt. Phys. 45, 085501 (2012). [20] N. Behzadi, B. Ahansaz and H. kasani, Phys. Rev. E, ——-, (2015). [21] M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay, and A. J. Landahl, Phys. Rev. A, 71, 032312 (2005). [22] E. M. Laine, J. Piilo and H. P. Breuer, Phys. Rev. A, 81, 062115 (2010). 8

text: References

label: reference

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text: Hartley,

text: and

text: J.

text: Eisert,

text: New

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text: Phys.

text: P.

text: J.

text: Pemberton-Ross,

text: and

text: Alastair

text: Kay,

text: Phys.

text: Rev.

text: Leet.

text: V.

text: karimipour,

text: M.

text: Sarmadi

text: Rad,

text: and

text: M.

text: Asoudeh,

text: Phys.

text: Rev.

text: A

text: N.

text: Behzadi,

text: S.

text: Kazemi

text: Rudsary,

text: B.

text: Ahansaz

text: Salmasi,

text: Eur.

text: Phys

text: J.

text: M.

text: Mohseni,

text: P.

text: Rebentrost,

text: S.

text: Lloyd

text: and

text: A.

text: Aspuru-Guzik,

text: J.

text: Chem.

text: Phys.

text: R.J.

text: Sension,

text: Nature

text: M.

text: B.

text: Plenio

text: and

text: S.

text: F.

text: Huelga.

text: New

text: J.

text: Phys.

text: F.

text: Caruso,

text: A.

text: W.

text: Chin,

text: A.

text: Datta,

text: S.

text: F.

text: Huelga,

text: and

text: M.

text: B.

text: Plenio,

text: J.

text: Chem.

text: Phys.

text: A.

text: W.

text: Chin,

text: A.

text: Datta,

text: F.

text: Caruso,

text: S.

text: F.

text: Huelga,

text: and

text: M.

text: B.

text: Plenio,

text: New

text: J.

text: Phys.

text: P.

text: Rebentrost,

text: M.

text: Mohseni,

text: I.

text: Kassal,

text: S.

text: Lloyd

text: and

text: A.

text: Aspuru-Guzik,

text: New

text: J.

text: Phys.

text: I.

text: Kassal

text: and

text: A.

text: Aspuru-Guzik,

text: New

text: J.

text: Phys.

text: I.

text: Sinayskiy,

text: A.

text: Marais,

text: F.

text: Petruccione

text: and

text: A.

text: Ekert,

text: Phys.

text: ev.

text: Lett.

text: J.

text: Lim,

text: M.

text: Tame,

text: K.

text: H.

text: Yee,

text: J-S.

text: Lee

text: and

text: J.

text: Lee,

text: New

text: J.

text: Phys.

text: E.

text: J.

text: OReilly

text: and

text: A.

text: Olaya-Castro,

text: Nature

text: Communications,——-,

text: E.

text: K.

text: Irish,

text: R.

text: Gomez-Bombarelli

text: and

text: B.

text: W.

text: Lovett,

text: Phys.

text: Rev.

text: A

text: F.

text: L.

text: Semi,

text: K.

text: Furuya

text: and

text: G.

text: J.

text: Milburn,

text: New

text: J.

text: Phys.

text: A.

text: Asadian,

text: M.

text: Tiersch,

text: G.

text: G.

text: Guerreschi,

text: J.

text: Cai,

text: S.

text: Popescu

text: and

text: H.

text: J.

text: Briegel,

text: New

text: J.

text: Phys.

text: F.

text: Caruso,

text: New

text: J.

text: Phys.

text: B.

text: Cui,

text: X.

text: X.

text: Yi

text: and

text: C.

text: H.

text: Oh,

text: J.

text: Phys.

text: B:

text: At.

text: Mol.

text: Opt.

text: Phys.

text: N.

text: Behzadi,

text: B.

text: Ahansaz

text: and

text: H.

text: kasani,

text: Phys.

text: Rev.

text: E,

text: M.

text: Christandl,

text: N.

text: Datta,

text: T.

text: C.

text: Dorlas,

text: A.

text: Ekert,

text: A.

text: Kay,

text: and

text: A.

text: J.

text: Landahl,

text: Phys.

text: Rev.

text: A,

text: E.

text: M.

text: Laine,

text: J.

text: Piilo

text: and

text: H.

text: P.

text: Breuer,

text: Phys.

text: Rev.

text: A,


text: Figure 1: Fig. 1. (a) Conﬁguration of four interacting two-level atoms with J = J = J = 1,2 1,3 2,4 J = J, irreversibly connected to the sink site. (b) The equivalent conﬁguration when the 3,4 set of basis introduced in Eqs. 2, is used. 9 ##LTFigure## ##LTFigure##

text: Figure

text: Fig.

text: (a)

text: Conﬁguration

text: of

text: four

text: interacting

text: two-level

text: atoms

text: with

text: J

text: J

text: J

text: J

text: J,

text: irreversibly

text: connected

text: to

text: the

text: sink

text: site.

text: (b)

text: The

text: equivalent

text: conﬁguration

text: when

text: the

text: set

text: of

text: basis

text: introduced

text: in

text: Eqs.

text: is

text: used.

text: ##LTFigure##

label: figure

text: ##LTFigure##


text: 11 ! $ 4!5 !""""#""""! & ()% $ % 9’"µ( 4="(3 ;)//-3,-."1%+)23-<2 $ ’ % " 48 4: % & 9: 98 $ % 46"#7 # (cid:95) 48 9: 4: 98 FIG. 11: (a) The tubular minima of the lattice potential are split into upper and lower sections with a cylindrically focused laser beam (light blue). (b) Band mapping image showing equally populated condensates at the minima of the second band X±. (c) Sketch of the atomic spatial distribution after a ballistic expansion of 25 ms. The grey area corresponds to the second ∈ { } ∈ {− } Brillouin zone. Each of the four Bragg maxima B (indexed i 1, 2 , σ , + ), corresponding to the X±-points in i,σ Fig. 9(a), carries an interference grating aligned with the z-axis. (d) Absorption image along the line of sight indicated by the ◦ (cid:13) vector r in (c), which lies in the xy-plane and encloses an angle of α = 13 with the y-axis. Reproduced from Ref. [54], c American Physical Society. such that tunneling between both lattices is completely suppressed (see Fig. 11(a)). Excitation of atoms into the second band and subsequent thermalisation is simultaneously and independently applied to both lattices. The initial BEC is positioned such that about half of the atoms are transferred to each side of the barrier. The lattice potential is adjusted to the case of degenerate condensation points X± such that equal populations of X± are prepared, as is conﬁrmed by the band mapping spectrum in Fig. 11(b) recorded with a line of sight parallel to the z-axis. Finally, ballistic expansion is applied during t = 25 ms until the atom clouds from each lattice overlap and interfere (see TOF Fig. 11(c)). Similar to the case of a single lattice in Fig. 10(d) (central panel), the density distribution in Fig. 11(c) represents an image of momentum space. In the ballistic expansion, the sub-sample in each condensation point X is decomposed σ into its two predominant momentum components with regard to the xy-plane, thus giving rise to four zero-order ∈ {− } Bragg peaks B (identiﬁed by the indices (i, σ) in Fig. 11(c)), where σ , + denotes the condensation point i,σ ∈ { } ≡ − (cid:126) that B is associated with and i 1, 2 numbers the two predominant momentum classes p ( 1)i 1 k(σ 1, 1) i,σ i,σ 2 associated with X (cf. Fig. 11(b)). Since the momentum spectrum arises from the interference of two lattices closely σ spaced on top of each other, in contrast to the case of a single lattice, each Bragg peak exhibits a density modulation along the z-direction. In the simpliﬁed picture that the interference results from two point sources at a distance d z observed after suﬃciently long time of ﬂight tTOF, the wavelength of the expected≈density grating is λz = hmtTdOzF , with h denoting Planck’s constant and m the atomic mass [55]. This amounts to λ 10 µm, close to what is observed z in the experiment. The density gratings B are recorded by absorption imaging with the line of sight indicated by i,σ ◦ the vector r in Fig. 11(c), which lies in the xy-plane and encloses an angle of α = 13 with the y-axis. An example of a single-shot image of these density gratings is shown in Fig. 11(d), where the new coordinates x˜, z˜ are used for the projected density distribution in the image plane. As is illustrated in Fig. 12(a), the density gratings B arise i,σ from the interference of sub-samples U and L , which denote the components with average momenta p of the i,σ i,σ i,σ condensates U and L formed in the condensation points X of the upper (U) and lower (L) lattice, respectively. σ σ σ Note that only sub-samples U and L associated with the same condensation point X yield interference, while σ σ σ sub-samples from diﬀerent condensation points are spatially separated in the ballistic expansion process. The global spatial phases ξ of the gratings B vary for diﬀerent experimental realizations, thus reﬂecting the i,σ i,σ complete independence of the atomic samples produced in the upper and lower lattice [56]. Only relative spatial *+,-.)/"0123-,$ ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: FIG.

text: (a)

text: The

text: tubular

text: minima

text: of

text: the

text: lattice

text: potential

text: are

text: split

text: into

text: upper

text: and

text: lower

text: sections

text: with

text: a

text: cylindrically

text: focused

text: laser

text: beam

text: (light

text: blue).

text: (b)

text: Band

text: mapping

text: image

text: showing

text: equally

text: populated

text: condensates

text: at

text: the

text: minima

text: of

text: the

text: second

text: band

text: X±.

text: (c)

text: Sketch

text: of

text: the

text: atomic

text: spatial

text: distribution

text: after

text: a

text: ballistic

text: expansion

text: of

text: ms.

text: The

text: grey

text: area

text: corresponds

text: to

text: the

text: second

text: Brillouin

text: zone.

text: Each

text: of

text: the

text: four

text: Bragg

text: maxima

text: B

text: (indexed

text: i

text: σ

text: corresponding

text: to

text: the

text: X±-points

text: in

text: i,σ

text: Fig.

text: carries

text: an

text: interference

text: grating

text: aligned

text: with

text: the

text: z-axis.

text: (d)

text: Absorption

text: image

text: along

text: the

text: line

text: of

text: sight

text: indicated

text: by

text: the

text: vector

text: r

text: in

text: (c),

text: which

text: lies

text: in

text: the

text: xy-plane

text: and

text: encloses

text: an

text: angle

text: of

text: α

text: with

text: the

text: y-axis.

text: Reproduced

text: from

text: Ref.

text: c

text: American

text: Physical

text: Society.

text: such

text: that

text: tunneling

text: between

text: both

text: lattices

text: is

text: completely

text: suppressed

text: (see

text: Fig.

text: Excitation

text: of

text: atoms

text: into

text: the

text: second

text: band

text: and

text: subsequent

text: thermalisation

text: is

text: simultaneously

text: and

text: independently

text: applied

text: to

text: both

text: lattices.

text: The

text: initial

text: BEC

text: is

text: positioned

text: such

text: that

text: about

text: half

text: of

text: the

text: atoms

text: are

text: transferred

text: to

text: each

text: side

text: of

text: the

text: barrier.

text: The

text: lattice

text: potential

text: is

text: adjusted

text: to

text: the

text: case

text: of

text: degenerate

text: condensation

text: points

text: X±

text: such

text: that

text: equal

text: populations

text: of

text: X±

text: are

text: prepared,

text: as

text: is

text: conﬁrmed

text: by

text: the

text: band

text: mapping

text: spectrum

text: in

text: Fig.

text: recorded

text: with

text: a

text: line

text: of

text: sight

text: parallel

text: to

text: the

text: z-axis.

text: Finally,

text: ballistic

text: expansion

text: is

text: applied

text: during

text: t

text: ms

text: until

text: the

text: atom

text: clouds

text: from

text: each

text: lattice

text: overlap

text: and

text: interfere

text: (see

text: TOF

text: Fig.

text: Similar

text: to

text: the

text: case

text: of

text: a

text: single

text: lattice

text: in

text: Fig.

text: (central

text: panel),

text: the

text: density

text: distribution

text: in

text: Fig.

text: represents

text: an

text: image

text: of

text: momentum

text: space.

text: In

text: the

text: ballistic

text: expansion,

text: the

text: sub-sample

text: in

text: each

text: condensation

text: point

text: X

text: is

text: decomposed

text: σ

text: into

text: its

text: two

text: predominant

text: momentum

text: components

text: with

text: regard

text: to

text: the

text: xy-plane,

text: thus

text: giving

text: rise

text: to

text: four

text: zero-order

text: Bragg

text: peaks

text: B

text: (identiﬁed

text: by

text: the

text: indices

text: (i,

text: σ)

text: in

text: Fig.

text: where

text: σ

text: denotes

text: the

text: condensation

text: point

text: i,σ

text: that

text: B

text: is

text: associated

text: with

text: and

text: i

text: numbers

text: the

text: two

text: predominant

text: momentum

text: classes

text: p

text: k(σ

text: i,σ

text: i,σ

text: associated

text: with

text: X

text: (cf.

text: Fig.

text: Since

text: the

text: momentum

text: spectrum

text: arises

text: from

text: the

text: interference

text: of

text: two

text: lattices

text: closely

text: σ

text: spaced

text: on

text: top

text: of

text: each

text: other,

text: in

text: contrast

text: to

text: the

text: case

text: of

text: a

text: single

text: lattice,

text: each

text: Bragg

text: peak

text: exhibits

text: a

text: density

text: modulation

text: along

text: the

text: z-direction.

text: In

text: the

text: simpliﬁed

text: picture

text: that

text: the

text: interference

text: results

text: from

text: two

text: point

text: sources

text: at

text: a

text: distance

text: d

text: z

text: observed

text: after

text: suﬃciently

text: long

text: time

text: of

text: ﬂight

text: tTOF,

text: the

text: wavelength

text: of

text: the

text: expected≈density

text: grating

text: is

text: λz

text: hmtTdOzF

text: with

text: h

text: denoting

text: Planck’s

text: constant

text: and

text: m

text: the

text: atomic

text: mass

text: This

text: amounts

text: to

text: λ

text: µm,

text: close

text: to

text: what

text: is

text: observed

text: z

text: in

text: the

text: experiment.

text: The

text: density

text: gratings

text: B

text: are

text: recorded

text: by

text: absorption

text: imaging

text: with

text: the

text: line

text: of

text: sight

text: indicated

text: by

text: i,σ

text: the

text: vector

text: r

text: in

text: Fig.

text: which

text: lies

text: in

text: the

text: xy-plane

text: and

text: encloses

text: an

text: angle

text: of

text: α

text: with

text: the

text: y-axis.

text: An

text: example

text: of

text: a

text: single-shot

text: image

text: of

text: these

text: density

text: gratings

text: is

text: shown

text: in

text: Fig.

text: where

text: the

text: new

text: coordinates

text: x˜,

text: z˜

text: are

text: used

text: for

text: the

text: projected

text: density

text: distribution

text: in

text: the

text: image

text: plane.

text: As

text: is

text: illustrated

text: in

text: Fig.

text: the

text: density

text: gratings

text: B

text: arise

text: i,σ

text: from

text: the

text: interference

text: of

text: sub-samples

text: U

text: and

text: L

text: which

text: denote

text: the

text: components

text: with

text: average

text: momenta

text: p

text: of

text: the

text: i,σ

text: i,σ

text: i,σ

text: condensates

text: U

text: and

text: L

text: formed

text: in

text: the

text: condensation

text: points

text: X

text: of

text: the

text: upper

text: (U)

text: and

text: lower

text: (L)

text: lattice,

text: respectively.

text: σ

text: σ

text: σ

text: Note

text: that

text: only

text: sub-samples

text: U

text: and

text: L

text: associated

text: with

text: the

text: same

text: condensation

text: point

text: X

text: yield

text: interference,

text: while

text: σ

text: σ

text: σ

text: sub-samples

text: from

text: diﬀerent

text: condensation

text: points

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text: Only

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