text: Figure 4: General ﬂowchart of a population-based search algorithm. The ParFrame can only inte- grate algorithms that follow this workﬂow. At present time, the P-EAJava is general enough to work with population-based algorithms that follow the workﬂow depicted in Figure 4. As such, the framework already provides all the necessary classes that deﬁne the working notions of Population and Stop Criteria. The evolution- ary algorithm has only to provide its own classes related with the operators responsible for the evolution of the population in each generation. The P-EAJava also uses the design pattern strategy (Gamma et al., 1995) to decouple as much as possible the implementation of a search algorithm from the remaining parameter-less strategy (see Figure 5). However, as expected, to create a new parameter-less algorithm is not as simple as plugging in a new problem. Let us look at the example of the parameter-less SGA to better understand all the necessary steps. Note that, the reading of the following instructions is best complemented with a good anal- ysis of the corresponding source code. 1. Deﬁne the class SGA that implements the interface IEAlgorithm. Here is the important part of this implementation: Note that, the function newIEASolver(int) has the return type IEASolver which is itself an- other interface. The class SGA is included in the ﬁle src/com/z PEA/SGA.java. 2. Deﬁne the class SGASolver that implements the interface IEASolver. This class includes the all-important function nextGeneration(), responsible for implementing the evolution of the population in each generation. Here is the main part of the implementation: 9 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Figure

text: General

text: ﬂowchart

text: of

text: a

text: population-based

text: search

text: algorithm.

text: The

text: ParFrame

text: can

text: only

text: inte-

text: grate

text: algorithms

text: that

text: follow

text: this

text: workﬂow.

text: At

text: present

text: time,

text: the

text: P-EAJava

text: is

text: general

text: enough

text: to

text: work

text: with

text: population-based

text: algorithms

text: that

text: follow

text: the

text: workﬂow

text: depicted

text: in

text: Figure

text: As

text: such,

text: the

text: framework

text: already

text: provides

text: all

text: the

text: necessary

text: classes

text: that

text: deﬁne

text: the

text: working

text: notions

text: of

text: Population

text: and

text: Stop

text: Criteria.

text: The

text: evolution-

text: ary

text: algorithm

text: has

text: only

text: to

text: provide

text: its

text: own

text: classes

text: related

text: with

text: the

text: operators

text: responsible

text: for

text: the

text: evolution

text: of

text: the

text: population

text: in

text: each

text: generation.

text: The

text: P-EAJava

text: also

text: uses

text: the

text: design

text: pattern

text: strategy

text: (Gamma

text: et

text: al.,

text: to

text: decouple

text: as

text: much

text: as

text: possible

text: the

text: implementation

text: of

text: a

text: search

text: algorithm

text: from

text: the

text: remaining

text: parameter-less

text: strategy

text: (see

text: Figure

text: However,

text: as

text: expected,

text: to

text: create

text: a

text: new

text: parameter-less

text: algorithm

text: is

text: not

text: as

text: simple

text: as

text: plugging

text: in

text: a

text: new

text: problem.

text: Let

text: us

text: look

text: at

text: the

text: example

text: of

text: the

text: parameter-less

text: SGA

text: to

text: better

text: understand

text: all

text: the

text: necessary

text: steps.

text: Note

text: that,

text: the

text: reading

text: of

text: the

text: following

text: instructions

text: is

text: best

text: complemented

text: with

text: a

text: good

text: anal-

text: ysis

text: of

text: the

text: corresponding

text: source

text: code.

text: Deﬁne

text: the

text: class

text: SGA

text: that

text: implements

text: the

text: interface

text: IEAlgorithm.

text: Here

text: is

text: the

text: important

text: part

text: of

text: this

text: implementation:

text: Note

text: that,

text: the

text: function

text: newIEASolver(int)

text: has

text: the

text: return

text: type

text: IEASolver

text: which

text: is

text: itself

text: an-

text: other

text: interface.

text: The

text: class

text: SGA

text: is

text: included

text: in

text: the

text: ﬁle

text: src/com/z

text: PEA/SGA.java.

text: Deﬁne

text: the

text: class

text: SGASolver

text: that

text: implements

text: the

text: interface

text: IEASolver.

text: This

text: class

text: includes

text: the

text: all-important

text: function

text: nextGeneration(),

text: responsible

text: for

text: implementing

text: the

text: evolution

text: of

text: the

text: population

text: in

text: each

text: generation.

text: Here

text: is

text: the

text: main

text: part

text: of

text: the

text: implementation:

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text: Figure 3: The evolution of the Hubble parameter (4.1) indicates an early cosmic inﬂation. Figure 4: the decay of the teleparallel torsion scalar ﬁeld (4.3). 14 ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

label: paragraph

text: The

label: caption

text: evolution

text: of

text: the

text: Hubble

text: parameter

text: indicates

text: an

text: early

text: cosmic

text: inﬂation.

text: Figure

text: the

text: decay

text: of

text: the

text: teleparallel

text: torsion

text: scalar

text: ﬁeld

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text: L. Nashed, Astrophys. Space Sci. 348 (2013), 591; G. Otalora, JCAP 1307 (2013), 044; J. Amoros, J. de Haro and S. D. Odintsov, Phys. Rev. D 87 (2013), 104037; G. Otalora, Phys. Rev. D 88 (2013), 063505; C. -Q. Geng, J. -A. Gu and C. -C. Lee, Phys. Rev. D 88 (2013), 024030; I. G. Salako, M. E. Rodrigues, A. V. Kpadonou, M. J. S. Houndjo and J. Tossa, JCAP 060 (2013), 1475; A. V. Astashenok, Astrophys. Space Sci. 351, (2014) 377; M. E. Rodrigues, I. G. Salako, M. J. S. Houndjo and J. Tossa, Int. J. Mod. Phys. D 23 (2014), 1450004; K. Bamba, S. D. Odintsov and D. Saez-Gomez, Phys. Rev. D 88 (2013), 084042; K. Bamba, S. Capozziello, M. De Laurentis, S. Nojiri and D. Saez-Gomez, Phys. Lett. B 727 (2013), 194; T. Harko, F. S. N. Lobo, G. Otalora and E. N. Saridakis, Phys. Rev. D89 (2014), 124036; G. Otalora, arXiv:1402.2256; K. Bamba, S. i. Nojiri and S. D. Odintsov, Phys. Lett. B 731 (2014) 257; G. Koﬁnas and E. N. Saridakis, Phys. Rev. D 90 (2014), 084044; K. Atazadeh and F. Darabi, Eur. Phys. J. C 72 (2012), 2016; A. Paliathanasis, S. Basilakos, E.N. Saridakis, S. Capozziello, K. Atazadeh, F. Darabi, M. Tsamparlis, Phys. Rev. D 89 (2014) 104042. [12] S. Camera, V. F. Cardone and N. Radicella, Phys. Rev. D 89 (2014), 083520. [13] C. M. Will, Living Rev. Rel. 9 (2006), 3. [14] L. Iorio and E. N. Saridakis, Mon. Not. Roy. Astron. Soc. 427 (2012), 1555. [15] Y. Xie and X. M. Deng, Mon. Not. Roy. Astron. Soc. 433 (2013), 3584. [16] N. Tamanini and C. G. Boehmer, Phys. Rev. D 86 (2012), 044009. [17] S. Capozziello, P. A. Gonza´lez, E. N. Saridakis, Y. Va´squez, JHEP 1302 (2013), 039. [18] J. B. Dent, S. Dutta, E. N. Saridakis, JCAP 1101 (2011), 009. [19] Y.-F. Cai, S.-H. Chen, J. B. Dent, S. Dutta, and E. N. Saridakis, Class. Quantum Grav. 28 (2011), 215011. [20] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl., (1928) 217 (1930) 401. [21] R. Aldrovandi and J. G. Pereira, An Introduction to Teleparallel Gravity, Instituto de Fisica Teorica, UNSEP, Sao Paulo (http://www.ift.unesp.br/gcg/tele.pdf). [22] M. H. Daouda, M. E. Rodrigues and M. J. S. Houndjo, Eur. Phys. J. C 72 (2012) 1890. [23] M. H. Daouda, M. E. Rodrigues and M. J. S. Houndjo, Phys. Lett. B 715 (2012), 241. [24] Di Liu and M. J. Reboucas, Phys. Rev. D 86 (2012) 083515. [25] K. Atazadeh, M. Mousavi, Eur. Phys. J. C 73 (2013), 2272. [26] H.P. Robertson, Ann. Math. 33 (1932), 496. [27] K. Karami and A. Abdolmaleki, JCAP 1204 (2012), 007. 16

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text: Phys.

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text: Saridakis,

text: Phys.

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text: Darabi,

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text: S.

text: Basilakos,

text: E.N.

text: Saridakis,

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text: Darabi,

text: M.

text: Tsamparlis,

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text: Bianchi Type I Cosmology in f (R, T ) Gravity ∗ M. Farasat Shamir Department of Sciences & Humanities, National University of Computer & Emerging Sciences, Lahore Campus, Pakistan. Abstract This paper is devoted to investigate the exact solutions of Bianchi type I spacetime in the context of f(R, T ) gravity [1], where f(R, T ) is an arbitrary function of Ricci scalar R and trace of the energy momentum tensor T . For this purpose, we ﬁnd two exact solutions using the assumption of constant deceleration parameter and variation law of Hubble parameter. The obtained solutions correspond to two diﬀerent models of the universe. The physical behavior of these models is also discussed. Keywords: f(R, T ) gravity, Bianchi type I, deceleration parameter. PACS: 04.50.Kd. 1 Introduction The most popular issue in the modern day cosmology is the current ex- pansion of universe. It is now evident from observational and theoretical facts that our universe is in the phase of accelerated expansion [2]-[11]. The phenomenon of dark energy and dark matter is another topic of discussion ∗ farasat.shamir@nu.edu.pk 1 ##LTLine##

text: Bianchi

label: title

text: Type

text: I

text: Cosmology

text: in

text: f

text: (R,

text: T

text: Gravity

label: author

text: M.

text: Farasat

text: Shamir

text: Department

text: of

text: Sciences

text: Humanities,

text: National

text: University

text: of

text: Computer

text: Emerging

text: Sciences,

text: Lahore

text: Campus,

text: Pakistan.

text: Abstract

text: This

label: abstract

text: paper

text: is

text: devoted

text: to

text: investigate

text: the

text: exact

text: solutions

text: of

text: Bianchi

text: type

text: I

text: spacetime

text: in

text: the

text: context

text: of

text: f(R,

text: T

text: gravity

text: where

text: f(R,

text: T

text: is

text: an

text: arbitrary

text: function

text: of

text: Ricci

text: scalar

text: R

text: and

text: trace

text: of

text: the

text: energy

text: momentum

text: tensor

text: T

text: For

text: this

text: purpose,

text: we

text: ﬁnd

text: two

text: exact

text: solutions

text: using

text: the

text: assumption

text: of

text: constant

text: deceleration

text: parameter

text: and

text: variation

text: law

text: of

text: Hubble

text: parameter.

text: The

text: obtained

text: solutions

text: correspond

text: to

text: two

text: diﬀerent

text: models

text: of

text: the

text: universe.

text: The

text: physical

text: behavior

text: of

text: these

text: models

text: is

text: also

text: discussed.

text: Keywords:

text: f(R,

text: T

text: gravity,

text: Bianchi

text: type

text: I,

text: deceleration

text: parameter.

text: PACS:

text: Introduction

text: The

text: most

text: popular

text: issue

text: in

text: the

text: modern

text: day

text: cosmology

text: is

text: the

text: current

text: ex-

text: pansion

text: of

text: universe.

text: It

text: is

text: now

text: evident

text: from

text: observational

text: and

text: theoretical

text: facts

text: that

text: our

text: universe

text: is

text: in

text: the

text: phase

text: of

text: accelerated

text: expansion

text: The

text: phenomenon

text: of

text: dark

text: energy

text: and

text: dark

text: matter

text: is

text: another

text: topic

text: of

text: discussion

text: farasat.shamir@nu.edu.pk

text: ##LTLine##


text: )k(f Under review as a conference paper at ICLR 2016 Probability density function on proportion of neurons dropped with p =0.5 dropout 8.0% random dropout 4.0% 0.0% 0 25 50 75 100 proportion of dropped neurons ( ) k Figure 1: The density function on proportion of neurons dropped is very peaked around p for dropout. This results in a low variance of the transformations magnitude induced by dropout. Ran- dom dropout on the other hand has a constant density function under pN, N being the number of ∼ U neurons, because ρ (0, p). It is thus more likely to see transformations closer to identity while it is very unlikely for standard dropout. This is intuitively desirable, as much as varying translation distances is preferable over constant large distances. A simple way to vary the transformation magnitude randomly is to replace p by a random variable. ∼ U ∼ B Let ρ (0, p ) and M (ρ ) where h deﬁnes the layer, i the sample, and j the layer’s h h hij h neuron. It is important to use the same ρ for all neurons of a layer, otherwise we would have ∼ B M (ph ). hij 2 To compensate for the change of level of activations during test, a scaling is normally applied. One could also simply apply the inverse scaling during training, turning equation 3 into 1 (cid:12) a˜(h) = − M rect (h) (9) 1 p To adapt the equation to random dropout level, we simply need to replace p with ρ 1 (cid:12) a˜(h) = − M rect (h) (10) 1 ρ No scaling needs to be done during test anymore. Figure 1 shows the differences between density function on proportion of neurons dropped for dropout and random dropout. Transformations induced by random dropout are clearly more diverse than those induced by dropout. 5 EXPERIMENTS 5.1 VISUALIZATIONS OF NOISE PROJECTED BACK INTO THE INPUT SPACE Visualizing the noise projected back into the input space helps to understand what kind of transfor- mations are induced by dropout. Unsupervised models learn more general features than supervised fully connected neural networks and produce thus more visually appealing transformations. For this reason, we trained autoencoders with dropout on the hidden layer to generate samples of transfor- mations. The autoencoder is very similar to denoising autoencoders, the only difference is that a Bernoulli mask is applied to the hidden activations rather than to the input. There is thus no noise applied to the input explicitly. Models are trained for 300 epochs, with mini-batch size of 100, p = 0.4, a learning rate of 0.001 on MNIST, 0.0001 on CIFAR-10 and a momentum of 0.7 on MNIST, 0.5 on CIFAR-10. For CIFAR-10, we do preprocessing with PCA dimensionality reduction and retain 781 features. 4 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )k(f

text: Under

text: review

text: as

text: a

text: conference

text: paper

text: at

text: ICLR

text: Probability

text: density

text: function

text: on

text: proportion

text: of

text: neurons

text: dropped

text: with

text: p

text: dropout

text: random

text: dropout

text: proportion

text: of

text: dropped

text: neurons

text: k

text: Figure

text: The

text: density

text: function

text: on

text: proportion

text: of

text: neurons

text: dropped

text: is

text: very

text: peaked

text: around

text: p

text: for

text: dropout.

text: This

text: results

text: in

text: a

text: low

text: variance

text: of

text: the

text: transformations

text: magnitude

text: induced

text: by

text: dropout.

text: Ran-

text: dom

text: dropout

text: on

text: the

text: other

text: hand

text: has

text: a

text: constant

text: density

text: function

text: under

text: pN,

text: N

text: being

text: the

text: number

text: of

text: U

text: neurons,

text: because

text: ρ

text: p).

text: It

text: is

text: thus

text: more

text: likely

text: to

text: see

text: transformations

text: closer

text: to

text: identity

text: while

text: it

text: is

text: very

text: unlikely

text: for

text: standard

text: dropout.

text: This

text: is

text: intuitively

text: desirable,

text: as

text: much

text: as

text: varying

text: translation

text: distances

text: is

text: preferable

text: over

text: constant

text: large

text: distances.

text: A

text: simple

text: way

text: to

text: vary

text: the

text: transformation

text: magnitude

text: randomly

text: is

text: to

text: replace

text: p

text: by

text: a

text: random

text: variable.

text: U

text: B

text: Let

text: ρ

text: p

text: and

text: M

text: (ρ

text: where

text: h

text: deﬁnes

text: the

text: layer,

text: i

text: the

text: sample,

text: and

text: j

text: the

text: layer’s

text: h

text: h

text: hij

text: h

text: neuron.

text: It

text: is

text: important

text: to

text: use

text: the

text: same

text: ρ

text: for

text: all

text: neurons

text: of

text: a

text: layer,

text: otherwise

text: we

text: would

text: have

text: B

text: M

text: (ph

text: hij

text: To

text: compensate

text: for

text: the

text: change

text: of

text: level

text: of

text: activations

text: during

text: test,

text: a

text: scaling

text: is

text: normally

text: applied.

text: One

text: could

text: also

text: simply

text: apply

text: the

text: inverse

text: scaling

text: during

text: training,

text: turning

text: equation

text: into

text: a˜(h)

text: M

text: rect

text: (h)

text: p

text: To

text: adapt

text: the

text: equation

text: to

text: random

text: dropout

text: level,

text: we

text: simply

text: need

text: to

text: replace

text: p

text: with

text: ρ

text: a˜(h)

text: M

text: rect

text: (h)

text: ρ

text: No

text: scaling

text: needs

text: to

text: be

text: done

text: during

text: test

text: anymore.

text: Figure

text: shows

text: the

text: differences

text: between

text: density

text: function

text: on

text: proportion

text: of

text: neurons

text: dropped

text: for

text: dropout

text: and

text: random

text: dropout.

text: Transformations

text: induced

text: by

text: random

text: dropout

text: are

text: clearly

text: more

text: diverse

text: than

text: those

text: induced

text: by

text: dropout.

text: EXPERIMENTS

text: VISUALIZATIONS

text: OF

text: NOISE

text: PROJECTED

text: BACK

text: INTO

text: THE

text: INPUT

text: SPACE

text: Visualizing

text: the

text: noise

text: projected

text: back

text: into

text: the

text: input

text: space

text: helps

text: to

text: understand

text: what

text: kind

text: of

text: transfor-

text: mations

text: are

text: induced

text: by

text: dropout.

text: Unsupervised

text: models

text: learn

text: more

text: general

text: features

text: than

text: supervised

text: fully

text: connected

text: neural

text: networks

text: and

text: produce

text: thus

text: more

text: visually

text: appealing

text: transformations.

text: For

text: this

text: reason,

text: we

text: trained

text: autoencoders

text: with

text: dropout

text: on

text: the

text: hidden

text: layer

text: to

text: generate

text: samples

text: of

text: transfor-

text: mations.

text: The

text: autoencoder

text: is

text: very

text: similar

text: to

text: denoising

text: autoencoders,

text: the

text: only

text: difference

text: is

text: that

text: a

text: Bernoulli

text: mask

text: is

text: applied

text: to

text: the

text: hidden

text: activations

text: rather

text: than

text: to

text: the

text: input.

text: There

text: is

text: thus

text: no

text: noise

text: applied

text: to

text: the

text: input

text: explicitly.

text: Models

text: are

text: trained

text: for

text: epochs,

text: with

text: mini-batch

text: size

text: of

text: p

text: a

text: learning

text: rate

text: of

text: on

text: MNIST,

text: on

text: and

text: a

text: momentum

text: of

text: on

text: MNIST,

text: on

text: For

text: we

text: do

text: preprocessing

text: with

text: PCA

text: dimensionality

text: reduction

text: and

text: retain

text: features.

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text: (cid:112) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) √ (cid:101) (cid:101) (cid:101) − − − 4E1/4 E U(E (cid:112)∆)1/4ei E(cid:112) U t(E) = (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) , (cid:101) (cid:101) d(E) (cid:101) − − − (cid:48) 2E1/4(E U)1/4((cid:112)E ∆ + (cid:112)E U) t (E) = (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) , (cid:101) d(E) √ (cid:101) (cid:101) (cid:101) − − − − − (cid:48) 2E1/4(E(cid:112) U)1(cid:112)/4( E U(cid:112) E ∆(cid:112))e2i E U r (E) = (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) , (cid:112) (cid:112) (cid:112)d(E) (cid:112) (cid:101) −(cid:101) (cid:101)− (cid:101) (cid:101)− (cid:101) (cid:101)− (cid:101) ( E E U)( E ∆ + E U√) (cid:101) (cid:101) − − −(cid:101) − − − (cid:112)( E(cid:113)+ E U(cid:113))( E ∆(cid:113) E U)e2i E U r(E) = , (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) d(E) (cid:112) (cid:113) (cid:113) (cid:113) − − − (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) d(E) = ( E + E U)( E ∆ + E U) √ (cid:101) (cid:101) − − − − − − − ( E E U)( E ∆ E U)e2i E U, (35) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:126) and E = E/E , U = U/E , ∆ = ∆/E , E⊥ = E⊥/E , E⊥ = 2k2/2m, d d d d i t = t/t , t = t /t , σ = σ/d, x = x/d, x = x /d. The conditions (31) and (33) d 0 0 d i i read in the dimensionless var(cid:101)iables, c(cid:101)or(cid:101)respondin(cid:101)gly, | | (cid:29) (cid:29) x 2σ, E⊥ 1/σ2. (36) i (cid:101) (cid:101) It is instructive to consider ﬁrst the limiting case deﬁned(cid:101)b(cid:101)y the sec(cid:101)on(cid:101)d inequality (36). In this case, the forw(cid:101)a(cid:101)rd-moving terms ψ (x, t) in Eqs. (34) > ≈ give the(cid:101)m(cid:101)ain contribution to the total wave function, i.e., ψ(x, t) ψ>(x, t). Also, the inte(cid:29)grals over energy(cid:101)in ψ>(cid:101)(x, t) (34) can be asymptotically evalua(cid:101)te(cid:101)d at λ = E σ2 1 due to the fact that the contribution to these integrals ma(cid:101)inly i ≈ c(cid:101)omes from the energy region E E⊥. In(cid:101)th(cid:101)(cid:101)is (cid:28)c(cid:101)ase(cid:101), the wave functions ψ>(x, t) reduce (in the ﬁrst approximation with 1 1) to the stationary (for E = (cid:101) (cid:101) −E⊥σ2 − E⊥) results, oscillating with time as exp[ iE⊥(t t )]. Thus, if we present Eqs. 0 (34) for ψ (x, t) as > (cid:90) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) ∞ − − ψ (x, t) = ϕ(x, x ; E) exp[ iE(cid:112)(t t )(cid:113)] exp[λf(E)]dE, > i 0 (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) 0 (cid:29) − − λ = E⊥σ2 1, f(E) = ( E E⊥)2/E⊥, (37) (cid:101) (cid:101) (cid:101) where ϕ(x, x ; E) stands for any integrand in (34) multiplied by exponentials i of (37), the asymptotic value of (37) is 18 ##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##

label: equation

text: E

text: U(E

text: U

text: t(E)

text: d(E)

text: U)

text: t

text: (E)

text: d(E)

text: E

text: E

text: E

text: U

text: r

text: (E)

text: E

text: E

text: U)(

text: E

text: E

text: U√)

text: E

text: E

text: E

text: E

text: U

text: r(E)

text: d(E)

text: d(E)

text: E

text: E

text: U)(

text: E

text: E

text: U)

text: E

text: E

text: U)(

text: E

text: E

text: E

text: U,

text: and

text: E

text: E/E

text: U

text: U/E

text: ∆/E

text: E⊥

text: E⊥/E

text: E⊥

text: d

text: d

text: d

text: d

text: i

text: t

text: t/t

text: t

text: t

text: /t

text: σ

text: σ/d,

text: x

text: x/d,

text: x

text: x

text: /d.

text: The

text: conditions

text: and

text: d

text: d

text: i

text: i

text: read

text: in

text: the

text: dimensionless

text: x

text: E⊥

text: i

text: It

text: is

text: instructive

text: to

text: consider

text: ﬁrst

text: the

text: limiting

text: case

text: the

text: inequality

text: In

text: this

text: case,

text: the

text: terms

text: ψ

text: (x,

text: t)

text: in

text: Eqs.

text: give

text: contribution

text: to

text: the

text: total

text: wave

text: function,

text: i.e.,

text: ψ(x,

text: t)

text: ψ>(x,

text: t).

text: Also,

text: the

text: over

text: t)

text: can

text: be

text: asymptotically

text: at

text: λ

text: E

text: σ2

text: due

text: to

text: the

text: fact

text: that

text: the

text: contribution

text: to

text: these

text: integrals

text: i

text: from

text: the

text: energy

text: region

text: E

text: E⊥.

text: the

text: wave

text: functions

text: ψ>(x,

text: t)

text: reduce

text: (in

text: the

text: ﬁrst

text: approximation

text: with

text: to

text: the

text: stationary

text: (for

text: E

text: −E⊥σ2

text: E⊥)

text: results,

text: oscillating

text: with

text: time

text: as

text: exp[

text: iE⊥(t

text: t

text: Thus,

text: if

text: we

text: present

text: Eqs.

text: for

text: ψ

text: (x,

text: t)

text: as

text: ψ

text: (x,

text: t)

text: ϕ(x,

text: x

text: E)

text: exp[

text: t

text: exp[λf(E)]dE,

text: i

text: λ

text: E⊥σ2

text: f(E)

text: E

text: where

text: ϕ(x,

text: x

text: E)

text: stands

text: for

text: any

text: integrand

text: in

text: multiplied

text: by

text: exponentials

text: i

text: of

text: the

text: asymptotic

text: value

text: of

text: is

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:112) at the potential left boundary x = 0 is transmitted at the resonance con- − (cid:126) dition 2m(E⊥ U)/ 2d = πn to the region x > d beyond the potential. Only for a symmetric rectangular potential (∆ = 0) the reﬂection amplitude r(E) = 0 for √the resonant energies and there is only the probability density | | ψ (x) 2 = 1/ 2πσ (x < 0) stemming from an incoming wave and arriving to > the x > d area. Thus, the dependence of the constant in space transmitted probability density (39) versus the potential amplitude U will exhibit the oscil- lating (at E⊥ > U) pattern beyond the barrier (x > d) with an amplitude which (cid:54) is greater for the asymmetric barrier (∆ = 0) as compared to the symmetric one | | (∆ = 0). The same is true for the oscillating x-dependence of ψ (x) 2 inside > the potential area (0 < x < d). | | The time dependence of the probability density ψ(x, t)(cid:101)2 e(cid:101)xhibits itself only when there is a suﬃcient momentum dispersion, as follows from Eqs. (34). On ∼ the other hand, a suﬃcient momentum dispersion, when E⊥σ2 1, leads to a nonnegligible counterintuitive contribution of the backward-moving components (cid:12) (cid:12) | | o(cid:12) f t(cid:101)he(cid:101) (cid:12)wave packet to ψ(x, t) 2. The spacetime evolution of the s(cid:101)ca(cid:101)ttering process can be visualized by numerical evaluation of the probability density 2 ψ(x, t) (34) of ﬁnding the particle in the scaled space-time point (x, t). We will focus on the inﬂuence of the wave packet backward-moving components and the potential asymmetry parameter ∆ on the particle dynamics. As mentioned earlier, the asymmetric rectangular potential can model the potential proﬁle of the magnetic threelayer when it is switched from the parallel conﬁguration of the magnetic layer (modelled by the symmetric potential proﬁle with ∆ = 0) to the antiparallel orientation. For the case under consideration, when the particle, associated with the Gaussian wave packet, moves towards the potential (1) from the left, one can expect that the inﬂuence of the asymmetry parameter ∆ (deﬁning the height of the right potential step of (1)) will be more pronounced if the contribution of the backward-moving components of the wave packet is essential (the numerical evaluation co(cid:101)nﬁrms this exp(cid:101)ectation). To make the dynamics of the wave packet more particle-like, we accept the condition of the narrow wave packet, σ < 1, and put t = 0. For an electron and − 0 ∼ · − the potential width d = 10(cid:101)7cm (1nm)(cid:101), the characteris(cid:101)tic energy E(cid:101) 3 10 2ev ∼ · − d and the(cid:101)characteristic time t 2 10 14s. In accordance with the accepted d − c(cid:101)onditions, we will posit E⊥ = 102, xi = 10, and σ = 1/(cid:101)3 or σ = 0.1. We choose U = 10 in the case of a potential barrier (over-barrier transmission), and − U = 102 for a potential well. We will compare two cases: (cid:101)σ = 1/3, when the second inequality (36) is satisﬁed and the backward-m(cid:101)oving positive energies components of the initial wave packet are not essential, and σ = 0.1, when their ÷ contribution matters. T(cid:101)he dimensionless time interval t = 0.1 1.5 is chosen (cid:113) from a simple estimation for the average scaled time t /t that it takes a particle (cid:101) (cid:101) (cid:101) i d with the initial energy E⊥ = 102 to reach th(cid:12)e poten(cid:12)tial starting from the point − | | (cid:126) | | (cid:12) (cid:101) (cid:101) (cid:12) x = 10: t /t = x m/ k t = x /2 E⊥ = 1/2. (cid:101)i i d i i d i (cid:101) (cid:101) 2 Figure 1 shows the probability density ψ(x, t) of ﬁnding the particle at x = 1, i.e. on the right-hand side of the barrier (1) (U > 0), as a function of t 20 ##LTLine## ##LTLine## ##LTLine##

text: at

text: the

text: potential

text: left

text: boundary

text: x

text: is

text: transmitted

text: at

text: the

text: resonance

text: con-

text: dition

text: U)/

text: πn

text: to

text: the

text: region

text: x

text: d

text: beyond

text: the

text: potential.

text: Only

text: for

text: a

text: symmetric

text: rectangular

text: potential

text: the

text: reﬂection

text: amplitude

text: r(E)

text: for

text: √the

text: resonant

text: energies

text: and

text: there

text: is

text: only

text: the

text: probability

text: density

text: ψ

text: (x)

text: 2πσ

text: (x

text: stemming

text: from

text: an

text: incoming

text: wave

text: and

text: arriving

text: to

text: the

text: x

text: d

text: area.

text: Thus,

text: the

text: dependence

text: of

text: the

text: constant

text: in

text: space

text: transmitted

text: probability

text: density

text: versus

text: the

text: potential

text: amplitude

text: U

text: will

text: exhibit

text: the

text: oscil-

text: lating

text: (at

text: E⊥

text: U)

text: pattern

text: beyond

text: the

text: barrier

text: (x

text: d)

text: with

text: an

text: amplitude

text: which

text: is

text: greater

text: for

text: the

text: asymmetric

text: barrier

text: as

text: compared

text: to

text: the

text: symmetric

text: one

text: The

text: same

text: is

text: true

text: for

text: the

text: oscillating

text: x-dependence

text: of

text: ψ

text: (x)

text: inside

text: the

text: potential

text: area

text: x

text: d).

text: The

text: time

text: dependence

text: of

text: the

text: probability

text: density

text: ψ(x,

text: itself

text: only

text: when

text: there

text: is

text: a

text: suﬃcient

text: momentum

text: dispersion,

text: as

text: follows

text: from

text: Eqs.

text: On

text: the

text: other

text: hand,

text: a

text: suﬃcient

text: momentum

text: dispersion,

text: when

text: E⊥σ2

text: leads

text: to

text: a

text: nonnegligible

text: counterintuitive

text: contribution

text: of

text: the

text: backward-moving

text: components

text: f

text: packet

text: to

text: ψ(x,

text: t)

text: The

text: spacetime

text: evolution

text: of

text: the

text: process

text: can

text: be

text: visualized

text: by

text: numerical

text: evaluation

text: of

text: the

text: probability

text: density

text: ψ(x,

text: t)

text: of

text: ﬁnding

text: the

text: particle

text: in

text: the

text: scaled

text: space-time

text: point

text: (x,

text: t).

text: We

text: will

text: focus

text: on

text: the

text: inﬂuence

text: of

text: the

text: wave

text: packet

text: backward-moving

text: components

text: and

text: the

text: potential

text: asymmetry

text: parameter

text: on

text: the

text: particle

text: dynamics.

text: As

text: mentioned

text: earlier,

text: the

text: asymmetric

text: rectangular

text: potential

text: can

text: model

text: the

text: potential

text: proﬁle

text: of

text: the

text: magnetic

text: threelayer

text: when

text: it

text: is

text: switched

text: from

text: the

text: parallel

text: conﬁguration

text: of

text: the

text: magnetic

text: layer

text: (modelled

text: by

text: the

text: symmetric

text: potential

text: proﬁle

text: with

text: to

text: the

text: antiparallel

text: orientation.

text: For

text: the

text: case

text: under

text: consideration,

text: when

text: the

text: particle,

text: associated

text: with

text: the

text: Gaussian

text: wave

text: packet,

text: moves

text: towards

text: the

text: potential

text: from

text: the

text: left,

text: one

text: can

text: expect

text: that

text: the

text: inﬂuence

text: of

text: the

text: asymmetry

text: parameter

text: (deﬁning

text: the

text: height

text: of

text: the

text: right

text: potential

text: step

text: of

text: will

text: be

text: more

text: pronounced

text: if

text: the

text: contribution

text: of

text: the

text: backward-moving

text: components

text: of

text: the

text: wave

text: packet

text: is

text: essential

text: (the

text: numerical

text: evaluation

text: this

text: To

text: make

text: the

text: dynamics

text: of

text: the

text: wave

text: packet

text: more

text: particle-like,

text: we

text: accept

text: the

text: condition

text: of

text: the

text: narrow

text: wave

text: packet,

text: σ

text: and

text: put

text: t

text: For

text: an

text: electron

text: and

text: the

text: potential

text: width

text: d

text: the

text: energy

text: d

text: and

text: time

text: t

text: In

text: accordance

text: with

text: the

text: accepted

text: d

text: we

text: will

text: posit

text: E⊥

text: xi

text: and

text: σ

text: or

text: σ

text: We

text: choose

text: U

text: in

text: the

text: case

text: of

text: a

text: potential

text: barrier

text: (over-barrier

text: transmission),

text: and

text: U

text: for

text: a

text: potential

text: well.

text: We

text: will

text: compare

text: two

text: cases:

text: when

text: the

text: second

text: inequality

text: is

text: satisﬁed

text: and

text: the

text: positive

text: energies

text: components

text: of

text: the

text: initial

text: wave

text: packet

text: are

text: not

text: essential,

text: and

text: σ

text: when

text: their

text: contribution

text: matters.

text: dimensionless

text: time

text: interval

text: t

text: is

text: chosen

text: from

text: a

text: simple

text: estimation

text: for

text: the

text: average

text: scaled

text: time

text: t

text: /t

text: that

text: it

text: takes

text: a

text: particle

text: i

text: d

text: with

text: the

text: initial

text: energy

text: E⊥

text: to

text: reach

text: starting

text: from

text: the

text: point

text: x

text: t

text: /t

text: x

text: m/

text: k

text: t

text: x

text: E⊥

text: i

text: d

text: i

text: i

text: d

text: i

text: Figure

text: shows

text: the

text: probability

text: density

text: ψ(x,

text: t)

text: of

text: ﬁnding

text: the

text: particle

text: at

text: x

text: i.e.

text: on

text: the

text: right-hand

text: side

text: of

text: the

text: barrier

text: (U

text: as

text: a

text: function

text: of

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ (Dated: May 27, 2020) We report the realization of a quantum degenerate mixture of bosonic 87Rb and fermionic 171Yb atoms in a hybrid optical dipole trap with a tunable, species-dependent trapping potential. 87Rb is shown to be a × viable refrigerant for the noninteracting 171Yb atoms, cooling up to 2.4 105 Yb atoms to a temperature × of T/T = 0.16(2) while simultaneously forming a 87Rb Bose-Einstein condensate of 3.5 105 atoms. F Furthermore we demonstrate our ability to independently tailor the potentials for each species, which paves the way for studying impurities immersed in a Bose gas. I. INTRODUCTION isotope cannot be evaporated by itself due to its small s-wave − scattering length of 3a , 87Rb serves as a coolant with a 0 − Mixtures of quantum degenerate gases provide a platform favorable interspecies scattering length of 59a [18] and we × 0 for conducting a variety of experiments, such as the creation are able to sympathetically cool 2.4 105 Yb atoms down to of polar molecules [1, 2], the study of impurities in a Bose T = 0.16T . This represents a tenfold improvement in num- F gas [3], and atom interferometry [4]. In particular, there has ber over previous methods [24] to cool 171Yb. Furthermore been recent interest in observing the behavior of polarons on we show that this mixture is long lived and not signiﬁcantly a lattice [5] and the implementation of novel lattice cooling limited by scattering from any dipole beams. schemes [6, 7] using a Bose-Einstein condensate (BEC) as a The BIODT consists of co-propagating, overlapped, 1064- bath. Both of these experiments require independent tuning nm (red) and 532-nm (green) dipole traps. While both beams of the trapping potential for each species. While the abil- provide an attractive potential for Yb, the red trap is attractive ity to create a species-dependent potential has been demon- for Rb while the green is repulsive, allowing for independent strated in a spin-mixture of 87Rb [8–12], a 23Na-6Li mixture control of trap depths for both species. A 423-nm crossed [13] and a 87Rb-41K mixture [14], these methods relied on the dipole beam provides longitudinal conﬁnement for ytterbium light shifts from near-resonant light, which led to heating from atoms while the magnetic quadrupole ﬁeld does the same for large photon scattering rates. In the ﬁrst case a spin-dependent Rb atoms, giving us the ability to create species-dependent lattice was created using the vector light shift of rubidium at potentials for efﬁcient sympathetic cooling of Yb atoms. 790 nm, while in the latter two, a near resonant 670.5-nm Our approach, outlined in Fig. 2, is to load laser-cooled (789.9-nm) beam created a large light shift for lithium (potas- nonmagnetic Yb atoms into the green dipole trap and hold sium) but a negligible one for sodium (rubidium). Mixtures them while a Rb magneto-optical trap (MOT) is prepared and of alkali-metal and alkaline-earth atoms [15–17] on the other transferred into the magnetic quadrupole trap for forced rf hand are ideal for such experiments because their atomic spec- evaporation. The two MOTs have to be loaded sequentially tra differ greatly, allowing one to use light shifts from substan- since light-assisted collisions in a two-species MOT severely tially different wavelengths to create species-dependent traps. limit its lifetime and size. The repulsive potential created by We demonstrate the realization of a degenerate mixture of the green beam prevents hot Rb atoms from heating the colder bosonic 87Rb and fermionic 171Yb atoms. Unlike the Li-Yb Yb atoms out of the dipole trap. After RF evaporation of the and Rb-Sr mixtures presented in [15, 16], cooling the Rb-Yb Rb to the temperature of the Yb cloud, the red beam is turned mixture to degeneracy has been challenging due to extremely on, changing the repulsive green potential into an attractive unfavorable s-wave scattering lengths [18] between 87Rb and BIODT potential and initiating thermal contact between the most Yb isotopes. 172Yb and 176Yb do not form large BECs two species. Evaporation of Rb in the BIODT is then per- due to negative self-scattering lengths, and 174Yb phase sepa- formed to cool the rubidium and ytterbium clouds to degener- rates from 87Rb well before the onset of degeneracy [19] due acy. to a large positive interspecies scattering length. While a de- II. APPARATUS generate 87Rb-170Yb mixture may be possible, the s-wave in- teraction between the two species is negligible, leaving 171Yb The apparatus for creating degenerate mixtures is an adap- and 173Yb as the only two viable candidates for a stable, in- tation of the Rb BEC system described in [25] with a Yb teracting mixture with 87Rb. 2D-MOT addition, similar to the setups described in [26, 27]. Our mixture is trapped and evaporated in a species-dependent The two-dimensional (2D) MOT uses a ﬁeld gradient of potential created by a bichromatic optical dipole trap (BIODT) 5.5 mT/cm (55 G/cm) provided by a pair of permanent mag- ◦ [17, 20, 21], an ytterbium-blind magnetic quadrupole trap nets and is fed by an Yb oven heated to 410 C. The 2D-MOT − → 1 and a rubidium-blind crossed-dipole trap at 423.018 nm, beams, detuned by 50 MHz from the 1S 1P transi- 0 1 where the light shift for 87Rb vanishes [23]. While this Yb tion in Yb, have an intensity of 1.2I and are retro-reﬂected s through the 2D-MOT beam viewports. A push beam detuned → by +10 MHz from the 1S 3P transition accelerates atoms 0 1 ∗ along the axis of the 2D-MOT into the experiment chamber 1 The ground state of 171Yb only has a nuclear spin and a weak magnetic where they are captured and cool→ed in a three-dimensional moment, which leaves it largely unaffected by the magnetic quadrupole (3D) MOT operating on the 1S 3P line as described in 0 1 trap. [28]. ##LTLine##

text: (Dated:

text: May

label: date

text: We

text: report

text: the

text: realization

text: of

text: a

text: quantum

text: degenerate

text: mixture

text: of

text: bosonic

text: and

text: fermionic

text: atoms

text: in

text: a

text: hybrid

text: optical

text: dipole

text: trap

text: with

text: a

text: tunable,

text: species-dependent

text: trapping

text: potential.

text: is

text: shown

text: to

text: be

text: a

text: viable

text: refrigerant

text: for

text: the

text: noninteracting

text: atoms,

text: cooling

text: up

text: to

text: Yb

text: atoms

text: to

text: a

text: temperature

text: of

text: T/T

text: while

text: simultaneously

text: forming

text: a

text: Bose-Einstein

text: condensate

text: of

text: atoms.

text: F

text: Furthermore

text: we

text: demonstrate

text: our

text: ability

text: to

text: independently

text: tailor

text: the

text: potentials

text: for

text: each

text: species,

text: which

text: paves

text: the

text: way

text: for

text: studying

text: impurities

text: immersed

text: in

text: a

text: Bose

text: gas.

text: I.

label: section

text: INTRODUCTION

text: isotope

text: cannot

text: be

text: evaporated

text: by

text: itself

text: due

text: to

text: its

text: small

text: s-wave

text: scattering

text: length

text: of

text: serves

text: as

text: a

text: coolant

text: with

text: a

text: Mixtures

text: of

text: quantum

text: degenerate

text: gases

text: provide

text: a

text: platform

text: favorable

text: interspecies

text: scattering

text: length

text: of

text: and

text: we

text: for

text: conducting

text: a

text: variety

text: of

text: experiments,

text: such

text: as

text: the

text: creation

text: are

text: able

text: to

text: sympathetically

text: cool

text: Yb

text: atoms

text: down

text: to

text: of

text: polar

text: molecules

text: the

text: study

text: of

text: impurities

text: in

text: a

text: Bose

text: T

text: This

text: represents

text: a

text: tenfold

text: improvement

text: in

text: num-

text: F

text: gas

text: and

text: atom

text: interferometry

text: In

text: particular,

text: there

text: has

text: ber

text: over

text: previous

text: methods

text: to

text: cool

text: Furthermore

text: been

text: recent

text: interest

text: in

text: observing

text: the

text: behavior

text: of

text: polarons

text: on

text: we

text: show

text: that

text: this

text: mixture

text: is

text: long

text: lived

text: and

text: not

text: signiﬁcantly

text: a

text: lattice

text: and

text: the

text: implementation

text: of

text: novel

text: lattice

text: cooling

text: limited

text: by

text: scattering

text: from

text: any

text: dipole

text: beams.

text: schemes

text: using

text: a

text: Bose-Einstein

text: condensate

text: (BEC)

text: as

text: a

text: The

text: BIODT

text: consists

text: of

text: co-propagating,

text: overlapped,

text: bath.

text: Both

text: of

text: these

text: experiments

text: require

text: independent

text: tuning

text: nm

text: (red)

text: and

text: (green)

text: dipole

text: traps.

text: While

text: both

text: beams

text: of

text: the

text: trapping

text: potential

text: for

text: each

text: species.

text: While

text: the

text: abil-

text: provide

text: an

text: attractive

text: potential

text: for

text: Yb,

text: the

text: red

text: trap

text: is

text: attractive

text: ity

text: to

text: create

text: a

text: species-dependent

text: potential

text: has

text: been

text: demon-

text: for

text: Rb

text: while

text: the

text: green

text: is

text: repulsive,

text: allowing

text: for

text: independent

text: strated

text: in

text: a

text: spin-mixture

text: of

text: a

text: mixture

text: control

text: of

text: trap

text: depths

text: for

text: both

text: species.

text: A

text: crossed

text: and

text: a

text: mixture

text: these

text: methods

text: relied

text: on

text: the

text: dipole

text: beam

text: provides

text: longitudinal

text: conﬁnement

text: for

text: ytterbium

text: light

text: shifts

text: from

text: near-resonant

text: light,

text: which

text: led

text: to

text: heating

text: from

text: atoms

text: while

text: the

text: magnetic

text: quadrupole

text: ﬁeld

text: does

text: the

text: same

text: for

text: large

text: photon

text: scattering

text: rates.

text: In

text: the

text: ﬁrst

text: case

text: a

text: spin-dependent

text: Rb

text: atoms,

text: giving

text: us

text: the

text: ability

text: to

text: create

text: species-dependent

text: lattice

text: was

text: created

text: using

text: the

text: vector

text: light

text: shift

text: of

text: rubidium

text: at

text: potentials

text: for

text: efﬁcient

text: sympathetic

text: cooling

text: of

text: Yb

text: atoms.

text: nm,

text: while

text: in

text: the

text: latter

text: two,

text: a

text: near

text: resonant

text: Our

text: approach,

text: outlined

text: in

text: Fig.

text: is

text: to

text: load

text: laser-cooled

text: beam

text: created

text: a

text: large

text: light

text: shift

text: for

text: lithium

text: (potas-

text: nonmagnetic

text: Yb

text: atoms

text: into

text: the

text: green

text: dipole

text: trap

text: and

text: hold

text: sium)

text: but

text: a

text: negligible

text: one

text: for

text: sodium

text: (rubidium).

text: Mixtures

text: them

text: while

text: a

text: Rb

text: magneto-optical

text: trap

text: (MOT)

text: is

text: prepared

text: and

text: of

text: alkali-metal

text: and

text: alkaline-earth

text: atoms

text: on

text: the

text: other

text: transferred

text: into

text: the

text: magnetic

text: quadrupole

text: trap

text: for

text: forced

text: rf

text: hand

text: are

text: ideal

text: for

text: such

text: experiments

text: because

text: their

text: atomic

text: spec-

text: evaporation.

text: The

text: two

text: MOTs

text: have

text: to

text: be

text: loaded

text: sequentially

text: tra

text: differ

text: greatly,

text: allowing

text: one

text: to

text: use

text: light

text: shifts

text: from

text: substan-

text: since

text: light-assisted

text: collisions

text: in

text: a

text: two-species

text: MOT

text: severely

text: tially

text: different

text: wavelengths

text: to

text: create

text: species-dependent

text: traps.

text: limit

text: its

text: lifetime

text: and

text: size.

text: The

text: repulsive

text: potential

text: created

text: by

text: We

text: demonstrate

text: the

text: realization

text: of

text: a

text: degenerate

text: mixture

text: of

text: the

text: green

text: beam

text: prevents

text: hot

text: Rb

text: atoms

text: from

text: heating

text: the

text: colder

text: bosonic

text: and

text: fermionic

text: atoms.

text: Unlike

text: the

text: Li-Yb

text: Yb

text: atoms

text: out

text: of

text: the

text: dipole

text: trap.

text: After

text: RF

text: evaporation

text: of

text: the

text: and

text: Rb-Sr

text: mixtures

text: presented

text: in

text: cooling

text: the

text: Rb-Yb

text: Rb

text: to

text: the

text: temperature

text: of

text: the

text: Yb

text: cloud,

text: the

text: red

text: beam

text: is

text: turned

text: mixture

text: to

text: degeneracy

text: has

text: been

text: challenging

text: due

text: to

text: extremely

text: on,

text: changing

text: the

text: repulsive

text: green

text: potential

text: into

text: an

text: attractive

text: unfavorable

text: s-wave

text: scattering

text: lengths

text: between

text: and

text: BIODT

text: potential

text: and

text: initiating

text: thermal

text: contact

text: between

text: the

text: most

text: Yb

text: isotopes.

text: and

text: do

text: not

text: form

text: large

text: BECs

text: two

text: species.

text: Evaporation

text: of

text: Rb

text: in

text: the

text: BIODT

text: is

text: then

text: per-

text: due

text: to

text: negative

text: self-scattering

text: lengths,

text: and

text: phase

text: sepa-

text: formed

text: to

text: cool

text: the

text: rubidium

text: and

text: ytterbium

text: clouds

text: to

text: degener-

text: rates

text: from

text: well

text: before

text: the

text: onset

text: of

text: degeneracy

text: due

text: acy.

text: to

text: a

text: large

text: positive

text: interspecies

text: scattering

text: length.

text: While

text: a

text: de-

text: II.

text: APPARATUS

text: generate

text: mixture

text: may

text: be

text: possible,

text: the

text: s-wave

text: in-

text: teraction

text: between

text: the

text: two

text: species

text: is

text: negligible,

text: leaving

text: The

text: apparatus

text: for

text: creating

text: degenerate

text: mixtures

text: is

text: an

text: adap-

text: and

text: as

text: the

text: only

text: two

text: viable

text: candidates

text: for

text: a

text: stable,

text: in-

text: tation

text: of

text: the

text: Rb

text: BEC

text: system

text: described

text: in

text: with

text: a

text: Yb

text: teracting

text: mixture

text: with

text: addition,

text: similar

text: to

text: the

text: setups

text: described

text: in

text: Our

text: mixture

text: is

text: trapped

text: and

text: evaporated

text: in

text: a

text: species-dependent

text: The

text: two-dimensional

text: MOT

text: uses

text: a

text: ﬁeld

text: gradient

text: of

text: potential

text: created

text: by

text: a

text: bichromatic

text: optical

text: dipole

text: trap

text: (BIODT)

text: mT/cm

text: G/cm)

text: provided

text: by

text: a

text: pair

text: of

text: permanent

text: mag-

text: an

text: ytterbium-blind

text: magnetic

text: quadrupole

text: trap

text: nets

text: and

text: is

text: fed

text: by

text: an

text: Yb

text: oven

text: heated

text: to

text: C.

text: The

text: and

text: a

text: rubidium-blind

text: crossed-dipole

text: trap

text: at

text: nm,

text: beams,

text: detuned

text: by

text: MHz

text: from

text: the

text: transi-

text: where

text: the

text: light

text: shift

text: for

text: vanishes

text: While

text: this

text: Yb

text: tion

text: in

text: Yb,

text: have

text: an

text: intensity

text: of

text: and

text: are

text: retro-reﬂected

text: s

text: through

text: the

text: beam

text: viewports.

text: A

text: push

text: beam

text: detuned

text: by

text: MHz

text: from

text: the

text: transition

text: accelerates

text: atoms

text: along

text: the

text: axis

text: of

text: the

text: into

text: the

text: experiment

text: chamber

text: The

text: ground

text: state

text: of

text: only

text: has

text: a

text: nuclear

text: spin

text: and

text: a

text: weak

text: magnetic

text: where

text: they

text: are

text: captured

text: and

text: cool→ed

text: in

text: a

text: three-dimensional

text: moment,

text: which

text: leaves

text: it

text: largely

text: unaffected

text: by

text: the

text: magnetic

text: quadrupole

text: MOT

text: operating

text: on

text: the

text: line

text: as

text: described

text: in

text: trap.

text: ##LTLine##


text: ≈ 3 Myr (Bricen˜o et al. 2005). Superimposed on top of et al. 2014). These are predicted to precess on timescales notable quasi-periodic variability, the star showed regu- of decades. But van Eyken et al. (2012) and Barnes et lar transit events with a period of just 0.45 days. But al. (2013) showed that, for PTFO 8-8695, the precession − unlike TW Hya, which is a classical T Tauri star with period was on the order of a 300 600 days – a timescale active accretion (Rucinski & Krautter 1983) and has pe- accessible to observation. ∼ riodic 2% dips likely caused by the disk occulting a hot Barnes et al. (2013) were able to simultaneously ﬁt spot (Siwak et al. 2014), PTFO 8-8695, being a WTTS, both years of the PTF Orion data set using a fully con- has no infrared excess and likely no disk to occult or veil sistent model which included gravitational darkening, the star. Finally, no evidence was found for a stellar oblateness of the host star, and an analytical model of companion. the precession of the system. The ﬁts yielded a planet ∼ The observed optical radial velocity variations (Keck mass M 3 M , and predicted rapid precession on p Jup HIRES and HET HRS) were found to be out of phase approximately year-long timescales. The ﬁts implied a ≈ ◦ with the transit events and were not of suﬃcient ampli- very high obliquity for the orbit ( 70 ), resulting in a tude to be caused by a stellar companion, arguing that prediction that the transits should change depth, shape, the observed transits were not the result of an eclipsing and disappear for periods of a few months to a year as binary (van Eyken et al. 2012). This is in contrast to the precession brings the planet’s orbit out of the line of V471 Tau which shows similar depth eclipses, but also sight to the star. The limited timespan of the data and 150 km/s amplitude radial velocity variations in phase the uncertainty in the stellar mass and radius left an un- with the eclipses caused by a stellar mass white dwarf resolved degeneracy in the solution. More observations orbiting a K-dwarf (Kamin´ski et al. 2007). The radial were needed to test the veracity of the model and better velocity variations observed in the optical for PTFO 8- determine the system parameters. In particular, obser- 8695 are likely caused by the rotation of the spotted stel- vations were needed to test that the transits changed lar surface (Prato et al. 2008; Hu´elamo et al. 2008; Huerta shape, disappeared, and reappeared in a predictable et al. 2008). The spot induced observed radial velocity manner. ∼ variations in the optical ( 2 km/s semi-amplitude) yield Towards this end, we obtained follow-up observations a candidate companion mass upper limit of M sin i < of PTFO 8-8695 which included Spitzer photometry to 4.8 M . Coupled with the transit modeling and the conﬁrm the transit events in the infrared; Keck NIR- Jup ≈ ◦ constraints on the line of sight inclination (i 60 , van SPEC infrared radial velocity measurements to mea- Eyken et al. 2012), the upper limit of the companion sure the radial velocity signature of the planet and the (cid:46) mass was found to be M 5 M – well within the Rossiter-McLaughlin eﬀect (Rossiter 1924; McLaughlin p Jup planetary mass regime. 1924; Queloz et al. 2000; Gaudi & Winn 2007) as the The transit light curves did show shape changes be- planet transited the star, hinted at in the original opti- tween the two years in which the PTF Orion observa- cal radial velocity data (van Eyken et al. 2012); optical tions were made. Barnes et al. (2013) showed that these photometric monitoring with the Las Cumbres Observa- eﬀects could result from planetary transits of a fast- tory Global telescope Network (LCOGT) to support the rotating, oblate, and signiﬁcantly gravitationally dark- radial velocity measurements and to better establish the ened star, which is consistent with the short rotation pe- transit parameters and orbit. riod and high projected rotational velocity of PTFO 8- The Spitzer data, the Keck NIRSPEC data, and early ≈ 8685 (v sin i 80 km/s, van Eyken et al. 2012). Be- optical LCOGT photometry data were acquired before ing non-spherical, and darker at the equator than at the the realization that the transits could disappear com- poles (von Zeipel 1924), oblique transits of such a rapidly pletely, but these data, in the end, provided evidence for rotating star can show unusual and asymmetric transit the changing and disappearing transits. Further optical shapes (Barnes 2009). Furthermore, if the planet orbit is LCOGT observations were obtained with the realization oblique to the stellar rotation axis, a torque is exerted on that the system could precess and that the transits could the orbit by the stellar oblateness, leading to precession disappear. The optical data show the return of the tran- of the orbital nodes. sits at the time predicted by propagating the transit mid- Orbital precession has been noted previously in other point ephemeris, but approximately 1 month prior to the systems and its eﬀect on observed planetary transits - prediction of the precession model. Given the complexity particularly in circumbinary systems (e.g., Leung & Lee of the system as described in van Eyken et al. (2012) and 2013; Martin & Triaud 2014). Recently, such behavior Barnes et al. (2013), we regard the partial matching of has previously been noted in the Kepler 13b planet sys- the observations to the model as an indication that the temDr(aSftzavbe´orseiotnalM. a2y01217,, 22002102, 2014; Barnes et al. 2011), planetary interpretation of the nature of the source of anPdrewpriitnht ttyhpeesectirucsuinmg bLAiTnEaXrystyplelaenmeutl,atKeaeppj lve.r124/1136b/11(Kostov the transits is still viable, but we are unable to conﬁrm the planet with these data and the current models. FOLLOW-UP OBSERVATIONS OF PTFO 8-8695: A 3 MYR OLD T-TAURI STAR HOSTING A JUPITER-MASS PLANETARY CANDIDATE David R. Ciardi1, Julian C. van Eyken1,2,3, Jason W. Barnes4, Charles A. Beichman1, Sean J. Carey5, Christopher J. Crockett6, Jason Eastman3,7, Christopher M. Johns-Krull8, Steve B. Howell9, Stephen R. Kane10, Jacob N. Mclane11, Peter Plavchan12, L. Prato11, John Stauffer5, Gerard T. van Belle11, Kaspar

text: Myr

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text: on

text: timescales

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text: quasi-periodic

text: variability,

text: the

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text: of

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text: Eyken

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text: period

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text: arguing

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text: prediction

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text: transits

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text: change

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text: shape,

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text: periods

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text: amplitude

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text: particular,

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text: are

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text: rotation

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text: the

text: spotted

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text: vations

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text: needed

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text: transits

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text: lar

text: surface

text: (Prato

text: et

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text: shape,

text: disappeared,

text: and

text: reappeared

text: in

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text: predictable

text: et

text: al.

text: The

text: spot

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text: radial

text: velocity

text: manner.

text: variations

text: in

text: the

text: optical

text: km/s

text: semi-amplitude)

text: yield

text: Towards

text: this

text: end,

text: we

text: obtained

text: follow-up

text: observations

text: a

text: candidate

text: companion

text: mass

text: upper

text: limit

text: of

text: M

text: sin

text: i

text: of

text: PTFO

text: which

text: included

text: Spitzer

text: photometry

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text: M

text: Coupled

text: with

text: the

text: transit

text: modeling

text: and

text: the

text: conﬁrm

text: the

text: transit

text: events

text: in

text: the

text: infrared;

text: Keck

text: NIR-

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text: constraints

text: on

text: the

text: line

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text: sight

text: inclination

text: (i

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text: infrared

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text: Eyken

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text: the

text: upper

text: limit

text: of

text: the

text: companion

text: sure

text: the

text: radial

text: velocity

text: signature

text: of

text: the

text: planet

text: and

text: the

text: mass

text: was

text: found

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text: be

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text: well

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text: the

text: Rossiter-McLaughlin

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text: McLaughlin

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text: planetary

text: mass

text: regime.

text: Queloz

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text: al.

text: Gaudi

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text: The

text: transit

text: light

text: curves

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text: show

text: shape

text: changes

text: be-

text: planet

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text: star,

text: hinted

text: at

text: in

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text: original

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text: tween

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text: two

text: years

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text: (van

text: Eyken

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text: made.

text: Barnes

text: et

text: al.

text: showed

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text: Las

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text: eﬀects

text: could

text: result

text: from

text: planetary

text: transits

text: of

text: a

text: fast-

text: tory

text: Global

text: telescope

text: Network

text: (LCOGT)

text: to

text: support

text: the

text: rotating,

text: oblate,

text: and

text: signiﬁcantly

text: gravitationally

text: dark-

text: radial

text: velocity

text: measurements

text: and

text: to

text: better

text: establish

text: the

text: ened

text: star,

text: which

text: is

text: consistent

text: with

text: the

text: short

text: rotation

text: pe-

text: transit

text: parameters

text: and

text: orbit.

text: riod

text: and

text: high

text: projected

text: rotational

text: velocity

text: of

text: PTFO

text: The

text: Spitzer

text: data,

text: the

text: Keck

text: NIRSPEC

text: data,

text: and

text: early

text: (v

text: sin

text: i

text: km/s,

text: van

text: Eyken

text: et

text: al.

text: Be-

text: optical

text: LCOGT

text: photometry

text: data

text: were

text: acquired

text: before

text: ing

text: non-spherical,

text: and

text: darker

text: at

text: the

text: equator

text: than

text: at

text: the

text: the

text: realization

text: that

text: the

text: transits

text: could

text: disappear

text: com-

text: poles

text: (von

text: Zeipel

text: oblique

text: transits

text: of

text: such

text: a

text: rapidly

text: pletely,

text: but

text: these

text: data,

text: in

text: the

text: end,

text: provided

text: evidence

text: for

text: rotating

text: star

text: can

text: show

text: unusual

text: and

text: asymmetric

text: transit

text: the

text: changing

text: and

text: disappearing

text: transits.

text: Further

text: optical

text: shapes

text: (Barnes

text: Furthermore,

text: if

text: the

text: planet

text: orbit

text: is

text: LCOGT

text: observations

text: were

text: obtained

text: with

text: the

text: realization

text: oblique

text: to

text: the

text: stellar

text: rotation

text: axis,

text: a

text: torque

text: is

text: exerted

text: on

text: that

text: the

text: system

text: could

text: precess

text: and

text: that

text: the

text: transits

text: could

text: the

text: orbit

text: by

text: the

text: stellar

text: oblateness,

text: leading

text: to

text: precession

text: disappear.

text: The

text: optical

text: data

text: show

text: the

text: return

text: of

text: the

text: tran-

text: of

text: the

text: orbital

text: nodes.

text: sits

text: at

text: the

text: time

text: predicted

text: by

text: propagating

text: the

text: transit

text: mid-

text: Orbital

text: precession

text: has

text: been

text: noted

text: previously

text: in

text: other

text: point

text: ephemeris,

text: but

text: approximately

text: month

text: prior

text: to

text: the

text: systems

text: and

text: its

text: eﬀect

text: on

text: observed

text: planetary

text: transits

text: prediction

text: of

text: the

text: precession

text: model.

text: Given

text: the

text: complexity

text: particularly

text: in

text: circumbinary

text: systems

text: (e.g.,

text: Leung

text: Lee

text: of

text: the

text: system

text: as

text: described

text: in

text: van

text: Eyken

text: et

text: al.

text: and

text: Martin

text: Triaud

text: Recently,

text: such

text: behavior

text: Barnes

text: et

text: al.

text: we

text: regard

text: the

text: partial

text: matching

text: of

text: has

text: previously

text: been

text: noted

text: in

text: the

text: Kepler

text: planet

text: sys-

text: the

text: observations

text: to

text: the

text: model

text: as

text: an

text: indication

text: that

text: the

text: temDr(aSftzavbe´orseiotnalM.

text: Barnes

text: et

text: al.

text: planetary

text: interpretation

text: of

text: the

text: nature

text: of

text: the

text: source

text: of

text: anPdrewpriitnht

text: ttyhpeesectirucsuinmg

text: bLAiTnEaXrystyplelaenmeutl,atKeaeppj

text: the

text: transits

text: is

text: still

text: viable,

text: but

text: we

text: are

text: unable

text: to

text: conﬁrm

text: the

text: planet

text: with

text: these

text: data

text: and

text: the

text: current

text: models.

text: FOLLOW-UP

text: OBSERVATIONS

text: OF

text: PTFO

text: A

text: MYR

text: OLD

text: T-TAURI

text: STAR

text: HOSTING

text: A

text: JUPITER-MASS

text: PLANETARY

text: CANDIDATE

text: David

text: R.

text: Julian

text: C.

text: van

text: Jason

text: W.

text: Charles

text: A.

text: Sean

text: J.

text: Christopher

text: J.

text: Jason

text: Christopher

text: M.

text: Steve

text: B.

text: Stephen

text: R.

text: Jacob

text: N.

text: Peter

text: L.

text: John

text: Gerard

text: T.

text: van

text: Kaspar


text: )Q(gol R xx (cid:105) (cid:104) 2r/ 2(cid:107)r 0 p 5 0.5 1 25 ξ=2.0, Re=5 0 0.8 20 ξ=2.5, Re=5 −0.5 ξ=3.0, Re=5 −1 0.6 15 −1.5 (a) 0.4 (b) 10 (c) −2 0.2 ξ=2.0 5 −2.5 fitted curve Re=1 ξ=2.5 fitted curve Re=5 ξ=3.0 −30 200 400 600 800 1000 00 200 400 600 800 00 100 200 300 400 γ˙τ γ˙t γ˙t FIG. 4. (a) Cumulative PDF, Q, of the residence times of spheres (γ˙τ) in a layer for φ = 0.3, Re = 1 (in blue) , and Re = 5 (in black) and ξ = 2 (circles), and ξ = 2.5 (squares). (b) The streamwise velocity-velocity auto-correlation function, R , for xx Re = 5 and φ = 0.3 for four diﬀerent values of ξ = 2(red triangles), 2.5 (blue squares), and 3 (black dots). (b) Evolution of (cid:104) (cid:105) averaged square of relative distance between pairs of sphere, r(cid:107)2(t, r ) which were at distance r at t = 0; for r = 4a, for 0 p 0 0 three diﬀerent values of ξ = 2 (red triangles), 2.5 (blue squares), and 3 (black dots). ≈ ≈ ξ = 2, 3 we estimate β 0.61 and 0.88 respectively. But for ξ = 2.5 , β 1 is obtained. The diﬀusive behaviour for − ξ = 2.5 can be further conﬁrmed by plotting the PDF of d . At intermediate times (t = 2.5γ˙ 1), for ξ = 2.5, the PDF z develops Gaussian tails, indicative of a diﬀusive process and at late times it approaches a constant (see Supplemental Material, Fig. (8)). To visualize the dynamics, we provide movies of the particles’ trajectories (available at https://www.youtube. com/watch?v=Qn4DiXZFsbw for the case ξ = 2 and at https://www.youtube.com/watch?v=AmNsAsY0eC8 for the case ξ = 2.5 ). These clearly demonstrate that for ξ = 2, each horizontal layer is structurally disordered but dynamically frozen. We quantify this phenomenon by three diﬀerent measurements: (A) We calculate the residence time (τ) of a sphere in a single layer. A sphere is considered to reside within a horizontal layer till the wall-normal coordinate of its center, z , is within a distance of 2a of its initial position. Instead c of calculating the PDF by histograms of τ, we calculate the cumulative PDF, Q(τ), by the rank-order method [21], as the latter is free from binning errors. The cumulative PDF, Q(τ), is displayed in Fig. (4a) as a function of τ for ξ = 2 and 2.5 for Re = 1 and 5. For ξ = 2, Q remains very close to unity during the whole duration of the simulation, i.e., very few spheres actually move from one layer to another. In other words, the layers are quite stable to perturbations in wall-normal directions. Conversely for ξ = 2.5, Q(τ) can be ﬁtted by a Gaussian. ≡ (cid:104) (cid:105) (B) The streamwise velocity auto-correlation function of the spheres, R Up(t)Up(0) , is shown in Fig. (4)b. ≈ xx x x For ξ = 2, R 1 for a very long time, implying that the temporal ﬂuctuations of the stream-wise velocity are xx negligible. This suggests that each sphere moves in a layer with a uniform stream-wise velocity keeping their relative distances practically constant. For the cases where the layering is not very strong, e.g., ξ = 2.5 the auto-correlation function decays rapidly. For ξ = 3, layering reappears and R again shows slow decay in time. xx (C) Let us deﬁne r(cid:107)(t; r ) to be the (horizontal) distance between a pair of spheres at time t, which were at a 0 (horizontal) distance r and t = 0. If the layers formed by the spheres were truly frozen-in-time we would obtain 0 (cid:104) (cid:105) r(cid:107)(t; r ) = r for all t and r . In Fig. (4c), we show the time evolution of r(cid:107)2(t; r ) , for r = 4a, where the symbol (cid:104)·(cid:105) 0 0 0 0 p 0 denotes averaging over all possible particle pairs 2. Had the spheres moved chaotically within a layer, r(cid:107) would p have grown exponentially with time. Clearly for all the cases shown in Fig. (4c), r(cid:107) grows at most linearly with time. ≥ − In particular, when layering occurs (ξ = 2, 3), it takes a long time (t 430γ˙ 1) for r(cid:107)(t; 4a) to grow by a factor of two. This quantiﬁes again the dynamical stability of the layer, the relative in-plane distance between pairs of spheres change slowly (linearly) with time. In conclusion, using numerical simulations, we demonstrate that the eﬀective viscosity of an extremely conﬁned non-Brownian suspension can exhibit a non-monotonic dependence on the channel width, in particular the eﬀective viscosity sharply decreases if the channel width is an (small) integer multiple of particle diameter. We demonstrate that this behavior is accompanied by a change in micro-structure, namely the formation of particle layers parallel to the conﬁning plates. The two-dimensional layers formed by the particles slide on each other, the layers are structurally liquid-like by evolve on very show time scales. Similar layering under shear, have been theoretically anticipated [23], 2 To take into account the movement of the spheres across lay- ers, we calculate r(cid:107) up to the time it takes for the wall-normal distance between the pair of spheres to become larger than 2a. ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )Q(gol

text: R

text: xx

text: p

text: −1

text: (a)

text: (b)

text: (c)

text: −2

text: fitted

text: curve

text: fitted

text: curve

text: γ˙τ

text: γ˙t

text: γ˙t

text: FIG.

text: (a)

text: Cumulative

text: PDF,

text: Q,

text: of

text: the

text: residence

text: times

text: of

text: spheres

text: (γ˙τ)

text: in

text: a

text: layer

text: for

text: φ

text: Re

text: (in

text: blue)

text: and

text: Re

text: (in

text: black)

text: and

text: ξ

text: (circles),

text: and

text: ξ

text: (squares).

text: (b)

text: The

text: streamwise

text: velocity-velocity

text: auto-correlation

text: function,

text: R

text: for

text: xx

text: Re

text: and

text: φ

text: for

text: four

text: diﬀerent

text: values

text: of

text: ξ

text: triangles),

text: (blue

text: squares),

text: and

text: (black

text: dots).

text: (b)

text: Evolution

text: of

text: averaged

text: square

text: of

text: relative

text: distance

text: between

text: pairs

text: of

text: sphere,

text: r

text: which

text: were

text: at

text: distance

text: r

text: at

text: t

text: for

text: r

text: for

text: p

text: three

text: diﬀerent

text: values

text: of

text: ξ

text: (red

text: triangles),

text: (blue

text: squares),

text: and

text: (black

text: dots).

text: ξ

text: we

text: estimate

text: β

text: and

text: respectively.

text: But

text: for

text: ξ

text: β

text: is

text: obtained.

text: The

text: diﬀusive

text: behaviour

text: for

text: ξ

text: can

text: be

text: further

text: conﬁrmed

text: by

text: plotting

text: the

text: PDF

text: of

text: d

text: At

text: intermediate

text: times

text: (t

text: for

text: ξ

text: the

text: PDF

text: z

text: develops

text: Gaussian

text: tails,

text: indicative

text: of

text: a

text: diﬀusive

text: process

text: and

text: at

text: late

text: times

text: it

text: approaches

text: a

text: constant

text: (see

text: Supplemental

text: Material,

text: Fig.

text: To

text: visualize

text: the

text: dynamics,

text: we

text: provide

text: movies

text: of

text: the

text: particles’

text: trajectories

text: (available

text: at

text: for

text: the

text: case

text: ξ

text: and

text: at

text: for

text: the

text: case

text: ξ

text: These

text: clearly

text: demonstrate

text: that

text: for

text: ξ

text: each

text: horizontal

text: layer

text: is

text: structurally

text: disordered

text: but

text: dynamically

text: frozen.

text: We

text: quantify

text: this

text: phenomenon

text: by

text: three

text: diﬀerent

text: measurements:

text: (A)

text: We

text: calculate

text: the

text: residence

text: time

text: (τ)

text: of

text: a

text: sphere

text: in

text: a

text: single

text: layer.

text: A

text: sphere

text: is

text: considered

text: to

text: reside

text: within

text: a

text: horizontal

text: layer

text: till

text: the

text: wall-normal

text: coordinate

text: of

text: its

text: center,

text: z

text: is

text: within

text: a

text: distance

text: of

text: of

text: its

text: initial

text: position.

text: Instead

text: c

text: of

text: calculating

text: the

text: PDF

text: by

text: histograms

text: of

text: τ,

text: we

text: calculate

text: the

text: cumulative

text: PDF,

text: Q(τ),

text: by

text: the

text: rank-order

text: method

text: as

text: the

text: latter

text: is

text: free

text: from

text: binning

text: errors.

text: The

text: cumulative

text: PDF,

text: Q(τ),

text: is

text: displayed

text: in

text: Fig.

text: as

text: a

text: function

text: of

text: τ

text: for

text: ξ

text: and

text: for

text: Re

text: and

text: For

text: ξ

text: Q

text: remains

text: very

text: close

text: to

text: unity

text: during

text: the

text: whole

text: duration

text: of

text: the

text: simulation,

text: i.e.,

text: very

text: few

text: spheres

text: actually

text: move

text: from

text: one

text: layer

text: to

text: another.

text: In

text: other

text: words,

text: the

text: layers

text: are

text: quite

text: stable

text: to

text: perturbations

text: in

text: wall-normal

text: directions.

text: Conversely

text: for

text: ξ

text: Q(τ)

text: can

text: be

text: ﬁtted

text: by

text: a

text: Gaussian.

text: (B)

text: The

text: streamwise

text: velocity

text: auto-correlation

text: function

text: of

text: the

text: spheres,

text: R

text: is

text: shown

text: in

text: Fig.

text: xx

text: x

text: x

text: For

text: ξ

text: R

text: for

text: a

text: very

text: long

text: time,

text: implying

text: that

text: the

text: temporal

text: ﬂuctuations

text: of

text: the

text: stream-wise

text: velocity

text: are

text: xx

text: negligible.

text: This

text: suggests

text: that

text: each

text: sphere

text: moves

text: in

text: a

text: layer

text: with

text: a

text: uniform

text: stream-wise

text: velocity

text: keeping

text: their

text: relative

text: distances

text: practically

text: constant.

text: For

text: the

text: cases

text: where

text: the

text: layering

text: is

text: not

text: very

text: strong,

text: e.g.,

text: ξ

text: the

text: auto-correlation

text: function

text: decays

text: rapidly.

text: For

text: ξ

text: layering

text: reappears

text: and

text: R

text: again

text: shows

text: slow

text: decay

text: in

text: time.

text: xx

text: (C)

text: Let

text: us

text: deﬁne

text: r

text: to

text: be

text: the

text: (horizontal)

text: distance

text: between

text: a

text: pair

text: of

text: spheres

text: at

text: time

text: t,

text: which

text: were

text: at

text: a

text: (horizontal)

text: distance

text: r

text: and

text: t

text: If

text: the

text: layers

text: formed

text: by

text: the

text: spheres

text: were

text: truly

text: frozen-in-time

text: we

text: would

text: obtain

text: r

text: r

text: for

text: all

text: t

text: and

text: r

text: In

text: Fig.

text: we

text: show

text: the

text: time

text: evolution

text: of

text: r

text: for

text: r

text: where

text: the

text: symbol

text: p

text: denotes

text: averaging

text: over

text: all

text: possible

text: particle

text: pairs

text: Had

text: the

text: spheres

text: moved

text: chaotically

text: within

text: a

text: layer,

text: would

text: p

text: have

text: grown

text: exponentially

text: with

text: time.

text: Clearly

text: for

text: all

text: the

text: cases

text: shown

text: in

text: Fig.

text: grows

text: at

text: most

text: linearly

text: with

text: time.

text: In

text: particular,

text: when

text: layering

text: occurs

text: (ξ

text: it

text: takes

text: a

text: long

text: time

text: (t

text: for

text: to

text: grow

text: by

text: a

text: factor

text: of

text: two.

text: This

text: quantiﬁes

text: again

text: the

text: dynamical

text: stability

text: of

text: the

text: layer,

text: the

text: relative

text: in-plane

text: distance

text: between

text: pairs

text: of

text: spheres

text: change

text: slowly

text: (linearly)

text: with

text: time.

text: In

text: conclusion,

text: using

text: numerical

text: simulations,

text: we

text: demonstrate

text: that

text: the

text: eﬀective

text: viscosity

text: of

text: an

text: extremely

text: conﬁned

text: non-Brownian

text: suspension

text: can

text: exhibit

text: a

text: non-monotonic

text: dependence

text: on

text: the

text: channel

text: width,

text: in

text: particular

text: the

text: eﬀective

text: viscosity

text: sharply

text: decreases

text: if

text: the

text: channel

text: width

text: is

text: an

text: (small)

text: integer

text: multiple

text: of

text: particle

text: diameter.

text: We

text: demonstrate

text: that

text: this

text: behavior

text: is

text: accompanied

text: by

text: a

text: change

text: in

text: micro-structure,

text: namely

text: the

text: formation

text: of

text: particle

text: layers

text: parallel

text: to

text: the

text: conﬁning

text: plates.

text: The

text: two-dimensional

text: layers

text: formed

text: by

text: the

text: particles

text: slide

text: on

text: each

text: other,

text: the

text: layers

text: are

text: structurally

text: liquid-like

text: by

text: evolve

text: on

text: very

text: show

text: time

text: scales.

text: Similar

text: layering

text: under

text: shear,

text: have

text: been

text: theoretically

text: anticipated

text: To

text: take

text: into

text: account

text: the

text: movement

text: of

text: the

text: spheres

text: across

text: lay-

text: ers,

text: we

text: calculate

text: up

text: to

text: the

text: time

text: it

text: takes

text: for

text: the

text: wall-normal

text: distance

text: between

text: the

text: pair

text: of

text: spheres

text: to

text: become

text: larger

text: than

text: ##LTFigure##

text: ##LTFigure##

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