text: Comparative study of nonclassicality, entanglement, and dimensionality of multimode noisy twin beams ∗ Ievgen I. Arkhipov, Jan Peˇrina Jr., and Jan Peˇrina RCPTM, Joint Laboratory of Optics of Palack´y University and Institute of Physics of AS CR, Palack´y University, 17. listopadu 12, 771 46 Olomouc, Czech Republic Adam Miranowicz Faculty of Physics, Adam Mickiewicz University, PL-61-614 Poznan, Poland The nonclassicality, entanglement, and dimensionality of a noisy twin beam are determined using a characteristic function of the beam written in the Fock basis. One-to-one correspondence between the negativity quantifying entanglement and the nonclassicality depth is revealed. Twin beams, which are either entangled or nonclassical (independent of their entanglement), are observed only for the limited degrees of noise, which degrades their quantumness. The dimensionality of the twin beam quantiﬁed by the participation ratio is compared with the dimensionality of entanglement determined from the negativity. Partitioning of the degrees of freedom of the twin beam into those related to entanglement and to noise is suggested. Both single-mode and multimode twin beams are analyzed. Weak nonclassicality based on integrated-intensity quasidistributions of multimode twin beams is studied. The relation of the model to the experimental twin beams is discussed. PACS numbers: 42.65.Lm,42.50.Ar,03.67.Mn,42.65.Yj I. INTRODUCTION tion [11, 23, 24] of the statistical operator of a given ﬁeld. The commonly accepted formal criterion for dis- tinguishing nonclassical states from classical ones is ex- The question whether a given state cannot be de- pressed as follows [10, 11, 25, 26]: A quantum state is scribed within a classical theory has been considered nonclassical if its Glauber-Sudarshan P function fails to one of the most serious since the early days of quantum have the properties of a probability density. Alterna- physics [1–3] (for a review see, e.g., Ref. [4]). Nonclassi- tively, several operational criteria for nonclassicality of cality and entanglement, which is one of the nonclassical- either single-mode [25–27] or multimode [28–30] ﬁelds ity manifestations, are the most important properties of have been revealed. Their derivations are based either on optical ﬁelds studied in quantum optics. Such ﬁelds have ﬁeld’s moments [28, 30, 31] or on direct reconstruction of no classical analogs and as such they have been found quasidistributions of integrated intensities [32–34]. Also, interesting for many reasons. Nonclassical properties of criteria derived from the majorization theory have been such ﬁelds have been found useful both for elucidating the found [35, 36]. principles of quantum mechanics and in various applica- tions including, e.g., quantum information processing [5], Entanglement (or inseparability) is a special nonclassi- quantum metrology [6–8], and highly-sensitive measure- cal property that describes quantum correlations among ments [9]. (in general) several subsystems that cannot be treated by the means of classical statistical theory [37]. Various ap- From both the theoretical and the experimental points proaches have been developed for discrete and continuous of view, the nonlinear process of parametric down- variables to reveal entanglement. This property has been conversion, in which photon pairs are generated, has exploited in suggesting an entanglement criterion and the played an important role here from the beginning of in- related entanglement measure (referred to as the negativ- vestigations [4, 10–12]. Its individual photon pairs have ity) based upon the partial transposition of a statistical been exploited in many fundamental experiments testing operator [38–41]. Another approach has been based on nonclassical behavior predicted by quantum physics [13, the violation of the Bell inequalities written for diﬀerent 14]. It has also allowed the generation of more intense mean values including the measurement on both parts of ﬁelds having their electric-ﬁeld amplitude quadratures a bipartite system [42]. Also, a method using positive squeezed below the vacuum level [15–17], exhibiting sub- semi-deﬁnite matrices of ﬁelds’ moments of diﬀerent or- shot-noise correlations [18, 19] or having sub-Poissonian ders [43, 44] has been found to be very powerful. We photon-number statistics [20–22]. would like to stress at this point that entanglement is In quantum optics, the deﬁnition of nonclassicality a very crucial tool in today’s quantum information pro- is based upon the Glauber-Sudarshan P representa- cessing. In this contribution, we study nonclassicality by apply- ing the Lee nonclassical depth [45] as well as entangle- ∗ ment via the negativity [40, 41] for (in general) noisy twin Electronic address: \markcomm{footercolor}{arkhipov@ jointlab.upol.cz} beams of diﬀerent intensities. Such ﬁelds occur under real ##LTLine##

text: Comparative

label: title

text: study

text: of

text: nonclassicality,

text: entanglement,

text: and

text: dimensionality

text: of

text: multimode

text: noisy

text: twin

text: beams

label: paragraph

text: Ievgen

label: author

text: I.

text: Arkhipov,

text: Jan

text: Peˇrina

text: Jr.,

text: and

text: Jan

text: Peˇrina

text: RCPTM,

text: Joint

text: Laboratory

text: of

text: Optics

text: of

text: Palack´y

text: University

text: and

text: Institute

text: of

text: Physics

text: of

text: AS

text: CR,

text: Palack´y

text: University,

text: listopadu

text: Olomouc,

text: Czech

text: Republic

text: Adam

text: Miranowicz

text: Faculty

text: of

text: Physics,

text: Adam

text: Mickiewicz

text: University,

text: Poznan,

text: Poland

text: The

label: abstract

text: nonclassicality,

text: entanglement,

text: and

text: dimensionality

text: of

text: a

text: noisy

text: twin

text: beam

text: are

text: determined

text: using

text: a

text: characteristic

text: function

text: of

text: the

text: beam

text: written

text: in

text: the

text: Fock

text: basis.

text: One-to-one

text: correspondence

text: between

text: the

text: negativity

text: quantifying

text: entanglement

text: and

text: the

text: nonclassicality

text: depth

text: is

text: revealed.

text: Twin

text: beams,

text: which

text: are

text: either

text: entangled

text: or

text: nonclassical

text: (independent

text: of

text: their

text: entanglement),

text: are

text: observed

text: only

text: for

text: the

text: limited

text: degrees

text: of

text: noise,

text: which

text: degrades

text: their

text: quantumness.

text: The

text: dimensionality

text: of

text: the

text: twin

text: beam

text: quantiﬁed

text: by

text: the

text: participation

text: ratio

text: is

text: compared

text: with

text: the

text: dimensionality

text: of

text: entanglement

text: determined

text: from

text: the

text: negativity.

text: Partitioning

text: of

text: the

text: degrees

text: of

text: freedom

text: of

text: the

text: twin

text: beam

text: into

text: those

text: related

text: to

text: entanglement

text: and

text: to

text: noise

text: is

text: suggested.

text: Both

text: single-mode

text: and

text: multimode

text: twin

text: beams

text: are

text: analyzed.

text: Weak

text: nonclassicality

text: based

text: on

text: integrated-intensity

text: quasidistributions

text: of

text: multimode

text: twin

text: beams

text: is

text: studied.

text: The

text: relation

text: of

text: the

text: model

text: to

text: the

text: experimental

text: twin

text: beams

text: is

text: discussed.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: tion

text: of

text: the

text: statistical

text: operator

text: of

text: a

text: given

text: ﬁeld.

text: The

text: commonly

text: accepted

text: formal

text: criterion

text: for

text: dis-

text: tinguishing

text: nonclassical

text: states

text: from

text: classical

text: ones

text: is

text: ex-

text: The

text: question

text: whether

text: a

text: given

text: state

text: cannot

text: be

text: de-

text: pressed

text: as

text: follows

text: A

text: quantum

text: state

text: is

text: scribed

text: within

text: a

text: classical

text: theory

text: has

text: been

text: considered

text: nonclassical

text: if

text: its

text: Glauber-Sudarshan

text: P

text: function

text: fails

text: to

text: one

text: of

text: the

text: most

text: serious

text: since

text: the

text: early

text: days

text: of

text: quantum

text: have

text: the

text: properties

text: of

text: a

text: probability

text: density.

text: Alterna-

text: physics

text: (for

text: a

text: review

text: see,

text: e.g.,

text: Ref.

text: Nonclassi-

text: tively,

text: several

text: operational

text: criteria

text: for

text: nonclassicality

text: of

text: cality

text: and

text: entanglement,

text: which

text: is

text: one

text: of

text: the

text: nonclassical-

text: either

text: single-mode

text: or

text: multimode

text: ﬁelds

text: ity

text: manifestations,

text: are

text: the

text: most

text: important

text: properties

text: of

text: have

text: been

text: revealed.

text: Their

text: derivations

text: are

text: based

text: either

text: on

text: optical

text: ﬁelds

text: studied

text: in

text: quantum

text: optics.

text: Such

text: ﬁelds

text: have

text: ﬁeld’s

text: moments

text: or

text: on

text: direct

text: reconstruction

text: of

text: no

text: classical

text: analogs

text: and

text: as

text: such

text: they

text: have

text: been

text: found

text: quasidistributions

text: of

text: integrated

text: intensities

text: Also,

text: interesting

text: for

text: many

text: reasons.

text: Nonclassical

text: properties

text: of

text: criteria

text: derived

text: from

text: the

text: majorization

text: theory

text: have

text: been

text: such

text: ﬁelds

text: have

text: been

text: found

text: useful

text: both

text: for

text: elucidating

text: the

text: found

text: principles

text: of

text: quantum

text: mechanics

text: and

text: in

text: various

text: applica-

text: tions

text: including,

text: e.g.,

text: quantum

text: information

text: processing

text: Entanglement

text: (or

text: inseparability)

text: is

text: a

text: special

text: nonclassi-

text: quantum

text: metrology

text: and

text: highly-sensitive

text: measure-

text: cal

text: property

text: that

text: describes

text: quantum

text: correlations

text: among

text: ments

text: (in

text: general)

text: several

text: subsystems

text: that

text: cannot

text: be

text: treated

text: by

text: the

text: means

text: of

text: classical

text: statistical

text: theory

text: Various

text: ap-

text: From

text: both

text: the

text: theoretical

text: and

text: the

text: experimental

text: points

text: proaches

text: have

text: been

text: developed

text: for

text: discrete

text: and

text: continuous

text: of

text: view,

text: the

text: nonlinear

text: process

text: of

text: parametric

text: down-

text: variables

text: to

text: reveal

text: entanglement.

text: This

text: property

text: has

text: been

text: conversion,

text: in

text: which

text: photon

text: pairs

text: are

text: generated,

text: has

text: exploited

text: in

text: suggesting

text: an

text: entanglement

text: criterion

text: and

text: the

text: played

text: an

text: important

text: role

text: here

text: from

text: the

text: beginning

text: of

text: in-

text: related

text: entanglement

text: measure

text: (referred

text: to

text: as

text: the

text: negativ-

text: vestigations

text: Its

text: individual

text: photon

text: pairs

text: have

text: ity)

text: based

text: upon

text: the

text: partial

text: transposition

text: of

text: a

text: statistical

text: been

text: exploited

text: in

text: many

text: fundamental

text: experiments

text: testing

text: operator

text: Another

text: approach

text: has

text: been

text: based

text: on

text: nonclassical

text: behavior

text: predicted

text: by

text: quantum

text: physics

text: the

text: violation

text: of

text: the

text: Bell

text: inequalities

text: written

text: for

text: diﬀerent

text: It

text: has

text: also

text: allowed

text: the

text: generation

text: of

text: more

text: intense

text: mean

text: values

text: including

text: the

text: measurement

text: on

text: both

text: parts

text: of

text: ﬁelds

text: having

text: their

text: electric-ﬁeld

text: amplitude

text: quadratures

text: a

text: bipartite

text: system

text: Also,

text: a

text: method

text: using

text: positive

text: squeezed

text: below

text: the

text: vacuum

text: level

text: exhibiting

text: sub-

text: semi-deﬁnite

text: matrices

text: of

text: ﬁelds’

text: moments

text: of

text: diﬀerent

text: or-

text: shot-noise

text: correlations

text: or

text: having

text: sub-Poissonian

text: ders

text: has

text: been

text: found

text: to

text: be

text: very

text: powerful.

text: We

text: photon-number

text: statistics

text: would

text: like

text: to

text: stress

text: at

text: this

text: point

text: that

text: entanglement

text: is

text: In

text: quantum

text: optics,

text: the

text: deﬁnition

text: of

text: nonclassicality

text: a

text: very

text: crucial

text: tool

text: in

text: today’s

text: quantum

text: information

text: pro-

text: is

text: based

text: upon

text: the

text: Glauber-Sudarshan

text: P

text: representa-

text: cessing.

text: In

text: this

text: contribution,

text: we

text: study

text: nonclassicality

text: by

text: apply-

text: ing

text: the

text: Lee

text: nonclassical

text: depth

text: as

text: well

text: as

text: entangle-

text: ment

text: via

text: the

text: negativity

text: for

text: (in

text: general)

text: noisy

text: twin

text: Electronic

text: address:

text: \markcomm{footercolor}{arkhipov@

text: jointlab.upol.cz}

text: beams

text: of

text: diﬀerent

text: intensities.

text: Such

text: ﬁelds

text: occur

text: under

text: real

text: ##LTLine##


text: 12 noise in independent spatiotemporal modes have been in- Education, Youth and Sports of the Czech Republic is vestigated from the point of view of their entanglement acknowledged. I.A. thanks project IGA PrF 2015004 and multimode nonclassicality including weak nonclassi- of IGA UP Olomouc. J.P. Jr. acknowledges cality and dimensionality. project P205/12/0382 of GA CˇR. A.M. was supported Acknowledgments by the Polish National Science Centre under Grants No. DEC-2011/03/B/ST2/01903 and No. DEC- I.A. and J.P. Jr. thank M. Bondani for discus- 2011/02/A/ST2/00305. sions. Support by project LO1305 of the Ministry of [1] A. Einstein, Ann. Phys. 17, 132 (1905). Weinheim, 2006). [2] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, [27] T. Richter and W. Vogel, Phys. Rev. Lett. 89, 283601 777 (1935). (2002). [3] E. Schro¨dinger, Naturwissenschaften 23, 807 (1935). [28] A. Miranowicz, M. Bartkowiak, X. Wang, Y.-X. Liu, and [4] J. Peˇrina, Z. Hradil, and B. 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text: noise

text: in

text: independent

text: spatiotemporal

text: modes

text: have

text: been

text: in-

text: Education,

text: Youth

text: and

text: Sports

text: of

text: the

text: Czech

text: Republic

text: is

text: vestigated

text: from

text: the

text: point

text: of

text: view

text: of

text: their

text: entanglement

text: acknowledged.

text: I.A.

text: thanks

text: project

text: IGA

text: PrF

text: and

text: multimode

text: nonclassicality

text: including

text: weak

text: nonclassi-

text: of

text: IGA

text: UP

text: Olomouc.

text: J.P.

text: Jr.

text: acknowledges

text: cality

text: and

text: dimensionality.

text: project

text: of

text: GA

text: CˇR.

text: A.M.

text: was

text: supported

text: Acknowledgments

text: by

text: the

text: Polish

text: National

text: Science

text: Centre

text: under

text: Grants

text: No.

text: and

text: No.

text: DEC-

text: I.A.

text: and

text: J.P.

text: Jr.

text: thank

text: M.

text: Bondani

text: for

text: discus-

text: sions.

text: Support

text: by

text: project

text: of

text: the

text: Ministry

text: of

label: reference

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text: P.

text: A.

text: Volkov,

text: E.

text: V.

text: of

text: Light

text: (Taylor

text: Francis,

text: New

text: York,

text: Moreva,

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text: Straupe,

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text: Kulik,

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text: 4 It can be shown that eigenvalues of a block 11..44 of dimension M + 1 can be expressed as − − νM, νM 1ν−, . . . , ν ν−M 1, ν−M using the eigenvalues + + + 1.2 ν and ν− of a(cid:32)block with d(cid:114)imension 2: (cid:33) + (cid:16) (cid:17) 1 − 1 ∓ − 2 | | ν± = 1 B˜ + B˜ B˜ B˜ + 4 D 2 . 0.8 2K˜ 1 2 2 1 12 N (13) 0.6 The negative eigenvalues can only be those containing 0.4 odd powers of ν−. They form a geometric progression 0.2 whose elements can be summed to arrive at the formula for the negativity N: (cid:113) 0 0 0.1 0.2 0.3 0.4 0.5 B p − | | − − 1 3(B˜1 + B˜2) + (B˜1 B˜2)2 + 4 D12 2 4K˜ 2 N = − . FIG. 1: (Color online) Negativity N as a function of the mean 2 4K˜ 2(B˜1 + B˜2) + 1 photon-pair number Bp for noiseless twin beams (i.e., Bs = (14) B = 0)) according to Eq. (16). i | | Expressing parameters B˜ , B˜ , and D 2 in Eq. (14) 1 2 12 in terms of parameters B , B , and B , we arrive at the p s i formula (cid:8) (cid:113) The behavior of the negativity N of noisy twin beams (cid:9) − − dependent on the noise parameters B and B is illus- N = 2B (B + B )(4B + 1) 4B B s i p s i p s i (cid:8) (cid:9) trated in Fig. 2 for several values of the mean photon- − + (B B )2 + 4B (B + 1) pair number B . It holds in general that the greater the s i p p p × −1 value of the mean photon-pair number Bp, the greater 4(Bs + Bi)(2Bp + 1) + 8BsBi + 2 .(15) the value of the negativity N. This can be explained as follows. The more intense twin beams, with their ther- mal statistics, are eﬀectively spread over a larger number Equation (15) simpliﬁes considerably for noiseless twin of the Fock states. This naturally results in the larger ef- beams: (cid:113) fectively populated Hilbert spaces used to describe the entanglement. The greater value of the negativity N means a greater eﬀective number of the paired modes N = B + B (B + 1). (16) p p p building the entanglement, i.e., a greater value of the en- tanglement dimensionality, as deﬁned in Sec. V. Also, the greater the value of the mean photon-pair number B , According to Eq. (16), all noiseless twin beams are p the larger the amount of overall noise B +B acceptable entangled. The more intense the noiseless twin beams s i in an entangled twin beam (see Fig. 3). The curves plot- are, the more entangled the signal and idler ﬁelds are ted in Fig. 3 indicate that entanglement is more resistent (see Fig. 1). The presence of noise in a twin beam can to noise when the noise is distributed in the signal and even completely destroy entanglement, as the analysis of idler ﬁelds asymmetrically. We note that separable states Eq. (15) shows. Indeed, the condition N > 0 for entan- (i.e., with N = 0) contain, in general, paired, signal, and glement can be rewritten using Eq. (15) as follows: idler noisy contributions. However, the noisy contribu- tions are suﬃciently strong to suppress the “entangling − power” of the photon-pair contribution and so the state B [1 (B + B )] > B B . (17) p s i s i eﬀectively behaves as a classical statistical mixture of the signal and idler ﬁelds. Condition (17) cannot be fulﬁlled for any value of B ≥ p provided that B + B 1. Thus, the twin beam can be The decomposition of the partially transposed statis- s i entangled only when tical operator ρˆΓ into blocks in its matrix representation and the fact that a block (subspace) with dimension M+1 describes only states with up to M photons in the signal B B B + B < 1 and B > − s i . (18) (and also idler) ﬁeld can be used to deﬁne the distribu- s i p 1 (Bs + Bi) tion dN of the negativity N fulﬁlling the normalization condition ##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##

text: It

text: can

text: be

text: shown

text: that

text: eigenvalues

text: of

text: a

text: block

font: default

text: of

text: dimension

text: M

text: can

text: be

text: expressed

text: as

text: νM,

text: νM

text: 1ν−,

text: ν

text: ν−M

text: ν−M

text: using

text: the

text: eigenvalues

text: ν

text: and

text: ν−

text: of

label: equation

text: with

text: ν±

text: B˜

text: B˜

text: B˜

text: B˜

text: D

text: N

text: The

text: negative

text: eigenvalues

text: can

text: only

text: be

text: those

text: containing

text: odd

text: powers

text: of

text: ν−.

text: They

text: form

text: a

text: geometric

text: progression

text: whose

text: elements

text: can

text: be

text: summed

text: to

text: arrive

text: at

text: the

text: formula

text: for

text: the

text: negativity

text: N:

text: B

text: p

text: N

text: FIG.

text: (Color

text: online)

text: Negativity

text: N

text: as

text: a

text: function

text: of

text: the

text: mean

text: photon-pair

text: number

text: Bp

text: for

text: noiseless

text: twin

text: beams

text: (i.e.,

text: Bs

text: B

text: according

text: to

text: Eq.

text: i

text: Expressing

text: parameters

text: B˜

text: B˜

text: and

text: D

text: in

text: Eq.

text: in

text: terms

text: of

text: parameters

text: B

text: B

text: and

text: B

text: we

text: arrive

text: at

text: the

text: p

text: s

text: i

text: formula

text: The

text: behavior

text: of

text: the

text: negativity

text: N

text: of

text: noisy

text: twin

text: beams

text: dependent

text: on

text: the

text: noise

text: parameters

text: B

text: and

text: B

text: is

text: illus-

text: N

text: (B

text: B

text: B

text: s

text: i

text: p

text: s

text: i

text: p

text: s

text: i

text: trated

text: in

text: Fig.

text: for

text: several

text: values

text: of

text: the

text: mean

text: photon-

text: (B

text: B

text: (B

text: pair

text: number

text: B

text: It

text: holds

text: in

text: general

text: that

text: the

text: greater

text: the

text: s

text: i

text: p

text: p

text: p

text: −1

text: value

text: of

text: the

text: mean

text: photon-pair

text: number

text: Bp,

text: the

text: greater

text: the

text: value

text: of

text: the

text: negativity

text: N.

text: This

text: can

text: be

text: explained

text: as

text: follows.

text: The

text: more

text: intense

text: twin

text: beams,

text: with

text: their

text: ther-

text: mal

text: statistics,

text: are

text: eﬀectively

text: spread

text: over

text: a

text: larger

text: number

text: Equation

text: simpliﬁes

text: considerably

text: for

text: noiseless

text: twin

text: of

text: the

text: Fock

text: states.

text: This

text: naturally

text: results

text: in

text: the

text: larger

text: ef-

text: beams:

text: fectively

text: populated

text: Hilbert

text: spaces

text: used

text: to

text: describe

text: the

text: entanglement.

text: The

text: greater

text: value

text: of

text: the

text: negativity

text: N

text: means

text: a

text: greater

text: eﬀective

text: number

text: of

text: the

text: paired

text: modes

text: N

text: B

text: B

text: (B

text: p

text: p

text: p

text: building

text: the

text: entanglement,

text: i.e.,

text: a

text: greater

text: value

text: of

text: the

text: en-

text: tanglement

text: dimensionality,

text: as

text: deﬁned

text: in

text: Sec.

text: V.

text: Also,

text: the

text: greater

text: the

text: value

text: of

text: the

text: mean

text: photon-pair

text: number

text: B

text: According

text: to

text: Eq.

text: all

text: noiseless

text: twin

text: beams

text: are

text: p

text: the

text: larger

text: the

text: amount

text: of

text: overall

text: noise

text: B

text: +B

text: acceptable

text: entangled.

text: The

text: more

text: intense

text: the

text: noiseless

text: twin

text: beams

text: s

text: i

text: in

text: an

text: entangled

text: twin

text: beam

text: (see

text: Fig.

text: The

text: curves

text: plot-

text: are,

text: the

text: more

text: entangled

text: the

text: signal

text: and

text: idler

text: ﬁelds

text: are

text: ted

text: in

text: Fig.

text: indicate

text: that

text: entanglement

text: is

text: more

text: resistent

text: (see

text: Fig.

text: The

text: presence

text: of

text: noise

text: in

text: a

text: twin

text: beam

text: can

text: to

text: noise

text: when

text: the

text: noise

text: is

text: distributed

text: in

text: the

text: signal

text: and

text: even

text: completely

text: destroy

text: entanglement,

text: as

text: the

text: analysis

text: of

text: idler

text: ﬁelds

text: asymmetrically.

text: We

text: note

text: that

text: separable

text: states

text: Eq.

text: shows.

text: Indeed,

text: the

text: condition

text: N

text: for

text: entan-

text: (i.e.,

text: with

text: N

text: contain,

text: in

text: general,

text: paired,

text: signal,

text: and

text: glement

text: can

text: be

text: rewritten

text: using

text: Eq.

text: as

text: follows:

text: idler

text: noisy

text: contributions.

text: However,

text: the

text: noisy

text: contribu-

text: tions

text: are

text: suﬃciently

text: strong

text: to

text: suppress

text: the

text: “entangling

text: power”

text: of

text: the

text: photon-pair

text: contribution

text: and

text: so

text: the

text: state

text: B

text: (B

text: B

text: B

text: B

text: p

text: s

text: i

text: s

text: i

text: eﬀectively

text: behaves

text: as

text: a

text: classical

text: statistical

text: mixture

text: of

text: the

text: signal

text: and

text: idler

text: ﬁelds.

text: Condition

text: cannot

text: be

text: fulﬁlled

text: for

text: any

text: value

text: of

text: B

text: p

text: provided

text: that

text: B

text: B

text: Thus,

text: the

text: twin

text: beam

text: can

text: be

text: The

text: decomposition

text: of

text: the

text: partially

text: transposed

text: statis-

text: s

text: i

text: entangled

text: only

text: when

text: tical

text: operator

text: ρˆΓ

text: into

text: blocks

text: in

text: its

text: matrix

text: representation

text: and

text: the

text: fact

text: that

text: a

text: block

text: (subspace)

text: with

text: dimension

text: describes

text: only

text: states

text: with

text: up

text: to

text: M

text: photons

text: in

text: the

text: signal

text: B

text: B

text: B

text: B

text: and

text: B

text: s

text: i

text: (and

text: also

text: idler)

text: ﬁeld

text: can

text: be

text: used

text: to

text: deﬁne

text: the

text: distribu-

text: s

text: i

text: p

text: (Bs

text: Bi)

text: tion

text: dN

text: of

text: the

text: negativity

text: N

text: fulﬁlling

text: the

text: normalization

text: condition

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: 7 statistical operators can be given in terms of the nega- 10 tivity N by a simple formula [49]: 8 || || K (ρˆ) = 2N(ρˆ) + 1 = ρˆΓ . (23) ent 1 6 ≥ Strictly speaking, it is the least integer K that N ent gives a lower bound to the number of entangled dimen- 4 sions between entangled subsystems (paired modes) of ρˆ [49]. According to Eq. (23), the entanglement dimen- sionality K equals 1 for separable states (N = 0). It ent 2 linearly increases with the negativity N. As the noise described by the mean noise photon numbers B and B s i decreases the values of the negativity N, it also decreases 0 0 0.1 0.2 τ 0.3 0.4 0.5 the values of the entanglement dimensionality Kent. We note that, for pure states, the Schmidt number is also a good quantiﬁer of the entanglement dimension K [63– ent FIG. 7: (Color online) Negativity N as a function of the non- 65]. The Schmidt decomposition of pure states accom- classical depth τ, according to Eq. (22). panied by convex optimization can even be applied for quantifying the entanglement dimension of mixed entan- gled states [37]. We note that inequality (17) represents the Simon cri- On the other hand, the noise present in the signal and terion for nonclassicality of Gaussian states as shown in idler ﬁelds requires additional degrees of freedom for its Ref. [62]. description. These degrees of freedom are, together with Comparison of Eqs. (16) and (21) made for noiseless those reserved for describing entanglement, determined twin beams reveals a simple relation between the nega- by the participation ratios R and R derived from the s i tivity N and the nonclassical depth τ: signal- and idler-ﬁeld reduced statistical operators ρˆ and s ρˆ , respectively [64, 66]: i τ N = − . (22) 1 2τ 1 R = , a = s, i. (24) a Tr [ρˆ2] a a Direct calculation based on Eqs. (15) and (20) then conﬁrms that relation (22) holds even for a general noisy Equation (10), giving the matrix elements of the sta- twin beam. We thus have a one-to-one correspondence tistical operator ρˆ, guarantees a diagonal form of the re- between the value of the negativity N and the value of duced statistical operators ρˆ and ρˆ of the signal and the nonclassical depth τ. Moreover, according to Eq. (22) s i idler ﬁelds, respectively. In this case, Eq. (24) can be the negativity N is an increasing function of the nonclas- rewritten in the form sical depth τ, and vice versa (see Fig. 7). There exists a (cid:80) deep physical reason for this correspondence. The non- linear process emits photons in pairs into the signal and 1 R = . (25) idler ﬁelds, which creates entanglement between these s ρ2 j s,jj ﬁelds. It is this entanglement that gives rise to nonclas- sical properties of twin beams, as the classical statistical Using Eq. (10) the matrix elements ρ can be written optics is unable to describe pairing of photons appropri- s,jj ately. as (cid:34)(cid:32) (cid:33) (cid:35) V. DIMENSIONALITY OF THE TWIN BEAM | | j 1 − B˜ D 2 Three diﬀerent numbers are needed to determine the ρ = 1 i + 12 . (26) dimensionality of a general noisy twin beam. The di- s,jj B˜s K˜ K˜ B˜s mensionality K of entanglement gives the number of ent degrees of freedom constituting the entangled (paired) Substituting Eq. (26) into Eq. (25) we obtain a simple part of the twin beam. We also need additional degrees of formula for the participation ratio R : s freedom to characterize the noisy parts of the twin beam. As the amount of noise is, in general, diﬀerent in the sig- nal and idler ﬁelds, we have independent participation R = 2(B + B ) + 1. (27) s p s ratios R and R for both ﬁelds. The entanglement di- s i mensionality K for bipartite states with axisymmetric ent ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: statistical

text: operators

text: can

text: be

text: given

text: in

text: terms

text: of

text: the

text: nega-

text: tivity

text: N

text: by

text: a

text: simple

text: formula

text: K

text: (ρˆ)

text: ρˆΓ

text: ent

text: Strictly

text: speaking,

text: it

text: is

text: the

text: least

text: integer

text: K

text: that

text: N

text: ent

text: gives

text: a

text: lower

text: bound

text: to

text: the

text: number

text: of

text: entangled

text: dimen-

text: sions

text: between

text: entangled

text: subsystems

text: (paired

text: modes)

text: of

text: ρˆ

text: According

text: to

text: Eq.

text: the

text: entanglement

text: dimen-

text: sionality

text: K

text: equals

text: for

text: separable

text: states

text: (N

text: It

text: ent

text: linearly

text: increases

text: with

text: the

text: negativity

text: N.

text: As

text: the

text: noise

text: described

text: by

text: the

text: mean

text: noise

text: photon

text: numbers

text: B

text: and

text: B

text: s

text: i

text: decreases

text: the

text: values

text: of

text: the

text: negativity

text: N,

text: it

text: also

text: decreases

text: τ

text: the

text: values

text: of

text: the

text: entanglement

text: dimensionality

text: Kent.

text: We

text: note

text: that,

text: for

text: pure

text: states,

text: the

text: Schmidt

text: number

text: is

text: also

text: a

text: good

text: quantiﬁer

text: of

text: the

text: entanglement

text: dimension

text: K

text: ent

text: FIG.

text: (Color

text: online)

text: Negativity

text: N

text: as

text: a

text: function

text: of

text: the

text: non-

text: The

text: Schmidt

text: decomposition

text: of

text: pure

text: states

text: accom-

text: classical

text: depth

text: τ,

text: according

text: to

text: Eq.

text: panied

text: by

text: convex

text: optimization

text: can

text: even

text: be

text: applied

text: for

text: quantifying

text: the

text: entanglement

text: dimension

text: of

text: mixed

text: entan-

text: gled

text: states

text: We

text: note

text: that

text: inequality

text: represents

text: the

text: Simon

text: cri-

text: On

text: the

text: other

text: hand,

text: the

text: noise

text: present

text: in

text: the

text: signal

text: and

text: terion

text: for

text: nonclassicality

text: of

text: Gaussian

text: states

text: as

text: shown

text: in

text: idler

text: ﬁelds

text: requires

text: additional

text: degrees

text: of

text: freedom

text: for

text: its

text: Ref.

text: description.

text: These

text: degrees

text: of

text: freedom

text: are,

text: together

text: with

text: Comparison

text: of

text: Eqs.

text: and

text: made

text: for

text: noiseless

text: those

text: reserved

text: for

text: describing

text: entanglement,

text: determined

text: twin

text: beams

text: reveals

text: a

text: simple

text: relation

text: between

text: the

text: nega-

text: by

text: the

text: participation

text: ratios

text: R

text: and

text: R

text: derived

text: from

text: the

text: s

text: i

text: tivity

text: N

text: and

text: the

text: nonclassical

text: depth

text: τ:

text: signal-

text: and

text: idler-ﬁeld

text: reduced

text: statistical

text: operators

text: ρˆ

text: and

text: s

text: ρˆ

text: respectively

text: i

text: τ

text: N

text: 2τ

text: R

text: a

text: s,

text: i.

text: a

text: Tr

text: a

text: a

text: Direct

text: calculation

text: based

text: on

text: Eqs.

text: and

text: then

text: conﬁrms

text: that

text: relation

text: holds

text: even

text: for

text: a

text: general

text: noisy

text: Equation

text: giving

text: the

text: matrix

text: elements

text: of

text: the

text: sta-

text: twin

text: beam.

text: We

text: thus

text: have

text: a

text: one-to-one

text: correspondence

text: tistical

text: operator

text: ρˆ,

text: guarantees

text: a

text: diagonal

text: form

text: of

text: the

text: re-

text: between

text: the

text: value

text: of

text: the

text: negativity

text: N

text: and

text: the

text: value

text: of

text: duced

text: statistical

text: operators

text: ρˆ

text: and

text: ρˆ

text: of

text: the

text: signal

text: and

text: the

text: nonclassical

text: depth

text: τ.

text: Moreover,

text: according

text: to

text: Eq.

text: s

text: i

text: idler

text: ﬁelds,

text: respectively.

text: In

text: this

text: case,

text: Eq.

text: can

text: be

text: the

text: negativity

text: N

text: is

text: an

text: increasing

text: function

text: of

text: the

text: nonclas-

text: rewritten

text: in

text: the

text: form

text: sical

text: depth

text: τ,

text: and

text: vice

text: versa

text: (see

text: Fig.

text: There

text: exists

text: a

text: deep

text: physical

text: reason

text: for

text: this

text: correspondence.

text: The

text: non-

text: linear

text: process

text: emits

text: photons

text: in

text: pairs

text: into

text: the

text: signal

text: and

text: R

text: idler

text: ﬁelds,

text: which

text: creates

text: entanglement

text: between

text: these

text: s

text: ρ2

text: j

text: s,jj

text: ﬁelds.

text: It

text: is

text: this

text: entanglement

text: that

text: gives

text: rise

text: to

text: nonclas-

text: sical

text: properties

text: of

text: twin

text: beams,

text: as

text: the

text: classical

text: statistical

text: Using

text: Eq.

text: the

text: matrix

text: elements

text: ρ

text: can

text: be

text: written

text: optics

text: is

text: unable

text: to

text: describe

text: pairing

text: of

text: photons

text: appropri-

text: s,jj

text: ately.

text: as

text: V.

text: DIMENSIONALITY

text: OF

text: THE

text: TWIN

text: BEAM

text: j

text: B˜

text: D

text: Three

text: diﬀerent

text: numbers

text: are

text: needed

text: to

text: determine

text: the

text: ρ

text: i

text: dimensionality

text: of

text: a

text: general

text: noisy

text: twin

text: beam.

text: The

text: di-

text: s,jj

text: B˜s

text: K˜

text: K˜

text: B˜s

text: mensionality

text: K

text: of

text: entanglement

text: gives

text: the

text: number

text: of

text: ent

text: degrees

text: of

text: freedom

text: constituting

text: the

text: entangled

text: (paired)

text: Substituting

text: Eq.

text: into

text: Eq.

text: we

text: obtain

text: a

text: simple

text: part

text: of

text: the

text: twin

text: beam.

text: We

text: also

text: need

text: additional

text: degrees

text: of

text: formula

text: for

text: the

text: participation

text: ratio

text: R

text: s

text: freedom

text: to

text: characterize

text: the

text: noisy

text: parts

text: of

text: the

text: twin

text: beam.

text: As

text: the

text: amount

text: of

text: noise

text: is,

text: in

text: general,

text: diﬀerent

text: in

text: the

text: sig-

text: nal

text: and

text: idler

text: ﬁelds,

text: we

text: have

text: independent

text: participation

text: R

text: B

text: s

text: p

text: s

text: ratios

text: R

text: and

text: R

text: for

text: both

text: ﬁelds.

text: The

text: entanglement

text: di-

text: s

text: i

text: mensionality

text: K

text: for

text: bipartite

text: states

text: with

text: axisymmetric

text: ent

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 D can be evaluated in terms of the derivatives ˆ γ¯ that have i jk already been computed for the connection coefﬁcients (7). where the polytropic constant κ is a measure of the entropy, D We evaluate the covariant derivatives ˆ in the ﬂuid equa- and where Γ is related to the polytropic index n by Γ = 1 + tions by expanding them in terms of connection coefﬁcients; 1/n. The speciﬁc internal energy density (cid:15) can then be found the continuity equation (35), for example, becomes from the ﬁrst law of thermodynamics to be (cid:112) − ∂t(e6φ γ¯/γˆD) + ∂j(fD)j = (fD)jΓˆkjk. (36) (cid:15) = −1 P . (40) Γ 1 ρ 0 The appearance of the new terms on the right-hand side are During the dynamical evolution of these initial data we a disadvantage of this approach. The vanishing of the right- adopt a Γ-law EOS, meaning that we use eq. (40) to ﬁnd P in hand side of the continuity equation in the original Valencia- terms of the conserved variables ρ and (cid:15). formulation meant that, in an HRSC implementation, the total 0 For all simulations in Section we will use units in which rest mass (cid:90) (cid:90) the polytropic constant κ in (39) is unity, κ = 1. However, all dimensional quantities can be rescaled for any other value √ √ of κ by recognizing that, in geometrized units, κn/2 has units M0 = d3x γαutρ0 = d3x γD, (37) of length. Density, for example, has units of inverse length squared, again in geometrized units. The star considered in Section has a central density of ρ = 0.2 in our units with is conserved exactly; this is no longer the case in the c κ = 1; for any other value of κ, the central density is then reference-metric formulation (see Section below for a nu- − κ nρ . merical example.) c 4. Numerical Implementation In [21] we therefore considered two different implemen- tations, a full approach, in which we adopted all three hy- We use a HRSC scheme to solve the equations of relativis- drodynamic equations in the reference-metric version, and a tic hydrodynamics in the above form (see, e.g. [33, 34] for partial approach, in which we adopted the Euler equation in text-book treatments). In particular, we have implemented a the reference-metric version, but left both the continuity and second-order slope limiter reconstruction scheme, namely the the energy equation in the original Valencia formulation. In monotonic centered limiter [35], to obtain the left and right [21] we found that the advantages of the partial approach out- states of the primitive hydrodynamic variables at each cell weighed those of the full approach; however, our implemen- interface. We have also adopted the Harten-Lax-van-Leer- tation there did not adopt a factoring of tensor components. Einfeld approximate Riemann solver [36, 37]. We also refer to 2. Factoring of Tensor Components [26, 38] for similar treatments and more detailed discussion. Following common practice we introduce an artiﬁcial at- In this paper we apply to all hydrodynamical variables the mosphere to deal with the vacuum regions of spacetime, same factoring of tensor components that we described in Sec- which would otherwise create numerical problems. We fol- tion . The momentum density S , for example, is written as i low the prescription of [26] we set the density ρ to ρ = 0 atm   f max(ρ ) wherever it would otherwise be less than a atm 0 sr threshold density ρthres = fthresρatm, or wherever (cid:15) < (cid:15)atm. Si = rsθ , (38) Here we compute (cid:15)atm from ρatm using a polytropic EOS. F−or r sin θsϕ the simulations presented in Section we used fatm = 10 8 and f = 10. thres III. OFF-CENTER SIMULATIONS (and similar for all tensorial variables), and the s , s and r θ s are then evolved as the dynamical variables. We found that ϕ A. Vacuum with this rescaling, and using the full approach, i.e. adopting the reference-metric approach for all hydrodynamical vari- ables, even a shock wave passing through the origin of the 1. Off-Centered Schwarzschild Black Holes coordinate system will not lead to the formation of spikes or other numerical artifacts – we will present examples in Sec- As a ﬁrst example of off-centered simulations we consider tion . a Schwarzschild black hole that is placed away from the ori- 3. Equation of State gin of the coordinate system. We express this black hole in maximally-sliced “trumpet” coordinates (see [39]), which can For the simulations presented in Section we construct the be expressed analytically in parametric form [40]. In these co- initial data using a polytropic equation of state (EOS) ordinates, the black-hole horizon is located at a coordinate dis- tance r = 0.779M away from the black hole’s center. We hor evolve these data with the non-advective 1+log slicing (17) P = κρΓ, (39) and Gamma driver condition (20), using the analytical values 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: D

text: can

text: be

text: evaluated

text: in

text: terms

text: of

text: the

text: derivatives

text: ˆ

text: γ¯

text: that

text: have

text: i

text: jk

text: already

text: been

text: computed

text: for

text: the

text: connection

text: coefﬁcients

text: where

text: the

text: polytropic

text: constant

text: κ

text: is

text: a

text: measure

text: of

text: the

text: entropy,

text: D

text: We

text: evaluate

text: the

text: covariant

text: derivatives

text: ˆ

text: in

text: the

text: ﬂuid

text: equa-

text: and

text: where

text: Γ

text: is

text: related

text: to

text: the

text: polytropic

text: index

text: n

text: by

text: Γ

text: tions

text: by

text: expanding

text: them

text: in

text: terms

text: of

text: connection

text: coefﬁcients;

text: The

text: speciﬁc

text: internal

text: energy

text: density

text: can

text: then

text: be

text: found

text: the

text: continuity

text: equation

text: for

text: example,

text: becomes

text: from

text: the

text: ﬁrst

text: law

text: of

text: thermodynamics

text: to

text: be

text: γ¯/γˆD)

text: ∂j(fD)j

text: (fD)jΓˆkjk.

text: −1

text: P

text: Γ

text: ρ

text: The

text: appearance

text: of

text: the

text: new

text: terms

text: on

text: the

text: right-hand

text: side

text: are

text: During

text: the

text: dynamical

text: evolution

text: of

text: these

text: initial

text: data

text: we

text: a

text: disadvantage

text: of

text: this

text: approach.

text: The

text: vanishing

text: of

text: the

text: right-

text: adopt

text: a

text: Γ-law

text: EOS,

text: meaning

text: that

text: we

text: use

text: eq.

text: to

text: ﬁnd

text: P

text: in

text: hand

text: side

text: of

text: the

text: continuity

text: equation

text: in

text: the

text: original

text: Valencia-

text: terms

text: of

text: the

text: conserved

text: variables

text: ρ

text: and

text: formulation

text: meant

text: that,

text: in

text: an

text: HRSC

text: implementation,

text: the

text: total

text: For

text: all

text: simulations

text: in

text: Section

text: we

text: will

text: use

text: units

text: in

text: which

text: rest

text: mass

text: the

text: polytropic

text: constant

text: κ

text: in

text: is

text: unity,

text: κ

text: However,

text: all

text: dimensional

text: quantities

text: can

text: be

text: rescaled

text: for

text: any

text: other

text: value

text: of

text: κ

text: by

text: recognizing

text: that,

text: in

text: geometrized

text: units,

text: has

text: units

text: γαutρ0

text: γD,

text: of

text: length.

text: Density,

text: for

text: example,

text: has

text: units

text: of

text: inverse

text: length

text: squared,

text: again

text: in

text: geometrized

text: units.

text: The

text: star

text: considered

text: in

text: Section

text: has

text: a

text: central

text: density

text: of

text: ρ

text: in

text: our

text: units

text: with

text: is

text: conserved

text: exactly;

text: this

text: is

text: no

text: longer

text: the

text: case

text: in

text: the

text: c

text: κ

text: for

text: any

text: other

text: value

text: of

text: κ,

text: the

text: central

text: density

text: is

text: then

text: reference-metric

text: formulation

text: (see

text: Section

text: below

text: for

text: a

text: nu-

text: κ

text: nρ

text: merical

text: example.)

text: c

text: Numerical

text: Implementation

text: In

text: we

text: therefore

text: considered

text: two

text: different

text: implemen-

text: tations,

text: a

text: full

text: approach,

text: in

text: which

text: we

text: adopted

text: all

text: three

text: hy-

text: We

text: use

text: a

text: HRSC

text: scheme

text: to

text: solve

text: the

text: equations

text: of

text: relativis-

text: drodynamic

text: equations

text: in

text: the

text: reference-metric

text: version,

text: and

text: a

text: tic

text: hydrodynamics

text: in

text: the

text: above

text: form

text: (see,

text: e.g.

text: for

text: partial

text: approach,

text: in

text: which

text: we

text: adopted

text: the

text: Euler

text: equation

text: in

text: text-book

text: treatments).

text: In

text: particular,

text: we

text: have

text: implemented

text: a

text: the

text: reference-metric

text: version,

text: but

text: left

text: both

text: the

text: continuity

text: and

text: second-order

text: slope

text: limiter

text: reconstruction

text: scheme,

text: namely

text: the

text: the

text: energy

text: equation

text: in

text: the

text: original

text: Valencia

text: formulation.

text: In

text: monotonic

text: centered

text: limiter

text: to

text: obtain

text: the

text: left

text: and

text: right

text: we

text: found

text: that

text: the

text: advantages

text: of

text: the

text: partial

text: approach

text: out-

text: states

text: of

text: the

text: primitive

text: hydrodynamic

text: variables

text: at

text: each

text: cell

text: weighed

text: those

text: of

text: the

text: full

text: approach;

text: however,

text: our

text: implemen-

text: interface.

text: We

text: have

text: also

text: adopted

text: the

text: Harten-Lax-van-Leer-

text: tation

text: there

text: did

text: not

text: adopt

text: a

text: factoring

text: of

text: tensor

text: components.

text: Einfeld

text: approximate

text: Riemann

text: solver

text: We

text: also

text: refer

text: to

text: Factoring

text: of

text: Tensor

text: Components

text: for

text: similar

text: treatments

text: and

text: more

text: detailed

text: discussion.

text: Following

text: common

text: practice

text: we

text: introduce

text: an

text: artiﬁcial

text: at-

text: In

text: this

text: paper

text: we

text: apply

text: to

text: all

text: hydrodynamical

text: variables

text: the

text: mosphere

text: to

text: deal

text: with

text: the

text: vacuum

text: regions

text: of

text: spacetime,

text: same

text: factoring

text: of

text: tensor

text: components

text: that

text: we

text: described

text: in

text: Sec-

text: which

text: would

text: otherwise

text: create

text: numerical

text: problems.

text: We

text: fol-

text: tion

text: The

text: momentum

text: density

text: S

text: for

text: example,

text: is

text: written

text: as

text: i

text: low

text: the

text: prescription

text: of

text: we

text: set

text: the

text: density

text: ρ

text: to

text: ρ

text: atm

text: f

text: max(ρ

text: wherever

text: it

text: would

text: otherwise

text: be

text: less

text: than

text: a

text: atm

text: sr

text: threshold

text: density

text: ρthres

text: fthresρatm,

text: or

text: wherever

text: Si

text: rsθ

text: Here

text: we

text: compute

text: from

text: ρatm

text: using

text: a

text: polytropic

text: EOS.

text: F−or

text: r

text: sin

text: θsϕ

text: the

text: simulations

text: presented

text: in

text: Section

text: we

text: used

text: fatm

text: and

text: f

text: thres

text: III.

text: OFF-CENTER

text: SIMULATIONS

text: (and

text: similar

text: for

text: all

text: tensorial

text: variables),

text: and

text: the

text: s

text: s

text: and

text: r

text: θ

text: s

text: are

text: then

text: evolved

text: as

text: the

text: dynamical

text: variables.

text: We

text: found

text: that

text: ϕ

text: A.

text: Vacuum

text: with

text: this

text: rescaling,

text: and

text: using

text: the

text: full

text: approach,

text: i.e.

text: adopting

text: the

text: reference-metric

text: approach

text: for

text: all

text: hydrodynamical

text: vari-

text: ables,

text: even

text: a

text: shock

text: wave

text: passing

text: through

text: the

text: origin

text: of

text: the

text: Off-Centered

text: Schwarzschild

text: Black

text: Holes

text: coordinate

text: system

text: will

text: not

text: lead

text: to

text: the

text: formation

text: of

text: spikes

text: or

text: other

text: numerical

text: artifacts

text: we

text: will

text: present

text: examples

text: in

text: Sec-

text: As

text: a

text: ﬁrst

text: example

text: of

text: off-centered

text: simulations

text: we

text: consider

text: tion

text: a

text: Schwarzschild

text: black

text: hole

text: that

text: is

text: placed

text: away

text: from

text: the

text: ori-

text: Equation

text: of

text: State

text: gin

text: of

text: the

text: coordinate

text: system.

text: We

text: express

text: this

text: black

text: hole

text: in

text: maximally-sliced

text: “trumpet”

text: coordinates

text: (see

text: which

text: can

text: For

text: the

text: simulations

text: presented

text: in

text: Section

text: we

text: construct

text: the

text: be

text: expressed

text: analytically

text: in

text: parametric

text: form

text: In

text: these

text: co-

text: initial

text: data

text: using

text: a

text: polytropic

text: equation

text: of

text: state

text: (EOS)

text: ordinates,

text: the

text: black-hole

text: horizon

text: is

text: located

text: at

text: a

text: coordinate

text: dis-

text: tance

text: r

text: away

text: from

text: the

text: black

text: hole’s

text: center.

text: We

text: hor

text: evolve

text: these

text: data

text: with

text: the

text: non-advective

text: slicing

text: P

text: κρΓ,

text: and

text: Gamma

text: driver

text: condition

text: using

text: the

text: analytical

text: values

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 FIG. 1. The lapse function α in a slice of constant azimuthal angle ϕ, evolved on a grid of size (192, 36, 2). The color-coded surface shows the data at time t = 10M, while the wireframe shows the initial data at t = 0 – the two surfaces are hardly distinguishable in FIG. 3. In the top panel we show the value of the lapse α at the the ﬁgure. The contour lines are drawn for α = j × 0.1, where j is origin of the coordinate system, r = 0, as a function of time for an integer. a Schwarzschild black hole whose center is located at r = 1M and θ = 0. We show results for three different grid resolutions (N96, N18, 2) with N = 1, 4/3 and 2. In the bottom panel we show the numerical errors ∆α rescaled with factors of N2, demon- strating at least second-order convergence. nate system in the positive z-direction (the direction towards the North Pole), and choose the numerical grid to extend to an outer boundary imposed at r = 16M. We also choose the max grid resolution such that the radial resolution ∆r = r /N max r is similar to the angular resolution r∆θ = rπ/N at the center θ r of the black hole, which implies center ∼ r N π center N . (41) θ r r max In the following we present results for grids of size (N96, N18, 2) with N = 1, 4/3 and 2. In Fig. 1 we show a “slice” of the lapse function α at a time t = 10M. Here and in the following, a slice, similar to a slice in an orange, represents the data as a function of r and θ for a given value of the azimuthal angle ϕ. Given that FIG. 2. The lapse function α (top panel) and the conformal exponent initial data for our simulations in this paper are axisymmetric, φ (bottom panel) as a function of radius r in the θ = 0 direction, the particular value of ϕ does not matter. We then graph the at t = 9.5M, for the same simulation as in Fig. 1. The insets show data as functions of Cartesian coordinates z = r cos θ and the difference of each function between t = 0 and t = 9.5M. The x = r sin θ. center of the black hole is located at r = 1M. In Fig. 1 we show the data for our highest-resolution run with N = 2 (i.e. on a grid of size (192, 36, 2)) at time t = 10M as the shaded surface. We also include the initial data at for the lapse and shift as initial data. Analytically, the result- t = 0 as a wireframe representing each grid point used in this ing evolution should result in all metric components remain- simulation. As expected, the difference between the two data ing time-independent; any evolution away from the initial data sets is very small, so that they cannot be distinguished in the is therefore caused by numerical error. ﬁgure. In particular, we do not observe any problems arising We place the center of the black hole at r = 1M and at the origin of the coordinate system at r = 0. In Fig. 2 we center θ = 0, i.e. at a distance of 1M from the origin of the coordi- show both the conformal exponent φ and the lapse function ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: The

label: caption

text: lapse

text: function

text: α

text: in

text: a

text: slice

text: of

text: constant

text: azimuthal

text: angle

text: ϕ,

text: evolved

text: on

text: a

text: grid

text: of

text: size

text: The

text: color-coded

text: surface

text: shows

text: the

text: data

text: at

text: time

text: t

text: while

text: the

text: wireframe

text: shows

text: the

text: initial

text: data

text: at

text: t

text: the

text: two

text: surfaces

text: are

text: hardly

text: distinguishable

text: in

text: FIG.

text: In

text: the

text: top

text: panel

text: we

text: show

text: the

text: value

text: of

text: the

text: lapse

text: α

text: at

text: the

text: the

text: ﬁgure.

text: The

text: contour

text: lines

text: are

text: drawn

text: for

text: α

text: j

text: where

text: j

text: is

text: origin

text: of

text: the

text: coordinate

text: system,

text: r

text: as

text: a

text: function

text: of

text: time

text: for

text: an

text: integer.

text: a

text: Schwarzschild

text: black

text: hole

text: whose

text: center

text: is

text: located

text: at

text: r

text: and

text: θ

text: We

text: show

text: results

text: for

text: three

text: different

text: grid

text: resolutions

text: with

text: N

text: and

text: In

text: the

text: bottom

text: panel

text: we

text: show

text: the

text: numerical

text: errors

text: ∆α

text: rescaled

text: with

text: factors

text: of

text: demon-

text: strating

text: at

text: least

text: second-order

text: convergence.

text: nate

text: system

text: in

text: the

text: positive

text: z-direction

text: (the

text: direction

text: towards

text: the

text: North

text: Pole),

text: and

text: choose

text: the

text: numerical

text: grid

text: to

text: extend

text: to

text: an

text: outer

text: boundary

text: imposed

text: at

text: r

text: We

text: also

text: choose

text: the

text: max

text: grid

text: resolution

text: such

text: that

text: the

text: radial

text: resolution

text: ∆r

text: r

text: /N

text: max

text: r

text: is

text: similar

text: to

text: the

text: angular

text: resolution

text: r∆θ

text: rπ/N

text: at

text: the

text: center

text: θ

text: r

text: of

text: the

text: black

text: hole,

text: which

text: implies

text: center

text: r

text: N

text: π

text: center

text: N

text: θ

text: r

text: r

text: max

text: In

text: the

text: following

text: we

text: present

text: results

text: for

text: grids

text: of

text: size

text: with

text: N

text: and

text: In

text: Fig.

text: we

text: show

text: a

text: “slice”

text: of

text: the

text: lapse

text: function

text: α

text: at

text: a

text: time

text: t

text: Here

text: and

text: in

text: the

text: following,

text: a

text: slice,

text: similar

text: to

text: a

text: slice

text: in

text: an

text: orange,

text: represents

text: the

text: data

text: as

text: a

text: function

text: of

text: r

text: and

text: θ

text: for

text: a

text: given

text: value

text: of

text: the

text: azimuthal

text: angle

text: ϕ.

text: Given

text: that

text: FIG.

text: The

text: lapse

text: function

text: α

text: (top

text: panel)

text: and

text: the

text: conformal

text: exponent

text: initial

text: data

text: for

text: our

text: simulations

text: in

text: this

text: paper

text: are

text: axisymmetric,

text: φ

text: (bottom

text: panel)

text: as

text: a

text: function

text: of

text: radius

text: r

text: in

text: the

text: θ

text: direction,

text: the

text: particular

text: value

text: of

text: ϕ

text: does

text: not

text: matter.

text: We

text: then

text: graph

text: the

text: at

text: t

text: for

text: the

text: same

text: simulation

text: as

text: in

text: Fig.

text: The

text: insets

text: show

text: data

text: as

text: functions

text: of

text: Cartesian

text: coordinates

text: z

text: r

text: cos

text: θ

text: and

text: the

text: difference

text: of

text: each

text: function

text: between

text: t

text: and

text: t

text: The

text: x

text: r

text: sin

text: θ.

text: center

text: of

text: the

text: black

text: hole

text: is

text: located

text: at

text: r

text: In

text: Fig.

text: we

text: show

text: the

text: data

text: for

text: our

text: highest-resolution

text: run

text: with

text: N

text: (i.e.

text: on

text: a

text: grid

text: of

text: size

text: at

text: time

text: t

text: as

text: the

text: shaded

text: surface.

text: We

text: also

text: include

text: the

text: initial

text: data

text: at

text: for

text: the

text: lapse

text: and

text: shift

text: as

text: initial

text: data.

text: Analytically,

text: the

text: result-

text: t

text: as

text: a

text: wireframe

text: representing

text: each

text: grid

text: point

text: used

text: in

text: this

text: ing

text: evolution

text: should

text: result

text: in

text: all

text: metric

text: components

text: remain-

text: simulation.

text: As

text: expected,

text: the

text: difference

text: between

text: the

text: two

text: data

text: ing

text: time-independent;

text: any

text: evolution

text: away

text: from

text: the

text: initial

text: data

text: sets

text: is

text: very

text: small,

text: so

text: that

text: they

text: cannot

text: be

text: distinguished

text: in

text: the

text: is

text: therefore

text: caused

text: by

text: numerical

text: error.

text: ﬁgure.

text: In

text: particular,

text: we

text: do

text: not

text: observe

text: any

text: problems

text: arising

text: We

text: place

text: the

text: center

text: of

text: the

text: black

text: hole

text: at

text: r

text: and

text: at

text: the

text: origin

text: of

text: the

text: coordinate

text: system

text: at

text: r

text: In

text: Fig.

text: we

text: center

text: θ

text: i.e.

text: at

text: a

text: distance

text: of

text: from

text: the

text: origin

text: of

text: the

text: coordi-

text: show

text: both

text: the

text: conformal

text: exponent

text: φ

text: and

text: the

text: lapse

text: function

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 This occurs when the phase ϕ in the order parameter be- the polymer structure ﬁlls the space [5–7]: Under poor (cid:104) (cid:105) comes disordered so that eiϕ = 0. This implies that solvent conditions or at low temperatures, when the at- the expectation value of the order parameter also van- tractive interactions between the monomers dominate, a (cid:104) (cid:105) ishes, Ψ = 0. The system is then in the pseudogap single polymer chain is presumed to collapse into a con- state [14–18]. The pseudogap state is a symmetric phase ﬁguration which is space ﬁlling. On the other hand, in a precursor state in the broken symmetry phase. In partic- good solvent or at high temperatures, when the repulsive ular, the transition between the broken symmetry phase interactions dominate and cause the chain to eﬀectively and the pseudogap state is not a phase transition but swell, its geometric structure resembles that of a self- a cross-over prelude to the fully symmetric state, that avoiding random walk (SAW). Between the two, there the system enters when the lowest energy state of the is a θ-regime (possibly a tri-critical θ-point) where the eﬀective potential is one where the modulus vanishes. attractive and repulsive interactions cancel each other. It is reminded that the general arguments due to In the θ-regime the polymer chain is presumed to have Kadanoﬀ and Wilson [19–23] imply, that in the ther- the characteristics of an ordinary random walk (RW). modynamical limit the phase transition properties of a Finally, some polymers such as collagen for example, are material system are commonly universal, i.e. indepen- more like straight, rigid rods. Each of these four phases dent of the atomic level details. From this perspective, – rigid rod, SAW, RW and the space ﬁlling one – can be the construction of the phase diagram of a linear and chi- characterised by the inverse of the Hausdorﬀ dimension ral structureless homopolymer presented here, should be of the chain, called the scaling exponent ν [5–7]. This relevant for the understanding of the phase diagram of quantity is deﬁned by (cid:115)the radius of gyration (cid:88) more elaborate linear chiral homopolymers, maybe even that of certain heteropolymers [5–7]. Indeed, linear poly- 1 − mers are presumed to have a very similar phase structure, R = (r r )2 (2) quite independently of their chemical composition [5–7] gyr 2N2 i j i,j even though the phase where a particular polymer resides depends on many factors such as concentration, the qual- where the r are the coordinates of the individual i ity of solvent, ambient temperature and pressure. monomers. When the number of monomers N becomes The article is organised as follows: The next Section very large, the radius of gyration has the asymptotic ex- describes the background and the methods. The stan- pansion [24–26, 28] dard phase structure of linear polymers is ﬁrst reviewed. The geometrical order parameter variables that are used −→ − ∼ to model the free energy of a homopolymer are deﬁned, R2 N large R2N2ν(1 + R N δ1 + ...) R2N2ν (3) gyr 0 1 0 followed by an outline how the ensuing universal Hamil- tonian emerges in the limit of low wavelength deforma- tions. The order parameter that detects the presence of Here the length scale R0, the Kuhn length, is the eﬀec- a pseudogap state is then deﬁned. The zero temperature tive distance between the monomers in the large-N limit. ground state is identiﬁed in the case of pure steric re- The Kuhn length is not a universal quantity, its value pulsion, and a universal attractive long-range interaction can in principle be computed from the atomic level de- is introduced to model the eﬀect of hydrophobic forces tails of the polymer and environment including pressure, in the homopolymer chain. A detailed analysis of vari- temperature and chemical microstructure of the solvent. ous Monte Carlo algorithms is presented, to identify one The dimensionless scaling exponent ν i.e. the inverse that is computationally most eﬀective in the case of a Hausdorﬀ dimension that governs the large-N asymp- single homopolymer chain. In the subsequent section totic form of equation (3), is presumed to be a universal the results are then described. The eﬀect of the vari- quantity. Its numerical value is independent of the lo- ous parameters to the phase structure is revealed, and cal atomic level structure of the polymer [5, 7, 24–26]. in particular the pseudogap state is identiﬁed. Finally, The δ1 etc. are critical exponents and the R1 etc. are the phase diagram is constructed as a function of the the corresponding amplitudes, and together they consti- various parameters. It is found that in the case of a chi- tute the ﬁnite-size corrections. The δ1 etc. are universal ral homopolymer the phase diagram has a much richer quantities [26], but the R1 etc. are not universal [26]. structure than in the case of a non-chiral homopolymer. The following mean ﬁeld values are conventionally as- II. METHODS signed to ν [5–7]:   A. Phases  1/3 collapsed 1/2 RW ν = (4) A review of the known phase structure of linear, non- 3/5 SAW chiral homopolymers is now presented, as a background 1 rod and motivation for the subsequent study. Three diﬀerent, universal phases are commonly iden- Under poor solvent conditions or at low temperatures, tiﬁed, and these phases are categorised by the way how the polymer collapses into a space-ﬁlling conformation ##LTLine## ##LTLine##

text: This

text: occurs

text: when

text: the

text: phase

text: ϕ

text: in

text: the

text: order

text: parameter

text: be-

text: the

text: polymer

text: structure

text: ﬁlls

text: the

text: space

text: Under

text: poor

text: comes

text: disordered

text: so

text: that

text: eiϕ

text: This

text: implies

text: that

text: solvent

text: conditions

text: or

text: at

text: low

text: temperatures,

text: when

text: the

text: at-

text: the

text: expectation

text: value

text: of

text: the

text: order

text: parameter

text: also

text: van-

text: tractive

text: interactions

text: between

text: the

text: monomers

text: dominate,

text: a

text: ishes,

text: Ψ

text: The

text: system

text: is

text: then

text: in

text: the

text: pseudogap

text: single

text: polymer

text: chain

text: is

text: presumed

text: to

text: collapse

text: into

text: a

text: con-

text: state

text: The

text: pseudogap

text: state

text: is

text: a

text: symmetric

text: phase

text: ﬁguration

text: which

text: is

text: space

text: ﬁlling.

text: On

text: the

text: other

text: hand,

text: in

text: a

text: precursor

text: state

text: in

text: the

text: broken

text: symmetry

text: phase.

text: In

text: partic-

text: good

text: solvent

text: or

text: at

text: high

text: temperatures,

text: when

text: the

text: repulsive

text: ular,

text: the

text: transition

text: between

text: the

text: broken

text: symmetry

text: phase

text: interactions

text: dominate

text: and

text: cause

text: the

text: chain

text: to

text: eﬀectively

text: and

text: the

text: pseudogap

text: state

text: is

text: not

text: a

text: phase

text: transition

text: but

text: swell,

text: its

text: geometric

text: structure

text: resembles

text: that

text: of

text: a

text: self-

text: a

text: cross-over

text: prelude

text: to

text: the

text: fully

text: symmetric

text: state,

text: that

text: avoiding

text: random

text: walk

text: (SAW).

text: Between

text: the

text: two,

text: there

text: the

text: system

text: enters

text: when

text: the

text: lowest

text: energy

text: state

text: of

text: the

text: is

text: a

text: θ-regime

text: (possibly

text: a

text: tri-critical

text: θ-point)

text: where

text: the

text: eﬀective

text: potential

text: is

text: one

text: where

text: the

text: modulus

text: vanishes.

text: attractive

text: and

text: repulsive

text: interactions

text: cancel

text: each

text: other.

text: It

text: is

text: reminded

text: that

text: the

text: general

text: arguments

text: due

text: to

text: In

text: the

text: θ-regime

text: the

text: polymer

text: chain

text: is

text: presumed

text: to

text: have

text: Kadanoﬀ

text: and

text: Wilson

text: imply,

text: that

text: in

text: the

text: ther-

text: the

text: characteristics

text: of

text: an

text: ordinary

text: random

text: walk

text: (RW).

text: modynamical

text: limit

text: the

text: phase

text: transition

text: properties

text: of

text: a

text: Finally,

text: some

text: polymers

text: such

text: as

text: collagen

text: for

text: example,

text: are

text: material

text: system

text: are

text: commonly

text: universal,

text: i.e.

text: indepen-

text: more

text: like

text: straight,

text: rigid

text: rods.

text: Each

text: of

text: these

text: four

text: phases

text: dent

text: of

text: the

text: atomic

text: level

text: details.

text: From

text: this

text: perspective,

text: rigid

text: rod,

text: SAW,

text: RW

text: and

text: the

text: space

text: ﬁlling

text: one

text: can

text: be

text: the

text: construction

text: of

text: the

text: phase

text: diagram

text: of

text: a

text: linear

text: and

text: chi-

text: characterised

text: by

text: the

text: inverse

text: of

text: the

text: Hausdorﬀ

text: dimension

text: ral

text: structureless

text: homopolymer

text: presented

text: here,

text: should

text: be

text: of

text: the

text: chain,

text: called

text: the

text: scaling

text: exponent

text: ν

text: This

text: relevant

text: for

text: the

text: understanding

text: of

text: the

text: phase

text: diagram

text: of

text: quantity

text: is

text: deﬁned

text: by

text: radius

text: of

text: gyration

text: more

text: elaborate

text: linear

text: chiral

text: homopolymers,

text: maybe

text: even

text: that

text: of

text: certain

text: heteropolymers

text: Indeed,

text: linear

text: poly-

text: mers

text: are

text: presumed

text: to

text: have

text: a

text: very

text: similar

text: phase

text: structure,

text: R

text: (r

text: r

text: quite

text: independently

text: of

text: their

text: chemical

text: composition

text: gyr

text: i

text: j

text: i,j

text: even

text: though

text: the

text: phase

text: where

text: a

text: particular

text: polymer

text: resides

text: depends

text: on

text: many

text: factors

text: such

text: as

text: concentration,

text: the

text: qual-

text: where

text: the

text: r

text: are

text: the

text: coordinates

text: of

text: the

text: individual

text: i

text: ity

text: of

text: solvent,

text: ambient

text: temperature

text: and

text: pressure.

text: monomers.

text: When

text: the

text: number

text: of

text: monomers

text: N

text: becomes

text: The

text: article

text: is

text: organised

text: as

text: follows:

text: The

text: next

text: Section

text: very

text: large,

text: the

text: radius

text: of

text: gyration

text: has

text: the

text: asymptotic

text: ex-

text: describes

text: the

text: background

text: and

text: the

text: methods.

text: The

text: stan-

text: pansion

text: dard

text: phase

text: structure

text: of

text: linear

text: polymers

text: is

text: ﬁrst

text: reviewed.

text: The

text: geometrical

text: order

text: parameter

text: variables

text: that

text: are

text: used

text: to

text: model

text: the

text: free

text: energy

text: of

text: a

text: homopolymer

text: are

text: deﬁned,

text: N

text: large

text: R

text: N

text: δ1

text: gyr

text: followed

text: by

text: an

text: outline

text: how

text: the

text: ensuing

text: universal

text: Hamil-

text: tonian

text: emerges

text: in

text: the

text: limit

text: of

text: low

text: wavelength

text: deforma-

text: tions.

text: The

text: order

text: parameter

text: that

text: detects

text: the

text: presence

text: of

text: Here

text: the

text: length

text: scale

text: the

text: Kuhn

text: length,

text: is

text: the

text: eﬀec-

text: a

text: pseudogap

text: state

text: is

text: then

text: deﬁned.

text: The

text: zero

text: temperature

text: tive

text: distance

text: between

text: the

text: monomers

text: in

text: the

text: large-N

text: limit.

text: ground

text: state

text: is

text: identiﬁed

text: in

text: the

text: case

text: of

text: pure

text: steric

text: re-

text: The

text: Kuhn

text: length

text: is

text: not

text: a

text: universal

text: quantity,

text: its

text: value

text: pulsion,

text: and

text: a

text: universal

text: attractive

text: long-range

text: interaction

text: can

text: in

text: principle

text: be

text: computed

text: from

text: the

text: atomic

text: level

text: de-

text: is

text: introduced

text: to

text: model

text: the

text: eﬀect

text: of

text: hydrophobic

text: forces

text: tails

text: of

text: the

text: polymer

text: and

text: environment

text: including

text: pressure,

text: in

text: the

text: homopolymer

text: chain.

text: A

text: detailed

text: analysis

text: of

text: vari-

text: temperature

text: and

text: chemical

text: microstructure

text: of

text: the

text: solvent.

text: ous

text: Monte

text: Carlo

text: algorithms

text: is

text: presented,

text: to

text: identify

text: one

text: The

text: dimensionless

text: scaling

text: exponent

text: ν

text: i.e.

text: the

text: inverse

text: that

text: is

text: computationally

text: most

text: eﬀective

text: in

text: the

text: case

text: of

text: a

text: Hausdorﬀ

text: dimension

text: that

text: governs

text: the

text: large-N

text: asymp-

text: single

text: homopolymer

text: chain.

text: In

text: the

text: subsequent

text: section

text: totic

text: form

text: of

text: equation

text: is

text: presumed

text: to

text: be

text: a

text: universal

text: the

text: results

text: are

text: then

text: described.

text: The

text: eﬀect

text: of

text: the

text: vari-

text: quantity.

text: Its

text: numerical

text: value

text: is

text: independent

text: of

text: the

text: lo-

text: ous

text: parameters

text: to

text: the

text: phase

text: structure

text: is

text: revealed,

text: and

text: cal

text: atomic

text: level

text: structure

text: of

text: the

text: polymer

text: in

text: particular

text: the

text: pseudogap

text: state

text: is

text: identiﬁed.

text: Finally,

text: The

text: δ1

text: etc.

text: are

text: critical

text: exponents

text: and

text: the

text: etc.

text: are

text: the

text: phase

text: diagram

text: is

text: constructed

text: as

text: a

text: function

text: of

text: the

text: the

text: corresponding

text: amplitudes,

text: and

text: together

text: they

text: consti-

text: various

text: parameters.

text: It

text: is

text: found

text: that

text: in

text: the

text: case

text: of

text: a

text: chi-

text: tute

text: the

text: ﬁnite-size

text: corrections.

text: The

text: δ1

text: etc.

text: are

text: universal

text: ral

text: homopolymer

text: the

text: phase

text: diagram

text: has

text: a

text: much

text: richer

text: quantities

text: but

text: the

text: etc.

text: are

text: not

text: universal

text: structure

text: than

text: in

text: the

text: case

text: of

text: a

text: non-chiral

text: homopolymer.

text: The

text: following

text: mean

text: ﬁeld

text: values

text: are

text: conventionally

text: as-

text: II.

text: METHODS

text: signed

text: to

text: ν

text: A.

text: Phases

text: collapsed

text: RW

text: ν

text: A

text: review

text: of

text: the

text: known

text: phase

text: structure

text: of

text: linear,

text: non-

text: SAW

text: chiral

text: homopolymers

text: is

text: now

text: presented,

text: as

text: a

text: background

text: rod

text: and

text: motivation

text: for

text: the

text: subsequent

text: study.

text: Three

text: diﬀerent,

text: universal

text: phases

text: are

text: commonly

text: iden-

text: Under

text: poor

text: solvent

text: conditions

text: or

text: at

text: low

text: temperatures,

text: tiﬁed,

text: and

text: these

text: phases

text: are

text: categorised

text: by

text: the

text: way

text: how

text: the

text: polymer

text: collapses

text: into

text: a

text: space-ﬁlling

text: conformation

text: ##LTLine##

text: ##LTLine##