text: 6 Arrasmith et. al. 4.1 A worked example: primordial power 0.12 spectrum generalization ‘ = 100 i 0.10 A particularly interesting application is to a general- ‘ = 900 i ization of the primordial power spectrum. As noted in e.g. 0.08 ‘ = 1700 Planck Collaboration et al. (2014), features in the inﬂaton i potential lead directly to small-scale features in the power 0.06 ‘i = 2500 spectrum. 0.04 A concordance model of inﬂation does not yet exist (Martin et al. 2013), so the possibility exists that some small- 0.02 scale feature, detectable through its eﬀect on the power spec- trum, can help narrow the zoo of inﬂationary models and 0.00 provide insight to cosmological initial conditions and funda- mental physics at the GUT scale. As was previously done −0.02 in Planck Collaboration et al. (2014), we can parameterize −0.04 these eﬀects on the power spectrum by considering a gener- 100 101 102 103 104 alized spectrum given by Multipole ‘ j ∆2R(k) = ∆2Λ(k)[1+ f(k)], (18) Figure 3. Columns from the covariance matrix ΣΞ(cid:96)i(cid:96)j for a di- agonal prior on the ξn of the Gaussian parameterization of equa- where f(k) is a parameterization of the fractional devi- tion 20 (solid) and sinusoidal parameterization (dotted). Evident ation from(cid:18)the (cid:19)ΛCDM expectation ∆2Λ(k), given by are the smoothing eﬀects, which cause increasing correlation at increasing multipole (cid:96)j. The overall structure is broadly similar between the two parameterizations, reﬂecting the importance of k ns−1 the transfer function in the structure of the response to varying ∆2Λ(k) = As k∗ . (19) the ξns. Here As is the amplitude of the spectrum at k = k∗ and ns 2000 controls the scale dependence (with higher-order evolution such as running suppressed by slow roll). In Planck Collab- (150 Mpc)−1 oration et al. (2014), the function f(k) is represented as a 1500 (33 Mpc)−1 cubic spline with knots separated by δ lnk = 0.25. The spline is ﬁt with a regularization applied to both zero f(k) outside (7 Mpc)−1 the observable range of k and to penalize high frequency 1000 oscillations. The strength of the regularization essentially introduces a minimum correlation scale in the space of al- lowed deviations f(k) in the power spectrum. Here we do a 500 similar analysis using the Planck 2015 binned high-(cid:96) temper- ature data (Planck Collaboration et al. 2015b), but rather 0 than ﬁtting for the presence of a feature in the model, we in- stead use our goodness-of-ﬁt statistic to search for evidence of features present in the data allowed by the generalized −500 power spectrum model of equation 18 but not by ΛCDM. 100 101 102 103 We expect our results to be more general in the sense that Multipole ‘ they provide not only a search for deviations exactly pa- rameterizable by equation 18, but any feature that shares signiﬁcant correlation with typical draws from equation 18. Figure 4. A characteristic subset of the ∂∂|sξΞn(cid:105), where |sΞ(cid:105) = More simply, through the use of goodness-of-ﬁt rather than [(cid:96)((cid:96)+1)/2π]C(cid:96) = D(cid:96), for the Gaussian parameterization of equa- parameter estimation, we trade the ability to identify ex- tion 20. Curves are labeled by the central wavenumber k of the actly the shape of f(k) we are detecting for a sensitivity to corresponding window wn. The majority of the power put into the a wider range of signals. localized bumps ∆2(k) translates to peaks in the D(cid:96)’s. Our choice to work with the Planck binned high-(cid:96) tem- perature data follows Planck Collaboration et al. (2015c) in binning the high-(cid:96) likelihood in bins suﬃcient to allow the implicit assumption in equation 16 that the structure of the folding of foreground uncertainty into the noise covariance. noise can be taken to be Gaussian. This choice eases both the computation and interpretation Though the features of the primordial power spectrum of the modes |vi(cid:105): N in equation 16 represents the total noise could be arbitrarily peaked, Hu & Okamoto (2004) show covariance–experimental, foreground, and cosmic variance– that when these features are transported to the observed present in the observation. In addition, since the cosmic vari- power spectra, projection from the surface of last scattering ance at (cid:96) < 30 is comparatively large, we discard the low-(cid:96) and gravitational lensing smooth features below a charac- data where non-Gaussianities play a role. This justiﬁes the teristic width in ∆lnk. This provides a minimum frequency MNRAS 000, 000–000 (0000) edutilpmA D∂ ‘ ξ∂ k ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Arrasmith

text: et.

text: al.

text: A

text: worked

text: example:

text: primordial

text: power

font: default

text: spectrum

text: generalization

text: i

text: A

text: particularly

text: interesting

text: application

text: is

text: to

text: a

text: general-

text: i

text: ization

text: of

text: the

text: primordial

text: power

text: spectrum.

text: As

text: noted

text: in

text: e.g.

text: Planck

text: Collaboration

text: et

text: al.

text: features

text: in

text: the

text: inﬂaton

text: i

text: potential

text: lead

text: directly

text: to

text: small-scale

text: features

text: in

text: the

text: power

text: ‘i

text: spectrum.

text: A

text: concordance

text: model

text: of

text: inﬂation

text: does

text: not

text: yet

text: exist

text: (Martin

text: et

text: al.

text: so

text: the

text: possibility

text: exists

text: that

text: some

text: small-

text: scale

text: feature,

text: detectable

text: through

text: its

text: eﬀect

text: on

text: the

text: power

text: spec-

text: trum,

text: can

text: help

text: narrow

text: the

text: zoo

text: of

text: inﬂationary

text: models

text: and

text: provide

text: insight

text: to

text: cosmological

text: initial

text: conditions

text: and

text: funda-

text: mental

text: physics

text: at

text: the

text: GUT

text: scale.

text: As

text: was

text: previously

text: done

text: in

text: Planck

text: Collaboration

text: et

text: al.

text: we

text: can

text: parameterize

text: these

text: eﬀects

text: on

text: the

text: power

text: spectrum

text: by

text: considering

text: a

text: gener-

text: alized

text: spectrum

text: given

text: by

text: Multipole

text: j

text: f(k)],

text: Figure

text: Columns

text: from

text: the

text: covariance

text: matrix

text: for

text: a

text: di-

text: agonal

text: prior

text: on

text: the

text: ξn

text: of

text: the

text: Gaussian

text: parameterization

text: of

text: equa-

text: where

text: f(k)

text: is

text: a

text: parameterization

text: of

text: the

text: fractional

text: devi-

text: tion

text: (solid)

text: and

text: sinusoidal

text: parameterization

text: (dotted).

text: Evident

text: ation

text: expectation

text: given

text: by

text: are

text: the

text: smoothing

text: eﬀects,

text: which

text: cause

text: increasing

text: correlation

text: at

text: increasing

text: multipole

text: The

text: overall

text: structure

text: is

text: broadly

text: similar

text: between

text: the

text: two

text: parameterizations,

text: reﬂecting

text: the

text: importance

text: of

text: k

text: the

text: transfer

text: function

text: in

text: the

text: structure

text: of

text: the

text: response

text: to

text: varying

text: As

text: k∗

text: the

text: ξns.

text: Here

text: As

text: is

text: the

text: amplitude

text: of

text: the

text: spectrum

text: at

text: k

text: k∗

text: and

text: ns

text: controls

text: the

text: scale

text: dependence

text: (with

text: higher-order

text: evolution

text: such

text: as

text: running

text: suppressed

text: by

text: slow

text: roll).

text: In

text: Planck

text: Collab-

text: oration

text: et

text: al.

text: the

text: function

text: f(k)

text: is

text: represented

text: as

text: a

text: cubic

text: spline

text: with

text: knots

text: separated

text: by

text: δ

text: lnk

text: The

text: spline

text: is

text: ﬁt

text: with

text: a

text: regularization

text: applied

text: to

text: both

text: zero

text: f(k)

text: outside

text: the

text: observable

text: range

text: of

text: k

text: and

text: to

text: penalize

text: high

text: frequency

text: oscillations.

text: The

text: strength

text: of

text: the

text: regularization

text: essentially

text: introduces

text: a

text: minimum

text: correlation

text: scale

text: in

text: the

text: space

text: of

text: al-

text: lowed

text: deviations

text: f(k)

text: in

text: the

text: power

text: spectrum.

text: Here

text: we

text: do

text: a

text: similar

text: analysis

text: using

text: the

text: Planck

text: binned

text: temper-

text: ature

text: data

text: (Planck

text: Collaboration

text: et

text: al.

text: but

text: rather

text: than

text: ﬁtting

text: for

text: the

text: presence

text: of

text: a

text: feature

text: in

text: the

text: model,

text: we

text: in-

text: stead

text: use

text: our

text: goodness-of-ﬁt

text: statistic

text: to

text: search

text: for

text: evidence

text: of

text: features

text: present

text: in

text: the

text: data

text: allowed

text: by

text: the

text: generalized

text: power

text: spectrum

text: model

text: of

text: equation

text: but

text: not

text: by

text: ΛCDM.

text: We

text: expect

text: our

text: results

text: to

text: be

text: more

text: general

text: in

text: the

text: sense

text: that

text: Multipole

text: they

text: provide

text: not

text: only

text: a

text: search

text: for

text: deviations

text: exactly

text: pa-

text: rameterizable

text: by

text: equation

text: but

text: any

text: feature

text: that

text: shares

text: signiﬁcant

text: correlation

text: with

text: typical

text: draws

text: from

text: equation

text: Figure

text: A

text: characteristic

text: subset

text: of

text: the

text: where

text: More

text: simply,

text: through

text: the

text: use

text: of

text: goodness-of-ﬁt

text: rather

text: than

text: for

text: the

text: Gaussian

text: parameterization

text: of

text: equa-

text: parameter

text: estimation,

text: we

text: trade

text: the

text: ability

text: to

text: identify

text: ex-

text: tion

text: Curves

text: are

text: labeled

text: by

text: the

text: central

text: wavenumber

text: k

text: of

text: the

text: actly

text: the

text: shape

text: of

text: f(k)

text: we

text: are

text: detecting

text: for

text: a

text: sensitivity

text: to

text: corresponding

text: window

text: wn.

text: The

text: majority

text: of

text: the

text: power

text: put

text: into

text: the

text: a

text: wider

text: range

text: of

text: signals.

text: localized

text: bumps

text: translates

text: to

text: peaks

text: in

text: the

text: Our

text: choice

text: to

text: work

text: with

text: the

text: Planck

text: binned

text: tem-

text: perature

text: data

text: follows

text: Planck

text: Collaboration

text: et

text: al.

text: in

text: binning

text: the

text: likelihood

text: in

text: bins

text: suﬃcient

text: to

text: allow

text: the

text: implicit

text: assumption

text: in

text: equation

text: that

text: the

text: structure

text: of

text: the

text: folding

text: of

text: foreground

text: uncertainty

text: into

text: the

text: noise

text: covariance.

text: noise

text: can

text: be

text: taken

text: to

text: be

text: Gaussian.

text: This

text: choice

text: eases

text: both

text: the

text: computation

text: and

text: interpretation

text: Though

text: the

text: features

text: of

text: the

text: primordial

text: power

text: spectrum

text: of

text: the

text: modes

text: N

text: in

text: equation

text: represents

text: the

text: total

text: noise

text: could

text: be

text: arbitrarily

text: peaked,

text: Hu

text: Okamoto

text: show

text: covariance–experimental,

text: foreground,

text: and

text: cosmic

text: variance–

text: that

text: when

text: these

text: features

text: are

text: transported

text: to

text: the

text: observed

text: present

text: in

text: the

text: observation.

text: In

text: addition,

text: since

text: the

text: cosmic

text: vari-

text: power

text: spectra,

text: projection

text: from

text: the

text: surface

text: of

text: last

text: scattering

text: ance

text: at

text: is

text: comparatively

text: large,

text: we

text: discard

text: the

text: and

text: gravitational

text: lensing

text: smooth

text: features

text: below

text: a

text: charac-

text: data

text: where

text: non-Gaussianities

text: play

text: a

text: role.

text: This

text: justiﬁes

text: the

text: teristic

text: width

text: in

text: ∆lnk.

text: This

text: provides

text: a

text: minimum

text: frequency

text: MNRAS

text: edutilpmA

text: D∂

text: ξ∂

text: k

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text: 1 Introduction Experimental and theoretical physicists have long sought to understand the proper- ties of the elementary particles (quarks and gluons) described by QCD at extreme temperatures and densities. Experimentalists are creating these extreme conditions by means of relativistic heavy-ion collisions performed at LHC and RHIC. During these collisions, a ﬁreball of deconﬁned quarks and gluons, known as the quark gluon plasma (QGP), is formed for a very brief time, which makes the investigation of the ﬁreball a challenging task. Joint experimental and theoretical eﬀorts have been carried out for about fourty years [1] to ﬁnd and analyse accessible observables for diagnosing this short-lived QGP. One of the promising candidates for diagnostics are heavy quarks produced and parte- cipating in the collisions. Due to the separation of scales between the heavy quarks and the main plasma constituents, the former can be used as dynamical probes of the surrounding medium. In particular, the seminal work of Matsui and Satz [2] suggested that the analysis of bound states of heavy quarks (quarkonia) could lead to a better understanding of the QGP, the reason being that, at suﬃciently high temperatures, colour screening weakens or even prevents a pair from binding in qq¯ the deconﬁned medium, resulting in a suppression of quarkonia (such as and ¯ cc¯ bb bound states) at the hadronization of the QGP. This paved the way for many quests for understanding the fate of quarkonia. Some of them made use of screened potential models [3], based on quantities like the free or internal energies. The screened potential, whose short- and medium-range parts can be obtained from resummed perturbative calculations, was also used to determine stationary states of a Schrödinger equation [4]. Non-perturbative studies on the lattice followed a diﬀerent line aiming at reconstructing spectral functions of quarkonia around the crossover temperature [5–8], whereas other approaches exploit the AdS/CFT correspondence [9–12]. Recent progresses showed that the inter-quark potential not only gets screened at high temperatures, but develops an imaginary part [13, 14], which reﬂects the Landau damping mechanism due to the collisions between the heavy quarks and plasma constituents [15–17]. The fact that the inter-quark potential is actually complex poses serious issues on the validity of the approaches relying on the standard Schrödinger equation, which is a good approximation only when the state lives long enough, that is when the deviation from the unitary evo- lution is small. This happens when the imaginary part of the potential (decay rate) is much smaller than the mass and binding energy of respectively a heavy quark and a heavy-quark bound state1. However, the imaginary part of the potential can be used to compute spectral functions of in-medium quarkonia or in the context of the stochastic Schrödinger equation (SSE) [18–20]. Both the real and the imaginary part of the potential have been calculated in perturbation theory, but there have 1Moreover the Schrödinger equation can not describe possible bound-state recombinations. – 2 – ##LTLine##

label: section

text: Introduction

text: Experimental

label: paragraph

text: and

text: theoretical

text: physicists

text: have

text: long

text: sought

text: to

text: understand

text: the

text: proper-

text: ties

text: of

text: the

text: elementary

text: particles

text: (quarks

text: and

text: gluons)

text: described

text: by

text: QCD

text: at

text: extreme

text: temperatures

text: and

text: densities.

text: Experimentalists

text: are

text: creating

text: these

text: extreme

text: conditions

text: by

text: means

text: of

text: relativistic

text: heavy-ion

text: collisions

text: performed

text: at

text: LHC

text: and

text: RHIC.

text: During

text: these

text: collisions,

text: a

text: ﬁreball

text: of

text: deconﬁned

text: quarks

text: and

text: gluons,

text: known

text: as

text: the

text: quark

text: gluon

text: plasma

text: (QGP),

text: is

text: formed

text: for

text: a

text: very

text: brief

text: time,

text: which

text: makes

text: the

text: investigation

text: of

text: the

text: ﬁreball

text: a

text: challenging

text: task.

text: Joint

text: experimental

text: and

text: theoretical

text: eﬀorts

text: have

text: been

text: carried

text: out

text: for

text: about

text: fourty

text: years

text: to

text: ﬁnd

text: and

text: analyse

text: accessible

text: observables

text: for

text: diagnosing

text: this

text: short-lived

text: QGP.

text: One

text: of

text: the

text: promising

text: candidates

text: for

text: diagnostics

text: are

text: heavy

text: quarks

text: produced

text: and

text: parte-

text: cipating

text: in

text: the

text: collisions.

text: Due

text: to

text: the

text: separation

text: of

text: scales

text: between

text: the

text: heavy

text: quarks

text: and

text: the

text: main

text: plasma

text: constituents,

text: the

text: former

text: can

text: be

text: used

text: as

text: dynamical

text: probes

text: of

text: the

text: surrounding

text: medium.

text: In

text: particular,

text: the

text: seminal

text: work

text: of

text: Matsui

text: and

text: Satz

text: suggested

text: that

text: the

text: analysis

text: of

text: bound

text: states

text: of

text: heavy

text: quarks

text: (quarkonia)

text: could

text: lead

text: to

text: a

text: better

text: understanding

text: of

text: the

text: QGP,

text: the

text: reason

text: being

text: that,

text: at

text: suﬃciently

text: high

text: temperatures,

text: colour

text: screening

text: weakens

text: or

text: even

text: prevents

text: a

text: pair

text: from

text: binding

text: in

text: qq¯

text: the

text: deconﬁned

text: medium,

text: resulting

text: in

text: a

text: suppression

text: of

text: quarkonia

text: (such

text: as

text: and

text: cc¯

text: bb

text: bound

text: states)

text: at

text: the

text: hadronization

text: of

text: the

text: QGP.

text: This

text: paved

text: the

text: way

text: for

text: many

text: quests

text: for

text: understanding

text: the

text: fate

text: of

text: quarkonia.

text: Some

text: of

text: them

text: made

text: use

text: of

text: screened

text: potential

text: models

text: based

text: on

text: quantities

text: like

text: the

text: free

text: or

text: internal

text: energies.

text: The

text: screened

text: potential,

text: whose

text: short-

text: and

text: medium-range

text: parts

text: can

text: be

text: obtained

text: from

text: resummed

text: perturbative

text: calculations,

text: was

text: also

text: used

text: to

text: determine

text: stationary

text: states

text: of

text: a

text: Schrödinger

text: equation

text: Non-perturbative

text: studies

text: on

text: the

text: lattice

text: followed

text: a

text: diﬀerent

text: line

text: aiming

text: at

text: reconstructing

text: spectral

text: functions

text: of

text: quarkonia

text: around

text: the

text: crossover

text: temperature

text: whereas

text: other

text: approaches

text: exploit

text: the

text: AdS/CFT

text: correspondence

text: Recent

text: progresses

text: showed

text: that

text: the

text: inter-quark

text: potential

text: not

text: only

text: gets

text: screened

text: at

text: high

text: temperatures,

text: but

text: develops

text: an

text: imaginary

text: part

text: which

text: reﬂects

text: the

text: Landau

text: damping

text: mechanism

text: due

text: to

text: the

text: collisions

text: between

text: the

text: heavy

text: quarks

text: and

text: plasma

text: constituents

text: The

text: fact

text: that

text: the

text: inter-quark

text: potential

text: is

text: actually

text: complex

text: poses

text: serious

text: issues

text: on

text: the

text: validity

text: of

text: the

text: approaches

text: relying

text: on

text: the

text: standard

text: Schrödinger

text: equation,

text: which

text: is

text: a

text: good

text: approximation

text: only

text: when

text: the

text: state

text: lives

text: long

text: enough,

text: that

text: is

text: when

text: the

text: deviation

text: from

text: the

text: unitary

text: evo-

text: lution

text: is

text: small.

text: This

text: happens

text: when

text: the

text: imaginary

text: part

text: of

text: the

text: potential

text: (decay

text: rate)

text: is

text: much

text: smaller

text: than

text: the

text: mass

text: and

text: binding

text: energy

text: of

text: respectively

text: a

text: heavy

text: quark

text: and

text: a

text: heavy-quark

text: bound

text: However,

text: the

text: imaginary

text: part

text: of

text: the

text: potential

text: can

text: be

text: used

text: to

text: compute

text: spectral

text: functions

text: of

text: in-medium

text: quarkonia

text: or

text: in

text: the

text: context

text: of

text: the

text: stochastic

text: Schrödinger

text: equation

text: (SSE)

text: Both

text: the

text: real

text: and

text: the

text: imaginary

text: part

text: of

text: the

text: potential

text: have

text: been

text: calculated

text: in

text: perturbation

text: theory,

text: but

text: there

text: have

text: the

text: Schrödinger

text: equation

text: can

text: not

text: describe

text: possible

text: bound-state

text: recombinations.

text: ##LTLine##


text: and similarly for the primed coordinates. The density matrix describing the dynamics of the relative qq¯ coordinate(cid:90)is d(cid:90)eﬁned as the trace of the density matrix with respect to the centre of mass coordinate, that is ρ˜(t, q , q(cid:48) ) = δ(q − q(cid:48) )ρ(t, q , q , q(cid:48) , q(cid:48) ) . (5.14) r r cm cm r cm r cm qcm qc(cid:48)m In the coincident limit q(cid:48) → q , the Lindblad operator does not depend on q cm cm cm anymore, therefor(cid:20)e one can carry out the trace over the centre of mass coordinate and get the Lindblad equation for the relative motion: (cid:18) (cid:18) i(cid:126) (cid:19) (cid:18)ig2 (cid:19) (cid:19) ∂ ρ˜(t, q , q(cid:48) ) = (∂2 − ∂2 ) − (cid:126) (V (q ) − V (q(cid:48) )) (5.15) t r r m qr qr(cid:48) r r g2 (cid:18) 1 (cid:18) (cid:19) 1 (cid:18) (cid:19) − (cid:126) 2W (q − q(cid:48) ) − 2W (q + q(cid:48) ) + W (q ) + W (q(cid:48) ) − 2W (0) 2 r r 2 r r r r (cid:126) (cid:18) (cid:19)(cid:19)(cid:21) g2β 1 1 − ∂ W (q − q(cid:48) ) · (∂ − ∂ ) − ∂ W (q + q(cid:48) ) · (∂ + ∂ ) m qr 2 r r qr qr(cid:48) qr 2 r r qr qr(cid:48) 1 −2∂2 W (q + q(cid:48) ) ρ˜(t, q , q(cid:48) ) . qr 2 r r r r Performing a change of variables analogous to (4.8) but with halved, that is y 1 1 r ≡ (q + q(cid:48) ) , y ≡ (q − q(cid:48) ) , (5.16) (cid:20) 2 r r 2 r r one ﬁnally obtains (cid:126) i ig2 (5.17) ∂ ρ˜(t, r, y) = ∂ · ∂ − (cid:126) (V (r + y) − V (r − y)) t m r y (cid:21) g2 (cid:0) (cid:1) − (cid:126) (2W (y) − 2W (r) + W (r + y) + W (r − y) − 2W (0)) (cid:126) g2β − ∂ W (y) · ∂ − ∂ W (r) · ∂ − ∂2W (r) ρ˜(t, r, y) . 2m y y r r r Observe that, as in the one particle case, the diﬀusive term (second line of (5.17)), responsible for quantum decoherence, aﬀects only the oﬀ-diagonal elements of the density matrix. Moreover, using integration by parts, it is immediate to check that the trace of the density matrix is preserved. 5.2 Fokker-Planck and Langevin dynamics in the classical limit In section it has been shown that the semiclassical limit of the master equation for a single heavy particle produces a classical stochastic dynamics. The same hap- pens when one takes(cid:20)the semiclassical limit of eq.(5.17) by expanding up to y2 (the expansion is actually in the dimensionless quantity ). The result is y/l env i(cid:126) 2ig2 g2 (cid:21) (cid:0) (cid:1) (5.18) ∂ ρ˜(t, r, y) = ∂ · ∂ − (cid:126) y · ∂ V (r) − (cid:126) y · (H(0) + H(r)) · y t m r y r (cid:126) g2β − y · H(0) · ∂ − ∂ W (r) · ∂ − ∂2W (r) ρ˜(t, r, y) , 2m y r r r – 26 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: similarly

text: for

text: the

text: primed

text: coordinates.

text: The

text: density

text: matrix

text: describing

text: the

text: dynamics

text: of

text: the

text: relative

text: qq¯

text: as

text: the

text: trace

text: of

text: the

text: density

text: matrix

text: with

text: respect

text: to

text: the

text: centre

text: of

text: mass

text: coordinate,

text: that

text: is

text: ρ˜(t,

text: q

text: δ(q

text: )ρ(t,

text: q

text: q

text: r

text: r

text: cm

text: cm

text: r

text: cm

text: r

text: cm

text: qcm

text: In

text: the

text: coincident

text: limit

text: q

text: the

text: Lindblad

text: operator

text: does

text: not

text: depend

text: on

text: q

text: cm

text: cm

text: cm

text: anymore,

text: one

text: can

text: carry

text: out

text: the

text: trace

text: over

text: the

text: centre

text: of

text: mass

text: coordinate

text: and

text: get

text: the

text: Lindblad

text: equation

text: for

text: the

text: relative

text: motion:

text: ρ˜(t,

text: q

text: ∂2

text: (V

text: (q

text: V

text: t

text: r

text: r

text: m

text: qr

text: r

text: r

text: (q

text: (q

text: W

text: (q

text: W

text: r

text: r

text: r

text: r

text: r

text: r

text: W

text: (q

text: W

text: (q

text: m

text: qr

text: r

text: r

text: qr

text: qr

text: r

text: r

text: qr

text: −2∂2

text: W

text: (q

text: ρ˜(t,

text: q

text: qr

text: r

text: r

text: r

text: r

text: Performing

text: a

text: change

text: of

text: variables

text: analogous

text: to

text: but

text: with

text: halved,

text: that

text: is

text: y

text: r

text: (q

text: y

text: (q

text: r

text: r

text: r

text: r

text: one

text: ﬁnally

text: obtains

text: i

text: ρ˜(t,

text: r,

text: y)

text: (V

text: (r

text: y)

text: V

text: (r

text: y))

text: t

text: m

text: r

text: y

text: (y)

text: (r)

text: W

text: (r

text: y)

text: W

text: (r

text: y)

text: W

text: (y)

text: W

text: (r)

text: (r)

text: ρ˜(t,

text: r,

text: y)

text: y

text: y

text: r

text: r

text: r

text: Observe

text: that,

text: as

text: in

text: the

text: one

text: particle

text: case,

text: the

text: diﬀusive

text: term

text: (second

text: line

text: of

text: responsible

text: for

text: quantum

text: decoherence,

text: aﬀects

text: only

text: the

text: oﬀ-diagonal

text: elements

text: of

text: the

text: density

text: matrix.

text: Moreover,

text: using

text: integration

text: by

text: parts,

text: it

text: is

text: immediate

text: to

text: check

text: that

text: the

text: trace

text: of

text: the

text: density

text: matrix

text: is

text: preserved.

text: Fokker-Planck

text: and

text: Langevin

text: dynamics

text: in

text: the

text: classical

text: limit

text: In

text: section

text: it

text: has

text: been

text: shown

text: that

text: the

text: semiclassical

text: limit

text: of

text: the

text: master

text: equation

text: for

text: a

text: single

text: heavy

text: particle

text: produces

text: a

text: classical

text: stochastic

text: dynamics.

text: The

text: same

text: hap-

text: pens

text: when

text: one

text: semiclassical

text: limit

text: of

text: by

text: expanding

text: up

text: to

text: (the

text: expansion

text: is

text: actually

text: in

text: the

text: dimensionless

text: quantity

text: The

text: result

text: is

text: y/l

text: env

text: ρ˜(t,

text: r,

text: y)

text: y

text: V

text: (r)

text: y

text: H(r))

text: y

text: t

text: m

text: r

text: y

text: r

text: y

text: W

text: (r)

text: (r)

text: ρ˜(t,

text: r,

text: y)

text: y

text: r

text: r

text: r

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: Another reason to consider the linear entropy is that it is a handy quantity for diag- nosing quantum decohe(cid:88)rence. Ind(cid:88)eed a pure state has S (cid:88)= 0 , being |ψ(t)(cid:105)(cid:104)ψ(t)| = L ρˆ(t) = ρˆ2(t) , whereas a mixed state has 0 < S ≤ 1 , since L 0 ≤ Tr ρˆ2 = P 2 ≤ P = 1 ; ρˆ = P |ψ (cid:105)(cid:104)ψ | . (6.1) mix n n mix n n n n n n If a pure state of the subsystem is plunged into the bath, the linear entropy will increase with a rapidity dependent on the ability of the medium to resolve the quan- tum system, that is according to the ratio that has been discussed in section l /l env sys . If this ratio is very small, the environment performs “invasive” measures on the subsystem (the momentum kicks of the bath are large), inducing a sudden raise of the linear entropy. In the opposite regime, the environment can not resolve the sys- tem and consequently the linear entropy remains close to zero. (cid:104) (cid:105) It is interesting to see how the oﬀ-diagonal elements of the density matrix aﬀect the (cid:38) time evolution of the linear entropy in the semiclassical limit ( ). Using that l l (cid:90) (cid:90) env sys ˙ Tr ˙ and inserting eq.(5.18), after some algebra one gets S = −2 ρˆρˆ L 8κ S˙ (t) ≈ (cid:126) y2|ρ(t, r, y)|2 ≥ 0 , (6.2) L 2 r y (cid:38) where it has been used that ∂2W (r)| − ∂2W (r) ≈ 0 when l l . It is r r=0 r env sys clear that in this regime the linear entropy rises as long as the density matrix has oﬀ-diagonal elements. This makes the role of oﬀ-diagonal terms in the process of quantum decoherence more transparent. 6.2 Model for simulations Instead of using the expressions in (3.8) for and , two similar expressions V W are used here for computational convenience. Th(cid:20)e(cid:114)real par(cid:21)t is taken to be the Pöschl- Teller potential (see Fig.2) ω µ ω V (x) = − j(j + 1) sech2 (cid:126) x , (6.3) 2 2 N where ω is a unit of energy, µ =(cid:16) m/(cid:17)2 is the(cid:18)reduce(cid:20)d(cid:114)mass o(cid:21)f(cid:19)the qq¯ pair and j ∈ . This potential has been chosen since it admits bound states, j µ ω 1/4 µ ω ψ (x) = Cj (cid:126) Pn tanh (cid:126) x , (6.4) n n 2 j 2 with n = 0, 1, . . . , j − 1, j , Cj a dimensionless normalisation factor and Pn(x) the n j associated Legendre polynomial. Therefore the maximum number of vacuum ( ) T = 0 bound states can be simply tuned by cha(cid:34)nging(cid:18)the (cid:19)nat(cid:35)ural number j , which is set to here. The imaginary part is taken to be (see Fig.2) 2 T 1 x 2 (6.5) W (x) = − exp − , 2 2 l env – 29 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Another

text: reason

text: to

text: consider

text: the

text: linear

text: entropy

text: is

text: that

text: it

text: is

text: a

text: handy

text: quantity

text: for

text: diag-

text: nosing

text: quantum

text: a

text: pure

text: state

text: has

text: S

text: being

text: L

text: ρˆ(t)

text: whereas

text: a

text: mixed

text: state

text: has

text: S

text: since

text: L

text: Tr

text: ρˆ2

text: P

text: P

text: ρˆ

text: P

text: |ψ

text: mix

text: n

text: n

text: mix

text: n

text: n

text: n

text: n

text: n

text: n

text: If

text: a

text: pure

text: state

text: of

text: the

text: subsystem

text: is

text: plunged

text: into

text: the

text: bath,

text: the

text: linear

text: entropy

text: will

text: increase

text: with

text: a

text: rapidity

text: dependent

text: on

text: the

text: ability

text: of

text: the

text: medium

text: to

text: resolve

text: the

text: quan-

text: tum

text: system,

text: that

text: is

text: according

text: to

text: the

text: ratio

text: that

text: has

text: been

text: discussed

text: in

text: section

text: l

text: /l

text: env

text: sys

text: If

text: this

text: ratio

text: is

text: very

text: small,

text: the

text: environment

text: performs

text: “invasive”

text: measures

text: on

text: the

text: subsystem

text: (the

text: momentum

text: kicks

text: of

text: the

text: bath

text: are

text: large),

text: inducing

text: a

text: sudden

text: raise

text: of

text: the

text: linear

text: entropy.

text: In

text: the

text: opposite

text: regime,

text: the

text: environment

text: can

text: not

text: resolve

text: the

text: sys-

text: tem

text: and

text: consequently

text: the

text: linear

text: entropy

text: remains

text: close

text: to

text: zero.

text: It

text: is

text: interesting

text: to

text: see

text: how

text: the

text: oﬀ-diagonal

text: elements

text: of

text: the

text: density

text: matrix

text: aﬀect

text: the

text: time

text: evolution

text: of

text: the

text: linear

text: entropy

text: in

text: the

text: semiclassical

text: limit

text: Using

text: that

text: l

text: l

text: env

text: sys

text: Tr

text: and

text: inserting

text: after

text: some

text: algebra

text: one

text: gets

text: S

text: −2

text: ρˆρˆ

text: L

text: 8κ

text: S˙

text: (t)

text: r,

text: L

text: r

text: y

text: where

text: it

text: has

text: been

text: used

text: that

text: (r)|

text: (r)

text: when

text: l

text: l

text: It

text: is

text: r

text: r

text: env

text: sys

text: clear

text: that

text: in

text: this

text: regime

text: the

text: linear

text: entropy

text: rises

text: as

text: long

text: as

text: the

text: density

text: matrix

text: has

text: oﬀ-diagonal

text: elements.

text: This

text: makes

text: the

text: role

text: of

text: oﬀ-diagonal

text: terms

text: in

text: the

text: process

text: of

text: quantum

text: decoherence

text: more

text: transparent.

text: Model

text: for

text: simulations

text: Instead

text: of

text: using

text: the

text: expressions

text: in

text: for

text: and

text: two

text: similar

text: expressions

text: V

text: W

text: are

text: used

text: here

text: for

text: computational

text: convenience.

text: is

text: taken

text: to

text: be

text: the

text: Pöschl-

text: Teller

text: potential

text: (see

text: ω

text: µ

text: ω

text: V

text: (x)

text: j(j

text: x

text: N

text: where

text: ω

text: is

text: a

text: unit

text: of

text: energy,

text: µ

text: is

text: qq¯

text: pair

text: and

text: j

text: This

text: potential

text: has

text: been

text: chosen

text: since

text: it

text: admits

text: bound

text: states,

text: j

text: µ

text: ω

text: µ

text: ω

text: ψ

text: (x)

text: Cj

text: Pn

text: tanh

text: x

text: n

text: n

text: j

text: with

text: n

text: j

text: j

text: Cj

text: a

text: dimensionless

text: normalisation

text: factor

text: and

text: Pn(x)

text: the

text: n

text: j

text: associated

text: Legendre

text: polynomial.

text: Therefore

text: the

text: maximum

text: number

text: of

text: vacuum

text: T

text: bound

text: states

text: can

text: be

text: simply

text: tuned

text: by

text: number

text: j

text: which

text: is

text: set

text: to

text: here.

text: The

text: imaginary

text: part

text: is

text: taken

text: to

text: be

text: (see

text: T

text: x

text: W

text: (x)

text: exp

text: l

text: env

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 1.25 1.5 EEgesx ==−−2ω2ω [[GGeeVV]] 1 WW˜ ((xx)) [[GGeeVV]] 1 V (x) [GeV] 0.5 ψgs(x) [fm−12] 0.75 WW˜ 00((xx)) [[GGfmeeVV]] 0 ψex(x) [fm−12] 0.5 fm −0.5 0.25 −1 0 −1.5 −0.25 −2 −2.5 −0.5 −2−1.5−1−0.5 0 0.5 1 1.5 2 0.25 0.5 0.75 1 x [fm] x [fm] Figure 2. On the left: Pöschl-Teller potential with ω = 0.8 GeV, µ = 0.6 GeV and j = 2 bound states. The ground state ψ and ﬁrst excited state ψ with their energies are gs ex also shown. On the right: Imaginary part of the potential (6.5) used in the simulations contrasted with W˜ (x) derived from the gauge theory in (3.8). Here T = 0.8 GeV and σ = 1 , (cid:126) hence l = /(σT ) ≈ 0.25 fm. env (cid:126) where the correlation length of the environment is and is a di- l = /(σ T ) σ env mensionless parameter that will be tuned to probe the diﬀerent physical regimes discussed in section . This form of the correlation length resembles the one of the (cid:82) (cid:126) Debye radius r = m−1 = /(αg T ) appearing in the original expression (3.8) of D D W , which is plotted in Fig.2 as W˜ (x) = −Tlenv 5 dz sin[2lezn|xv|] . Comparing with the |x| 0 (z2+1)2 expression derived from the gauge theory, a factor 1 has been omitted in front of 4π ˜ . This has been done since the model is one dimensional anyway, and the form W (x) (3.8) is strictly valid only in three dimensions. In the numerical simulations, the bound-state probabilities ( and for the ground and excited state respec- P (t) P (t) 0 1 tively) and the linear entropy will be extracted from the solution of the following (cid:20) one-dimensional Lindblad equation for the relative motion of a pair: qq¯ (cid:126) i i (6.6) ∂ ρ(t, r, y) = ∂ ∂ − (cid:126) (V (r + y) − V (r − y)) t m r y (cid:21) 1 (cid:0) (cid:1) −(cid:126) (2W (y) − 2W (r) + W (r + y) + W (r − y) − 2W (0)) (cid:126) − ∂ W (y)∂ − ∂ W (r) ∂ − ∂2W (r) ρ(t, r, y) . 2mT y y r r r This equation is the one-dimensional analog of eq.(5.17) with g2 = 1 15 and V (x), W (x) deﬁned as in (6.3),(6.5) respectively. The ﬁrst part of the analysis assumes that ini- tially the pair is in one of the two bound states of the Pöschl-Teller potential, that qq¯ 15Notice that the perturbative approximation is still valid since the expansion is actually in powers of α = g2 . Moreover T = 0.8 GeV and m = 1.2 GeV have been used here, so that the s 4π nonrelativistic limit is valid to a good degree of accuracy. – 30 – ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EEgesx

text: ==−−2ω2ω

text: [[GGeeVV]]

text: WW˜

text: ((xx))

text: [[GGeeVV]]

text: V

text: (x)

text: [GeV]

text: ψgs(x)

text: WW˜

text: [[GGfmeeVV]]

text: ψex(x)

text: fm

text: −1

text: −2

text: x

text: [fm]

text: x

text: [fm]

text: Figure

text: On

text: the

text: left:

text: Pöschl-Teller

text: potential

text: with

text: ω

text: GeV,

text: µ

text: GeV

text: and

text: j

text: bound

text: states.

text: The

text: ground

text: state

text: ψ

text: and

text: ﬁrst

text: excited

text: state

text: ψ

text: with

text: their

text: energies

text: are

text: gs

text: ex

text: also

text: shown.

text: On

text: the

text: right:

text: Imaginary

text: part

text: of

text: the

text: potential

text: used

text: in

text: the

text: simulations

text: contrasted

text: with

text: W˜

text: (x)

text: derived

text: from

text: the

text: gauge

text: theory

text: in

text: Here

text: T

text: GeV

text: and

text: σ

text: hence

text: l

text: /(σT

text: fm.

text: env

text: where

text: the

text: correlation

text: length

text: of

text: the

text: environment

text: is

text: and

text: is

text: a

text: di-

text: l

text: /(σ

text: T

text: σ

text: env

text: mensionless

text: parameter

text: that

text: will

text: be

text: tuned

text: to

text: probe

text: the

text: diﬀerent

text: physical

text: regimes

text: discussed

text: in

text: section

text: This

text: form

text: of

text: the

text: correlation

text: length

text: resembles

text: the

text: one

text: of

text: the

text: Debye

text: radius

text: r

text: /(αg

text: T

text: appearing

text: in

text: the

text: original

text: expression

text: of

text: D

text: D

text: W

text: which

text: is

text: plotted

text: in

text: as

text: W˜

text: (x)

text: −Tlenv

text: dz

text: Comparing

text: with

text: the

text: |x|

text: expression

text: derived

text: from

text: the

text: gauge

text: theory,

text: a

text: factor

text: has

text: been

text: omitted

text: in

text: front

text: of

text: 4π

text: This

text: has

text: been

text: done

text: since

text: the

text: model

text: is

text: one

text: dimensional

text: anyway,

text: and

text: the

text: form

text: W

text: (x)

text: is

text: strictly

text: valid

text: only

text: in

text: three

text: dimensions.

text: In

text: the

text: numerical

text: simulations,

text: the

text: bound-state

text: probabilities

text: and

text: for

text: the

text: ground

text: and

text: excited

text: state

text: respec-

text: P

text: (t)

text: P

text: (t)

text: tively)

text: and

text: the

text: linear

text: entropy

text: will

text: be

text: extracted

text: from

text: the

text: solution

text: of

text: the

text: following

text: one-dimensional

text: Lindblad

text: equation

text: for

text: the

text: relative

text: motion

text: of

text: a

text: pair:

text: qq¯

text: i

text: i

text: ρ(t,

text: r,

text: y)

text: (V

text: (r

text: y)

text: V

text: (r

text: y))

text: t

text: m

text: r

text: y

text: (y)

text: (r)

text: W

text: (r

text: y)

text: W

text: (r

text: y)

text: W

text: (y)∂

text: W

text: (r)

text: (r)

text: ρ(t,

text: r,

text: y)

text: y

text: y

text: r

text: r

text: r

text: This

text: equation

text: is

text: the

text: one-dimensional

text: analog

text: of

text: with

text: and

text: V

text: (x),

text: W

text: (x)

text: deﬁned

text: as

text: in

text: respectively.

text: The

text: ﬁrst

text: part

text: of

text: the

text: analysis

text: assumes

text: that

text: ini-

text: tially

text: the

text: pair

text: is

text: in

text: one

text: of

text: the

text: two

text: bound

text: states

text: of

text: the

text: Pöschl-Teller

text: potential,

text: that

text: qq¯

text: that

text: the

text: perturbative

text: approximation

text: is

text: still

text: valid

text: since

text: the

text: expansion

text: is

text: actually

text: in

text: powers

text: of

text: α

text: Moreover

text: T

text: GeV

text: and

text: m

text: GeV

text: have

text: been

text: used

text: here,

text: so

text: that

text: the

text: s

text: 4π

text: nonrelativistic

text: limit

text: is

text: valid

text: to

text: a

text: good

text: degree

text: of

text: accuracy.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: (cid:88) expression (4.3) as a result. The computation of the last term in (4.1) reads (cid:90) (cid:90) (cid:90) (cid:90) (cid:18) (cid:19) −i (cid:88)(cid:104)q|Lˆ ρˆLˆ† |q(cid:48)(cid:105) (cid:16) (cid:17) µ µ µ (cid:90) (cid:90) (cid:90) (cid:90) p l = −i ρ(t, 2x − q, 2y − q(cid:48)) L x, L∗ y, e(cid:126)i p·(q−x)e−(cid:126)i l·(q(cid:48)−y) µ 2 µ 2 x y p l µ (cid:32) (cid:33) (cid:20) (cid:21) (3=.18) −i ρ(t, 2x − q, 2y − q(cid:48)) e(cid:88)(cid:126)i p·((cid:2)q−x)−(cid:126)i l·(q(cid:48)−y) (cid:3) x y p l (cid:90) (cid:90) p l 1 (cid:90) (cid:90) (cid:20) 1 (cid:21) iϕ x, ; y, − D2[x; y] + |Λ (x)|2 + |Λ (y)|2 τ 2 2 2 τ 2 µ µ µ (cid:32) p(cid:33) l = ρ(t, 2x −(cid:88)q, 2(cid:2)y − q(cid:48)) ϕ x, (cid:3) ; y, e(cid:126)i p·(q−x)−(cid:126)i l·(q(cid:48)−y) τ 2 2 x y p l i + D2[q; q(cid:48)] − |Λ (q)|2 + |Λ (q(cid:48))|2 ρ(t, q, q(cid:48)) . (A.2) 2 τ µ µ µ Collecting all the results as shown in section , one ﬁnally gets to the Lindblad equation (4.6). B Derivation of the Ehrenfest equations of motion Here the proof of the ﬁrst two equations of motion in (4.27) is shown. The derivation of the other relations is very similar and is therefore omitted here. The Ehrenfest equation of motion for the ﬁrst momentum of the position operator is obtained through the following steps (hats on operators have been removed for ease (cid:20) (cid:21) of notation): d d Tr (cid:104)q(cid:105) = q ρ(t) dt dt i iγ (cid:2) (cid:2) (cid:3)(cid:3) κ (4.26) Tr Tr Tr = −(cid:126) [q [H, ρ(t)]] − (cid:126) [q [q, {p, ρ(t)}]] − (cid:126) [q [q, [q, ρ(t)]]] 2 2 2 i i = −(cid:126)Tr [q [H, ρ(t)]] = − (cid:126)Tr q p2, ρ(t) , (B.1) 2m where in the last line the cyclicity of the trace has been exploited to cancel the friction and diﬀusion terms. The fact that the position operator commutes with the (cid:126) potential operator has been also used. The commutator then gives [q, p] = i d i (cid:126) Tr Tr Tr (cid:104)q(cid:105) = − (cid:126) (i [p ρ(t)] + [pqp ρ(t)] − [q ρ(t)pp]) dt 2m i (cid:126) Tr Tr Tr = − (cid:126) (2i [p ρ(t)] + [ppq ρ(t)] − [q ρ(t)pp]) 2m 1 (cid:104)p(cid:105) Tr (B.2) = [p ρ(t)] = . m m – 42 – ##LTLine## ##LTLine## ##LTLine## ##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: expression

text: as

text: a

text: result.

text: The

text: computation

text: of

text: the

text: last

text: term

text: in

text: reads

text: −i

text: ρˆLˆ†

text: µ

text: µ

text: µ

text: p

text: l

text: −i

text: ρ(t,

text: q,

text: L

text: x,

text: L∗

text: y,

text: µ

text: µ

text: x

text: y

text: p

text: l

text: µ

text: −i

text: ρ(t,

text: q,

text: x

text: y

text: p

text: l

text: p

text: l

text: iϕ

text: x,

text: y,

text: y]

text: |Λ

text: |Λ

text: τ

text: τ

text: µ

text: µ

text: µ

text: l

text: ρ(t,

text: ϕ

text: x,

text: y,

text: τ

text: x

text: y

text: p

text: l

text: i

text: |Λ

text: |Λ

text: ρ(t,

text: q,

text: τ

text: µ

text: µ

text: µ

text: Collecting

text: all

text: the

text: results

text: as

text: shown

text: in

text: section

text: one

text: ﬁnally

text: gets

text: to

text: the

text: Lindblad

text: equation

text: B

text: Derivation

text: of

text: the

text: Ehrenfest

text: equations

text: of

text: motion

text: Here

text: the

text: proof

text: of

text: the

text: ﬁrst

text: two

text: equations

text: of

text: motion

text: in

text: is

text: shown.

text: The

text: derivation

text: of

text: the

text: other

text: relations

text: is

text: very

text: similar

text: and

text: is

text: therefore

text: omitted

text: here.

text: The

text: Ehrenfest

text: equation

text: of

text: motion

text: for

text: the

text: ﬁrst

text: momentum

text: of

text: the

text: position

text: operator

text: is

text: obtained

text: through

text: the

text: following

text: steps

text: (hats

text: on

text: operators

text: have

text: been

text: removed

text: for

text: ease

text: of

text: notation):

text: d

text: d

text: Tr

text: q

text: ρ(t)

text: dt

text: dt

text: i

text: iγ

text: κ

text: Tr

text: Tr

text: Tr

text: [q

text: [H,

text: ρ(t)]]

text: [q

text: [q,

text: {p,

text: ρ(t)}]]

text: [q

text: [q,

text: [q,

text: ρ(t)]]]

text: i

text: i

text: [q

text: [H,

text: ρ(t)]]

text: q

text: ρ(t)

text: where

text: in

text: the

text: last

text: line

text: the

text: cyclicity

text: of

text: the

text: trace

text: has

text: been

text: exploited

text: to

text: cancel

text: the

text: friction

text: and

text: diﬀusion

text: terms.

text: The

text: fact

text: that

text: the

text: position

text: operator

text: commutes

text: with

text: the

text: potential

text: operator

text: has

text: been

text: also

text: used.

text: The

text: commutator

text: then

text: gives

text: [q,

text: p]

text: i

text: d

text: i

text: Tr

text: Tr

text: Tr

text: (i

text: [p

text: ρ(t)]

text: [pqp

text: ρ(t)]

text: [q

text: ρ(t)pp])

text: dt

text: i

text: Tr

text: Tr

text: Tr

text: [p

text: ρ(t)]

text: [ppq

text: ρ(t)]

text: [q

text: ρ(t)pp])

text: Tr

text: [p

text: ρ(t)]

text: m

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: expansions. In section the associated Lindblad equation is then read oﬀ from the in- ﬂuence functional. The Lindbladian terms will result in an expression written solely in terms of the imaginary part of the potential. From the structure of the Lindblad equation, it will be shown that diﬀerent types of dynamics of the heavy particle are obtained according to the ratio between the correlation length of the environment and the typical size of the heavy probe. In particular, it will be proven that the clas- sical Langevin dynamics stems from a semiclassical approximation of the Lindblad equation (see lower part of Fig.1). Section contains the computation of the Lindblad equation for a heavy quark-antiquark pair, which could in principle be generalised to heavy quarks and antiquarks. Section shows how quantum decoherence and N the probability of having a bound state at a certain time can be calculated from qq¯ the density matrix. This section also contains the setup for the one-dimensional numerical simulation of the Lindblad equation for the relative motion of the pair, qq¯ after the motion of the centre of mass has been traced out. The results are displayed in section , with speciﬁc focus on the linear entropy (a measure of decoherence) and bound-state probabilities, which can be used to extract dissociation and recombina- tion probabilities for diﬀerent correlation lengths of the thermal bath. Conclusions and perspectives are found in section , followed by appendices , , . 2 Density matrix and propagator of an open quantum system The goal of this work is to study a probe propagating through a medium in thermal equilibrium. The probe is treated as an open quantum system (also called reduced system or subsystem) that can dissipate or gain energy from the plasma environment. The environment and the probe form a closed quantum system. A master equation for the density matrix describing the probe will be derived in order to study its real- time dynamics. Of particular interest is the probability to ﬁnd the reduced system P (ψ, t|ψ , t ) 0 0 in a quantum state at time , given that it was in a state at time . |ψ(cid:105) t |ψ (cid:105) t 0 0 This quantity can be written in terms of the density operator of the subsystem, Tr , where is the density operator of the total (closed) system ρˆ(t) ≡ {ρˆ (t)} ρˆ env tot tot and the trace is taken over the degrees of freedom of the environment. It is assumed that the subsystem has not yet interacted with the medium at time , so that the t 0 initial total density operator takes the factorized form (2.1) ρˆ (t ) = ρˆ(t ) ⊗ ρˆ (t ) , tot 0 0 env 0 with ρˆ(t ) = |ψ (cid:105)(cid:104)ψ | and ρˆ (t ) = 1 e−βHˆenv is the density operator of the 0 0 0 env 0 Zenv environment, whose Hamiltonian and partition function are respectively ˆ and H env Tr ˆ . The total density operator at time is given by Z = exp(−βH ) t env env env ρˆ (t) = e−iHˆtot(t−t0)ρˆ (t )eiHˆtot(t−t0) , (2.2) tot tot 0 – 5 – ##LTLine##

text: expansions.

text: In

text: section

text: the

text: associated

text: Lindblad

text: equation

text: is

text: then

text: read

text: oﬀ

text: from

text: the

text: in-

text: ﬂuence

text: functional.

text: The

text: Lindbladian

text: terms

text: will

text: result

text: in

text: an

text: expression

text: written

text: solely

text: in

text: terms

text: of

text: the

text: imaginary

text: part

text: of

text: the

text: potential.

text: From

text: the

text: structure

text: of

text: the

text: Lindblad

text: equation,

text: it

text: will

text: be

text: shown

text: that

text: diﬀerent

text: types

text: of

text: dynamics

text: of

text: the

text: heavy

text: particle

text: are

text: obtained

text: according

text: to

text: the

text: ratio

text: between

text: the

text: correlation

text: length

text: of

text: the

text: environment

text: and

text: the

text: typical

text: size

text: of

text: the

text: heavy

text: probe.

text: In

text: particular,

text: it

text: will

text: be

text: proven

text: that

text: the

text: clas-

text: sical

text: Langevin

text: dynamics

text: stems

text: from

text: a

text: semiclassical

text: approximation

text: of

text: the

text: Lindblad

text: equation

text: (see

text: lower

text: part

text: of

text: Section

text: contains

text: the

text: computation

text: of

text: the

text: Lindblad

text: equation

text: for

text: a

text: heavy

text: quark-antiquark

text: pair,

text: which

text: could

text: in

text: principle

text: be

text: generalised

text: to

text: heavy

text: quarks

text: and

text: antiquarks.

text: Section

text: shows

text: how

text: quantum

text: decoherence

text: and

text: N

text: the

text: probability

text: of

text: having

text: a

text: bound

text: state

text: at

text: a

text: certain

text: time

text: can

text: be

text: calculated

text: from

text: qq¯

text: the

text: density

text: matrix.

text: This

text: section

text: also

text: contains

text: the

text: setup

text: for

text: the

text: one-dimensional

text: numerical

text: simulation

text: of

text: the

text: Lindblad

text: equation

text: for

text: the

text: relative

text: motion

text: of

text: the

text: pair,

text: qq¯

text: after

text: the

text: motion

text: of

text: the

text: centre

text: of

text: mass

text: has

text: been

text: traced

text: out.

text: The

text: results

text: are

text: displayed

text: in

text: section

text: with

text: speciﬁc

text: focus

text: on

text: the

text: linear

text: entropy

text: (a

text: measure

text: of

text: decoherence)

text: and

text: bound-state

text: probabilities,

text: which

text: can

text: be

text: used

text: to

text: extract

text: dissociation

text: and

text: recombina-

text: tion

text: probabilities

text: for

text: diﬀerent

text: correlation

text: lengths

text: of

text: the

text: thermal

text: bath.

text: Conclusions

text: and

text: perspectives

text: are

text: found

text: in

text: section

text: followed

text: by

text: appendices

text: Density

text: matrix

text: and

text: propagator

text: of

text: an

text: open

text: quantum

text: system

text: The

text: goal

text: of

text: this

text: work

text: is

text: to

text: study

text: a

text: probe

text: propagating

text: through

text: a

text: medium

text: in

text: thermal

text: equilibrium.

text: The

text: probe

text: is

text: treated

text: as

text: an

text: open

text: quantum

text: system

text: (also

text: called

text: reduced

text: system

text: or

text: subsystem)

text: that

text: can

text: dissipate

text: or

text: gain

text: energy

text: from

text: the

text: plasma

text: environment.

text: The

text: environment

text: and

text: the

text: probe

text: form

text: a

text: closed

text: quantum

text: system.

text: A

text: master

text: equation

text: for

text: the

text: density

text: matrix

text: describing

text: the

text: probe

text: will

text: be

text: derived

text: in

text: order

text: to

text: study

text: its

text: real-

text: time

text: dynamics.

text: Of

text: particular

text: interest

text: is

text: the

text: probability

text: to

text: ﬁnd

text: the

text: reduced

text: system

text: P

text: (ψ,

text: t|ψ

text: t

text: in

text: a

text: quantum

text: state

text: at

text: time

text: given

text: that

text: it

text: was

text: in

text: a

text: state

text: at

text: time

text: t

text: |ψ

text: t

text: This

text: quantity

text: can

text: be

text: written

text: in

text: terms

text: of

text: the

text: density

text: operator

text: of

text: the

text: subsystem,

text: Tr

text: where

text: is

text: the

text: density

text: operator

text: of

text: the

text: total

text: (closed)

text: system

text: ρˆ(t)

text: {ρˆ

text: (t)}

text: ρˆ

text: env

text: tot

text: tot

text: and

text: the

text: trace

text: is

text: taken

text: over

text: the

text: degrees

text: of

text: freedom

text: of

text: the

text: environment.

text: It

text: is

text: assumed

text: that

text: the

text: subsystem

text: has

text: not

text: yet

text: interacted

text: with

text: the

text: medium

text: at

text: time

text: so

text: that

text: the

text: t

text: initial

text: total

text: density

text: operator

text: takes

text: the

text: factorized

text: form

text: ρˆ

text: (t

text: ρˆ(t

text: ρˆ

text: (t

text: tot

text: env

text: with

text: ρˆ(t

text: |ψ

text: and

text: ρˆ

text: (t

text: e−βHˆenv

text: is

text: the

text: density

text: operator

text: of

text: the

text: env

text: Zenv

text: environment,

text: whose

text: Hamiltonian

text: and

text: partition

text: function

text: are

text: respectively

text: ˆ

text: and

text: H

text: env

text: Tr

text: ˆ

text: The

text: total

text: density

text: operator

text: at

text: time

text: is

text: given

text: by

text: Z

text: exp(−βH

text: t

text: env

text: env

text: env

text: ρˆ

text: (t)

text: (t

text: tot

text: tot

text: ##LTLine##


text: (cid:90) (cid:18) (cid:19) and the identity q + q(cid:48) O(q, q(cid:48)) = O , p e(cid:126)i p(q−q(cid:48)) , (2.14) 2 (cid:90)p where the Wigner transform is deﬁned as q(cid:48) q(cid:48) O(q, p) ≡ (cid:104)q − |Oˆ|q + (cid:105) e(cid:126)i pq(cid:48) . (2.15) 2 2 (cid:90) (cid:90) (cid:90) (cid:90) q(cid:48) (cid:20) The right hand side of eq.(2.12) then reads (cid:32) i∆t (cid:33)(cid:35) (cid:104)q |ρˆ(t)|q(cid:48) (cid:105) e(cid:126)i (p(q−q0(cid:88))−p(cid:48)(q(cid:48)−q0(cid:48) )) 1 + (cid:126) · 0 0 q0 q0(cid:48) p p(cid:48) · −H (q¯, p) + H∗ (q¯(cid:48), p(cid:48)) − i L (q¯, p) L∗ (q¯(cid:48), p(cid:48)) , eﬀ eﬀ µ µ µ with the midpoints q¯ ≡(cid:90) q+(cid:90)q0 a(cid:90)nd(cid:90) q¯(cid:48) ≡ q(cid:48)+q0(cid:48) . The(cid:20)refore, up to ﬁrst order in ∆t , 2 2 eq.(2.12) re-exponentiated becomes i (cid:35) i∆t (cid:104)q|ρˆ(t + ∆t)|q(cid:48)(cid:105) = (cid:104)q |ρˆ(t)(cid:88)|q(cid:48) (cid:105) exp (cid:126)p(q − q ) − (cid:126) H (q¯, p) 0 0 0 eﬀ q0 q0(cid:48) p p(cid:48) i i∆t ∆t −(cid:126)p(cid:48)(q(cid:48) − q(cid:48) ) − (cid:126) H∗ (q¯(cid:48), p(cid:48)) + (cid:126) L (q¯, p) L∗ (q¯(cid:48), p(cid:48)) . (2.16) 0 eﬀ µ µ µ The next step consists in deriving the ﬁnal density matrix ρ(t, q, q(cid:48)) from the initial density matrix ρ(t , q, q(cid:48)) . To(cid:90)ac(cid:90)complish this, the time interval [t , t] is partitioned 0 0 in small steps of size . Then, after using eq.(2.16) iteratively, one is left with N ∆t ρ(t, q, q(cid:48)) = P (q, q(cid:48), t|q , q(cid:48) , t )ρ(t , q , q(cid:48) ) , (2.17) 0 0 0 0 0 0 (cid:90) q0(cid:90) q0(cid:48) (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) with the propagator (cid:34) (cid:32) (cid:33)(cid:35) P (q, q(cid:48), t|q(cid:88), q(cid:48) , t ) = lim · · · · · · (cid:88) 0 0 0 N→∞ p1 q1 qN−1 pN p(cid:48)1 q1(cid:48) qN(cid:48) −1 p(cid:48)N N i∆t exp (cid:126) p q˙ − H (q¯ , p ) − p(cid:48)q˙(cid:48) + H∗ (q¯(cid:48), p(cid:48)) − i L (q¯ , p )L∗(q¯(cid:48), p(cid:48)) , i i eﬀ i i i i eﬀ i i µ i i µ i i i=1 µ and (q , p ) = (q, p) , (q(cid:48) , p(cid:48) ) = (q(cid:48), p(cid:48)) . One should notice that the appearance N N N N of the midpoints q¯ = qi+qi−1 is a result of the use of the Wigner representation i 2 (2.15). On the other hand the velocities are deﬁned through the retarded rule q˙ = i . (qi − qi−1)/∆t (cid:90) (cid:90) (cid:20) (cid:21) In the continuous limit the propagator (2.18) can be formally written as P (q , q(cid:48) , t|q , q(cid:48) , t ) = (qf,tD) qDp (qf(cid:48) ,tD) q(cid:48)Dp(cid:48) exp (cid:126)i S[q, p; q(cid:48), p(cid:48)] , (2.18) f f 0 0 0 (q0,t0) (q0(cid:48) ,t0) – 8 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: the

text: identity

text: q

text: O(q,

text: O

text: p

text: where

text: the

text: Wigner

text: transform

text: is

text: deﬁned

text: as

text: O(q,

text: p)

text: |Oˆ|q

text: The

text: right

text: hand

text: side

text: of

text: then

text: reads

text: i∆t

text: p

text: −H

text: (q¯,

text: p)

text: H∗

text: i

text: L

text: (q¯,

text: p)

text: L∗

text: eﬀ

text: eﬀ

text: µ

text: µ

text: µ

text: with

text: the

text: midpoints

text: q¯

text: up

text: to

text: ﬁrst

text: order

text: in

text: ∆t

text: re-exponentiated

text: becomes

text: i

text: i∆t

text: exp

text: q

text: H

text: (q¯,

text: p)

text: eﬀ

text: p

text: i

text: i∆t

text: ∆t

text: H∗

text: L

text: (q¯,

text: p)

text: L∗

text: eﬀ

text: µ

text: µ

text: µ

text: The

text: next

text: step

text: consists

text: in

text: deriving

text: the

text: ﬁnal

text: density

text: matrix

text: ρ(t,

text: q,

text: from

text: the

text: initial

text: density

text: matrix

text: ρ(t

text: q,

text: this,

text: the

text: time

text: interval

text: [t

text: t]

text: is

text: partitioned

text: in

text: small

text: steps

text: of

text: size

text: Then,

text: after

text: using

text: iteratively,

text: one

text: is

text: left

text: with

text: N

text: ∆t

text: ρ(t,

text: q,

text: P

text: (q,

text: t|q

text: t

text: )ρ(t

text: q

text: with

text: the

text: propagator

text: P

text: (q,

text: t

text: lim

text: N→∞

text: pN

text: −1

text: N

text: i∆t

text: exp

text: p

text: q˙

text: H

text: (q¯

text: p

text: H∗

text: i

text: L

text: (q¯

text: p

text: i

text: i

text: eﬀ

text: i

text: i

text: i

text: i

text: eﬀ

text: i

text: i

text: µ

text: i

text: i

text: µ

text: i

text: i

text: µ

text: and

text: (q

text: p

text: (q,

text: p)

text: One

text: should

text: notice

text: that

text: the

text: appearance

text: N

text: N

text: N

text: N

text: of

text: the

text: midpoints

text: q¯

text: is

text: a

text: result

text: of

text: the

text: use

text: of

text: the

text: Wigner

text: representation

text: i

text: On

text: the

text: other

text: hand

text: the

text: velocities

text: are

text: deﬁned

text: through

text: the

text: retarded

text: rule

text: q˙

text: i

text: (qi

text: In

text: the

text: continuous

text: limit

text: the

text: propagator

text: can

text: be

text: formally

text: written

text: as

text: P

text: (q

text: t|q

text: t

text: (qf,tD)

text: qDp

text: ,tD)

text: exp

text: S[q,

text: p;

text: f

text: f

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: (cid:34) (cid:90) with the phase space action functional t (cid:35) S[q, p; q(cid:48), p(cid:48)] = (cid:88)dτ q˙p − H (q, p) − q˙(cid:48)p(cid:48) + H∗ (q(cid:48), p(cid:48)) eﬀ eﬀ t0 −i L (q, p)L∗(q(cid:48), p(cid:48)) . (2.19) µ µ µ The appearance of the primed and unprimed coordinates can be traced back to the way the density operator evolves with time (see eq.(2.2)), that is with a time evolution from to and one in the opposite direction. The unprimed coordinates t t 0 live in the former interval, whereas the primed ones live in the latter branch. These two oriented time branches form the well-known Keldysh contour. Observe that the primed and unprimed coordinates decouple in the absence of Lindblad operators. 2.2 Equations of motion and quantum decoherence (cid:126) When the semiclassical limit of expression (2.19) is taken, the variational → 0 principle δS[q, p; q(cid:48), p(cid:48)] = 0 (2.20) gives the complex equation of motions corresponding to the stationary paths of the (cid:88) subsystem, that is, ∂H (q, p) ∂L (q, p) q˙ = eﬀ + i (cid:88) µ L∗(q(cid:48), p(cid:48)) , ∂p ∂p µ µ ∂H (q, p) ∂L (q, p) p˙ = − eﬀ − i(cid:88) µ L∗(q(cid:48), p(cid:48)) , ∂q ∂q µ µ ∂H∗ (q(cid:48), p(cid:48)) ∂L∗(q(cid:48), p(cid:48)) q˙(cid:48) = eﬀ − i (cid:88)L (q, p) µ , ∂p(cid:48) µ ∂p(cid:48) µ ∂H∗ (q(cid:48), p(cid:48)) ∂L∗(q(cid:48), p(cid:48)) p˙(cid:48) = − eﬀ + i L (q, p) µ , (2.21) ∂q(cid:48) µ ∂q(cid:48) µ with the boundary conditions q(t ) = q , q(cid:48)(t ) = q(cid:48) , q(t) = q , q(cid:48)(t) = q(cid:48) , (2.22) 0 0 0 0 f f The equations of motion for one heavy quark propagating through a medium will be derived in section . Notice the remarkable fact that dissipative dynamics can be derived from a variational principle. Following [49], the propagator (2.18) can be rewritten in a more transparent way, which separates the reversible dynamics generated by from the irreversible contribution coming from the Lindbladian H(q, p) – 9 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: the

text: phase

text: space

text: action

text: functional

text: t

text: S[q,

text: p;

text: q˙p

text: H

text: (q,

text: p)

text: H∗

text: eﬀ

text: eﬀ

text: −i

text: L

text: (q,

text: µ

text: µ

text: µ

text: The

text: appearance

text: of

text: the

text: primed

text: and

text: unprimed

text: coordinates

text: can

text: be

text: traced

text: back

text: to

text: the

text: way

text: the

text: density

text: operator

text: evolves

text: with

text: time

text: (see

text: that

text: is

text: with

text: a

text: time

text: evolution

text: from

text: to

text: and

text: one

text: in

text: the

text: opposite

text: direction.

text: The

text: unprimed

text: coordinates

text: t

text: t

text: live

text: in

text: the

text: former

text: interval,

text: whereas

text: the

text: primed

text: ones

text: live

text: in

text: the

text: latter

text: branch.

text: These

text: two

text: oriented

text: time

text: branches

text: form

text: the

text: well-known

text: Keldysh

text: contour.

text: Observe

text: that

text: the

text: primed

text: and

text: unprimed

text: coordinates

text: decouple

text: in

text: the

text: absence

text: of

text: Lindblad

text: operators.

text: Equations

text: of

text: motion

text: and

text: quantum

text: decoherence

text: When

text: the

text: semiclassical

text: limit

text: of

text: expression

text: is

text: taken,

text: the

text: variational

text: principle

text: δS[q,

text: p;

text: gives

text: the

text: complex

text: equation

text: of

text: motions

text: corresponding

text: to

text: the

text: stationary

text: paths

text: of

text: the

text: subsystem,

text: that

text: is,

text: ∂H

text: (q,

text: p)

text: ∂L

text: (q,

text: p)

text: q˙

text: eﬀ

text: i

text: µ

text: ∂p

text: ∂p

text: µ

text: µ

text: ∂H

text: (q,

text: p)

text: ∂L

text: (q,

text: p)

text: p˙

text: eﬀ

text: µ

text: ∂q

text: ∂q

text: µ

text: µ

text: ∂H∗

text: eﬀ

text: i

text: (q,

text: p)

text: µ

text: µ

text: µ

text: ∂H∗

text: eﬀ

text: i

text: L

text: (q,

text: p)

text: µ

text: µ

text: µ

text: with

text: the

text: boundary

text: conditions

text: q(t

text: q

text: q(t)

text: q

text: f

text: f

text: The

text: equations

text: of

text: motion

text: for

text: one

text: heavy

text: quark

text: propagating

text: through

text: a

text: medium

text: will

text: be

text: derived

text: in

text: section

text: Notice

text: the

text: remarkable

text: fact

text: that

text: dissipative

text: dynamics

text: can

text: be

text: derived

text: from

text: a

text: variational

text: principle.

text: Following

text: the

text: propagator

text: can

text: be

text: rewritten

text: in

text: a

text: more

text: transparent

text: way,

text: which

text: separates

text: the

text: reversible

text: dynamics

text: generated

text: by

text: from

text: the

text: irreversible

text: contribution

text: coming

text: from

text: the

text: Lindbladian

text: H(q,

text: p)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: to the extraction and analysis of large amounts of quantitative features from medical imaging data, radiomics3 allows for the creation of a high-dimensional abstract feature space that can be uti- lized for cancer detection via the detailed characterization of cancer phenotypes. The prognostic potential of radiomics has previously been demonstrated in studies on lung and head-and-neck cancer patients.4,5 Aerts et al.4 introduced a comprehensive study spanning over 1000 patients across seven datasets to demonstrate the application of radiomics towards differentiating between tumour phenotypes, indicating clinical and prognostic implications. In addition, radiomics has shown promise in combination with multi-parametric magnetic resonance imaging for breast can- cer detection6 and prostate cancer detection.7,8 Radiomics-driven methods have previously been developed for malignant lung nodule detec- tion using computed tomography (CT) images.9–12 Anirudh et al.9 used weakly labelled lung data from the SPIE-LUNGx dataset to train a 3D convolutional neural network (CNN) and generate ra- diomic sequences for lung nodule detection. In contrast, Orozco et al.10 generated wavelet-based radiomic sequences and demonstrated the effectiveness of wavelet-based features using a subset of images from the early lung cancer action project (ELCAP) and lung image database consortium (LIDC) datasets. Shen et al.11 proposed multi-scale convolutional neural networks (MCNN), a hierarchical framework for extracting discriminative features from lung nodules. Speciﬁcally, the framework comprises of alternating, stacked layers, and uses multi-scale nodule patches to learn class-speciﬁc features. More recently, Shen et al. extended their previous work to malignancy suspiciousness classiﬁcation.12 In addition, the extension simpliﬁed the training process via a multi-crop pool- ing architecture. An important aspect of these aforementioned radiomics-driven methods is that they leverage radiologist-driven nodule annotations for predicting the malignancy of lung nodules, 2

text: to

text: the

text: extraction

text: and

text: analysis

text: of

text: large

text: amounts

text: of

text: quantitative

text: features

text: from

text: medical

text: imaging

text: data,

text: allows

text: for

text: the

text: creation

text: of

text: a

text: high-dimensional

text: abstract

text: feature

text: space

text: that

text: can

text: be

text: uti-

text: lized

text: for

text: cancer

text: detection

text: via

text: the

text: detailed

text: characterization

text: of

text: cancer

text: phenotypes.

text: The

text: prognostic

text: potential

text: of

text: radiomics

text: has

text: previously

text: been

text: demonstrated

text: in

text: studies

text: on

text: lung

text: and

text: head-and-neck

text: cancer

text: Aerts

text: et

text: introduced

text: a

text: comprehensive

text: study

text: spanning

text: over

text: patients

text: across

text: seven

text: datasets

text: to

text: demonstrate

text: the

text: application

text: of

text: radiomics

text: towards

text: differentiating

text: between

text: tumour

text: phenotypes,

text: indicating

text: clinical

text: and

text: prognostic

text: implications.

text: In

text: addition,

text: radiomics

text: has

text: shown

text: promise

text: in

text: combination

text: with

text: multi-parametric

text: magnetic

text: resonance

text: imaging

text: for

text: breast

text: can-

text: cer

text: and

text: prostate

text: cancer

text: Radiomics-driven

text: methods

text: have

text: previously

text: been

text: developed

text: for

text: malignant

text: lung

text: nodule

text: detec-

text: tion

text: using

text: computed

text: tomography

text: (CT)

text: Anirudh

text: et

text: used

text: weakly

text: labelled

text: lung

text: data

text: from

text: the

text: SPIE-LUNGx

text: dataset

text: to

text: train

text: a

text: convolutional

text: neural

text: network

text: (CNN)

text: and

text: generate

text: ra-

text: diomic

text: sequences

text: for

text: lung

text: nodule

text: detection.

text: In

text: contrast,

text: Orozco

text: et

text: generated

text: wavelet-based

text: radiomic

text: sequences

text: and

text: demonstrated

text: the

text: effectiveness

text: of

text: wavelet-based

text: features

text: using

text: a

text: subset

text: of

text: images

text: from

text: the

text: early

text: lung

text: cancer

text: action

text: project

text: (ELCAP)

text: and

text: lung

text: image

text: database

text: consortium

text: (LIDC)

text: datasets.

text: Shen

text: et

text: proposed

text: multi-scale

text: convolutional

text: neural

text: networks

text: (MCNN),

text: a

text: hierarchical

text: framework

text: for

text: extracting

text: discriminative

text: features

text: from

text: lung

text: nodules.

text: Speciﬁcally,

text: the

text: framework

text: comprises

text: of

text: alternating,

text: stacked

text: layers,

text: and

text: uses

text: multi-scale

text: nodule

text: patches

text: to

text: learn

text: class-speciﬁc

text: features.

text: More

text: recently,

text: Shen

text: et

text: al.

text: extended

text: their

text: previous

text: work

text: to

text: malignancy

text: suspiciousness

text: In

text: addition,

text: the

text: extension

text: simpliﬁed

text: the

text: training

text: process

text: via

text: a

text: multi-crop

text: pool-

text: ing

text: architecture.

text: An

text: important

text: aspect

text: of

text: these

text: aforementioned

text: radiomics-driven

text: methods

text: is

text: that

text: they

text: leverage

text: radiologist-driven

text: nodule

text: annotations

text: for

text: predicting

text: the

text: malignancy

text: of

text: lung

text: nodules,


text: (cid:89) (cid:104) (cid:105) H F E · | P ( ) = ( ) P (Sc W − ) . (3) g c g g 1 ∈ c C F E More speciﬁcally, the environmental factor ( ) is formulated such that the offspring ra- c diomic sequencer is limited to 80% of the total number of synapses in its direct ancestor sequencer. 3 Results 3.1 Experimental Setup The proposed evolutionary deep radiomic sequencer discovery framework was examined using the pathology-proven subset of the LIDC-IDRI21,22 dataset, and was compared to state-of-the-art methods. In this section, the conﬁguration of dataset, the underlying network architecture of the discovered radiomic sequencers, and the competing methods are explained. 3.1.1 Lung Dataset In this study, we used the subset of the LIDC-IDRI21,22 dataset that had corresponding pathology- proven diagnostic data. The dataset is a public dataset provide by Cancer Imaging Archive21,22 consists of diagnostic and lung cancer screening thoracic computed tomography (CT) scans with marked-up annotated lesions. The CT images were captured using a broad range of scanner models from different manufacturers by applying the following tube peak potential energies for acquiring the scans: 120kV (n = 818), 130kV (n = 31), 135kV (n = 69), and 140kV (n = 100). A subset of 93 patient cases which have deﬁnite diagnostic results was selected from the LIDC-IDRI. While pathology data was used to generate labels for the CT images in the LIDC-IDRI dataset, the pathol- 12

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text: 0.8 0.7 0.6 0.5 0.4 0.3 0 2 4 6 8 10 12 Generation Fig 5: Running time evaluation of synthesized radiomic sequencer at each generation is evaluated by 1500 sample inputs. As seen the subsequent radiomic sequencers perform faster than their ancestors which shows the efﬁciency of the synthesized sequencer discovered via the proposed evolutionary deep radiomic sequencer discovery process. tion of the sequencers are examined at each generation with 1500 sample inputs. Figure 5 demon- strates the running time performance of synthesized radiomic sequencers through generations. As seen the subsequent generations performs faster than their ancestors which shows the efﬁciency of the proposed evolutionary deep intelligence framework. Decreasing the number of ﬁlters in the model decreases the length of the radiomic sequence. As shown in Table 1, the length of the radiomic sequence is shortened generation by generation and the length of radiomic sequence in the last generation is about half size of the radiomic sequence of the ﬁrst generation, demonstrating that it is possible to increase the concision of the radiomic sequence while simultaneously increasing the modeling accuracy. Figure 6 demonstrates the sensitivity of the evolved radiomic sequencers overlaid with the stan- dard deviation across different folds of cross validation over multiple generations. By evolving the 16 )S( emiT gnissecorP ##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##

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text: efﬁciency

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text: tion

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text: sequencers

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text: examined

text: at

text: each

text: generation

text: with

text: sample

text: inputs.

text: Figure

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text: strates

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text: generations.

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text: 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.8 0.78 0.76 1 2 3 4 5 6 7 8 9 10 11 Generation Fig 8: Radiomic sequencers modeling accuracy over generations. radiomic sequencer. The ultimate goal here is to synthesize a new deep neural network as the underlying core of radiomic sequencer with a fewer number of parameters which produce more concise radiomic sequences that can better capture the differences between healthy and cancerous lung tissue. Results show that by evolving and discovering more efﬁcient radiomic sequencers, the diagnostic accuracy can be increased. Experimental results demonstrate that the evolved deep ra- diomic sequencer discovered using the proposed evolutionary deep radiomic sequencer discovery approach can outperform other state-of-the-art radiomics-driven methods, achieving a sensitivity of 93.42%, a speciﬁcity of 82.39%, and an accuracy of 88.78%. It has been showed in the literature that there is a direct relation between the number of parameters and the need for training data. As a future work it is suggested to study the effect of evolutionary deep intelligence framework when a limited training data is available. Disclosures The authors have no relevant ﬁnancial interests in the manuscript and no other potential conﬂicts 20 ycaruccA ##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##

text: Generation

text: Fig

text: Radiomic

text: sequencers

text: modeling

text: accuracy

text: over

text: generations.

text: radiomic

text: sequencer.

text: The

text: ultimate

text: goal

text: here

text: is

text: to

text: synthesize

text: a

text: new

text: deep

text: neural

text: network

text: as

text: the

text: underlying

text: core

text: of

text: radiomic

text: sequencer

text: with

text: a

text: fewer

text: number

text: of

text: parameters

text: which

text: produce

text: more

text: concise

text: radiomic

text: sequences

text: that

text: can

text: better

text: capture

text: the

text: differences

text: between

text: healthy

text: and

text: cancerous

text: lung

text: tissue.

text: Results

text: show

text: that

text: by

text: evolving

text: and

text: discovering

text: more

text: efﬁcient

text: radiomic

text: sequencers,

text: the

text: diagnostic

text: accuracy

text: can

text: be

text: increased.

text: Experimental

text: results

text: demonstrate

text: that

text: the

text: evolved

text: deep

text: ra-

text: diomic

text: sequencer

text: discovered

text: using

text: the

text: proposed

text: evolutionary

text: deep

text: radiomic

text: sequencer

text: discovery

text: approach

text: can

text: outperform

text: other

text: state-of-the-art

text: radiomics-driven

text: methods,

text: achieving

text: a

text: sensitivity

text: of

text: a

text: speciﬁcity

text: of

text: and

text: an

text: accuracy

text: of

text: It

text: has

text: been

text: showed

text: in

text: the

text: literature

text: that

text: there

text: is

text: a

text: direct

text: relation

text: between

text: the

text: number

text: of

text: parameters

text: and

text: the

text: need

text: for

text: training

text: data.

text: As

text: a

text: future

text: work

text: it

text: is

text: suggested

text: to

text: study

text: the

text: effect

text: of

text: evolutionary

text: deep

text: intelligence

text: framework

text: when

text: a

text: limited

text: training

text: data

text: is

text: available.

text: Disclosures

text: The

text: authors

text: have

text: no

text: relevant

text: ﬁnancial

text: interests

text: in

text: the

text: manuscript

text: and

text: no

text: other

text: potential

text: conﬂicts

text: ycaruccA

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