text: ∼ − ∼ − a high H O gas abundance ( 10 4) extends to a au) are not so small ( 10 20% of total emission 2 ∼ ∼ radius of r 10 au at z/r 0.1 compared with ﬂuxes) compared to the region with a high H O ∼ ∼ ∼ − 2 that at z 0 (r 7-8 au), and is reﬂected in the gas abundance ( 10 4, r <8 au). Although the emissivity distribution. However, since the radial H O 933.28 µm line resides in the sub-millimeter 2 shift in the distribution is small (a few au), its in- region, this line has a larger E (=1861.2K) than up ﬂuence is not so serious for obtaining information other lines, and thus most of the emission ﬂux is on the general trend of the H O distribution in emitted from the region with high temperature 2 ∼ − the inner disk within the H O snowline. and a high H O gas abundance ( 10 4, r <8 au). 2 2 The diﬀerences in the properties of the line proﬁles Since the radial diﬀerence of the positions be- (see Figures 4, 5, 6, and ??) come from the diﬀer- tween the exact H O snowline location and the 2 ences in A , E , and wavelengths among lines. outer edge of the hot H O gas region is not so ul up 2 For the lines with similar wavelengths, the values large (several au), the inﬂuence is not so serious of optical depth tend to be larger as values of A when we want to get information on the overall ul of the lines become larger, since the absorption by H O distribution of the inner disk and roughly 2 excited H O molecules increases. In addition, the estimate the position of the H O snowline. How- 2 2 values of optical depth become larger as values of ever, if we observe several candidate H O lines ∼ − − − 2 E become smaller, since the line absorption be- with small A ( 10 6 10 3), various E (e.g., up ∼ − ul up come stronger even in the colder region of the disk. 700 2100K), and at various wavelengths be- tween mid-infrared and sub-millimeter, there is For the lines at shorter wavelengths, the opacity the possibility to conﬁne the detailed distribution, of the dust grains becomes larger (e.g., ?). In the not only the position of the H O snowline, but 2 cases of shorter wavelength (mid- and far-infrared) also the H O gas abundance and the gas temper- 2 lines, the absorption by the dust grains mainly de- ature in the disk midplane. In addition, we could termines total optical depth proﬁles (including gas trace the water reservoir within the H O snowline 2 and dust components) and line emitting regions from the Keplerian line proﬁles independently re- in the both inner and outer disk. In contrast, for gardless of the assumption of the relation between the lines with longer wavelengths, the values of the disk gas temperature and radius, as adopted in line absorption by excited molecules become larger previous works to get the H O distributions (e.g., 2 (see also Equation (10) of Paper I (?)) even if the ??). We note that the previous observations of ∼ − − − values of A and E are similar. In the case of H O lines with large A ( 10 1 100 s 1) and ul up 2 ul longer wavelength lines (sub-millimeter lines), the very high E (>3000K) in many cases mainly up line absorption by excited molecules mainly de- trace the emission from the disk surface (e.g., termines the total optical depth proﬁles and line ????????, see also Section 3.2.2). emitting regions in the inner disk midplane with ∼ − − a high H2O gas abundance ( 10 4, r < 7 8 au), 3.2.2. H2O lines which are less suited to trace although the absorption by dust grains mainly de- emission from water reservoir within the termines those in the disk surface and outer colder H O snowline 2 disk midplane. In the top left panel of Figure 7, we show the line proﬁle for the H O 63.32µm line. The contribution The H O 482.99µm and 682.66µm lines have rel- 2 2 from the optically thin surface layer of the outer atively smaller values of E (<1100K), and thus up disk (r =14-30 au) is large (three times larger in they can also trace outer colder region compared ﬂux density) compared with that of the optically to lines with larger values of E . In addition, up thick region near the midplane of inner disk (r <14 they have longer wavelengths (> 400µm) com- au), and the shape of the line proﬁle is a narrower pared with other lines, thus the dust opacity is by around a factor of two compared to those of smaller and they can trace the regions closer to the candidate H O lines which trace the emission from midplane in the outer disk. These are the reasons 2 the hot water vapor within the H O snowline. why emission ﬂuxes from the region with a rela- 2 tively high H2O gas abundance (∼10−8, r =8-14 This is because−the H2O 63.32µm line has a large A (=1.772 s 1), although E (=1070.6K) is ul up 10

text: a

text: high

text: H

text: O

text: gas

text: abundance

text: extends

text: to

text: a

text: au)

text: are

text: not

text: so

text: small

text: of

text: total

text: emission

text: radius

text: of

text: r

text: au

text: at

text: z/r

text: compared

text: with

text: ﬂuxes)

text: compared

text: to

text: the

text: region

text: with

text: a

text: high

text: H

text: O

text: that

text: at

text: z

text: (r

text: au),

text: and

text: is

text: reﬂected

text: in

text: the

text: gas

text: abundance

text: r

text: au).

text: Although

text: the

text: emissivity

text: distribution.

text: However,

text: since

text: the

text: radial

text: H

text: O

text: µm

text: line

text: resides

text: in

text: the

text: sub-millimeter

text: shift

text: in

text: the

text: distribution

text: is

text: small

text: (a

text: few

text: au),

text: its

text: in-

text: region,

text: this

text: line

text: has

text: a

text: larger

text: E

text: than

text: up

text: ﬂuence

text: is

text: not

text: so

text: serious

text: for

text: obtaining

text: information

text: other

text: lines,

text: and

text: thus

text: most

text: of

text: the

text: emission

text: ﬂux

text: is

text: on

text: the

text: general

text: trend

text: of

text: the

text: H

text: O

text: distribution

text: in

text: emitted

text: from

text: the

text: region

text: with

text: high

text: temperature

text: the

text: inner

text: disk

text: within

text: the

text: H

text: O

text: snowline.

text: and

text: a

text: high

text: H

text: O

text: gas

text: abundance

text: r

text: au).

text: The

text: diﬀerences

text: in

text: the

text: properties

text: of

text: the

text: line

text: proﬁles

text: Since

text: the

text: radial

text: diﬀerence

text: of

text: the

text: positions

text: be-

text: (see

text: Figures

text: and

text: come

text: from

text: the

text: diﬀer-

text: tween

text: the

text: exact

text: H

text: O

text: snowline

text: location

text: and

text: the

text: ences

text: in

text: A

text: E

text: and

text: wavelengths

text: among

text: lines.

text: outer

text: edge

text: of

text: the

text: hot

text: H

text: O

text: gas

text: region

text: is

text: not

text: so

text: ul

text: up

text: For

text: the

text: lines

text: with

text: similar

text: wavelengths,

text: the

text: values

text: large

text: (several

text: au),

text: the

text: inﬂuence

text: is

text: not

text: so

text: serious

text: of

text: optical

text: depth

text: tend

text: to

text: be

text: larger

text: as

text: values

text: of

text: A

text: when

text: we

text: want

text: to

text: get

text: information

text: on

text: the

text: overall

text: ul

text: of

text: the

text: lines

text: become

text: larger,

text: since

text: the

text: absorption

text: by

text: H

text: O

text: distribution

text: of

text: the

text: inner

text: disk

text: and

text: roughly

text: excited

text: H

text: O

text: molecules

text: increases.

text: In

text: addition,

text: the

text: estimate

text: the

text: position

text: of

text: the

text: H

text: O

text: snowline.

text: How-

text: values

text: of

text: optical

text: depth

text: become

text: larger

text: as

text: values

text: of

text: ever,

text: if

text: we

text: observe

text: several

text: candidate

text: H

text: O

text: lines

text: E

text: become

text: smaller,

text: since

text: the

text: line

text: absorption

text: be-

text: with

text: small

text: A

text: various

text: E

text: (e.g.,

text: up

text: ul

text: up

text: come

text: stronger

text: even

text: in

text: the

text: colder

text: region

text: of

text: the

text: disk.

text: and

text: at

text: various

text: wavelengths

text: be-

text: tween

text: mid-infrared

text: and

text: sub-millimeter,

text: there

text: is

text: For

text: the

text: lines

text: at

text: shorter

text: wavelengths,

text: the

text: opacity

text: the

text: possibility

text: to

text: conﬁne

text: the

text: detailed

text: distribution,

text: of

text: the

text: dust

text: grains

text: becomes

text: larger

text: (e.g.,

text: In

text: the

text: not

text: only

text: the

text: position

text: of

text: the

text: H

text: O

text: snowline,

text: but

text: cases

text: of

text: shorter

text: wavelength

text: (mid-

text: and

text: far-infrared)

text: also

text: the

text: H

text: O

text: gas

text: abundance

text: and

text: the

text: gas

text: temper-

text: lines,

text: the

text: absorption

text: by

text: the

text: dust

text: grains

text: mainly

text: de-

text: ature

text: in

text: the

text: disk

text: midplane.

text: In

text: addition,

text: we

text: could

text: termines

text: total

text: optical

text: depth

text: proﬁles

text: (including

text: gas

text: trace

text: the

text: water

text: reservoir

text: within

text: the

text: H

text: O

text: snowline

text: and

text: dust

text: components)

text: and

text: line

text: emitting

text: regions

text: from

text: the

text: Keplerian

text: line

text: proﬁles

text: independently

text: re-

text: in

text: the

text: both

text: inner

text: and

text: outer

text: disk.

text: In

text: contrast,

text: for

text: gardless

text: of

text: the

text: assumption

text: of

text: the

text: relation

text: between

text: the

text: lines

text: with

text: longer

text: wavelengths,

text: the

text: values

text: of

text: the

text: disk

text: gas

text: temperature

text: and

text: radius,

text: as

text: adopted

text: in

text: line

text: absorption

text: by

text: excited

text: molecules

text: become

text: larger

text: previous

text: works

text: to

text: get

text: the

text: H

text: O

text: distributions

text: (e.g.,

text: (see

text: also

text: Equation

text: of

text: Paper

text: I

text: even

text: if

text: the

text: We

text: note

text: that

text: the

text: previous

text: observations

text: of

text: values

text: of

text: A

text: and

text: E

text: are

text: similar.

text: In

text: the

text: case

text: of

text: H

text: O

text: lines

text: with

text: large

text: A

text: s

text: and

text: ul

text: up

text: ul

text: longer

text: wavelength

text: lines

text: (sub-millimeter

text: lines),

text: the

text: very

text: high

text: E

text: in

text: many

text: cases

text: mainly

text: up

text: line

text: absorption

text: by

text: excited

text: molecules

text: mainly

text: de-

text: trace

text: the

text: emission

text: from

text: the

text: disk

text: surface

text: (e.g.,

text: termines

text: the

text: total

text: optical

text: depth

text: proﬁles

text: and

text: line

text: see

text: also

text: Section

text: emitting

text: regions

text: in

text: the

text: inner

text: disk

text: midplane

text: with

text: a

text: high

text: gas

text: abundance

text: r

text: au),

text: lines

text: which

text: are

text: less

text: suited

text: to

text: trace

text: although

text: the

text: absorption

text: by

text: dust

text: grains

text: mainly

text: de-

text: emission

text: from

text: water

text: reservoir

text: within

text: the

text: termines

text: those

text: in

text: the

text: disk

text: surface

text: and

text: outer

text: colder

text: H

text: O

text: snowline

text: disk

text: midplane.

text: In

text: the

text: top

text: left

text: panel

text: of

text: Figure

text: we

text: show

text: the

text: line

text: proﬁle

text: for

text: the

text: H

text: O

text: line.

text: The

text: contribution

text: The

text: H

text: O

text: and

text: lines

text: have

text: rel-

text: from

text: the

text: optically

text: thin

text: surface

text: layer

text: of

text: the

text: outer

text: atively

text: smaller

text: values

text: of

text: E

text: and

text: thus

text: up

text: disk

text: (r

text: au)

text: is

text: large

text: (three

text: times

text: larger

text: in

text: they

text: can

text: also

text: trace

text: outer

text: colder

text: region

text: compared

text: ﬂux

text: density)

text: compared

text: with

text: that

text: of

text: the

text: optically

text: to

text: lines

text: with

text: larger

text: values

text: of

text: E

text: In

text: addition,

text: up

text: thick

text: region

text: near

text: the

text: midplane

text: of

text: inner

text: disk

text: (r

text: they

text: have

text: longer

text: wavelengths

text: com-

text: au),

text: and

text: the

text: shape

text: of

text: the

text: line

text: proﬁle

text: is

text: a

text: narrower

text: pared

text: with

text: other

text: lines,

text: thus

text: the

text: dust

text: opacity

text: is

text: by

text: around

text: a

text: factor

text: of

text: two

text: compared

text: to

text: those

text: of

text: smaller

text: and

text: they

text: can

text: trace

text: the

text: regions

text: closer

text: to

text: the

text: candidate

text: H

text: O

text: lines

text: which

text: trace

text: the

text: emission

text: from

text: midplane

text: in

text: the

text: outer

text: disk.

text: These

text: are

text: the

text: reasons

text: the

text: hot

text: water

text: vapor

text: within

text: the

text: H

text: O

text: snowline.

text: why

text: emission

text: ﬂuxes

text: from

text: the

text: region

text: with

text: a

text: rela-

text: tively

text: high

text: gas

text: abundance

text: r

text: This

text: is

text: because−the

text: line

text: has

text: a

text: large

text: A

text: s

text: although

text: E

text: is

text: ul

text: up


text: Abstract It is known that, for the isolated dielectric cylinder waveguide, there exists the cutoﬀ frequency ω∗ below which there are no guided (radiationless) modes. It is shown in the paper that the inﬁnite plane periodic array of such waveguides possesses guided modes in the frequency domain which is below the frequency ω∗. So far as the ﬁnite array is concerned, the modes in this frequency ∼ domain are weakly radiating ones, but their quality factor Q increases as Q(N) N3, N being the number of the waveguides in the array. This dependence is obtained both numerically, using the multiple scattering formalism, and is justiﬁed within the framework a simple analytical model. 1

text: Abstract

label: paragraph

text: It

label: abstract

text: is

text: known

text: that,

text: for

text: the

text: isolated

text: dielectric

text: cylinder

text: waveguide,

text: there

text: exists

text: the

text: cutoﬀ

text: frequency

text: ω∗

text: below

text: which

text: there

text: are

text: no

text: guided

text: (radiationless)

text: modes.

text: It

text: is

text: shown

text: in

text: the

text: paper

text: that

text: the

text: inﬁnite

text: plane

text: periodic

text: array

text: of

text: such

text: waveguides

text: possesses

text: guided

text: modes

text: in

text: the

text: frequency

text: domain

text: which

text: is

text: below

text: the

text: frequency

text: ω∗.

text: So

text: far

text: as

text: the

text: ﬁnite

text: array

text: is

text: concerned,

text: the

text: modes

text: in

text: this

text: frequency

text: domain

text: are

text: weakly

text: radiating

text: ones,

text: but

text: their

text: quality

text: factor

text: Q

text: increases

text: as

text: Q(N)

text: N

text: being

text: the

text: number

text: of

text: the

text: waveguides

text: in

text: the

text: array.

text: This

text: dependence

text: is

text: obtained

text: both

text: numerically,

text: using

text: the

text: multiple

text: scattering

text: formalism,

text: and

text: is

text: justiﬁed

text: within

text: the

text: framework

text: a

text: simple

text: analytical

text: model.


text: (cid:80) (cid:80) a H (κ R) + a U − (ω, β) J (κ R) = c J (κR), j 0 0 (cid:54) l l j 0 0 j 0 l=j (16) (cid:48) (cid:48) (cid:48) (cid:48) a n iω0 H (κ R) + a n U − (ω, β)iω0 J (κ R) = c n iω J (κR). j 0 κ0 0 0 (cid:54) l 0 l j κ0 0 0 j κ 0 l=j Then the system of equations (16) is red(cid:88)uced to the form a − j U − (ω, β) a = 0, (17) l j l a¯(ω, β) (cid:54) l=j c = c¯(ω, β) a , (18) j j where (cid:48) − (cid:48) n2κ J (κR) J (κ R) n2κ J (κR) J (κ R) a¯(ω, β) = 0 0 (cid:48) 0 0 − 0 0(cid:48) 0 0 , (19) n2κ J (κR) H (κ R) n2κ J (κR) H (κ R) 0 0 0 0 0 0 0 0 { (cid:48) − (cid:48) } n2κ H (κ R)J (κ R) H (κ R)J (κ R) c¯(ω, β) = 0 (cid:48)0 0 0 0 − 0 0 0 (cid:48) 0 . (20) n2κJ (κ R)J (κR) n2κJ (κ R)J (κR) 0 0 0 0 0 0 0 The terms U − (ω, β) in Eq.(17) desribe the interaction between the waveguides. If the l j terms U − (ω, β) in Eq.(17) are neglected, the poles of a¯(ω, β) or c¯(ω, β) determines the l j guied modes for the isolated wavegide. (cid:13) (cid:13) System of equations (17) possesses nontrivial solutions if (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) δ − det jl U − (ω, β) = 0. (21) l j a¯(ω, β) This equation relates the frequency of the guided mode ω and its propagation constant β implicitly. For the inﬁnite periodical array of identical waveguides, the solution of Eq.(17) reads − ≤ a = a eikaj, π/a < k π/a. (22) j 0 In this case, the nontrivial solution exists if 1 − U(ω, β, k) = 0, (23) a¯(ω, β) 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: a

text: H

text: (κ

text: R)

text: a

text: U

text: (ω,

text: β)

text: J

text: (κ

text: R)

text: c

text: J

text: (κR),

text: j

font: default

text: l

text: l

text: j

text: j

text: l=j

text: a

text: n

text: H

text: (κ

text: R)

text: a

text: n

text: U

text: (ω,

text: J

text: (κ

text: R)

text: c

text: n

text: iω

text: J

text: (κR).

text: j

text: κ0

text: l

text: l

text: j

text: κ0

text: j

text: κ

text: l=j

text: Then

text: the

text: system

text: of

text: equations

text: is

text: to

text: the

text: form

text: a

text: j

text: U

text: (ω,

text: β)

text: a

text: l

text: j

text: l

text: a¯(ω,

text: β)

text: l=j

text: c

text: c¯(ω,

text: β)

text: a

text: j

text: j

text: where

text: J

text: (κR)

text: J

text: (κ

text: R)

text: J

text: (κR)

text: J

text: (κ

text: R)

text: a¯(ω,

text: β)

text: J

text: (κR)

text: H

text: (κ

text: R)

text: J

text: (κR)

text: H

text: (κ

text: R)

text: H

text: (κ

text: R)J

text: (κ

text: R)

text: H

text: (κ

text: R)J

text: (κ

text: R)

text: c¯(ω,

text: β)

text: (κ

text: R)J

text: (κR)

text: (κ

text: R)J

text: (κR)

text: The

text: terms

text: U

text: (ω,

text: β)

text: in

text: desribe

text: the

text: interaction

text: between

text: the

text: waveguides.

text: If

text: the

text: l

text: j

text: terms

text: U

text: (ω,

text: β)

text: in

text: are

text: neglected,

text: the

text: poles

text: of

text: a¯(ω,

text: β)

text: or

text: c¯(ω,

text: β)

text: determines

text: the

text: l

text: j

text: guied

text: modes

text: for

text: the

text: isolated

text: wavegide.

text: System

text: of

text: equations

text: possesses

text: nontrivial

text: solutions

text: if

text: δ

text: det

text: jl

text: U

text: (ω,

text: β)

text: l

text: j

text: a¯(ω,

text: β)

text: This

text: equation

text: relates

text: the

text: frequency

text: of

text: the

text: guided

text: mode

text: ω

text: and

text: its

text: propagation

text: constant

text: β

text: implicitly.

text: For

text: the

text: inﬁnite

text: periodical

text: array

text: of

text: identical

text: waveguides,

text: the

text: solution

text: of

text: reads

text: a

text: a

text: eikaj,

text: π/a

text: k

text: π/a.

text: j

text: In

text: this

text: case,

text: the

text: nontrivial

text: solution

text: exists

text: if

text: U(ω,

text: β,

text: k)

text: a¯(ω,

text: β)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 = 0.7, T = 0.2, = 10, Ampl = 0.3, = 0.2 α ωc ∆ 58 Here we studied the complete ﬁrst passage time prob- = 0.25 Ω 56 Ωε = 10 ability density for general time-dependent driving of the noεise only two energy parameters characterizing the spin-boson sys- 54 static tem. Speciﬁc driving mechanisms involve a modulation 52 in terms of a stationary two-state process with exponen- tially correlated noise or also an external deterministic 50 periodic driving of those parameters, including combina- 48 tions of both driving mechanisms. In contrast to the case of stationary noise driving, the passage time dynamics 46 for deterministic driving is cumbersome as it involves ex- 44 plicit time-dependent transition rates with corresponding 10-4 10-3 10-2 10-1 100 101 102 103 time-dependent boundary conditions for reﬂection and absorption. Particularly, the role of periodic driving ν results in a decaying ﬁrst passage time probability den- sities which exhibits multiple peaks. These peaks reﬂect FIG. 13. (Color online) Mean ﬁrst passage time t1 (averaged over an initial phase-dependent quantum synchronization fea- initial phase φ) versus the two-state noise switching rate ν acting ture [6, 35, 38]. This latter feature is absent when the on the tunneling matrix element in the presence of a simultaneous transition rates are time-independent (stationary noise periodic driving of the bias (cid:15)(t). The bias amplitude is held at driving), resulting now in a monotonic decay of the ﬁrst A(cid:15) = 0.3 while the two chosen angular frequencies values for Ω(cid:15) passage time pdf. are indicated in the ﬁgure. In addition, a comparison is made with the case in which the deterministic drive for the bias is switched oﬀ; This ﬁrst passage time pdf allows for the evaluation of i.e. dichotomous noise is solely modulating the tunneling matrix all its moments. Of particular interest is its ﬁrst mean, element. The dotted line marks the static case. The amplitude for the MFPT. This quantity displays the typical signatures the modulation of the tunneling matrix amplitude is set at ∆ = 0.2. of resonant activation, i.e. the existence of an intermedi- Other parameters are as in Fig. 2. ate modulation regime where the MFPT undrerruns the values assumed in the opposite limits of adiabatic slow driving and high frequency modulation. In the limit of very high frequency modulation one approaches the non- ducing a periodically driven bias with frequency Ω con- (cid:15) driven MFPT value. sists in shifting downwards to smaller values the curves Our ﬁndings for various modulation settings corrobo- of the MFPT versus the switching rate ν. Speciﬁcally, rate the universal behavior [9] found for classical over- t assumes systematically lower values with a bias (cid:15)(t) 1 the-barrier resonant activation, where (i) at low frequen- periodically driven at Ω = 0.25. At this angular driv- (cid:15) → ∞ cies the MFPT is dominated by the adiabatic conﬁgu- ing frequency, t (ν) converges in the limit ν to the 1 ration, with the largest possible passage time ruling the value highlighted by the full circle located at the mini- overall escape, while (ii) for high frequency modulations mum of t in Fig. 11, as to be expected. Likewise, for the 1 the MFPT is governed by the value of the time-averaged case of a large angular driving frequency, i.e., Ω = 10, (cid:15) energy proﬁle – yielding typically the static MFPT value the line t (ν) virtually coincides with the analytical re- 1 –; (iii) for modulations at intermediate time-scales (of sult obtained with (cid:15)(t) = 0 and dichotomous noise on the the order of the system dynamics time scale) the regime tunneling matrix element. This is due to the fact that, with minimal MFPT values emerges (resonant activation for such a large deterministic driving frequency, one ap- regime) where the MFPT underruns both limits (i) and proaches the situation discussed in Fig. 3 (green line) for (ii). The wide parameter region for the quantum tun- ∆ = 0.2. neling rate in the modulated TSS allows one to engineer the regime of resonant activation towards either smaller or also – more interestingly – much wider modulation V. CONCLUSIONS regimes. This feature becomes apparent by supplement- ing the information contained in Figs. 3 and 7 with those With this work we studied, by the use of analytical of Figs. 5 and 8. and numerical means, the phenomenon of resonant acti- A further interesting feature we detected with this vation, occurring for a dissipative two-state quantum sys- study is the approximate, although nearly exact, cross- tem (spin-boson system) which is modulated by periodic ing behavior (as demonstrated analytically and validated deterministic driving and/or via telegraphic two-state numerically in Sec. IV.A) of the nonadiabatic MFPT noise. At strong system-bath coupling the quantum dy- entering the resonant activation regime at some critical namics proceeds incoherently so that an eﬀective classical ∗ frequency ν , being only weakly dependent on the driv- description in terms of a master equation with incoherent ing amplitude. quantum rates becomes feasible. This in turn allows for The experimental implementation of an absorbing studying the detailed ﬁrst passage time statistics when state may not always be straightforward. In such cases, starting out at one of the two metastable states, with the pdf of residence times provided by Eq. (37), or also absorption occurring at the neighboring state. t 1 ##LTFigure##

text: T

text: Ampl

text: α

text: ωc

text: Here

text: we

text: studied

text: the

text: complete

text: ﬁrst

text: passage

text: time

text: prob-

text: Ω

text: Ωε

text: ability

text: density

text: for

text: general

text: time-dependent

text: driving

text: of

text: the

text: noεise

text: only

text: two

text: energy

text: parameters

text: characterizing

text: the

text: spin-boson

text: sys-

text: static

text: tem.

text: Speciﬁc

text: driving

text: mechanisms

text: involve

text: a

text: modulation

text: in

text: terms

text: of

text: a

text: stationary

text: two-state

text: process

text: with

text: exponen-

text: tially

text: correlated

text: noise

text: or

text: also

text: an

text: external

text: deterministic

text: periodic

text: driving

text: of

text: those

text: parameters,

text: including

text: combina-

text: tions

text: of

text: both

text: driving

text: mechanisms.

text: In

text: contrast

text: to

text: the

text: case

text: of

text: stationary

text: noise

text: driving,

text: the

text: passage

text: time

text: dynamics

text: for

text: deterministic

text: driving

text: is

text: cumbersome

text: as

text: it

text: involves

text: ex-

text: plicit

text: time-dependent

text: transition

text: rates

text: with

text: corresponding

text: time-dependent

text: boundary

text: conditions

text: for

text: reﬂection

text: and

text: absorption.

text: Particularly,

text: the

text: role

text: of

text: periodic

text: driving

text: ν

text: results

text: in

text: a

text: decaying

text: ﬁrst

text: passage

text: time

text: probability

text: den-

text: sities

text: which

text: exhibits

text: multiple

text: peaks.

text: These

text: peaks

text: reﬂect

text: FIG.

text: (Color

text: online)

text: Mean

text: ﬁrst

text: passage

text: time

text: (averaged

text: over

text: an

text: initial

text: phase-dependent

text: quantum

text: synchronization

text: fea-

text: initial

text: phase

text: φ)

text: versus

text: the

text: two-state

text: noise

text: switching

text: rate

text: ν

text: acting

text: ture

text: This

text: latter

text: feature

text: is

text: absent

text: when

text: the

text: on

text: the

text: tunneling

text: matrix

text: element

text: in

text: the

text: presence

text: of

text: a

text: simultaneous

text: transition

text: rates

text: are

text: time-independent

text: (stationary

text: noise

text: periodic

text: driving

text: of

text: the

text: bias

text: The

text: bias

text: amplitude

text: is

text: held

text: at

text: driving),

text: resulting

text: now

text: in

text: a

text: monotonic

text: decay

text: of

text: the

text: ﬁrst

text: while

text: the

text: two

text: chosen

text: angular

text: frequencies

text: values

text: for

text: passage

text: time

text: pdf.

text: are

text: indicated

text: in

text: the

text: ﬁgure.

text: In

text: addition,

text: a

text: comparison

text: is

text: made

text: with

text: the

text: case

text: in

text: which

text: the

text: deterministic

text: drive

text: for

text: the

text: bias

text: is

text: switched

text: oﬀ;

text: This

text: ﬁrst

text: passage

text: time

text: pdf

text: allows

text: for

text: the

text: evaluation

text: of

text: i.e.

text: dichotomous

text: noise

text: is

text: solely

text: modulating

text: the

text: tunneling

text: matrix

text: all

text: its

text: moments.

text: Of

text: particular

text: interest

text: is

text: its

text: ﬁrst

text: mean,

text: element.

text: The

text: dotted

text: line

text: marks

text: the

text: static

text: case.

text: The

text: amplitude

text: for

text: the

text: MFPT.

text: This

text: quantity

text: displays

text: the

text: typical

text: signatures

text: the

text: modulation

text: of

text: the

text: tunneling

text: matrix

text: amplitude

text: is

text: set

text: at

text: of

text: resonant

text: activation,

text: i.e.

text: the

text: existence

text: of

text: an

text: intermedi-

text: Other

text: parameters

text: are

text: as

text: in

text: Fig.

text: ate

text: modulation

text: regime

text: where

text: the

text: MFPT

text: undrerruns

text: the

text: values

text: assumed

text: in

text: the

text: opposite

text: limits

text: of

text: adiabatic

text: slow

text: driving

text: and

text: high

text: frequency

text: modulation.

text: In

text: the

text: limit

text: of

text: very

text: high

text: frequency

text: modulation

text: one

text: approaches

text: the

text: non-

text: ducing

text: a

text: periodically

text: driven

text: bias

text: with

text: frequency

text: Ω

text: con-

text: driven

text: MFPT

text: value.

text: sists

text: in

text: shifting

text: downwards

text: to

text: smaller

text: values

text: the

text: curves

text: Our

text: ﬁndings

text: for

text: various

text: modulation

text: settings

text: corrobo-

text: of

text: the

text: MFPT

text: versus

text: the

text: switching

text: rate

text: ν.

text: Speciﬁcally,

text: rate

text: the

text: universal

text: behavior

text: found

text: for

text: classical

text: over-

text: t

text: assumes

text: systematically

text: lower

text: values

text: with

text: a

text: bias

text: the-barrier

text: resonant

text: activation,

text: where

text: (i)

text: at

text: low

text: frequen-

text: periodically

text: driven

text: at

text: Ω

text: At

text: this

text: angular

text: driv-

text: cies

text: the

text: MFPT

text: is

text: dominated

text: by

text: the

text: adiabatic

text: conﬁgu-

text: ing

text: frequency,

text: t

text: (ν)

text: converges

text: in

text: the

text: limit

text: ν

text: to

text: the

text: ration,

text: with

text: the

text: largest

text: possible

text: passage

text: time

text: ruling

text: the

text: value

text: highlighted

text: by

text: the

text: full

text: circle

text: located

text: at

text: the

text: mini-

text: overall

text: escape,

text: while

text: (ii)

text: for

text: high

text: frequency

text: modulations

text: mum

text: of

text: t

text: in

text: Fig.

text: as

text: to

text: be

text: expected.

text: Likewise,

text: for

text: the

text: the

text: MFPT

text: is

text: governed

text: by

text: the

text: value

text: of

text: the

text: time-averaged

text: case

text: of

text: a

text: large

text: angular

text: driving

text: frequency,

text: i.e.,

text: Ω

text: energy

text: proﬁle

text: yielding

text: typically

text: the

text: static

text: MFPT

text: value

text: the

text: line

text: t

text: (ν)

text: virtually

text: coincides

text: with

text: the

text: analytical

text: re-

text: (iii)

text: for

text: modulations

text: at

text: intermediate

text: time-scales

text: (of

text: sult

text: obtained

text: with

text: and

text: dichotomous

text: noise

text: on

text: the

text: the

text: order

text: of

text: the

text: system

text: dynamics

text: time

text: scale)

text: the

text: regime

text: tunneling

text: matrix

text: element.

text: This

text: is

text: due

text: to

text: the

text: fact

text: that,

text: with

text: minimal

text: MFPT

text: values

text: emerges

text: (resonant

text: activation

text: for

text: such

text: a

text: large

text: deterministic

text: driving

text: frequency,

text: one

text: ap-

text: regime)

text: where

text: the

text: MFPT

text: underruns

text: both

text: limits

text: (i)

text: and

text: proaches

text: the

text: situation

text: discussed

text: in

text: Fig.

text: (green

text: line)

text: for

text: (ii).

text: The

text: wide

text: parameter

text: region

text: for

text: the

text: quantum

text: tun-

text: neling

text: rate

text: in

text: the

text: modulated

text: TSS

text: allows

text: one

text: to

text: engineer

text: the

text: regime

text: of

text: resonant

text: activation

text: towards

text: either

text: smaller

text: or

text: also

text: more

text: interestingly

text: much

text: wider

text: modulation

text: V.

text: CONCLUSIONS

text: regimes.

text: This

text: feature

text: becomes

text: apparent

text: by

text: supplement-

text: ing

text: the

text: information

text: contained

text: in

text: Figs.

text: and

text: with

text: those

text: With

text: this

text: work

text: we

text: studied,

text: by

text: the

text: use

text: of

text: analytical

text: of

text: Figs.

text: and

text: and

text: numerical

text: means,

text: the

text: phenomenon

text: of

text: resonant

text: acti-

text: A

text: further

text: interesting

text: feature

text: we

text: detected

text: with

text: this

text: vation,

text: occurring

text: for

text: a

text: dissipative

text: two-state

text: quantum

text: sys-

text: study

text: is

text: the

text: approximate,

text: although

text: nearly

text: exact,

text: cross-

text: tem

text: (spin-boson

text: system)

text: which

text: is

text: modulated

text: by

text: periodic

text: ing

text: behavior

text: (as

text: demonstrated

text: analytically

text: and

text: validated

text: deterministic

text: driving

text: and/or

text: via

text: telegraphic

text: two-state

text: numerically

text: in

text: Sec.

text: IV.A)

text: of

text: the

text: nonadiabatic

text: MFPT

text: noise.

text: At

text: strong

text: system-bath

text: coupling

text: the

text: quantum

text: dy-

text: entering

text: the

text: resonant

text: activation

text: regime

text: at

text: some

text: critical

text: namics

text: proceeds

text: incoherently

text: so

text: that

text: an

text: eﬀective

text: classical

text: frequency

text: ν

text: being

text: only

text: weakly

text: dependent

text: on

text: the

text: driv-

text: description

text: in

text: terms

text: of

text: a

text: master

text: equation

text: with

text: incoherent

text: ing

text: amplitude.

text: quantum

text: rates

text: becomes

text: feasible.

text: This

text: in

text: turn

text: allows

text: for

text: The

text: experimental

text: implementation

text: of

text: an

text: absorbing

text: studying

text: the

text: detailed

text: ﬁrst

text: passage

text: time

text: statistics

text: when

text: state

text: may

text: not

text: always

text: be

text: straightforward.

text: In

text: such

text: cases,

text: starting

text: out

text: at

text: one

text: of

text: the

text: two

text: metastable

text: states,

text: with

text: the

text: pdf

text: of

text: residence

text: times

text: provided

text: by

text: Eq.

text: or

text: also

text: absorption

text: occurring

text: at

text: the

text: neighboring

text: state.

text: t

text: ##LTFigure##


text: 6 Before showing the results of our analysis of the reso- In this case Eq. (19) for the noise-averaged population nant activation, it is important to note that the theory reads explicitly developed above, aside from the speciﬁc expressions of (cid:0) (cid:1) the spin-boson NIBA rates, is completely general and (cid:104) (cid:105) − (cid:104) (cid:105) − applies to generic systems where the rates of incoherent P˙L(t) η = −W0+(cid:104)PL(t)(cid:105)η − W1+y(t) tunneling are subject to periodic modulations and/or to y˙(t) = W+ P (t) W+ + ν y(t), (25) 1 L η 0 dichotomous noise. IV. RESULTS where the time-independent transition rates are given (cid:48) This section reports the ﬁndings for the resonant ac- by Eq. (20) with ζ(t, t ) = 0. Using Eqs. (20) and (21), tivation occurring in an incoherent spin-boson system the rates W+ are explicitly given by i with modulated tunneling matrix element and/or oscil- lating bias. Speciﬁcally, we consider for the tunneling matrix element ∆(t) separate modulations: Either an W0+ = ∆20a0 + ∆2aν unbiased two-state noise η(t) or a deterministic periodic W+ = ∆ ∆(a + a ), (26) 1 0 0 ν driving. Only afterwards we consider a more general case for which the tunneling matrix element is ﬂuctu- ating while a periodically oscillating ﬁeld drives the bias where (cid:90) (cid:15)(t). Throughout the remaining parts all quantities are scaled in terms of the bare tunneling frequency ∆ ; i.e., ∞ • 0 1 − − (cid:48) (cid:48)(cid:48) Fνraeqreu−einncuiensitΩsdo, fΩ∆(cid:15),0a.ndTiωmceantdisnomiseeassuwrietdchiinngurnaittes aν = 2 0 dτ e ντ Q (τ) cos[Q (τ)] (27) of ∆ 1. 0 ≡ • Temperature T is measured in units of (cid:126)∆ /k . and a0 aν=0. 0 B Due to the fact that the transition rates are time- independent, Eq. (25) can be solved analytically with the As can be deduced from Hamiltonian (1), noise and driv- (cid:104) (cid:105) boundary conditions P (0) = 1 and y(0) = 0. The so- ing amplitudes are frequencies and are thus given in units L η lution of Eq. (25) for the noise-averaged population of of ∆ . Moreover, in what follows cutoﬀ frequency and 0 the left state is temperature are held ﬁxed, assuming the values ω = 10 c and T = 0.2. Finally, with the exception of the results in Figs. 5 and 8, the dimensionless coupling strength is (cid:104) (cid:105) − − P (t) = C e γ1t + C e γ2t , (28a) always set to the value α = 0.7. L η 1 2 where A. Dichotomously ﬂuctuating tunneling matrix element in absence of a bias energy: Analytical treatment ± ∓ d ν 2W+ + ν d C = and γ = 0 , (28b) 1/2 2d 1/2 2 This situation with stationary telegraphic noise mod- ulations can be treated analytically. We ﬁrst address the setting described in Sec. in this analytically solvable case and further (cid:113) in which the bias is vanishing, i.e., A = 0, and the tun- (cid:15) neling matrix element ﬂuctuates around its static value ∆ , namely d = ν2 + 4(W+)2 . (28c) 0 1 (cid:15)(t) = 0 Note that, for vanishing noise amplitude, i.e., ∆ = 0, the rate W+ vanishes identically. In this latter case ∆(t) = ∆0 + ∆η(t) . (24) 1 Eq. (28) renders a strictly single-exponential decay with (cid:104) (cid:105) − P (t) = exp( ∆2a t). L η 0 0 The noise ξ(t) = ∆η(t) denotes the Markovian two- state noise of vanishing average, as discussed in Sec. . The FPT pdf, i.e., This setting corresponds to a TSS version of the clas- sical Brownian particle in a piecewise linear ﬂuctuating (cid:104) (cid:105) g(t) = g[t; η(t)] potential considered by Doering and Gadoua in Ref. [8], η −(cid:104) (cid:105) but here the dynamics is governed by quantum tunneling = P˙ (t) (29) L η transition rates rather than by classical (Arrhenius-like) over-barrier escape rates. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Before

text: showing

text: the

text: results

text: of

text: our

text: analysis

text: of

text: the

text: reso-

text: In

text: this

text: case

text: Eq.

text: for

text: the

text: noise-averaged

text: population

text: nant

text: activation,

text: it

text: is

text: important

text: to

text: note

text: that

text: the

text: theory

text: reads

text: explicitly

text: developed

text: above,

text: aside

text: from

text: the

text: speciﬁc

text: expressions

text: of

text: the

text: spin-boson

text: NIBA

text: rates,

text: is

text: completely

text: general

text: and

text: applies

text: to

text: generic

text: systems

text: where

text: the

text: rates

text: of

text: incoherent

text: P˙L(t)

text: η

text: tunneling

text: are

text: subject

text: to

text: periodic

text: modulations

text: and/or

text: to

text: y˙(t)

text: W+

text: P

text: (t)

text: W+

text: ν

text: y(t),

text: L

text: η

text: dichotomous

text: noise.

text: IV.

label: section

text: RESULTS

text: where

text: the

text: time-independent

text: transition

text: rates

text: are

text: given

text: This

text: section

text: reports

text: the

text: ﬁndings

text: for

text: the

text: resonant

text: ac-

text: by

text: Eq.

text: with

text: ζ(t,

text: t

text: Using

text: Eqs.

text: and

text: tivation

text: occurring

text: in

text: an

text: incoherent

text: spin-boson

text: system

text: the

text: rates

text: W+

text: are

text: explicitly

text: given

text: by

text: i

text: with

text: modulated

text: tunneling

text: matrix

text: element

text: and/or

text: oscil-

text: lating

text: bias.

text: Speciﬁcally,

text: we

text: consider

text: for

text: the

text: tunneling

text: matrix

text: element

text: ∆(t)

text: separate

text: modulations:

text: Either

text: an

text: ∆2aν

text: unbiased

text: two-state

text: noise

text: η(t)

text: or

text: a

text: deterministic

text: periodic

text: W+

text: ∆(a

text: a

text: ν

text: driving.

text: Only

text: afterwards

text: we

text: consider

text: a

text: more

text: general

text: case

text: for

text: which

text: the

text: tunneling

text: matrix

text: element

text: is

text: ﬂuctu-

text: ating

text: while

text: a

text: periodically

text: oscillating

text: ﬁeld

text: drives

text: the

text: bias

text: where

text: Throughout

text: the

text: remaining

text: parts

text: all

text: quantities

text: are

text: scaled

text: in

text: terms

text: of

text: the

text: bare

text: tunneling

text: frequency

text: i.e.,

text: Fνraeqreu−einncuiensitΩsdo,

text: aν

text: dτ

text: e

text: ντ

text: Q

text: (τ)

text: cos[Q

text: (τ)]

text: of

text: Temperature

text: T

text: is

text: measured

text: in

text: units

text: of

text: /k

text: and

text: B

text: Due

text: to

text: the

text: fact

text: that

text: the

text: transition

text: rates

text: are

text: time-

text: independent,

text: Eq.

text: can

text: be

text: solved

text: analytically

text: with

text: the

text: As

text: can

text: be

text: deduced

text: from

text: Hamiltonian

text: noise

text: and

text: driv-

text: boundary

text: conditions

text: P

text: and

text: The

text: so-

text: ing

text: amplitudes

text: are

text: frequencies

text: and

text: are

text: thus

text: given

text: in

text: units

text: L

text: η

text: lution

text: of

text: Eq.

text: for

text: the

text: noise-averaged

text: population

text: of

text: of

text: Moreover,

text: in

text: what

text: follows

text: cutoﬀ

text: frequency

text: and

text: the

text: left

text: state

text: is

text: temperature

text: are

text: held

text: ﬁxed,

text: assuming

text: the

text: values

text: ω

text: c

text: and

text: T

text: Finally,

text: with

text: the

text: exception

text: of

text: the

text: results

text: in

text: Figs.

text: and

text: the

text: dimensionless

text: coupling

text: strength

text: is

text: P

text: (t)

text: C

text: e

text: C

text: e

text: always

text: set

text: to

text: the

text: value

text: α

text: L

text: η

text: where

text: A.

text: Dichotomously

text: ﬂuctuating

text: tunneling

text: matrix

text: element

text: in

text: absence

text: of

text: a

text: bias

text: energy:

text: Analytical

text: treatment

text: d

text: ν

text: ν

text: d

text: C

text: and

text: γ

text: This

text: situation

text: with

text: stationary

text: telegraphic

text: noise

text: mod-

text: ulations

text: can

text: be

text: treated

text: analytically.

text: We

text: ﬁrst

text: address

text: the

text: setting

text: described

text: in

text: Sec.

text: in

text: this

text: analytically

text: solvable

text: case

text: and

text: further

text: in

text: which

text: the

text: bias

text: is

text: vanishing,

text: i.e.,

text: A

text: and

text: the

text: tun-

text: neling

text: matrix

text: element

text: ﬂuctuates

text: around

text: its

text: static

text: value

text: namely

text: d

text: ν2

text: Note

text: that,

text: for

text: vanishing

text: noise

text: amplitude,

text: i.e.,

text: the

text: rate

text: W+

text: vanishes

text: identically.

text: In

text: this

text: latter

text: case

text: ∆(t)

text: ∆0

text: ∆η(t)

text: Eq.

text: renders

text: a

text: strictly

text: single-exponential

text: decay

text: with

text: P

text: (t)

text: exp(

text: t).

text: L

text: η

text: The

text: noise

text: ξ(t)

text: ∆η(t)

text: denotes

text: the

text: Markovian

text: two-

text: state

text: noise

text: of

text: vanishing

text: average,

text: as

text: discussed

text: in

text: Sec.

text: The

text: FPT

text: pdf,

text: i.e.,

text: This

text: setting

text: corresponds

text: to

text: a

text: TSS

text: version

text: of

text: the

text: clas-

text: sical

text: Brownian

text: particle

text: in

text: a

text: piecewise

text: linear

text: ﬂuctuating

text: g(t)

text: g[t;

text: η(t)]

text: potential

text: considered

text: by

text: Doering

text: and

text: Gadoua

text: in

text: Ref.

text: η

text: but

text: here

text: the

text: dynamics

text: is

text: governed

text: by

text: quantum

text: tunneling

text: P˙

text: (t)

text: L

text: η

text: transition

text: rates

text: rather

text: than

text: by

text: classical

text: (Arrhenius-like)

text: over-barrier

text: escape

text: rates.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 ANDREAS ALPERS AND PETER GRITZMANN (a) (b) ? (d) (c) 3 4 2 4 2 3 1 0 3 0 3 2 2 2 4 1 Figure 1. The super-resolution imaging task DR. (a) Original (unknown) high- resolution image, (b) the corresponding low-resolution grayscale image, (c) gray levels converted into block constraints, (d) taken in combination with high-resolution row and column sum data. The task is to reconstruct from (d) the original binary image shown in (a). From a combinatorial point of view, they can be viewed as reconstructing binary matrices from given row and column sums and some additional constraints. Unexpected complexity jumps based on the results of the present paper are discussed in [6] and, to some extent, at the end of this section. Background information on such problems involving diﬀerent kinds of additional constraints can be found in [11, Sect. 4]. Other applications belong to the realm of dynamic discrete tomography [5, 6, 7, 49]. For instance, in plasma particle tracking, some particles are reconstructed at time t. Between t and the next time step t + 1 the particles may have moved to other positions. The task is to reconstruct their new positions, again from few projections. One way of incorporating additional prior knowledge about the movement leads to block constraints of the kind discussed in this paper; see [5]. As the task of reconstructing a binary matrix from its row and column sums can be formulated as the task of ﬁnding a b-matching in a bipartite graph, we remark that our theorems can also be seen as results on ﬁnding b-matchings that are subject to speciﬁc additional constraints. Related (but intrinsically quite diﬀerent) results on matchings subject to so-called budget constraints can be found, e.g., in [21]. For super-resolution imaging the two main contributions of this paper are as follows. On the one hand, we show that reliable bimodal tomographic data can be utilized eﬃciently, i.e., the resolution, in each dimension, can be doubled in polynomial time if the gray levels of the low-resolution image have been determined precisely (Theorem 1). On the other hand, the task becomes already intractable (unless P = NP) if the gray levels of some pixels come with some small error of ±1 (Theorem 3 and Corollary 2). This proves that noise does not only aﬀect the quality of the reconstructed image but also the algorithmic tractability of the inverse problem itself. Hence the possibility of compensating noisy imaging by including bimodal information may in practice be jeopardized by its algorithmic complexity. While this is not the focus of the present paper let us point out that, even in the presence of (a reasonably restricted degree of) noise, the approach leading to Theorem 1 can, in principle, be extended to a fast heuristic for doubling the resolution of such images in each dimension. At present, however, we do not know how such an approach would compare with other heuristics on real-world data. From the perspective of discrete tomography, the contributions of this paper can be interpreted as follows. It is well known that the problem of reconstructing binary images from X-ray data taken from two directions can be solved in polynomial time [16, 41]. Typically, this information does not determine the image uniquely (see, e.g., [23] and the papers quoted there). Hence, one would like to take and utilize additional measurements. If, however, we add additional constraints that enforce that the solutions satisfy the X-ray data taken from a third direction, then the problem becomes NP-hard, ##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##

text: ANDREAS

text: ALPERS

text: AND

text: PETER

text: GRITZMANN

text: (a)

text: (b)

text: (d)

text: (c)

text: Figure

text: The

label: caption

text: super-resolution

text: imaging

text: task

text: DR.

text: (a)

text: Original

text: (unknown)

text: high-

text: resolution

text: image,

text: (b)

text: the

text: corresponding

text: low-resolution

text: grayscale

text: image,

text: (c)

text: gray

text: levels

text: converted

text: into

text: block

text: constraints,

text: (d)

text: taken

text: in

text: combination

text: with

text: high-resolution

text: row

text: and

text: column

text: sum

text: data.

text: The

text: task

text: is

text: to

text: reconstruct

text: from

text: (d)

text: the

text: original

text: binary

text: image

text: shown

text: in

text: (a).

text: From

text: a

text: combinatorial

text: point

text: of

text: view,

text: they

text: can

text: be

text: viewed

text: as

text: reconstructing

text: binary

text: matrices

text: from

text: given

text: row

text: and

text: column

text: sums

text: and

text: some

text: additional

text: constraints.

text: Unexpected

text: complexity

text: jumps

text: based

text: on

text: the

text: results

text: of

text: the

text: present

text: paper

text: are

text: discussed

text: in

text: and,

text: to

text: some

text: extent,

text: at

text: the

text: end

text: of

text: this

text: section.

text: Background

text: information

text: on

text: such

text: problems

text: involving

text: diﬀerent

text: kinds

text: of

text: additional

text: constraints

text: can

text: be

text: found

text: in

text: Sect.

text: Other

text: applications

text: belong

text: to

text: the

text: realm

text: of

text: dynamic

text: discrete

text: tomography

text: For

text: instance,

text: in

text: plasma

text: particle

text: tracking,

text: some

text: particles

text: are

text: reconstructed

text: at

text: time

text: t.

text: Between

text: t

text: and

text: the

text: next

text: time

text: step

text: t

text: the

text: particles

text: may

text: have

text: moved

text: to

text: other

text: positions.

text: The

text: task

text: is

text: to

text: reconstruct

text: their

text: new

text: positions,

text: again

text: from

text: few

text: projections.

text: One

text: way

text: of

text: incorporating

text: additional

text: prior

text: knowledge

text: about

text: the

text: movement

text: leads

text: to

text: block

text: constraints

text: of

text: the

text: kind

text: discussed

text: in

text: this

text: paper;

text: see

text: As

text: the

text: task

text: of

text: reconstructing

text: a

text: binary

text: matrix

text: from

text: its

text: row

text: and

text: column

text: sums

text: can

text: be

text: formulated

text: as

text: the

text: task

text: of

text: ﬁnding

text: a

text: b-matching

text: in

text: a

text: bipartite

text: graph,

text: we

text: remark

text: that

text: our

text: theorems

text: can

text: also

text: be

text: seen

text: as

text: results

text: on

text: ﬁnding

text: b-matchings

text: that

text: are

text: subject

text: to

text: speciﬁc

text: additional

text: constraints.

text: Related

text: (but

text: intrinsically

text: quite

text: diﬀerent)

text: results

text: on

text: matchings

text: subject

text: to

text: so-called

text: budget

text: constraints

text: can

text: be

text: found,

text: e.g.,

text: in

text: For

text: super-resolution

text: imaging

text: the

text: two

text: main

text: contributions

text: of

text: this

text: paper

text: are

text: as

text: follows.

text: On

text: the

text: one

text: hand,

text: we

text: show

text: that

text: reliable

text: bimodal

text: tomographic

text: data

text: can

text: be

text: utilized

text: eﬃciently,

text: i.e.,

text: the

text: resolution,

text: in

text: each

text: dimension,

text: can

text: be

text: doubled

text: in

text: polynomial

text: time

text: if

text: the

text: gray

text: levels

text: of

text: the

text: low-resolution

text: image

text: have

text: been

text: determined

text: precisely

text: (Theorem

text: On

text: the

text: other

text: hand,

text: the

text: task

text: becomes

text: already

text: intractable

text: (unless

text: P

text: NP)

text: if

text: the

text: gray

text: levels

text: of

text: some

text: pixels

text: come

text: with

text: some

text: small

text: error

text: of

text: ±1

text: (Theorem

text: and

text: Corollary

text: This

text: proves

text: that

text: noise

text: does

text: not

text: only

text: aﬀect

text: the

text: quality

text: of

text: the

text: reconstructed

text: image

text: but

text: also

text: the

text: algorithmic

text: tractability

text: of

text: the

text: inverse

text: problem

text: itself.

text: Hence

text: the

text: possibility

text: of

text: compensating

text: noisy

text: imaging

text: by

text: including

text: bimodal

text: information

text: may

text: in

text: practice

text: be

text: jeopardized

text: by

text: its

text: algorithmic

text: complexity.

text: While

text: this

text: is

text: not

text: the

text: focus

text: of

text: the

text: present

text: paper

text: let

text: us

text: point

text: out

text: that,

text: even

text: in

text: the

text: presence

text: of

text: (a

text: reasonably

text: restricted

text: degree

text: of)

text: noise,

text: the

text: approach

text: leading

text: to

text: Theorem

text: can,

text: in

text: principle,

text: be

text: extended

text: to

text: a

text: fast

text: heuristic

text: for

text: doubling

text: the

text: resolution

text: of

text: such

text: images

text: in

text: each

text: dimension.

text: At

text: present,

text: however,

text: we

text: do

text: not

text: know

text: how

text: such

text: an

text: approach

text: would

text: compare

text: with

text: other

text: heuristics

text: on

text: real-world

text: data.

text: From

text: the

text: perspective

text: of

text: discrete

text: tomography,

text: the

text: contributions

text: of

text: this

text: paper

text: can

text: be

text: interpreted

text: as

text: follows.

text: It

text: is

text: well

text: known

text: that

text: the

text: problem

text: of

text: reconstructing

text: binary

text: images

text: from

text: X-ray

text: data

text: taken

text: from

text: two

text: directions

text: can

text: be

text: solved

text: in

text: polynomial

text: time

text: Typically,

text: this

text: information

text: does

text: not

text: determine

text: the

text: image

text: uniquely

text: (see,

text: e.g.,

text: and

text: the

text: papers

text: quoted

text: there).

text: Hence,

text: one

text: would

text: like

text: to

text: take

text: and

text: utilize

text: additional

text: measurements.

text: If,

text: however,

text: we

text: add

text: additional

text: constraints

text: that

text: enforce

text: that

text: the

text: solutions

text: satisfy

text: the

text: X-ray

text: data

text: taken

text: from

text: a

text: third

text: direction,

text: then

text: the

text: problem

text: becomes

text: NP-hard,

text: ##LTFigure##

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text: 12 ANDREAS ALPERS AND PETER GRITZMANN Class Turns Into Illustration (cid:104)1(cid:105) (B3,2, Ai,1) → (B3,3, Ai,2) (cid:104)2(cid:105) (B3,1, Ai,2) → (B3,3, Ai,1) (cid:104)3(cid:105) (B3,2, B3,1) → (B3,3, B3,3) (cid:104)4(cid:105) (B3,2, Ci,2) → (B3,3, Ci,1) (cid:104)5(cid:105) (B3,1, Ci,1) → (B3,3, Ci,2) (cid:104)6(cid:105) (Ci(cid:48),1, Ai,1) → (Ci(cid:48),2, Ai,2) (cid:104)7(cid:105) (B3,4) → (B3,3) (cid:48) Table 2. Vertical local switches. The parameters i and i are elements of [2]. that, in particular, the number of type B and type B blocks in each horizontal and vertical strip, 1 3,1 respectively, equals some prescribed cardinality. Type B Type B Type B 1 3,1 3,3 Figure 8. The three potential block types in solutions of DR(2) instances considered in the proof of Lemma 4. Value 1 and 0 are indicated in black and white, respectively. Now, we introduce 0/1-variables ζ , η , λ , (i, j) ∈ I, that are one if, and only if, B(i, j) is of i,j i,j i,j type B , B , B , respectively. Then we ask for 0/1-solutions that satisfy 1 3,1 3,3 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ANDREAS

text: ALPERS

text: AND

text: PETER

text: GRITZMANN

text: Class

label: table

text: Turns

text: Into

text: Illustration

text: Table

text: Vertical

text: local

text: switches.

text: The

text: parameters

text: i

text: and

text: i

text: are

text: elements

text: of

text: that,

text: in

text: particular,

text: the

text: number

text: of

text: type

text: B

text: and

text: type

text: B

text: blocks

text: in

text: each

text: horizontal

text: and

text: vertical

text: strip,

text: respectively,

text: equals

text: some

text: prescribed

text: cardinality.

text: Type

text: B

text: Type

text: B

text: Type

text: B

text: Figure

text: The

text: three

text: potential

text: block

text: types

text: in

text: solutions

text: of

text: instances

text: considered

text: in

text: the

text: proof

text: of

text: Lemma

text: Value

text: and

text: are

text: indicated

text: in

text: black

text: and

text: white,

text: respectively.

text: Now,

text: we

text: introduce

text: ζ

text: η

text: λ

text: (i,

text: j)

text: I,

text: that

text: are

text: one

text: if,

text: and

text: only

text: if,

text: B(i,

text: j)

text: is

text: of

text: i,j

text: i,j

text: i,j

text: type

text: B

text: B

text: B

text: respectively.

text: Then

text: we

text: ask

text: for

text: that

text: satisfy

text: ##LTFigure##

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text: ON DOUBLE-RESOLUTION IMAGING AND DISCRETE TOMOGRAPHY 13 (cid:80) (cid:80) ζ = (r − r )/2, j ∈ Π (I), ∈ p,j j j+1 y p:(p,j) I (cid:80) (η + λ ) = ρ (m) − (r − r )/2, j ∈ Π (I), ∈ p,j p,j j j j+1 y p:(p,j) I (cid:80) η = (c − c )/2, i ∈ Π (I), ∈ i,q i i+1 x q:(i,q) I (ζ + λ ) = σ (n) − (c − c )/2, i ∈ Π (I), ∈ i,q i,q i i i+1 x q:(i,q) I ζ + η + λ = 1, (i, j) ∈ I. i,j i,j i,j Eliminating λ = 1−ζ −η , (i, j) ∈ I, we see that the 0/1-solutions of this system are in one-to-one i,j i,j i,j correspondence with the 0/1 so(cid:80)lution of (cid:80) ζ = (r − r )/2, j ∈ Π (I), ∈ p,j j j+1 y p:(p,j) I η = (c − c )/2, i ∈ Π (I), (3.4) ∈ i,q i i+1 x q:(i,q) I ζ + η ≤ 1, (i, j) ∈ I. i,j i,j As we assume that our DR(2) instance is feasible (hence containing two ones in each block), we have r + r = 2ρ (m) and c + c = 2σ (n) for every j ∈ Π (I) and i ∈ Π (I). Hence, if a solution j j+1 j i i+∗1 ∗ i ∗ ∗ y x to (3.4) is expande(cid:80)d, replacing the ζ , η , 1−ζ −η , (i, j) ∈ I, by the respective block types, then, i,j i,j i,j i,j indeed (cid:80) ∗ ξ = 2 · (r − r )/2 + ρ (m) − (r − r )/2 = r , ∈ p,j j j+1 j j j+1 j p:(p,j) G(I) (cid:80) ∗ ξ = ρ (m) − (r − r )/2 = r , ∈ p,j+1 j j j+1 j+1 p:(p,j) G(I) (cid:80) ∗ ξ = 2 · (c − c )/2 + σ (n) − (c − c )/2 = c , ∈ i,q i i+1 i i i+1 i q:(i,q) G(I) ∗ ξ = σ (n) − (c − c )/2 = c , ∈ i+1,q i i i+1 i+1 q:(i,q) G(I) for every j ∈ Π (I) and i ∈ Π (I). Any 0/1-solution to (3.4) provides thus a solution to the given y x DR(2) instance. Since 0/1-solutions to (3.4) can be found in polynomial time (see Lemma 2), we have hence determined a solution to our instance in polynomial time. Note that this solution is reduced as it does not contain blocks of type B , B , and B . 2 3,2 3,4 It remains to consider the case that the given DR(2) instance is infeasible. Then one of the previous steps must fail: either (3.4) admits no 0/1-solution or the expanded “solution” does not satisfy all row (cid:3) and column sums. Both can be detected in polynomial time (see also Lemma 2). Let us now turn to the task of solving DR. Suppose, we are given a feasible instance of DR that does not contain any block constraints of the form v(i, j) = ν, ν ∈ {0, 4}, and whose row and column sums satisfy r ≥ r and c ≥ c , (i, j) ∈ C(m, n, 2). (3.5) j j+1 i i+1 We call such an instance a proper instance. Considering a horizontal strip [m] × {j, j + 1} with j ∈ [n] ∩ (2N + 1), we set 0 a := |{i ∈ [m] ∩ (2N + 1) : v(i, j) = 1}|, j 0 b := |{i ∈ [m] ∩ (2N + 1) : v(i, j) = 2}|, j 0 c := |{i ∈ [m] ∩ (2N + 1) : v(i, j) = 3}|. j 0

text: ON

text: DOUBLE-RESOLUTION

text: IMAGING

text: AND

text: DISCRETE

text: TOMOGRAPHY

text: ζ

text: (r

text: r

text: j

text: Π

text: (I),

text: p,j

text: j

text: y

text: p:(p,j)

text: I

text: (η

text: λ

text: ρ

text: (m)

text: (r

text: r

text: j

text: Π

text: (I),

text: p,j

text: p,j

text: j

text: j

text: y

text: p:(p,j)

text: I

text: η

text: (c

text: c

text: i

text: Π

text: (I),

text: i,q

text: i

text: x

text: q:(i,q)

text: I

text: (ζ

text: λ

text: σ

text: (n)

text: (c

text: c

text: i

text: Π

text: (I),

text: i,q

text: i,q

text: i

text: i

text: x

text: q:(i,q)

text: I

text: ζ

text: η

text: λ

text: (i,

text: j)

text: I.

text: i,j

text: i,j

text: i,j

text: Eliminating

text: λ

text: 1−ζ

text: −η

text: (i,

text: j)

text: I,

text: we

text: see

text: that

text: the

text: of

text: this

text: system

text: are

text: in

text: one-to-one

text: i,j

text: i,j

text: i,j

text: correspondence

text: with

text: the

text: of

text: ζ

text: (r

text: r

text: j

text: Π

text: (I),

text: p,j

text: j

text: y

text: p:(p,j)

text: I

text: η

text: (c

text: c

text: i

text: Π

text: (I),

text: i,q

text: i

text: x

text: q:(i,q)

text: I

text: ζ

text: η

text: (i,

text: j)

text: I.

text: i,j

text: i,j

text: As

text: we

text: assume

text: that

text: our

text: instance

text: is

text: feasible

text: (hence

text: containing

text: two

text: ones

text: in

text: each

text: block),

text: we

text: have

text: r

text: r

text: 2ρ

text: (m)

text: and

text: c

text: c

text: 2σ

text: (n)

text: for

text: every

text: j

text: Π

text: (I)

text: and

text: i

text: Π

text: (I).

text: Hence,

text: if

text: a

text: solution

text: j

text: j

text: i

text: i

text: y

text: x

text: to

text: is

text: replacing

text: the

text: ζ

text: η

text: 1−ζ

text: −η

text: (i,

text: j)

text: I,

text: by

text: the

text: respective

text: block

text: types,

text: then,

text: i,j

text: i,j

text: i,j

text: i,j

text: indeed

text: ξ

text: (r

text: r

text: ρ

text: (m)

text: (r

text: r

text: r

text: p,j

text: j

text: j

text: j

text: j

text: p:(p,j)

text: G(I)

text: ξ

text: ρ

text: (m)

text: (r

text: r

text: r

text: j

text: j

text: p:(p,j)

text: G(I)

text: ξ

text: (c

text: c

text: σ

text: (n)

text: (c

text: c

text: c

text: i,q

text: i

text: i

text: i

text: i

text: q:(i,q)

text: G(I)

text: ξ

text: σ

text: (n)

text: (c

text: c

text: c

text: i

text: i

text: q:(i,q)

text: G(I)

text: for

text: every

text: j

text: Π

text: (I)

text: and

text: i

text: Π

text: (I).

text: Any

text: to

text: provides

text: thus

text: a

text: solution

text: to

text: the

text: given

text: y

text: x

text: instance.

text: Since

text: to

text: can

text: be

text: found

text: in

text: polynomial

text: time

text: (see

text: Lemma

text: we

text: have

text: hence

text: determined

text: a

text: solution

text: to

text: our

text: instance

text: in

text: polynomial

text: time.

text: Note

text: that

text: this

text: solution

text: is

text: reduced

text: as

text: it

text: does

text: not

text: contain

text: blocks

text: of

text: type

text: B

text: B

text: and

text: B

text: It

text: remains

text: to

text: consider

text: the

text: case

text: that

text: the

text: given

text: instance

text: is

text: infeasible.

text: Then

text: one

text: of

text: the

text: previous

text: steps

text: must

text: fail:

text: either

text: admits

text: no

text: or

text: the

text: expanded

text: “solution”

text: does

text: not

text: satisfy

text: all

text: row

text: and

text: column

text: sums.

text: Both

text: can

text: be

text: detected

text: in

text: polynomial

text: time

text: (see

text: also

text: Lemma

text: Let

text: us

text: now

text: turn

text: to

text: the

text: task

text: of

text: solving

text: DR.

text: Suppose,

text: we

text: are

text: given

text: a

text: feasible

text: instance

text: of

text: DR

text: that

text: does

text: not

text: contain

text: any

text: block

text: constraints

text: of

text: the

text: form

text: v(i,

text: j)

text: ν,

text: ν

text: and

text: whose

text: row

text: and

text: column

text: sums

text: satisfy

text: r

text: r

text: and

text: c

text: c

text: (i,

text: j)

text: C(m,

text: n,

text: j

text: i

text: We

text: call

text: such

text: an

text: instance

text: a

text: proper

text: instance.

text: Considering

text: a

text: horizontal

text: strip

text: [m]

text: {j,

text: j

text: with

text: j

text: [n]

text: we

text: set

text: a

text: |{i

text: [m]

text: v(i,

text: j)

text: j

text: b

text: |{i

text: [m]

text: v(i,

text: j)

text: j

text: c

text: |{i

text: [m]

text: v(i,

text: j)

text: j


text: ON DOUBLE-RESOLUTION IMAGING AND DISCRETE TOMOGRAPHY 5 (a) (b) (c) Figure 4. Alg. 1 applied to a phantom. (a) Low-resolution 100×100 grayscale image, (b) magniﬁcation of the part contained in the red rectangle, (c) magniﬁcation of the corresponding part of the 200 × 200 high-resolution binary image obtained by the algorithm. (a) (b) (c) Figure 5. Alg. 1 applied to a phantom. (a) Low-resolution 50 × 50 grayscale im- age, (b) high-resolution 100 × 100 binary image obtained by the algorithm, (c) high- resolution 100 × 100 binary image minimizing the total variation. Theorem 2. For every instance of DR it can be decided in polynomial time whether the instance admits a unique solution. While the additional X-ray information (at double-resolution) does in general reduce the ambiguity, typically it will still not lead to uniqueness. Of course, a standard way of dealing with non-uniqueness in practice is regularization [8, 9, 24, 38]. To illustrate the possibility of adapting a regularization scheme to our context, we show in Figure 5(c) a binary solution minimizing the total variation (TV). In fact, the solution is obtained from Figu(cid:88)re 5((cid:88)a) by applying 15 so-called local switches, each of which is strictly decreasing the value of the total variation functional J(x) := ||(∇x(1), ∇x(2))T || , p,q p,q 2 ∈ ∈ p [m] q [n] (cid:26) where (cid:26) ξ − ξ : p < m, (∇x)(1) := p+1,q p,q p,q 0 : otherwise, ξ − ξ : q < n, (∇x)(2) := p,q+1 p,q p,q 0 : otherwise, ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: ON

text: DOUBLE-RESOLUTION

text: IMAGING

text: AND

text: DISCRETE

text: TOMOGRAPHY

text: (a)

text: (b)

text: (c)

text: Figure

text: Alg.

text: applied

text: to

text: a

text: phantom.

text: (a)

text: Low-resolution

text: grayscale

text: image,

text: (b)

text: magniﬁcation

text: of

text: the

text: part

text: contained

text: in

text: the

text: red

text: rectangle,

text: (c)

text: magniﬁcation

text: of

text: the

text: corresponding

text: part

text: of

text: the

text: high-resolution

text: binary

text: image

text: obtained

text: by

text: the

text: algorithm.

text: (a)

text: (b)

text: (c)

text: Figure

text: Alg.

text: applied

text: to

text: a

text: phantom.

text: (a)

text: Low-resolution

text: grayscale

text: im-

text: age,

text: (b)

text: high-resolution

text: binary

text: image

text: obtained

text: by

text: the

text: algorithm,

text: (c)

text: high-

text: resolution

text: binary

text: image

text: minimizing

text: the

text: total

text: variation.

text: Theorem

text: For

text: every

text: instance

text: of

text: DR

text: it

text: can

text: be

text: decided

text: in

text: polynomial

text: time

text: whether

text: the

text: instance

text: admits

text: a

text: unique

text: solution.

text: While

text: the

text: additional

text: X-ray

text: information

text: (at

text: double-resolution)

text: does

text: in

text: general

text: reduce

text: the

text: ambiguity,

text: typically

text: it

text: will

text: still

text: not

text: lead

text: to

text: uniqueness.

text: Of

text: course,

text: a

text: standard

text: way

text: of

text: dealing

text: with

text: non-uniqueness

text: in

text: practice

text: is

text: regularization

text: To

text: illustrate

text: the

text: possibility

text: of

text: adapting

text: a

text: regularization

text: scheme

text: to

text: our

text: context,

text: we

text: show

text: in

text: Figure

text: a

text: binary

text: solution

text: minimizing

text: the

text: total

text: variation

text: (TV).

text: In

text: fact,

text: the

text: solution

text: is

text: obtained

text: from

text: by

text: applying

text: so-called

text: local

text: switches,

text: each

text: of

text: which

text: is

text: strictly

text: decreasing

text: the

text: value

text: of

text: the

text: total

text: variation

text: functional

text: J(x)

text: p,q

text: p,q

text: p

text: [m]

text: q

text: [n]

text: where

text: ξ

text: ξ

text: p

text: m,

text: p,q

text: p,q

text: otherwise,

text: ξ

text: ξ

text: q

text: n,

text: p,q

text: p,q

text: otherwise,

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: Quark-Nuclear Hybrid Star EoS with Excluded Volume Effects Mark Alexander Randolph Kaltenborn,1,2 Niels-Uwe Friedrich Bastian,3 and David Bernhard Blaschke3,4,5 1Department of Physics, The George Washington University, Washington DC 20052, USA 2Los Alamos National Laboratory, Los Alamos, NM 87545, USA 3Institute of Theoretical Physics, University of Wrocław, 50-204 Wrocław, Poland 4Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, 141980 Dubna, Russia 5National Research Nuclear University (MEPhI), 115409 Moscow, Russia (Dated: May 29, 2020) A two-phase description of the quark-nuclear matter hybrid equation of state that takes into account the effect of excluded volume in both the hadronic and the quark-matter phases is introduced. The nuclear phase manifests a reduction of the available volume as density increases, leading to a stiffening of the matter. The quark-matter phase displays a reduction of the effective string-tension in the conﬁning density-functional from available volume contributions. The nuclear equation of state is based upon the relativistic density functional model DD2 with excluded volume. The quark-matter EoS is based upon a quasiparticle model derived from a relativistic density-functional approach and will be discussed in greater detail. The interactions are decomposed into mean scalar and vector components. The scalar interaction is motivated by a string potential between quarks, whereas the vector interaction potential is motivated by higher-order interactions of quarks leading to an increased stiffening at high densities. As an application, we consider matter under compact star constraints of electric neutrality and β-equilibrium. We obtain mass-radius relations for hybrid stars that form a third family, disconnected from the purely hadronic star branch, and fulﬁll the 2M(cid:12) constraint. PACS numbers: 12.38.Lg, 25.75.Ag, 26.60.Kp, 97.60.Jd I. INTRODUCTION string length between quarks in uniform matter is related to − the scalar number density n by being proportional to n 1/3, s s Theoretical investigations into compact star matter have one obtains a corresponding contribution to the energy density seen a boom over the past decade, spurred forward by the functional of quark matter, see also [6]. Adopting the con- plethora of observational data collected for the masses and ﬁning interaction as a scalar potential, this density functional radii of pulsars. The mass-radius (M-R) data has been par- describes the medium modiﬁcation of the quark mass, being ticularly useful for constraining the dense matter equation of a Dirac scalar. At low densities the quark mass diverges thus state (EoS) at zero temperature. For a recent review see, e.g., mimicking conﬁnement by removing quarks as dynamical de- Ref. [1]. These constraints provide an experimental sandbox grees of freedom. At low density the quarks are clustered into for probing the properties of matter that is inaccessible in ter- hadrons, and this quark cluster phase is energetically prefer- restrial experiments. The question investigated in this work is able over the uniform quark matter with diverging scalar con- whether or not quark matter is manifest in the interior of com- ﬁning interaction. Increasing the density, quark exchange ef- pact stars by a phase transition from hadronic matter to quark fects between hadrons (Pauli blocking) lead to strong repul- matter and, if quark matter is indeed manifest, what would the sion (that can be described by an excluded hadron volume implications be on the observables, as well as the QCD phase model) and to delocalization of the quark wave function, driv- diagram. ing the system into the homogeneous quark-matter phase. In In order to answer this question, we employ an effective, the phase transition region, the reduction of the available vol- relativistic density-functional approach to both the hadronic ume (in which the nonperturbative dual superconductor vac- and the quark-matter equation of state (EoS) at zero temper- uum of QCD persists which expells the color ﬁeld lines by ature. A method for implementing quark conﬁnement in a the dual Meissner effect) leads to a reduction of the density chiral quark-matter model is based on reviving the idea of of color ﬁeld lines and thus to a “screening” of the string ten- the string-ﬂip model (SFM) [2, 3]. According to the SFM, sion by the available volume fraction factor. At still higher the conﬁning interactions between colored constituents (mod- densities higher order quark interactions adopted according to eled, e.g., through a Cornell-type conﬁning potential) get satu- [7, 8] lead to a stiffening of quark matter that allows for a rated within nearest neighbors. In hadronic matter these quark stable branch of hybrid stars with quark-matter core. neighbors are the ones bound in a hadron, while at high den- In order to obtain the EoS for hybrid star matter we have sities in a multi-quark system one has to draw string connec- developed a relativistic, self-consistent mean-ﬁeld approach tions between quarks so as to neutralize the color while keep- within the path integral approach, based on an ansatz for the ing the total length of all strings at a minimum. This task can potential energy density functional from which density depen- be solved by simulation [4] or by effective quantum statistical dent self-energies follow as derivatives. The resulting quasi- methods [3, 5]. One can obtain the string length distribution particle model necessarily maintains thermodynamic consis- function for a given density and fold a given interaction poten- tency, since all approximation steps are done on the level of tial model with this distribution in order to obtain the correla- the partition function from which the expressions for thermo- tion energy in a meanﬁeld approximation. Since the average dynamical functions follow by derivation.

text: Quark-Nuclear

label: title

text: Hybrid

text: Star

text: EoS

text: with

text: Excluded

text: Volume

text: Effects

text: Mark

label: author

text: Alexander

text: Randolph

text: Niels-Uwe

text: Friedrich

text: and

text: David

text: Bernhard

text: of

text: Physics,

text: The

text: George

text: Washington

text: University,

text: Washington

text: DC

text: USA

text: Alamos

text: National

text: Laboratory,

text: Los

text: Alamos,

text: NM

text: USA

text: of

text: Theoretical

text: Physics,

text: University

text: of

text: Wrocław,

text: Wrocław,

text: Poland

text: Laboratory

text: for

text: Theoretical

text: Physics,

text: JINR

text: Dubna,

text: Dubna,

text: Russia

text: Research

text: Nuclear

text: University

text: (MEPhI),

text: Moscow,

text: Russia

text: (Dated:

text: May

label: date

text: A

text: two-phase

text: description

text: of

text: the

text: quark-nuclear

text: matter

text: hybrid

text: equation

text: of

text: state

text: that

text: takes

text: into

text: account

text: the

text: effect

text: of

text: excluded

text: volume

text: in

text: both

text: the

text: hadronic

text: and

text: the

text: quark-matter

text: phases

text: is

text: introduced.

text: The

text: nuclear

text: phase

text: manifests

text: a

text: reduction

text: of

text: the

text: available

text: volume

text: as

text: density

text: increases,

text: leading

text: to

text: a

text: stiffening

text: of

text: the

text: matter.

text: The

text: quark-matter

text: phase

text: displays

text: a

text: reduction

text: of

text: the

text: effective

text: string-tension

text: in

text: the

text: conﬁning

text: density-functional

text: from

text: available

text: volume

text: contributions.

text: The

text: nuclear

text: equation

text: of

text: state

text: is

text: based

text: upon

text: the

text: relativistic

text: density

text: functional

text: model

text: with

text: excluded

text: volume.

text: The

text: quark-matter

text: EoS

text: is

text: based

text: upon

text: a

text: quasiparticle

text: model

text: derived

text: from

text: a

text: relativistic

text: density-functional

text: approach

text: and

text: will

text: be

text: discussed

text: in

text: greater

text: detail.

text: The

text: interactions

text: are

text: decomposed

text: into

text: mean

text: scalar

text: and

text: vector

text: components.

text: The

text: scalar

text: interaction

text: is

text: motivated

text: by

text: a

text: string

text: potential

text: between

text: quarks,

text: whereas

text: the

text: vector

text: interaction

text: potential

text: is

text: motivated

text: by

text: higher-order

text: interactions

text: of

text: quarks

text: leading

text: to

text: an

text: increased

text: stiffening

text: at

text: high

text: densities.

text: As

text: an

text: application,

text: we

text: consider

text: matter

text: under

text: compact

text: star

text: constraints

text: of

text: electric

text: neutrality

text: and

text: β-equilibrium.

text: We

text: obtain

text: mass-radius

text: relations

text: for

text: hybrid

text: stars

text: that

text: form

text: a

text: third

text: family,

text: disconnected

text: from

text: the

text: purely

text: hadronic

text: star

text: branch,

text: and

text: fulﬁll

text: the

text: constraint.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: string

text: length

text: between

text: quarks

text: in

text: uniform

text: matter

text: is

text: related

text: to

text: the

text: scalar

text: number

text: density

text: n

text: by

text: being

text: proportional

text: to

text: n

text: s

text: s

text: Theoretical

text: investigations

text: into

text: compact

text: star

text: matter

text: have

text: one

text: obtains

text: a

text: corresponding

text: contribution

text: to

text: the

text: energy

text: density

text: seen

text: a

text: boom

text: over

text: the

text: past

text: decade,

text: spurred

text: forward

text: by

text: the

text: functional

text: of

text: quark

text: matter,

text: see

text: also

text: Adopting

text: the

text: con-

text: plethora

text: of

text: observational

text: data

text: collected

text: for

text: the

text: masses

text: and

text: ﬁning

text: interaction

text: as

text: a

text: scalar

text: potential,

text: this

text: density

text: functional

text: radii

text: of

text: pulsars.

text: The

text: mass-radius

text: (M-R)

text: data

text: has

text: been

text: par-

text: describes

text: the

text: medium

text: modiﬁcation

text: of

text: the

text: quark

text: mass,

text: being

text: ticularly

text: useful

text: for

text: constraining

text: the

text: dense

text: matter

text: equation

text: of

text: a

text: Dirac

text: scalar.

text: At

text: low

text: densities

text: the

text: quark

text: mass

text: diverges

text: thus

text: state

text: (EoS)

text: at

text: zero

text: temperature.

text: For

text: a

text: recent

text: review

text: see,

text: e.g.,

text: mimicking

text: conﬁnement

text: by

text: removing

text: quarks

text: as

text: dynamical

text: de-

text: Ref.

text: These

text: constraints

text: provide

text: an

text: experimental

text: sandbox

text: grees

text: of

text: freedom.

text: At

text: low

text: density

text: the

text: quarks

text: are

text: clustered

text: into

text: for

text: probing

text: the

text: properties

text: of

text: matter

text: that

text: is

text: inaccessible

text: in

text: ter-

text: hadrons,

text: and

text: this

text: quark

text: cluster

text: phase

text: is

text: energetically

text: prefer-

text: restrial

text: experiments.

text: The

text: question

text: investigated

text: in

text: this

text: work

text: is

text: able

text: over

text: the

text: uniform

text: quark

text: matter

text: with

text: diverging

text: scalar

text: con-

text: whether

text: or

text: not

text: quark

text: matter

text: is

text: manifest

text: in

text: the

text: interior

text: of

text: com-

text: ﬁning

text: interaction.

text: Increasing

text: the

text: density,

text: quark

text: exchange

text: ef-

text: pact

text: stars

text: by

text: a

text: phase

text: transition

text: from

text: hadronic

text: matter

text: to

text: quark

text: fects

text: between

text: hadrons

text: (Pauli

text: blocking)

text: lead

text: to

text: strong

text: repul-

text: matter

text: and,

text: if

text: quark

text: matter

text: is

text: indeed

text: manifest,

text: what

text: would

text: the

text: sion

text: (that

text: can

text: be

text: described

text: by

text: an

text: excluded

text: hadron

text: volume

text: implications

text: be

text: on

text: the

text: observables,

text: as

text: well

text: as

text: the

text: QCD

text: phase

text: model)

text: and

text: to

text: delocalization

text: of

text: the

text: quark

text: wave

text: function,

text: driv-

text: diagram.

text: ing

text: the

text: system

text: into

text: the

text: homogeneous

text: quark-matter

text: phase.

text: In

text: In

text: order

text: to

text: answer

text: this

text: question,

text: we

text: employ

text: an

text: effective,

text: the

text: phase

text: transition

text: region,

text: the

text: reduction

text: of

text: the

text: available

text: vol-

text: relativistic

text: density-functional

text: approach

text: to

text: both

text: the

text: hadronic

text: ume

text: (in

text: which

text: the

text: nonperturbative

text: dual

text: superconductor

text: vac-

text: and

text: the

text: quark-matter

text: equation

text: of

text: state

text: (EoS)

text: at

text: zero

text: temper-

text: uum

text: of

text: QCD

text: persists

text: which

text: expells

text: the

text: color

text: ﬁeld

text: lines

text: by

text: ature.

text: A

text: method

text: for

text: implementing

text: quark

text: conﬁnement

text: in

text: a

text: the

text: dual

text: Meissner

text: effect)

text: leads

text: to

text: a

text: reduction

text: of

text: the

text: density

text: chiral

text: quark-matter

text: model

text: is

text: based

text: on

text: reviving

text: the

text: idea

text: of

text: of

text: color

text: ﬁeld

text: lines

text: and

text: thus

text: to

text: a

text: “screening”

text: of

text: the

text: string

text: ten-

text: the

text: string-ﬂip

text: model

text: (SFM)

text: According

text: to

text: the

text: SFM,

text: sion

text: by

text: the

text: available

text: volume

text: fraction

text: factor.

text: At

text: still

text: higher

text: the

text: conﬁning

text: interactions

text: between

text: colored

text: constituents

text: (mod-

text: densities

text: higher

text: order

text: quark

text: interactions

text: adopted

text: according

text: to

text: eled,

text: e.g.,

text: through

text: a

text: Cornell-type

text: conﬁning

text: potential)

text: get

text: satu-

text: lead

text: to

text: a

text: stiffening

text: of

text: quark

text: matter

text: that

text: allows

text: for

text: a

text: rated

text: within

text: nearest

text: neighbors.

text: In

text: hadronic

text: matter

text: these

text: quark

text: stable

text: branch

text: of

text: hybrid

text: stars

text: with

text: quark-matter

text: core.

text: neighbors

text: are

text: the

text: ones

text: bound

text: in

text: a

text: hadron,

text: while

text: at

text: high

text: den-

text: In

text: order

text: to

text: obtain

text: the

text: EoS

text: for

text: hybrid

text: star

text: matter

text: we

text: have

text: sities

text: in

text: a

text: multi-quark

text: system

text: one

text: has

text: to

text: draw

text: string

text: connec-

text: developed

text: a

text: relativistic,

text: self-consistent

text: mean-ﬁeld

text: approach

text: tions

text: between

text: quarks

text: so

text: as

text: to

text: neutralize

text: the

text: color

text: while

text: keep-

text: within

text: the

text: path

text: integral

text: approach,

text: based

text: on

text: an

text: ansatz

text: for

text: the

text: ing

text: the

text: total

text: length

text: of

text: all

text: strings

text: at

text: a

text: minimum.

text: This

text: task

text: can

text: potential

text: energy

text: density

text: functional

text: from

text: which

text: density

text: depen-

text: be

text: solved

text: by

text: simulation

text: or

text: by

text: effective

text: quantum

text: statistical

text: dent

text: self-energies

text: follow

text: as

text: derivatives.

text: The

text: resulting

text: quasi-

text: methods

text: One

text: can

text: obtain

text: the

text: string

text: length

text: distribution

text: particle

text: model

text: necessarily

text: maintains

text: thermodynamic

text: consis-

text: function

text: for

text: a

text: given

text: density

text: and

text: fold

text: a

text: given

text: interaction

text: poten-

text: tency,

text: since

text: all

text: approximation

text: steps

text: are

text: done

text: on

text: the

text: level

text: of

text: tial

text: model

text: with

text: this

text: distribution

text: in

text: order

text: to

text: obtain

text: the

text: correla-

text: the

text: partition

text: function

text: from

text: which

text: the

text: expressions

text: for

text: thermo-

text: tion

text: energy

text: in

text: a

text: meanﬁeld

text: approximation.

text: Since

text: the

text: average

text: dynamical

text: functions

text: follow

text: by

text: derivation.