text: 4 tably, the highest order in the polynomial for EP3 is four. However, in the close ring form trimer shown in √ Fig. 1(b), two types of coalescence may occur: an EP3 ∈ Z when Φ = 2nπ + π/2 (n ) and γ = J2 + 2κ2, or an (cid:54) ∈ Z c EP2 when Φ = 2nπ +π/2 (n ). Critical gain and loss for the EP2 is (cid:114) (cid:113) − γ = 2κ2 + J2 3 3 (Jκ2 cos Φ)2, (5) c which is plotted in Fig. 3(a) for coupling strengths κ = 1 and J = 1/2 as a function of the eﬀective magnetic ﬂux Φ. The energy spectrum at EP2 is plotted in Fig. 3(b), where two states of three are coalesced; however, at mag- netic ﬂux Φ = π/2, all three states coalesced. The spec- trum is central symmetric at zero energy. When magnetic − ﬂux varies from Φ to π Φ , the spectrum E is in- − 0 0 1,2 versed to E . As depicted in Fig. 2(a), the boundaries 1,2 between regions with diﬀerent colors indicate when EP2 occurs, whereas the boundaries (black curves) between the colored regions and the white region indicate when EP3 occurs. The eigenenergies are solved from the cubic equation, and the expressions are complicated. We depict the real and imaginary part of the three band in Fig. 4 for sev- eral cases. Speciﬁcally, the bands are symmetric with respect to Φ = π. In Figs. 4(a) and 4(d), γ = J = κ = 1 PT [along κ/J = 1 in Fig. 2(c)], the -symmetric regions ∈ − − in Φ [0, 2π] are 1 < Φ < π 1 and π + 1 < Φ < 2π 1. As eﬀective magnetic ﬂux changes from Φ = 0 to Φ = 1, FIG. 4. (Color online) (a-c) Real part (in blue) of the trimer the two energies with higher real part are the complex ring spectrum. (d-f) Imaginary part (in red) of the trimer conjugate pairs; they change to real energy at Φ = 1 and PT ring spectrum. The parameters are (a,d) J = κ = 1, (b,e) − the system is in the exact -symmetric region during − γ/J = 1 + cos Φ, κ/J = cos Φ, and (c,f) γ/J = 1 cos Φ, the shift from Φ = 1 to Φ = π 1 with three real en- κ/J = cos Φ. ergies. The states with the highest and lowest energies − continue increasing in this region until Φ = π 1, where the state with the mediate real energy coalesces with the The energy degeneracy is the traditional degeneracy in state with the lowest real energy. Subsequently, from − Hermitian systems, the same situation which occurs in Φ = π 1 to Φ = π, the higher real energy increases, but Figs. 4(c) and 4(f) at Φ = 0. the lower two real energies decrease. The conjugate pair III. DYNAMICS AT EXCEPTIONAL POINT is the two states with lower real energy. As Φ changes in the region [0, π], the trimer ring transitions from bro- PT PT ken -symmetric to exact -symmetric and back to The exceptional points universally exist in non-Hermitian PT PT broken -symmetric again. The symmetry transi- systems [43, 44]. Their existence and the role they play tion point is an EP2, and the coalesced energy is nonzero for the dynamics of open quantum systems, were inves- and Φ dependent. In Figs. 4(b) and 4(e), the parameters tigated by studying the eﬀective Hamiltonian using the are given as γ/J = 1 + cos Φ and κ/J = cos Φ; in this Feshbach projection technique. Speciﬁcally, the topo- PT situation, the trimer ring is in the exact -symmetric logical structures of the exceptional points signiﬁcantly phase for π/2 < Φ < 3π/2. By contrast, in Figs. 4(c) aﬀect the dynamical properties [45, 46], and the dynam- − and 4(f), the parameters are given as γ/J = 1 cos Φ ics at the exceptional points have been investigated in and κ/J = cos Φ, and the trimer ring is in the broken many quantum systems [47–49]. Because of the unique PT -symmetric phase for π/2 < Φ < 3π/2. features of the exceptional points, they have been widely As all of these ﬁgures [Figs. 4(b), 4(c), 4(e), and 4(f)] applied to sensitivity enhancement [50], parameter esti- reveal, the trimer ring experiences three states coa- mation [51, 52], and topological energy transfer [53]. PT lesced symmetry transition points as the parameters Calculation of time evolution.—The Hamiltonian is change. For Φ = π in Figs. 4(b) and 4(e), the trimer nondiagonalizable at the exceptional point; instead, ring is a Hermitian trimer ring with antisymmetric pe- changed into a Jordan block form after transformation, − riodical boundary condition at the gain and loss γ = 0. namely, V HV 1 = h, where h is a Jordan block. Gen- ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: tably,

text: the

text: highest

text: order

text: in

text: the

text: polynomial

text: for

text: is

text: four.

text: However,

text: in

text: the

text: close

text: ring

text: form

text: trimer

text: shown

text: in

text: Fig.

text: two

text: types

text: of

text: coalescence

text: may

text: occur:

text: an

text: Z

text: when

text: Φ

text: (n

text: and

text: γ

text: or

text: an

text: Z

text: c

text: when

text: Φ

text: (n

text: Critical

text: gain

text: and

text: loss

text: for

text: the

text: is

text: γ

text: cos

text: c

text: which

text: is

text: plotted

text: in

text: Fig.

text: for

text: coupling

text: strengths

text: κ

text: and

text: J

text: as

text: a

text: function

text: of

text: the

text: eﬀective

text: magnetic

text: ﬂux

text: Φ.

text: The

text: energy

text: spectrum

text: at

text: is

text: plotted

text: in

text: Fig.

text: where

text: two

text: states

text: of

text: three

text: are

text: coalesced;

text: however,

text: at

text: mag-

text: netic

text: ﬂux

text: Φ

text: all

text: three

text: states

text: coalesced.

text: The

text: spec-

text: trum

text: is

text: central

text: symmetric

text: at

text: zero

text: energy.

text: When

text: magnetic

text: ﬂux

text: varies

text: from

text: Φ

text: to

text: π

text: Φ

text: the

text: spectrum

text: E

text: is

text: in-

text: versed

text: to

text: E

text: As

text: depicted

text: in

text: Fig.

text: the

text: boundaries

text: between

text: regions

text: with

text: diﬀerent

text: colors

text: indicate

text: when

text: occurs,

text: whereas

text: the

text: boundaries

text: (black

text: curves)

text: between

text: the

text: colored

text: regions

text: and

text: the

text: white

text: region

text: indicate

text: when

text: occurs.

text: The

text: eigenenergies

text: are

text: solved

text: from

text: the

text: cubic

text: equation,

text: and

text: the

text: expressions

text: are

text: complicated.

text: We

text: depict

text: the

text: real

text: and

text: imaginary

text: part

text: of

text: the

text: three

text: band

text: in

text: Fig.

text: for

text: sev-

text: eral

text: cases.

text: Speciﬁcally,

text: the

text: bands

text: are

text: symmetric

text: with

text: respect

text: to

text: Φ

text: π.

text: In

text: Figs.

text: and

text: γ

text: J

text: κ

text: PT

text: [along

text: κ/J

text: in

text: Fig.

text: the

text: -symmetric

text: regions

text: in

text: Φ

text: are

text: Φ

text: π

text: and

text: π

text: Φ

text: 2π

text: As

text: eﬀective

text: magnetic

text: ﬂux

text: changes

text: from

text: Φ

text: to

text: Φ

text: FIG.

text: (Color

text: online)

text: (a-c)

text: Real

text: part

text: (in

text: blue)

text: of

text: the

text: trimer

text: the

text: two

text: energies

text: with

text: higher

text: real

text: part

text: are

text: the

text: complex

text: ring

text: spectrum.

text: (d-f)

text: Imaginary

text: part

text: (in

text: red)

text: of

text: the

text: trimer

text: conjugate

text: pairs;

text: they

text: change

text: to

text: real

text: energy

text: at

text: Φ

text: and

text: PT

text: ring

text: spectrum.

text: The

text: parameters

text: are

text: (a,d)

text: J

text: κ

text: (b,e)

text: the

text: system

text: is

text: in

text: the

text: exact

text: -symmetric

text: region

text: during

text: γ/J

text: cos

text: Φ,

text: κ/J

text: cos

text: Φ,

text: and

text: (c,f)

text: γ/J

text: cos

text: Φ,

text: the

text: shift

text: from

text: Φ

text: to

text: Φ

text: π

text: with

text: three

text: real

text: en-

text: κ/J

text: cos

text: Φ.

text: ergies.

text: The

text: states

text: with

text: the

text: highest

text: and

text: lowest

text: energies

text: continue

text: increasing

text: in

text: this

text: region

text: until

text: Φ

text: π

text: where

text: the

text: state

text: with

text: the

text: mediate

text: real

text: energy

text: coalesces

text: with

text: the

text: The

text: energy

text: degeneracy

text: is

text: the

text: traditional

text: degeneracy

text: in

text: state

text: with

text: the

text: lowest

text: real

text: energy.

text: Subsequently,

text: from

text: Hermitian

text: systems,

text: the

text: same

text: situation

text: which

text: occurs

text: in

text: Φ

text: π

text: to

text: Φ

text: π,

text: the

text: higher

text: real

text: energy

text: increases,

text: but

text: Figs.

text: and

text: at

text: Φ

text: the

text: lower

text: two

text: real

text: energies

text: decrease.

text: The

text: conjugate

text: pair

text: III.

text: DYNAMICS

text: AT

text: EXCEPTIONAL

text: POINT

text: is

text: the

text: two

text: states

text: with

text: lower

text: real

text: energy.

text: As

text: Φ

text: changes

text: in

text: the

text: region

text: π],

text: the

text: trimer

text: ring

text: transitions

text: from

text: bro-

text: PT

text: PT

text: ken

text: -symmetric

text: to

text: exact

text: -symmetric

text: and

text: back

text: to

text: The

text: exceptional

text: points

text: universally

text: exist

text: in

text: non-Hermitian

text: PT

text: PT

text: broken

text: -symmetric

text: again.

text: The

text: symmetry

text: transi-

text: systems

text: Their

text: existence

text: and

text: the

text: role

text: they

text: play

text: tion

text: point

text: is

text: an

text: and

text: the

text: coalesced

text: energy

text: is

text: nonzero

text: for

text: the

text: dynamics

text: of

text: open

text: quantum

text: systems,

text: were

text: inves-

text: and

text: Φ

text: dependent.

text: In

text: Figs.

text: and

text: the

text: parameters

text: tigated

text: by

text: studying

text: the

text: eﬀective

text: Hamiltonian

text: using

text: the

text: are

text: given

text: as

text: γ/J

text: cos

text: Φ

text: and

text: κ/J

text: cos

text: Φ;

text: in

text: this

text: Feshbach

text: projection

text: technique.

text: Speciﬁcally,

text: the

text: topo-

text: PT

text: situation,

text: the

text: trimer

text: ring

text: is

text: in

text: the

text: exact

text: -symmetric

text: logical

text: structures

text: of

text: the

text: exceptional

text: points

text: signiﬁcantly

text: phase

text: for

text: Φ

text: By

text: contrast,

text: in

text: Figs.

text: aﬀect

text: the

text: dynamical

text: properties

text: and

text: the

text: dynam-

text: and

text: the

text: parameters

text: are

text: given

text: as

text: γ/J

text: cos

text: Φ

text: ics

text: at

text: the

text: exceptional

text: points

text: have

text: been

text: investigated

text: in

text: and

text: κ/J

text: cos

text: Φ,

text: and

text: the

text: trimer

text: ring

text: is

text: in

text: the

text: broken

text: many

text: quantum

text: systems

text: Because

text: of

text: the

text: unique

text: PT

text: -symmetric

text: phase

text: for

text: Φ

text: features

text: of

text: the

text: exceptional

text: points,

text: they

text: have

text: been

text: widely

text: As

text: all

text: of

text: these

text: ﬁgures

text: [Figs.

text: and

text: applied

text: to

text: sensitivity

text: enhancement

text: parameter

text: esti-

text: reveal,

text: the

text: trimer

text: ring

text: experiences

text: three

text: states

text: coa-

text: mation

text: and

text: topological

text: energy

text: transfer

text: PT

text: lesced

text: symmetry

text: transition

text: points

text: as

text: the

text: parameters

text: Calculation

text: of

text: time

text: evolution.—The

text: Hamiltonian

text: is

text: change.

text: For

text: Φ

text: π

text: in

text: Figs.

text: and

text: the

text: trimer

text: nondiagonalizable

text: at

text: the

text: exceptional

text: point;

text: instead,

text: ring

text: is

text: a

text: Hermitian

text: trimer

text: ring

text: with

text: antisymmetric

text: pe-

text: changed

text: into

text: a

text: Jordan

text: block

text: form

text: after

text: transformation,

text: riodical

text: boundary

text: condition

text: at

text: the

text: gain

text: and

text: loss

text: γ

text: namely,

text: V

text: HV

text: h,

text: where

text: h

text: is

text: a

text: Jordan

text: block.

text: Gen-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 such as Φ = 0, the chiral symmetry of the trimer ring breaks, and the scattering probabilities are asymmetric (cid:0) (cid:1) at k = π/2 [Fig. 9(d-f)]. When k = π/2, the reﬂection and transmission are identical, both symmetrical and γ- − independent, T = R = T = R = cos2 Φ + 1 1. L L R R The divergence of the scattering coeﬃcients occurs only for wave vectors k = π/4 and 3π/4. At k = π/4, the transmissions (t and t ) are larger than zero under any L R balanced trimer gain and loss γ or when the eﬀective magnetic ﬂux Φ enclosed. The scattering coeﬃcients di- vergence emerges at √ γ2 = 2 cos Φ + 2, (15) (cid:112) (cid:112) and √the spectral singularit√y emerges in the region − (cid:54) | | (cid:54) 2 2 γ/κ 2 + 2. Notably, the trimer acts as a wave emitter at the spectral singularities. The wave emission is asymmetric, and the contrast ratio of √ √ left-traveling wave emission over right-traveling wave − emission is ( 2γ 1)/( 2γ + 1), which is obtained by comparing the transmission and reﬂection probabilities. The contrast ratio less than one is related to left side with √ the gain resonator having a larger emission rate. For ex- ample, at γ = 2, Φ = π/2, we have T /R = 1/3, L L T /R = 3. This indicates that the lasing to diﬀerent R R directions have diﬀerent intensities, similar to the trimer chain. The lasing generates three times as many left- traveling waves than does the right-traveling wave. For either the trimer chain or the trimer ring, the asymmetric FIG. 9. (Color online) Transmission and reﬂection probability of the uniform trimer ring (J = κ) for (a-c) Φ = π/2 and (d-f) lasing intensity are both stronger right-traveling waves. Φ = 0. V. CONCLUSION PT (cid:0) (cid:1) Two -symmetric trimer systems, comprising an open trimer chain and a closed trimer ring were exam- i sin k(cid:0) e−iΦ + 2 cos k (cid:1) ined in this study. First, we investigatePdTthe spectra tL = ie−2ik sin k + cos Φ + cos k (1 − γ2), (11) of the trimer systems and obtained the -symmetric phase diagram. we found that the nontrivial magnetic − − PT i sin k e 2ik 2iγ cos k − ﬂux suppresses the transition. For the trimer chain, rL = ie−2ik sin k + cos Φ + cos k (1 − γ2) 1, (12) the PT -symmetric phase transition point is an EP3 with three states coalescence; for the trimer ring, except for PT (cid:0) (cid:1) the EP3, the -symmetric phase transition point can be an EP2 with two states coalescence as long as an i sin(cid:0)k eiΦ + 2 cos k (cid:1) eﬀective magnetic ﬂux breaks the chiral symmetry of the tR = ie−2ik sin k + cos Φ + cos k (1 − γ2), (13) trimer ring. Second, we studied the time evolution at the exceptional point in detail, and determined that the state − i sin k e 2ik + 2iγ cos k − probability can change in four ways when the trimer ring rR = ie−2ik sin k + cos Φ + cos k (1 − γ2) 1. (14) is at EP2: (i) unchanged, (ii) oscillation, (iii) power law increase, and (iv) power law increase with oscillation. where the coupling strengths satisfy J = κ. From the ex- These occur because of the nonzero coalesced eigenstate pressions of the scattering coeﬃcients, the transmission and the third normal eigenstate. Finally, we investigated is symmetric (i.e., T = T = T ). Figure 9 illustrates the the scattering properties of the trimer systems with the L R plotted transmission and reﬂection probabilities. In the gain and loss sites embedded in the lead, and calculated ∈ region k [0, π], the scattering probability is symmetri- the transmissions and reﬂections for the left- and right- ∈ cal at k = π/2 for eﬀective magnetic Φ = 2nπ +π/2 (n side inputs. Notably, the transmission is symmetric and Z ) [Fig. 9(a-c)], where the trimer ring possesses chiral the reﬂection is asymmetric. We also found the spectral symmetry and its spectrum is symmetrical at zero energy singularities, at which wave emission is asymmetric. The − (i.e., the energy 2 cos(π/2)). In other magnetic ﬂuxes, critical dynamics at exceptional points and the scatter- ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: such

text: as

text: Φ

text: the

text: chiral

text: symmetry

text: of

text: the

text: trimer

text: ring

text: breaks,

text: and

text: the

text: scattering

text: probabilities

text: are

text: asymmetric

text: at

text: k

text: [Fig.

text: When

text: k

text: the

text: reﬂection

text: and

text: transmission

text: are

text: identical,

text: both

text: symmetrical

text: and

text: γ-

text: independent,

text: T

text: R

text: T

text: R

text: Φ

text: L

text: L

text: R

text: R

text: The

text: divergence

text: of

text: the

text: scattering

text: coeﬃcients

text: occurs

text: only

text: for

text: wave

text: vectors

text: k

text: and

text: At

text: k

text: the

text: transmissions

text: (t

text: and

text: t

text: are

text: larger

text: than

text: zero

text: under

text: any

text: L

text: R

text: balanced

text: trimer

text: gain

text: and

text: loss

text: γ

text: or

text: when

text: the

text: eﬀective

text: magnetic

text: ﬂux

text: Φ

text: enclosed.

text: The

text: scattering

text: coeﬃcients

text: di-

text: vergence

text: emerges

text: at

text: γ2

text: cos

text: Φ

text: and

text: √the

text: spectral

text: singularit√y

text: emerges

text: in

text: the

text: region

text: γ/κ

text: Notably,

text: the

text: trimer

text: acts

text: as

text: a

text: wave

text: emitter

text: at

text: the

text: spectral

text: singularities.

text: The

text: wave

text: emission

text: is

text: asymmetric,

text: and

text: the

text: contrast

text: ratio

text: of

text: left-traveling

text: wave

text: emission

text: over

text: right-traveling

text: wave

text: emission

text: is

text: 2γ

text: 2γ

text: which

text: is

text: obtained

text: by

text: comparing

text: the

text: transmission

text: and

text: reﬂection

text: probabilities.

text: The

text: contrast

text: ratio

text: less

text: than

text: one

text: is

text: related

text: to

text: left

text: side

text: with

text: the

text: gain

text: resonator

text: having

text: a

text: larger

text: emission

text: rate.

text: For

text: ex-

text: ample,

text: at

text: γ

text: Φ

text: we

text: have

text: T

text: /R

text: L

text: L

text: T

text: /R

text: This

text: indicates

text: that

text: the

text: lasing

text: to

text: diﬀerent

text: R

text: R

text: directions

text: have

text: diﬀerent

text: intensities,

text: similar

text: to

text: the

text: trimer

text: chain.

text: The

text: lasing

text: generates

text: three

text: times

text: as

text: many

text: left-

text: traveling

text: waves

text: than

text: does

text: the

text: right-traveling

text: wave.

text: For

text: either

text: the

text: trimer

text: chain

text: or

text: the

text: trimer

text: ring,

text: the

text: asymmetric

text: FIG.

text: (Color

text: online)

text: Transmission

text: and

text: reﬂection

text: probability

text: of

text: the

text: uniform

text: trimer

text: ring

text: (J

text: κ)

text: for

text: (a-c)

text: Φ

text: and

text: (d-f)

text: lasing

text: intensity

text: are

text: both

text: stronger

text: right-traveling

text: waves.

text: Φ

text: V.

text: CONCLUSION

text: PT

text: Two

text: -symmetric

text: trimer

text: systems,

text: comprising

text: an

text: open

text: trimer

text: chain

text: and

text: a

text: closed

text: trimer

text: ring

text: were

text: exam-

text: i

text: sin

text: e−iΦ

text: cos

text: k

text: ined

text: in

text: this

text: study.

text: First,

text: we

text: investigatePdTthe

text: spectra

text: tL

text: sin

text: k

text: cos

text: Φ

text: cos

text: k

text: of

text: the

text: trimer

text: systems

text: and

text: obtained

text: the

text: -symmetric

text: phase

text: diagram.

text: we

text: found

text: that

text: the

text: nontrivial

text: magnetic

text: PT

text: i

text: sin

text: k

text: e

text: cos

text: k

text: ﬂux

text: suppresses

text: the

text: transition.

text: For

text: the

text: trimer

text: chain,

text: rL

text: sin

text: k

text: cos

text: Φ

text: cos

text: k

text: the

text: PT

text: -symmetric

text: phase

text: transition

text: point

text: is

text: an

text: with

text: three

text: states

text: coalescence;

text: for

text: the

text: trimer

text: ring,

text: except

text: for

text: PT

text: the

text: the

text: -symmetric

text: phase

text: transition

text: point

text: can

text: be

text: an

text: with

text: two

text: states

text: coalescence

text: as

text: long

text: as

text: an

text: i

text: eiΦ

text: cos

text: k

text: eﬀective

text: magnetic

text: ﬂux

text: breaks

text: the

text: chiral

text: symmetry

text: of

text: the

text: tR

text: sin

text: k

text: cos

text: Φ

text: cos

text: k

text: trimer

text: ring.

text: Second,

text: we

text: studied

text: the

text: time

text: evolution

text: at

text: the

text: exceptional

text: point

text: in

text: detail,

text: and

text: determined

text: that

text: the

text: state

text: i

text: sin

text: k

text: e

text: cos

text: k

text: probability

text: can

text: change

text: in

text: four

text: ways

text: when

text: the

text: trimer

text: ring

text: rR

text: sin

text: k

text: cos

text: Φ

text: cos

text: k

text: is

text: at

text: (i)

text: unchanged,

text: (ii)

text: oscillation,

text: (iii)

text: power

text: law

text: increase,

text: and

text: (iv)

text: power

text: law

text: increase

text: with

text: oscillation.

text: where

text: the

text: coupling

text: strengths

text: satisfy

text: J

text: κ.

text: From

text: the

text: ex-

text: These

text: occur

text: because

text: of

text: the

text: nonzero

text: coalesced

text: eigenstate

text: pressions

text: of

text: the

text: scattering

text: coeﬃcients,

text: the

text: transmission

text: and

text: the

text: third

text: normal

text: eigenstate.

text: Finally,

text: we

text: investigated

text: is

text: symmetric

text: (i.e.,

text: T

text: T

text: T

text: Figure

text: illustrates

text: the

text: the

text: scattering

text: properties

text: of

text: the

text: trimer

text: systems

text: with

text: the

text: L

text: R

text: plotted

text: transmission

text: and

text: reﬂection

text: probabilities.

text: In

text: the

text: gain

text: and

text: loss

text: sites

text: embedded

text: in

text: the

text: lead,

text: and

text: calculated

text: region

text: k

text: π],

text: the

text: scattering

text: probability

text: is

text: symmetri-

text: the

text: transmissions

text: and

text: reﬂections

text: for

text: the

text: left-

text: and

text: right-

text: cal

text: at

text: k

text: for

text: eﬀective

text: magnetic

text: Φ

text: (n

text: side

text: inputs.

text: Notably,

text: the

text: transmission

text: is

text: symmetric

text: and

text: Z

text: [Fig.

text: where

text: the

text: trimer

text: ring

text: possesses

text: chiral

text: the

text: reﬂection

text: is

text: asymmetric.

text: We

text: also

text: found

text: the

text: spectral

text: symmetry

text: and

text: its

text: spectrum

text: is

text: symmetrical

text: at

text: zero

text: energy

text: singularities,

text: at

text: which

text: wave

text: emission

text: is

text: asymmetric.

text: The

text: (i.e.,

text: the

text: energy

text: In

text: other

text: magnetic

text: ﬂuxes,

text: critical

text: dynamics

text: at

text: exceptional

text: points

text: and

text: the

text: scatter-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 ing properties at spectral singularities may be useful in which are quantum metrology in the future. Note added: We become aware of a related work, the PT idψ dynamics in the exact and broken -symmetric phases 1 = λψ + ψ , (A7) dt 1 2 were investigated in an open trimer chain [57]. idψ ACKNOWLEDGMENTS 2 = λψ + ψ , (A8) dt 2 3 idψ We acknowledge the support of National Natural Sci- 3 = λψ . (A9) dt 3 ence Foundation of China (Grant No. 11605094) and the Tianjin Natural Science Foundation (Grant No. 16JCY- From the last equation, we get BJC40800). − ψ = c e iλt, (A10) 3 3 Appendix A: Dynamics at exceptional points − thus, we have idψ /dt = λψ + c e iλt, and then we get 2 2 3 √ For the trimer ring at EP3, the Hamiltonian at critical − − − gain and loss γc = J2 + 2κ2 is in the formof ψ2 = c2e iλt + ( it) c3e iλt, (A11)  √  − − −consequ−ently, we have idψ1/dt = λψ1 + c2e iλt + i J2−+ 2κ2 κ √ −iJ ( it) c3e iλt and H = κ 0 κ . (A1) − − − iJ κ i J2 + 2κ2 − − − − − ψ = c e iλt + ( it) c e iλt (t2/2)c e iλt. (A12) − 1 1 2 3 The Hamiltonian can be tranformed to H = V hV 1 with the transformation  The obtained wave function ψ (t) is  (cid:0) (cid:1)  √  −   √κ2 i J2 + 2κ2 1 − − − − c + ( it) c (t2/2)c V = iκ J + J2 + 2κ2 κ 0 , (A2) − 1 −2 3 ψ (t) = e iλt c + ( it) c . (A13) κ2 iJ 0 2 3 c 3 × and h is a 3 3 Jordan block with diagonal elements Thus, the time evolution state is Ψ (t) = V ψ (t). The λ = 0,   coeﬃcients c are determined from the initial state. 1,2,3   At t = 0, we have Ψ (0) = V ψ (0), therefore, the initial state is expressed as     λ 1 0 h = 0 λ 1 . (A3)     0 0 λ Ψ c 1 1 Ψ (0) = Ψ = V c , (A14) 2 2 The Schr¨odingerequations are  Ψ c 3 3     − the coeﬃcients satisfy ( c c c )T = V 1( Ψ Ψ Ψ )T , 1 2 3 1 2 3 id Ψ1 Ψ1 and the time evolution state Ψ (t) is determined, as Ψ = H Ψ , (A4) 2 2 dt   Ψ3 Ψ3 − − c + ( it) c (t2/2)c substituting H = V hV −1, we obtain  Ψ (t) = e−iλtV 1 c2 + (−2 it) c3 3 . (A15) c 3     Ψ Ψ Here, λ = 0 for the three states coalescence in the trimer id − 1 − 1 V 1 Ψ = hV 1 Ψ , (A5) ring. 2 2 dt Ψ Ψ For the trimer ring at EP2 at magnetic ﬂux Φ = π/3, 3 3 √ the coupling strengths are κ = 1 and J = 1/2, at critical − by setting ψ = V 1Ψ, the Schr¨odinger equations are gain and loss γ = 3/2, the Hamiltonian is in the form c reduced to diﬀerential equationsof ψ,in the form of of        √  − − ψ ψ i 3/2 1/2 eiπ/3 id 1 1 − − √ ψ = h ψ , (A6) H = 1/2 0 1/2 , (A16) dt 2 2 − − − − ψ ψ e iπ/3 1/2 i 3/2 3 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ing

text: properties

text: at

text: spectral

text: singularities

text: may

text: be

text: useful

text: in

text: which

text: are

text: quantum

text: metrology

text: in

text: the

text: future.

text: Note

text: added:

text: We

text: become

text: aware

text: of

text: a

text: related

text: work,

text: the

text: PT

text: idψ

label: equation

text: dynamics

text: in

text: the

text: exact

text: and

text: broken

text: -symmetric

text: phases

text: λψ

text: ψ

text: dt

text: were

text: investigated

text: in

text: an

text: open

text: trimer

text: chain

text: idψ

text: ACKNOWLEDGMENTS

text: λψ

text: ψ

text: dt

text: idψ

text: We

text: acknowledge

text: the

text: support

text: of

text: National

text: Natural

text: Sci-

text: λψ

text: dt

text: ence

text: Foundation

text: of

text: China

text: (Grant

text: No.

text: and

text: the

text: Tianjin

text: Natural

text: Science

text: Foundation

text: (Grant

text: No.

text: From

text: the

text: last

text: equation,

text: we

text: get

text: ψ

text: c

text: e

text: iλt,

text: Appendix

text: A:

text: Dynamics

text: at

text: exceptional

text: points

text: thus,

text: we

text: have

text: idψ

text: /dt

text: λψ

text: c

text: e

text: iλt,

text: and

text: then

text: we

text: get

text: For

text: the

text: trimer

text: ring

text: at

text: the

text: Hamiltonian

text: at

text: critical

text: gain

text: and

text: loss

text: γc

text: is

text: in

text: the

text: formof

text: ψ2

text: iλt

text: it)

text: iλt,

text: −consequ−ently,

text: we

text: have

text: λψ1

text: iλt

text: i

text: κ

text: −iJ

text: it)

text: iλt

text: and

text: H

text: κ

text: κ

text: iJ

text: κ

text: i

text: ψ

text: c

text: e

text: iλt

text: it)

text: c

text: e

text: iλt

text: e

text: iλt.

text: The

text: Hamiltonian

text: can

text: be

text: tranformed

text: to

text: H

text: V

text: hV

text: with

text: the

text: transformation

text: The

text: obtained

text: wave

text: function

text: ψ

text: (t)

text: is

text: √κ2

text: i

text: c

text: it)

text: c

text: V

text: iκ

text: J

text: κ

text: −2

text: ψ

text: (t)

text: e

text: iλt

text: c

text: it)

text: c

text: κ2

text: iJ

text: c

text: and

text: h

text: is

text: a

text: Jordan

text: block

text: with

text: diagonal

text: elements

text: Thus,

text: the

text: time

text: evolution

text: state

text: is

text: Ψ

text: (t)

text: V

text: ψ

text: (t).

text: The

text: λ

text: coeﬃcients

text: c

text: are

text: determined

text: from

text: the

text: initial

text: state.

text: At

text: t

text: we

text: have

text: Ψ

text: V

text: ψ

text: therefore,

text: the

text: initial

text: state

text: is

text: expressed

text: as

text: λ

text: h

text: λ

text: λ

text: Ψ

text: c

text: Ψ

text: Ψ

text: V

text: c

text: The

text: Schr¨odingerequations

text: are

text: Ψ

text: c

text: the

text: coeﬃcients

text: satisfy

text: c

text: c

text: c

text: )T

text: V

text: Ψ

text: Ψ

text: Ψ

text: )T

text: id

text: Ψ1

text: Ψ1

text: and

text: the

text: time

text: evolution

text: state

text: Ψ

text: (t)

text: is

text: determined,

text: as

text: Ψ

text: H

text: Ψ

text: dt

text: Ψ3

text: Ψ3

text: c

text: it)

text: c

text: substituting

text: H

text: V

text: hV

text: −1,

text: we

text: obtain

text: Ψ

text: (t)

text: e−iλtV

text: it)

text: c

text: Ψ

text: Ψ

text: Here,

text: λ

text: for

text: the

text: three

text: states

text: coalescence

text: in

text: the

text: trimer

text: id

text: V

text: Ψ

text: hV

text: Ψ

text: ring.

text: dt

text: Ψ

text: Ψ

text: For

text: the

text: trimer

text: ring

text: at

text: at

text: magnetic

text: ﬂux

text: Φ

text: the

text: coupling

text: strengths

text: are

text: κ

text: and

text: J

text: at

text: critical

text: by

text: setting

text: ψ

text: V

text: the

text: Schr¨odinger

text: equations

text: are

text: gain

text: and

text: loss

text: γ

text: the

text: Hamiltonian

text: is

text: in

text: the

text: form

text: c

text: reduced

text: to

text: diﬀerential

text: equationsof

text: ψ,in

text: the

text: form

text: of

text: of

text: ψ

text: ψ

text: i

text: id

text: ψ

text: h

text: ψ

text: H

text: dt

text: ψ

text: ψ

text: e

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 spectra, resulting from the inability of stellar photons to reach depths in the atmosphere below the cloud and haze layers (Gibson et al. 2012, 2013; Deming et al. 2013; Jord´an et al. 2013; Mandell et al. 2013; Sing et al. 2013; Chen et al. 2014; Schlawin et al. 2014; Wilkins et al. 2014; Fukui et al. 2014; Mallonn & Strassmeier 2016). Such ﬂat transmis- sion spectra have been seen across many exoplanets of diﬀerent sizes, eﬀective temperatures, and stellar irradiation levels (e.g. Crossﬁeld et al. 2013; Kreidberg et al. 2014; Knutson et al. 2014a,b; Sing et al. 2016), suggesting that the processes governing cloud and haze formation in exoplanet atmospheres are as complex as they are ubiquitous. Clouds and hazes have also been inferred from optical phase curves, where higher than expected albedoes have been attributed to reﬂective clouds (e.g. Demory et al. 2013), and dust clouds have been invoked to ﬁt spectra of directly imaged exoplanets (Barman et al. 2011; Madhusudhan et al. 2011; Marley et al. 2012; Currie et al. 2014; Skemer et al. 2014; Chilcote et al. 2015; Macintosh et al. 2015). One key unknown is whether the aerosols blocking the stellar photons are part of a cloud, which condense from atmospheric gases and are typically supported in an atmosphere by turbulent mixing, or part of a haze, whose particles are usually produced by photochemistry and often found at high altitudes. Morley et al. (2013) showed that, for the super Earth GJ 1214 b, photochemical hazes may be preferred as a solution to its ﬂat transmission spectra, as they are formed high up in the atmosphere. By contrast, cloud particles must be lofted by turbulent mixing, and high metallicity may be required to ensure that enough material gets to the pressure levels probed by transmission spectroscopy. The composition and vertical structure of atmospheric aerosols, whether clouds or hazes, is also of ﬁrst order importance for understanding the reﬂected light spectra of a planet (see Marley et al. (2013) for a review). Cloudless extrasolar giant planets, for example, are expected to be relatively bright at blue wavelengths but dark in the red optical and near infrared where molecular absorption dominates over Rayleigh scattering; cloudy planets, by scattering more photons before they can be absorbed, can be much brighter (Marley et al. 1999; Sudarsky et al. 2000). Likewise for terrestrial planets Morley et al. (2015) and Charnay et al. (2015) showed that there are large diﬀerences in the reﬂected light spectra of super Earths depending on whether the planet is cloudy or hazy, and what kind of aerosols are present. Speciﬁcally, cooler planets with KCl and water clouds may be more reﬂective than clear planets in the near-infrared, while ZnS clouds, forming at similar temperatures, are more absorbing. In contrast, planets with complex hydrocarbon, “soot” hazes resulting from methane photolysis and polymerization could be very dark, whereas hazes composed of tholins–nitrogen rich organics thought to comprise Titan’s hazes (Khare et al. 1984)–are dark at the blue end of the visible range and more reﬂective at the red end. The reﬂectivity of clouds and hazes also impacts the amount of compositional information that can be retrieved from direct imaging, as reﬂected light is only able to sample the atmosphere above the cloud/haze deck. If the cloud/haze deck is at high altitudes, then the column optical depth of absorbing gases that direct imaging can sample is small, thereby reducing the magnitude of their spectral features in these planets’ reﬂected light spectra (Marley et al. 1999; Sudarsky et al. 2000). Meanwhile, if the cloud/haze is absorbing, then very few photons can sample the chemical composition of the atmosphere and escape. Such considerations are particularly relevant, as NASA is now studying future space based telescopes capable of characterizing extrasolar planets through high-contrast direct imaging (Krist et al. 2007; Boccaletti et al. 2015; Robinson et al. 2016). An alternative set of compounds that are known to form hazes in planetary atmospheres are those derived from sulfur chemistry. In oxidizing atmospheres, SO , OCS, and H S from volcanic outgassing is transformed into sulfuric 2 2 acid through photolysis and reactions with water. Condensation of sulfuric acid can then form clouds and hazes, such as the global cloud deck of Venus (Hansen & Hovenier 1974) and the Junge layer in the upper stratosphere of Earth (Junge 1963). In reducing atmospheres, photolysis of H S can lead to the formation of elemental sulfur. For example, 2 the existence of sulfur aerosols has been proposed for the Venus atmosphere, especially in regions depleted in oxidants (Toon et al. 1982; Zhang et al. 2012). Precipitation of elemental sulfur is the leading hypothesis for the preservation of mass independent fractionation of sulfur isotopes found in sediments on Earth older than 2.4 billion years (Pavlov & Kasting 2002). Hu et al. (2013) showed that terrestrial exoplanets with H –dominated atmospheres could be enveloped 2 in optically thick sulfur hazes resulting from volcanic outgassing of H S. 2 More recent work by Zahnle et al. (2016) showed that rich sulfur photochemistry could potentially take place in the atmospheres of temperate giant exoplanets (250 K < T < 700 K), generating sulfur allotropes that may condense to eq form hazes at lower temperatures. As these planets are the targets of current and planned direct imaging campaigns, it is essential that the optical characteristics of elemental sulfur hazes be known in order to inform these future observations. In this paper we investigate the geometric albedo spectra of giant planets hosting elemental sulfur hazes and their variations with haze properties, such as the location of the haze in the atmosphere and the haze optical depth. We also address the observability of a sulfur haze and its impact on inferences of atmospheric composition for

text: spectra,

text: resulting

text: from

text: the

text: inability

text: of

text: stellar

text: photons

text: to

text: reach

text: depths

text: in

text: the

text: atmosphere

text: below

text: the

text: cloud

text: and

text: haze

text: layers

text: (Gibson

text: et

text: al.

text: Deming

text: et

text: al.

text: Jord´an

text: et

text: al.

text: Mandell

text: et

text: al.

text: Sing

text: et

text: al.

text: Chen

text: et

text: al.

text: Schlawin

text: et

text: al.

text: Wilkins

text: et

text: al.

text: Fukui

text: et

text: al.

text: Mallonn

text: Strassmeier

text: Such

text: ﬂat

text: transmis-

text: sion

text: spectra

text: have

text: been

text: seen

text: across

text: many

text: exoplanets

text: of

text: diﬀerent

text: sizes,

text: eﬀective

text: temperatures,

text: and

text: stellar

text: irradiation

text: levels

text: (e.g.

text: Crossﬁeld

text: et

text: al.

text: Kreidberg

text: et

text: al.

text: Knutson

text: et

text: al.

text: Sing

text: et

text: al.

text: suggesting

text: that

text: the

text: processes

text: governing

text: cloud

text: and

text: haze

text: formation

text: in

text: exoplanet

text: atmospheres

text: are

text: as

text: complex

text: as

text: they

text: are

text: ubiquitous.

text: Clouds

text: and

text: hazes

text: have

text: also

text: been

text: inferred

text: from

text: optical

text: phase

text: curves,

text: where

text: higher

text: than

text: expected

text: albedoes

text: have

text: been

text: attributed

text: to

text: reﬂective

text: clouds

text: (e.g.

text: Demory

text: et

text: al.

text: and

text: dust

text: clouds

text: have

text: been

text: invoked

text: to

text: ﬁt

text: spectra

text: of

text: directly

text: imaged

text: exoplanets

text: (Barman

text: et

text: al.

text: Madhusudhan

text: et

text: al.

text: Marley

text: et

text: al.

text: Currie

text: et

text: al.

text: Skemer

text: et

text: al.

text: Chilcote

text: et

text: al.

text: Macintosh

text: et

text: al.

text: One

text: key

text: unknown

text: is

text: whether

text: the

text: aerosols

text: blocking

text: the

text: stellar

text: photons

text: are

text: part

text: of

text: a

text: cloud,

text: which

text: condense

text: from

text: atmospheric

text: gases

text: and

text: are

text: typically

text: supported

text: in

text: an

text: atmosphere

text: by

text: turbulent

text: mixing,

text: or

text: part

text: of

text: a

text: haze,

text: whose

text: particles

text: are

text: usually

text: produced

text: by

text: photochemistry

text: and

text: often

text: found

text: at

text: high

text: altitudes.

text: Morley

text: et

text: al.

text: showed

text: that,

text: for

text: the

text: super

text: Earth

text: GJ

text: b,

text: photochemical

text: hazes

text: may

text: be

text: preferred

text: as

text: a

text: solution

text: to

text: its

text: ﬂat

text: transmission

text: spectra,

text: as

text: they

text: are

text: formed

text: high

text: up

text: in

text: the

text: atmosphere.

text: By

text: contrast,

text: cloud

text: particles

text: must

text: be

text: lofted

text: by

text: turbulent

text: mixing,

text: and

text: high

text: metallicity

text: may

text: be

text: required

text: to

text: ensure

text: that

text: enough

text: material

text: gets

text: to

text: the

text: pressure

text: levels

text: probed

text: by

text: transmission

text: spectroscopy.

text: The

text: composition

text: and

text: vertical

text: structure

text: of

text: atmospheric

text: aerosols,

text: whether

text: clouds

text: or

text: hazes,

text: is

text: also

text: of

text: ﬁrst

text: order

text: importance

text: for

text: understanding

text: the

text: reﬂected

text: light

text: spectra

text: of

text: a

text: planet

text: (see

text: Marley

text: et

text: al.

text: for

text: a

text: review).

text: Cloudless

text: extrasolar

text: giant

text: planets,

text: for

text: example,

text: are

text: expected

text: to

text: be

text: relatively

text: bright

text: at

text: blue

text: wavelengths

text: but

text: dark

text: in

text: the

text: red

text: optical

text: and

text: near

text: infrared

text: where

text: molecular

text: absorption

text: dominates

text: over

text: Rayleigh

text: scattering;

text: cloudy

text: planets,

text: by

text: scattering

text: more

text: photons

text: before

text: they

text: can

text: be

text: absorbed,

text: can

text: be

text: much

text: brighter

text: (Marley

text: et

text: al.

text: Sudarsky

text: et

text: al.

text: Likewise

text: for

text: terrestrial

text: planets

text: Morley

text: et

text: al.

text: and

text: Charnay

text: et

text: al.

text: showed

text: that

text: there

text: are

text: large

text: diﬀerences

text: in

text: the

text: reﬂected

text: light

text: spectra

text: of

text: super

text: Earths

text: depending

text: on

text: whether

text: the

text: planet

text: is

text: cloudy

text: or

text: hazy,

text: and

text: what

text: kind

text: of

text: aerosols

text: are

text: present.

text: Speciﬁcally,

text: cooler

text: planets

text: with

text: KCl

text: and

text: water

text: clouds

text: may

text: be

text: more

text: reﬂective

text: than

text: clear

text: planets

text: in

text: the

text: near-infrared,

text: while

text: ZnS

text: clouds,

text: forming

text: at

text: similar

text: temperatures,

text: are

text: more

text: absorbing.

text: In

text: contrast,

text: planets

text: with

text: complex

text: hydrocarbon,

text: “soot”

text: hazes

text: resulting

text: from

text: methane

text: photolysis

text: and

text: polymerization

text: could

text: be

text: very

text: dark,

text: whereas

text: hazes

text: composed

text: of

text: tholins–nitrogen

text: rich

text: organics

text: thought

text: to

text: comprise

text: Titan’s

text: hazes

text: (Khare

text: et

text: al.

text: dark

text: at

text: the

text: blue

text: end

text: of

text: the

text: visible

text: range

text: and

text: more

text: reﬂective

text: at

text: the

text: red

text: end.

text: The

text: reﬂectivity

text: of

text: clouds

text: and

text: hazes

text: also

text: impacts

text: the

text: amount

text: of

text: compositional

text: information

text: that

text: can

text: be

text: retrieved

text: from

text: direct

text: imaging,

text: as

text: reﬂected

text: light

text: is

text: only

text: able

text: to

text: sample

text: the

text: atmosphere

text: above

text: the

text: cloud/haze

text: deck.

text: If

text: the

text: cloud/haze

text: deck

text: is

text: at

text: high

text: altitudes,

text: then

text: the

text: column

text: optical

text: depth

text: of

text: absorbing

text: gases

text: that

text: direct

text: imaging

text: can

text: sample

text: is

text: small,

text: thereby

text: reducing

text: the

text: magnitude

text: of

text: their

text: spectral

text: features

text: in

text: these

text: planets’

text: reﬂected

text: light

text: spectra

text: (Marley

text: et

text: al.

text: Sudarsky

text: et

text: al.

text: Meanwhile,

text: if

text: the

text: cloud/haze

text: is

text: absorbing,

text: then

text: very

text: few

text: photons

text: can

text: sample

text: the

text: chemical

text: composition

text: of

text: the

text: atmosphere

text: and

text: escape.

text: Such

text: considerations

text: are

text: particularly

text: relevant,

text: as

text: NASA

text: is

text: now

text: studying

text: future

text: space

text: based

text: telescopes

text: capable

text: of

text: characterizing

text: extrasolar

text: planets

text: through

text: high-contrast

text: direct

text: imaging

text: (Krist

text: et

text: al.

text: Boccaletti

text: et

text: al.

text: Robinson

text: et

text: al.

text: An

text: alternative

text: set

text: of

text: compounds

text: that

text: are

text: known

text: to

text: form

text: hazes

text: in

text: planetary

text: atmospheres

text: are

text: those

text: derived

text: from

text: sulfur

text: chemistry.

text: In

text: oxidizing

text: atmospheres,

text: SO

text: OCS,

text: and

text: H

text: S

text: from

text: volcanic

text: outgassing

text: is

text: transformed

text: into

text: sulfuric

text: acid

text: through

text: photolysis

text: and

text: reactions

text: with

text: water.

text: Condensation

text: of

text: sulfuric

text: acid

text: can

text: then

text: form

text: clouds

text: and

text: hazes,

text: such

text: as

text: the

text: global

text: cloud

text: deck

text: of

text: Venus

text: (Hansen

text: Hovenier

text: and

text: the

text: Junge

text: layer

text: in

text: the

text: upper

text: stratosphere

text: of

text: Earth

text: (Junge

text: In

text: reducing

text: atmospheres,

text: photolysis

text: of

text: H

text: S

text: can

text: lead

text: to

text: the

text: formation

text: of

text: elemental

text: sulfur.

text: For

text: example,

text: the

text: existence

text: of

text: sulfur

text: aerosols

text: has

text: been

text: proposed

text: for

text: the

text: Venus

text: atmosphere,

text: especially

text: in

text: regions

text: depleted

text: in

text: oxidants

text: (Toon

text: et

text: al.

text: Zhang

text: et

text: al.

text: Precipitation

text: of

text: elemental

text: sulfur

text: is

text: the

text: leading

text: hypothesis

text: for

text: the

text: preservation

text: of

text: mass

text: independent

text: fractionation

text: of

text: sulfur

text: isotopes

text: found

text: in

text: sediments

text: on

text: Earth

text: older

text: than

text: billion

text: years

text: (Pavlov

text: Kasting

text: Hu

text: et

text: al.

text: showed

text: that

text: terrestrial

text: exoplanets

text: with

text: H

text: –dominated

text: atmospheres

text: could

text: be

text: enveloped

text: in

text: optically

text: thick

text: sulfur

text: hazes

text: resulting

text: from

text: volcanic

text: outgassing

text: of

text: H

text: S.

text: More

text: recent

text: work

text: by

text: Zahnle

text: et

text: al.

text: showed

text: that

text: rich

text: sulfur

text: photochemistry

text: could

text: potentially

text: take

text: place

text: in

text: the

text: atmospheres

text: of

text: temperate

text: giant

text: exoplanets

text: K

text: T

text: K),

text: generating

text: sulfur

text: allotropes

text: that

text: may

text: condense

text: to

text: eq

text: form

text: hazes

text: at

text: lower

text: temperatures.

text: As

text: these

text: planets

text: are

text: the

text: targets

text: of

text: current

text: and

text: planned

text: direct

text: imaging

text: campaigns,

text: it

text: is

text: essential

text: that

text: the

text: optical

text: characteristics

text: of

text: elemental

text: sulfur

text: hazes

text: be

text: known

text: in

text: order

text: to

text: inform

text: these

text: future

text: observations.

text: In

text: this

text: paper

text: we

text: investigate

text: the

text: geometric

text: albedo

text: spectra

text: of

text: giant

text: planets

text: hosting

text: elemental

text: sulfur

text: hazes

text: and

text: their

text: variations

text: with

text: haze

text: properties,

text: such

text: as

text: the

text: location

text: of

text: the

text: haze

text: in

text: the

text: atmosphere

text: and

text: the

text: haze

text: optical

text: depth.

text: We

text: also

text: address

text: the

text: observability

text: of

text: a

text: sulfur

text: haze

text: and

text: its

text: impact

text: on

text: inferences

text: of

text: atmospheric

text: composition

text: for


text: that the exoplanetary system is located within about 10 pc and that the planet–star angular separation is within the coronagraph inner and outer working angles. Lupu et al. (2016) expanded beyond these constraints and gave a ∼ preliminary target list for WFIRST (see their Figure 1), which showed several objects ( 1/3) within the temperature § range given in . Therefore, we can expect a signiﬁcant fraction of WFIRST targets to exhibit atmospheric conditions favorable to sulfur haze formation. We investigate the observability of a sulfur haze and its impact on the detectability of various atmospheric chemical species using an updated version of the noise model described in Robinson et al. (2016), which includes speciﬁc coronagraph designs that have been proposed for WFIRST, such as the Shaped–Pupil Coronagraph (SPC; capable of both broadband imaging and spectroscopy; Kasdin et al. 2004) and the Hybrid Lyot Coronagraph (HLC; imaging mode only; Trauger et al. 2016). The baseline telescope and instrument parameter values corresponding to these coronagraphs are diﬀerent from that of Robinson et al. (2016) (see their Table 3). We show these updated values in Table 1. Table 1. Changes to Telescope and Instrument Parameter Values Value from Value of Value of Parameter Description Robinson et al. (2016) Current Work (HLC) Current Work (SPC) D telescope diameter 2 m 2.4 m 2.4 m R − read noise counts per pixel 0.1 0.2 0.2 e T telescope and instrument throughput 0.05 0.074 0.037 θ (λ/D) coronagraph inner working angle 2 2.8 2.7 IWA Note—Only parameters updated for this work are shown. To fully characterize the impact of sulfur hazes on the geometric albedo spectrum, we position our exoplanetary system, composed of the K1IVe subgiant host star and the warm giant exoplanet at 8 pc distance so that the full wavelength range under consideration can be observed without going beyond the inner or outer working angles (the ∼ real Gamma Cephei Ab is located 13.79 pc away (Hatzes et al. 2003), such that wavelengths longer than 0.6 µm would not be observable due to the planet–star on–sky separation for those wavelengths being within WFIRST’s inner working angle). In addition, we assume that (1) the exozodi brightness is the same as zodiacal light in our own Solar System and (2) our model planet has the same radius as Jupiter. As the observation would likely be made while the exoplanet is at quadrature rather than at full phase, we multiply the contrast ratio calculated from the (full phase) geometric albedo by a correction factor 1/π, roughly simulating the drop in brightness at quadrature versus full phase. However, this does not take into account the changes in geometric albedo due to non–uniform scattering phase functions of the clouds and hazes in the atmosphere. It should also be noted that the real Gamma Cephei A is Draft version May 29, 2020 part of a binary system, which would reduce WFIRST’s ability to reach its designed planet–star raw contrasts. For Preprint typeset using LATEX style AASTeX6 v. 1.0 simplicity, we assume that our model host star does not have a stellar companion. SULFUR HAZES IN GIANT EXOPLANET ATMOSPHERES: IMPACTS ON REFLECTED LIGHT SPECTRA Peter Gao1,2,3, Mark S. Marley, and Kevin Zahnle 0 NASA Ames Research Center Moﬀett Field, CA 94035, USA Tyler D. Robinson4,5 0 NASA Ames Research Center Moﬀett Field, CA 94035, USA Department of Astronomy and Astrophysics University of California Santa Cruz Santa Cruz, CA 95064, USA ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: that

text: the

text: exoplanetary

text: system

text: is

text: located

text: within

text: about

text: pc

text: and

text: that

text: the

text: planet–star

text: angular

text: separation

text: is

text: within

text: the

text: coronagraph

text: inner

text: and

text: outer

text: working

text: angles.

text: Lupu

text: et

text: al.

text: expanded

text: beyond

text: these

text: constraints

text: and

text: gave

text: a

text: preliminary

text: target

text: list

text: for

text: WFIRST

text: (see

text: their

text: Figure

text: which

text: showed

text: several

text: objects

text: within

text: the

text: temperature

text: range

text: given

text: in

text: Therefore,

text: we

text: can

text: expect

text: a

text: signiﬁcant

text: fraction

text: of

text: WFIRST

text: targets

text: to

text: exhibit

text: atmospheric

text: conditions

text: favorable

text: to

text: sulfur

text: haze

text: formation.

text: We

text: investigate

text: the

text: observability

text: of

text: a

text: sulfur

text: haze

text: and

text: its

text: impact

text: on

text: the

text: detectability

text: of

text: various

text: atmospheric

text: chemical

text: species

text: using

text: an

text: updated

text: version

text: of

text: the

text: noise

text: model

text: described

text: in

text: Robinson

text: et

text: al.

text: which

text: includes

text: speciﬁc

text: coronagraph

text: designs

text: that

text: have

text: been

text: proposed

text: for

text: WFIRST,

text: such

text: as

text: the

text: Shaped–Pupil

text: Coronagraph

text: (SPC;

text: capable

text: of

text: both

text: broadband

text: imaging

text: and

text: spectroscopy;

text: Kasdin

text: et

text: al.

text: and

text: the

text: Hybrid

text: Lyot

text: Coronagraph

text: (HLC;

text: imaging

text: mode

text: only;

text: Trauger

text: et

text: al.

text: The

text: baseline

text: telescope

text: and

text: instrument

text: parameter

text: values

text: corresponding

text: to

text: these

text: coronagraphs

text: are

text: diﬀerent

text: from

text: that

text: of

text: Robinson

text: et

text: al.

text: (see

text: their

text: Table

text: We

text: show

text: these

text: updated

text: values

text: in

text: Table

text: Table

text: Changes

text: to

text: Telescope

text: and

text: Instrument

text: Parameter

text: Values

text: Value

text: from

text: Value

text: of

text: Value

text: of

text: Parameter

text: Description

text: Robinson

text: et

text: al.

text: Current

text: Work

text: (HLC)

text: Current

text: Work

text: (SPC)

text: D

text: telescope

text: diameter

text: m

text: m

text: m

text: R

text: read

text: noise

text: counts

text: per

text: pixel

text: e

text: T

text: telescope

text: and

text: instrument

text: throughput

text: θ

text: (λ/D)

text: coronagraph

text: inner

text: working

text: angle

text: IWA

text: Note—Only

text: parameters

text: updated

text: for

text: this

text: work

text: are

text: shown.

text: To

text: fully

text: characterize

text: the

text: impact

text: of

text: sulfur

text: hazes

text: on

text: the

text: geometric

text: albedo

text: spectrum,

text: we

text: position

text: our

text: exoplanetary

text: system,

text: composed

text: of

text: the

text: subgiant

text: host

text: star

text: and

text: the

text: warm

text: giant

text: exoplanet

text: at

text: pc

text: distance

text: so

text: that

text: the

text: full

text: wavelength

text: range

text: under

text: consideration

text: can

text: be

text: observed

text: without

text: going

text: beyond

text: the

text: inner

text: or

text: outer

text: working

text: angles

text: (the

text: real

text: Gamma

text: Cephei

text: Ab

text: is

text: located

text: pc

text: away

text: (Hatzes

text: et

text: al.

text: such

text: that

text: wavelengths

text: longer

text: than

text: µm

text: would

text: not

text: be

text: observable

text: due

text: to

text: the

text: planet–star

text: on–sky

text: separation

text: for

text: those

text: wavelengths

text: being

text: within

text: WFIRST’s

text: inner

text: working

text: angle).

text: In

text: addition,

text: we

text: assume

text: that

text: the

text: exozodi

text: brightness

text: is

text: the

text: same

text: as

text: zodiacal

text: light

text: in

text: our

text: own

text: Solar

text: System

text: and

text: our

text: model

text: planet

text: has

text: the

text: same

text: radius

text: as

text: Jupiter.

text: As

text: the

text: observation

text: would

text: likely

text: be

text: made

text: while

text: the

text: exoplanet

text: is

text: at

text: quadrature

text: rather

text: than

text: at

text: full

text: phase,

text: we

text: multiply

text: the

text: contrast

text: ratio

text: calculated

text: from

text: the

text: (full

text: phase)

text: geometric

text: albedo

text: by

text: a

text: correction

text: factor

text: roughly

text: simulating

text: the

text: drop

text: in

text: brightness

text: at

text: quadrature

text: versus

text: full

text: phase.

text: However,

text: this

text: does

text: not

text: take

text: into

text: account

text: the

text: changes

text: in

text: geometric

text: albedo

text: due

text: to

text: non–uniform

text: scattering

text: phase

text: functions

text: of

text: the

text: clouds

text: and

text: hazes

text: in

text: the

text: atmosphere.

text: It

text: should

text: also

text: be

text: noted

text: that

text: the

text: real

text: Gamma

text: Cephei

text: A

text: is

text: Draft

text: version

text: May

text: part

text: of

text: a

text: binary

text: system,

text: which

text: would

text: reduce

text: WFIRST’s

text: ability

text: to

text: reach

text: its

text: designed

text: planet–star

text: raw

text: contrasts.

text: For

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: v.

text: simplicity,

text: we

text: assume

text: that

text: our

text: model

text: host

text: star

text: does

text: not

text: have

text: a

text: stellar

text: companion.

text: SULFUR

text: HAZES

text: IN

text: GIANT

text: EXOPLANET

text: ATMOSPHERES:

text: IMPACTS

text: ON

text: REFLECTED

text: LIGHT

text: SPECTRA

text: Peter

text: Mark

text: S.

text: Marley,

text: and

text: Kevin

text: Zahnle

text: NASA

text: Ames

text: Research

text: Center

text: Moﬀett

text: Field,

text: CA

text: USA

text: Tyler

text: D.

text: NASA

text: Ames

text: Research

text: Center

text: Moﬀett

text: Field,

text: CA

text: USA

text: Department

text: of

text: Astronomy

text: and

text: Astrophysics

text: University

text: of

text: California

text: Santa

text: Cruz

text: Santa

text: Cruz,

text: CA

text: USA

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ ∈ = ≡ for some k [1,m] and x has another neighbor x V such that X (x ) m. This yields X (x ) + ≡ − ∈ − 1 = + 0 t 0 < t =2 2m k k 1 [0,m 1] modulo κ 2m 1. Since x cannot be excited when its color is b(κ) m, = = 2 it follows that X − + (x ) 0 and X − 2m. On the other hand, note that x has no neighbors t k 1 2 t k − − 1 besides x and x , neither of which blinks during the time interval [t k,t 1]. Hence it follows that =0 2 − = X − (x ) m. But then x excites x at time t k, so we must have X − + (x ) 2m, a contradiction. t k 1 1 2 t k 1 2 = ≥ Now suppose κ 2m for some integer m 2. We use a similar backtracking argument to rule out = − ﬂipping particles after a constant time depending on m. Recall that b(2m) m 1 is the blinking color = here. Suppose, for the sake of contradiction, we have a ﬂipping particle on edge x x at time t N for > = − − 1 2 ≥ some integer N 10m. We ﬁrst claim that {X (x ), X (x )} {m 1,2m 1} for some t 9m. Without t1 =1 −t1 2 1 (cid:54)= loss of generality, we may assume dX (x ,x ) m 1 and x is excited by a blinking neighbor x x = = − N =1 2− + 1 ∈ ≡ − + ≡0 −2 at time t N. So X (x ) m 1, X (x ) m 1 k for some k [1,m], and X (x ) 2m 2 k k 2 = N 0 N 1 − N 2 − modulo κ 2m. As before, none of x and x blink during time interval (N k,N], so backtracking k 2 = + 0 2 = = iterations gives X − − (x ) m 1 and X − − (x ) 0 and one more iteration gives X − − (x ) m =N k −2 1 N k 2= 2 − = − N k 1 1 and X − − (x ) 2m 1. This yields X − (x ) m 1 and X − − (x ) 2m 1 since x excites x at N =k 1 −2 N k 1 N k 2 2 1 2 time t : N k. This shows the claim. 1 Next, we show that as we further backtrack the dynamics, the same “opposite” local conﬁguration = − − + ≥ ≥ + {X (x ), X (x )} {m 1,2m 1} appears with period m 1. Since t 9m 6(m 1), this yields that t 1 t 2 1 the same pattern appears at least seven times during [0,t ]. We will then obtain a contradiction. For = 1 simplicity, we work with κ 6 case but generalizing to any even κ is immediate. In the following backtracking tables, time increases from right to left and each column gives a local conﬁguration on x , x and their neighbors. The ﬁrst column shows the opposite local conﬁgu- = 1 2 ration at t t . 1 x a a a a 0 1 2 3 4 x 2 1 0 5 1 ∗ (16) x 5 2 x b b b b 3 1 2 3 4 (cid:54)= ∗ = While x may or may not have a neighbor x x , observe that 2 in the above table contradicts 2 3 1 the transition rule, which is the case if x is absent. Hence x must be present and we can extend the 3 3 table as follows: x a a a a a a a a 0 1 2 3 4 5 6 7 8 x 2 1 0 5 5 1 (17) x 5 2 1 0 5 2 x b b b b b b b b 3 1 2 3 4 5 6 7 8 ∈ (cid:54)= = = with 2 {b ,b ,b }. But note that b 2 since otherwise we have b 0 and b 5, which contradict 2 3 4 2 4 5 = the transition rule since then x would excite x at the column containing b so that b 5. Notice 2 3 5 4 that the column containing a is identical to the ﬁrst one with the role of x and x exchanged. Hence 5 = + 1 2 the opposite local conﬁguration appears with period 4 m 1. ≥ + Since t 6(m 1), we further extend our backtracking table as follow: 1 11

font: LVRXUO+Fourier-Math-Symbols

text: for

font: TVTNZO+Utopia-Regular

text: some

text: k

font: GIRVSK+Utopia-Italic

text: and

text: x

text: has

text: another

text: neighbor

text: x

text: V

text: such

text: that

text: X

text: (x

text: m.

text: This

text: yields

text: X

text: (x

text: t

text: t

text: k

text: k

text: modulo

text: κ

font: LWJQKE+Fourier-Math-Letters-Italic

text: Since

text: x

text: cannot

text: be

text: excited

text: when

text: its

text: color

text: is

text: b(κ)

text: m,

text: it

text: follows

text: that

text: X

text: (x

text: and

text: X

text: On

text: the

text: other

text: hand,

text: note

text: that

text: x

text: has

text: no

text: neighbors

text: t

text: k

text: t

text: k

text: besides

text: x

text: and

text: x

text: neither

text: of

text: which

text: blinks

text: during

text: the

text: time

text: interval

text: [t

text: k,t

text: Hence

text: it

text: follows

text: that

text: X

text: (x

text: m.

text: But

text: then

text: x

text: excites

text: x

text: at

text: time

text: t

text: k,

text: so

text: we

text: must

text: have

text: X

text: (x

text: a

text: contradiction.

text: t

text: k

text: t

text: k

text: Now

text: suppose

text: κ

text: for

text: some

text: integer

text: m

text: We

text: use

text: a

text: similar

text: backtracking

text: argument

text: to

text: rule

text: out

text: ﬂipping

text: particles

text: after

text: a

text: constant

text: time

text: depending

text: on

text: m.

text: Recall

text: that

text: m

text: is

text: the

text: blinking

text: color

text: here.

text: Suppose,

text: for

text: the

text: sake

text: of

text: contradiction,

text: we

text: have

text: a

text: ﬂipping

text: particle

text: on

text: edge

text: x

text: x

text: at

text: time

text: t

text: N

text: for

text: some

text: integer

text: N

text: We

text: ﬁrst

text: claim

text: that

text: {X

text: (x

text: X

text: (x

text: {m

text: for

text: some

text: t

text: Without

text: loss

text: of

text: generality,

text: we

text: may

text: assume

text: dX

text: (x

text: ,x

text: m

text: and

text: x

text: is

text: excited

text: by

text: a

text: blinking

text: neighbor

text: x

text: x

text: N

text: 2−

text: ≡0

text: −2

text: at

text: time

text: t

text: N.

text: So

text: X

text: (x

text: m

text: X

text: (x

text: m

text: k

text: for

text: some

text: k

text: and

text: X

text: (x

text: k

text: k

text: N

text: N

text: N

text: modulo

text: κ

text: As

text: before,

text: of

text: x

text: and

text: x

text: blink

text: during

text: time

text: interval

text: (N

text: k,N],

text: so

text: backtracking

text: k

text: iterations

text: gives

text: X

text: (x

text: m

text: and

text: X

text: (x

text: and

text: one

text: more

text: iteration

text: gives

text: X

text: (x

text: m

text: =N

text: k

text: −2

text: N

text: k

text: N

text: k

text: and

text: X

text: (x

text: This

text: yields

text: X

text: (x

text: m

text: and

text: X

text: (x

text: since

text: x

text: excites

text: x

text: at

text: N

text: =k

text: −2

text: N

text: k

text: N

text: k

text: time

text: t

text: N

text: k.

text: This

text: shows

text: the

text: claim.

text: Next,

text: we

text: show

text: that

text: as

text: we

text: further

text: backtrack

text: the

text: dynamics,

text: the

text: same

text: “opposite”

text: local

text: conﬁguration

text: {X

text: (x

text: X

text: (x

text: {m

text: appears

text: with

text: period

text: m

text: Since

text: t

text: this

text: yields

text: that

text: t

text: t

text: the

text: same

text: pattern

text: appears

text: at

text: least

text: seven

text: times

text: during

text: We

text: will

text: then

text: obtain

text: a

text: contradiction.

text: For

text: simplicity,

text: we

text: work

text: with

text: κ

text: case

text: but

text: generalizing

text: to

text: any

text: even

text: κ

text: is

text: immediate.

text: In

text: the

text: following

text: backtracking

text: tables,

text: time

text: increases

text: from

text: right

text: to

text: left

text: and

text: each

text: column

text: gives

text: a

text: local

text: conﬁguration

text: on

text: x

text: x

text: and

text: their

text: neighbors.

text: The

text: ﬁrst

text: column

text: shows

text: the

text: opposite

text: local

text: conﬁgu-

text: ration

text: at

text: t

text: t

text: x

text: a

text: a

text: a

text: a

text: x

text: x

text: x

text: b

text: b

text: b

text: b

text: While

text: x

text: may

text: or

text: may

text: not

text: have

text: a

text: neighbor

text: x

text: x

text: observe

text: that

text: in

text: the

text: above

text: table

text: contradicts

text: the

text: transition

text: rule,

text: which

text: is

text: the

text: case

text: if

text: x

text: is

text: absent.

text: Hence

text: x

text: must

text: be

text: present

text: and

text: we

text: can

text: extend

text: the

text: table

text: as

text: follows:

text: x

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: x

text: x

text: x

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: b

text: with

text: {b

text: ,b

text: ,b

text: But

text: note

text: that

text: b

text: since

text: otherwise

text: we

text: have

text: b

text: and

text: b

text: which

text: contradict

text: the

text: transition

text: rule

text: since

text: then

text: x

text: would

text: excite

text: x

text: at

text: the

text: column

text: containing

text: b

text: so

text: that

text: b

text: Notice

text: that

text: the

text: column

text: containing

text: a

text: is

text: identical

text: to

text: the

text: ﬁrst

text: one

text: with

text: the

text: role

text: of

text: x

text: and

text: x

text: exchanged.

text: Hence

text: the

text: opposite

text: local

text: conﬁguration

text: appears

text: with

text: period

text: m

text: Since

text: t

text: we

text: further

text: extend

text: our

text: backtracking

text: table

text: as

text: follow:


text: We ﬁrst verify that the chunked increments ζ ’s are i.i.d. with ﬁnite moments. i Proposition 3.3. ζ ’s are i.i.d. with mean zero and ﬁnite variance. i Proof. That ζ ’s are identically distributed is clear from the construction, and that they have mean i zero follows from symmetry. Note that each random interval I has geometrically distributed length. i Since dX is uniformly bounded, it follows that each ζ has ﬁnite variance. 5κ i It remains to show the independence. To this end, we trace information propagation back in = = ∈ time from t 5κ to t 0. Namely, for each x (cid:90), we are going to deﬁne an ascending sets of integers ⊆ ⊆ ··· ⊆ = N(0)(x) N(1)(x) N(5κ)(x) recursively as follows: N(0)(x) {x} and + = ∪ ∈ = − − N(i 1)(x) N(i)(x) {y (cid:90) : y excites any site in N(i)(x) at time t 5κ i 1}. (24) In words, N(5κ)(x) contains all sites whose initial color could affect X (x). Note that X (x) is com- 5κ 5κ = pletely determined by the initial colors of sites in N(5κ)(x). Moreover, since dX (B , A + ) 0 for all ∈ 5κ i i 1 ∈ i (cid:90), ζ depends only on the initial colors in I . Hence it is enough to show that for any i (cid:90) and ∈ i ⊆ i x I , we have N(5κ)(x) I . i i = ∈ Let I [A ,B ]’s be the random intervals as before. Fix i (cid:90). By construction, the six sites in + i i i = = ∈ [A , A 6] have initial color 0 so they all have color b(κ) at time t b(κ). So X (x) 0 for all x i i + κ [A , A 5] and none of the x’s gets excited during time interval [0,κ]. Repeating the same argument, i i we see that the site A is not excited during the time interval [0,5κ]. By symmetry and translation i = − ∈ invariance, the same observation holds for B A 1 and for all i (cid:90). i i ∈ ∈ ∩ (cid:54)= (cid:59) (cid:48) Now ﬁx i (cid:90) and x I . Suppose for contrary that I − N(5κ)(x) and let x be in the intersec- i i 1 (cid:48) ≤ < ≤ tion. Since every site can only excite its neighbors at any given time and since x B − A x, this i 1 i implies that A is excited by B − at some time during [0,5κ], which contradicts the observation in the i i 1∩ = (cid:59) ∩ = (cid:59) ⊆ previous paragraph. Hence I − N(5κ)(x) . Similarly N(5κ)(x) I + . This shows N(5κ)(x) I i 1 i 1 i as desired. Next, we recap some results on survival probabilities of random walks with i.i.d. increments. For ··· = ∈ its ∞statement, let X1,X2 , be i.i.d. integer v=alued+random va≥riables suc=h that (cid:69)[X1] 0 and (cid:69)[X21] ≥ − (0, ). Deﬁne a random walk (Sn)n 0 by Sn Sn 1 Xn for n 1 and S0 x is to be speciﬁed. Deﬁne survival probabilities = ≥ ≥ ··· ≥ | = ∀ ≥ Q(x,t) (cid:80)(S1 0,S2 0, ,St 0 S0 x) x,t 0. (25) ≥ + ≥ Note that Q(x,t) Q(x,t 1) for all x,t 0. We are interested in the asymptotic behavior of the se- ≥ ≥ quences (Q(x,t))t 0 for each x 0. This is closely related to the asymptotic behavior of their generat- ing functions (cid:88) (cid:101) = Qx(u) : ≥ utQ(x,t) (26) t 0 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: ﬁrst

text: verify

text: that

text: the

text: chunked

text: increments

text: ζ

text: ’s

text: are

text: i.i.d.

text: with

text: ﬁnite

text: moments.

text: i

text: Proposition

font: VPHQXA+Utopia-Bold

text: ζ

text: ’s

text: are

text: i.i.d.

text: with

text: mean

text: zero

text: and

text: ﬁnite

text: variance.

text: i

text: Proof.

text: That

text: ζ

text: ’s

text: are

text: identically

text: distributed

text: is

text: clear

text: from

text: the

text: construction,

text: and

text: that

text: they

text: have

text: mean

text: i

text: zero

text: follows

text: from

text: symmetry.

text: Note

text: that

text: each

text: random

text: interval

text: I

text: has

text: geometrically

text: distributed

text: length.

text: i

text: Since

text: dX

text: is

text: uniformly

text: bounded,

text: it

text: follows

text: that

text: each

text: ζ

text: has

text: ﬁnite

text: variance.

text: 5κ

text: i

text: It

text: remains

text: to

text: show

text: the

text: independence.

text: To

text: this

text: end,

text: we

text: trace

text: information

text: propagation

text: back

text: in

text: time

text: from

text: t

text: 5κ

text: to

text: t

text: Namely,

text: for

text: each

text: x

text: we

text: are

text: going

text: to

text: deﬁne

text: an

text: ascending

text: sets

text: of

text: integers

text: recursively

text: as

text: follows:

text: {x}

text: and

text: N(i

text: N(i)(x)

text: {y

text: y

text: excites

text: any

text: site

text: in

text: N(i)(x)

text: at

text: time

text: t

text: 5κ

text: i

text: In

text: words,

text: contains

text: all

text: sites

text: whose

text: initial

text: color

text: could

text: affect

text: X

text: (x).

text: Note

text: that

text: X

text: (x)

text: is

text: com-

text: 5κ

text: 5κ

text: pletely

text: determined

text: by

text: the

text: initial

text: colors

text: of

text: sites

text: in

text: Moreover,

text: since

text: dX

text: (B

text: A

text: for

text: all

text: 5κ

text: i

text: i

text: i

text: ζ

text: depends

text: only

text: on

text: the

text: initial

text: colors

text: in

text: I

text: Hence

text: it

text: is

text: enough

text: to

text: show

text: that

text: for

text: any

text: i

text: and

text: i

text: i

text: x

text: I

text: we

text: have

text: I

text: i

text: i

text: Let

text: I

text: [A

text: ,B

text: ]’s

text: be

text: the

text: random

text: intervals

text: as

text: before.

text: Fix

text: i

text: By

text: construction,

text: the

text: six

text: sites

text: in

text: i

text: i

text: i

text: [A

text: A

text: have

text: initial

text: color

text: so

text: they

text: all

text: have

text: color

text: b(κ)

text: at

text: time

text: t

text: b(κ).

text: So

text: X

text: (x)

text: for

text: all

text: x

text: i

text: i

text: κ

text: [A

text: A

text: and

text: of

text: the

text: x’s

text: gets

text: excited

text: during

text: time

text: interval

text: Repeating

text: the

text: same

text: argument,

text: i

text: i

text: we

text: see

text: that

text: the

text: site

text: A

text: is

text: not

text: excited

text: during

text: the

text: time

text: interval

text: By

text: symmetry

text: and

text: translation

text: i

text: invariance,

text: the

text: same

text: observation

text: holds

text: for

text: B

text: A

text: and

text: for

text: all

text: i

text: i

text: i

text: Now

text: ﬁx

text: i

text: and

text: x

text: I

text: Suppose

text: for

text: contrary

text: that

text: I

text: and

text: let

text: x

text: be

text: in

text: the

text: intersec-

text: i

text: i

text: tion.

text: Since

text: every

text: site

text: can

text: only

text: excite

text: its

text: neighbors

text: at

text: any

text: given

text: time

text: and

text: since

text: x

text: B

text: A

text: x,

text: this

text: i

text: i

text: implies

text: that

text: A

text: is

text: excited

text: by

text: B

text: at

text: some

text: time

text: during

text: which

text: contradicts

text: the

text: observation

text: in

text: the

text: i

text: i

text: 1∩

text: previous

text: paragraph.

text: Hence

text: I

text: Similarly

text: I

text: This

text: shows

text: I

text: i

text: i

text: i

text: as

text: desired.

text: Next,

text: we

text: recap

text: some

text: results

text: on

text: survival

text: probabilities

text: of

text: random

text: walks

text: with

text: i.i.d.

text: increments.

text: For

text: its

text: ∞statement,

text: let

text: be

text: i.i.d.

text: integer

text: v=alued+random

text: va≥riables

text: suc=h

text: that

text: and

text: Deﬁne

text: a

text: random

text: walk

text: (Sn)n

text: by

text: Sn

text: Sn

text: Xn

text: for

text: n

text: and

text: x

text: is

text: to

text: be

text: speciﬁed.

text: Deﬁne

text: survival

text: probabilities

text: Q(x,t)

text: ,St

text: x)

text: x,t

text: Note

text: that

text: Q(x,t)

text: Q(x,t

text: for

text: all

text: x,t

text: We

text: are

text: interested

text: in

text: the

text: asymptotic

text: behavior

text: of

text: the

text: se-

text: quences

text: (Q(x,t))t

text: for

text: each

text: x

text: This

text: is

text: closely

text: related

text: to

text: the

text: asymptotic

text: behavior

text: of

text: their

text: generat-

text: ing

text: functions

text: Qx(u)

text: utQ(x,t)

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:37) as u 1 by a Tauberian theorem (see e.g., Theorem 5 of Chapter XIII in [6]). ≥ ∈ Lemma 3.2. For each x 0 and u [0,1), we have (cid:101) (cid:101) ≤ + Qx(u) (x 1)Q0(u). (27) Proof. Let = = | = M M(x,t) : m≤ i≤n {Sk S0 x} (28) 0 k t and = = ≤ ≤ = τ τ(x,t) : min{0 k t : Sk M(x,t)}. (29) ≤ ≤ ≤ ≤ For each 0 i x and 0 j t, den(cid:161)ote (cid:162) = ≥ ≥ ··· ≥ = | = a(i, j) (cid:80) S0 i,S1 i, ,Sj−1 i, Sj i S0 x . (30) ≤ ≤ ∈ Note that for each 0 i x and u [0,1), we have (cid:88) (cid:88) ∞ ∞ ≤ = = ≥ ≤ a(i, j)uj a(i, j) (cid:80)[M(x,t) i for some t 0] 1. (31) = = j 0 j 0 Now by Markov property of Sn, (cid:88) (cid:88) = ≥ = x t − Q(x,t) (cid:80)(M(x,t) 0) a(i, j)Q(0,t j). (32) = = i 0 j 0 ≥ Multiplying by ut and summing over all t 0, we obtain 15

text: as

text: u

text: by

text: a

text: Tauberian

text: theorem

text: (see

text: e.g.,

text: Theorem

text: of

text: Chapter

text: XIII

text: in

text: Lemma

text: For

text: each

text: x

text: and

text: u

text: we

text: have

font: ILXBYS+Fourier-Math-Extension

text: Qx(u)

text: (x

text: Proof.

text: Let

text: M

text: M(x,t)

text: m≤

text: i≤n

text: {Sk

text: x}

text: k

text: t

text: and

text: τ

text: τ(x,t)

text: k

text: t

text: Sk

text: M(x,t)}.

text: For

text: each

text: i

text: x

text: and

text: j

text: t,

text: a(i,

text: j)

text: i,

text: i,

text: Sj

text: i

text: x

text: Note

text: that

text: for

text: each

text: i

text: x

text: and

text: u

text: we

text: have

text: a(i,

text: j)uj

text: a(i,

text: j)

text: i

text: for

text: some

text: t

text: j

text: j

text: Now

text: by

text: Markov

text: property

text: of

text: Sn,

text: x

text: t

text: Q(x,t)

text: a(i,

text: j).

text: i

text: j

text: Multiplying

text: by

text: ut

text: and

text: summing

text: over

text: all

text: t

text: we

text: obtain


text: (cid:88) (cid:88) ∞ ∞ ≤ − + (cid:98) (cid:99) F(u) (cid:88) e cs (cid:88) unQ(b s, n/2µ ) (45) =− = s ∞b n∞0 ≤ − (cid:98) (cid:99) + (cid:98) (cid:99) (cid:88) e cs (cid:88) u n/2µ Q(b s, n/2µ ) (46) =− = s ∞b n∞0 ≤ − + + e cs(cid:88) (2µ 1)utQ(b s,t) (47) =− = (cid:101) s b t∞0 = + −(cid:195) (cid:33) (2µ 1) =− e c(cid:88)sQb+s(u) (48) (cid:101) s b ∞ ≤ + − + + (2µ 1)Q (u) e cs(b s 1) (49) 0 =− s b ■ This shows (42) as desired, and completes the proof of the assertion. 4. Tournament expansion of the FCA dynamics on (cid:90) Particle system expansion of the FCA dynamics has been successful in getting an upper bound on the (cid:54)= + probability (cid:80)(X (x) X (x 1)) of local disagreement at time t. We handled the local dependence on t t the edge particles developed during the burn-in period by coarse-graining our associated locally cor- related random walk into a “chunked” random walk by giving up bounded increments but retrieving independent increments in return. However, this technique does not readily produce the matching = lower bound; see Section 6, wherein we obtain precise asymptotics for κ 3. In this section, we in- troduce another comparison process for FCA with a slightly different perspective, which we call the tournament expansion. We originally developed the tournament expansion with Gravner in [10] in order to better understand the CCA and GHM dynamics, and a corresponding expansion of the FCA dynamics has some extra details. We begin with an instructive toy model called a tournament process. Instead of coloring nodes = → of a graph G (V,E) by using m+od κ colors, consider a map Rk0 : V (cid:90) called a ranking on G. The transition map from time t to t 1 is given by = | ∈ ∪ + Rkt 1(x) max{Rkt(y) y N(x) {x}}. (50) (cid:55)→ + In words, in each transition Rkt Rkt 1, each node simultaneously copies the maximum rank among it and its neighbors. Observe that if G is ﬁnite, then for any initial ranking Rk on G there is a global 0 maximum, and each node will eventually adopt the global maximum. In general, locally maximum rank propagates with unit speed across the graph until it is overcome by a higher ranker. The basic idea of relating a similar process to the FCA dynamics on (cid:90) is the following: we con- struct an accompanying tournament-like system where each site increases its rank if and only if it gets excited. To give a detailed construction, we ﬁrst deﬁne a path integral of the 1-form dX . A directed t 18

text: F(u)

text: e

text: cs

text: unQ(b

text: s,

text: s

text: ∞b

text: e

text: cs

text: u

text: Q(b

text: s,

text: s

text: ∞b

text: e

text: s,t)

text: s

text: b

text: e

text: s

text: b

text: (u)

text: e

text: cs(b

text: s

text: s

text: b

text: This

text: shows

text: as

text: desired,

text: and

text: completes

text: the

text: proof

text: of

text: the

text: assertion.

text: Tournament

text: expansion

text: of

text: the

text: FCA

text: dynamics

text: on

text: Particle

text: system

text: expansion

text: of

text: the

text: FCA

text: dynamics

text: has

text: been

text: successful

text: in

text: getting

text: an

text: upper

text: bound

text: on

text: the

text: probability

text: (x)

text: X

text: (x

text: of

text: local

text: disagreement

text: at

text: time

text: t.

text: We

text: handled

text: the

text: local

text: dependence

text: on

text: t

text: t

text: the

text: edge

text: particles

text: developed

text: during

text: the

text: burn-in

text: period

text: by

text: coarse-graining

text: our

text: associated

text: locally

text: cor-

text: related

text: random

text: walk

text: into

text: a

text: “chunked”

text: random

text: walk

text: by

text: giving

text: up

text: bounded

text: increments

text: but

text: retrieving

text: independent

text: increments

text: in

text: return.

text: However,

text: this

text: technique

text: does

text: not

text: readily

text: produce

text: the

text: matching

text: lower

text: bound;

text: see

text: Section

text: wherein

text: we

text: obtain

text: precise

text: asymptotics

text: for

text: κ

text: In

text: this

text: section,

text: we

text: in-

text: troduce

text: another

text: comparison

text: process

text: for

text: FCA

text: with

text: a

text: slightly

text: different

text: perspective,

text: which

text: we

text: call

text: the

text: tournament

text: expansion.

text: We

text: originally

text: developed

text: the

text: tournament

text: expansion

text: with

text: Gravner

text: in

text: in

text: order

text: to

text: better

text: understand

text: the

text: CCA

text: and

text: GHM

text: dynamics,

text: and

text: a

text: corresponding

text: expansion

text: of

text: the

text: FCA

text: dynamics

text: has

text: some

text: extra

text: details.

text: We

text: begin

text: with

text: an

text: instructive

text: toy

text: model

text: called

text: a

text: tournament

text: process.

text: Instead

text: of

text: coloring

text: nodes

text: of

text: a

text: graph

text: G

text: (V,E)

text: by

text: using

text: m+od

text: κ

text: colors,

text: consider

text: a

text: map

text: V

text: called

text: a

text: ranking

text: on

text: G.

text: The

text: transition

text: map

text: from

text: time

text: t

text: to

text: t

text: is

text: given

text: by

text: Rkt

text: max{Rkt(y)

text: y

text: N(x)

text: {x}}.

text: In

text: words,

text: in

text: each

text: transition

text: Rkt

text: Rkt

text: each

text: node

text: simultaneously

text: copies

text: the

text: maximum

text: rank

text: among

text: it

text: and

text: its

text: neighbors.

text: Observe

text: that

text: if

text: G

text: is

text: ﬁnite,

text: then

text: for

text: any

text: initial

text: ranking

text: Rk

text: on

text: G

text: there

text: is

text: a

text: global

text: maximum,

text: and

text: each

text: node

text: will

text: eventually

text: adopt

text: the

text: global

text: maximum.

text: In

text: general,

text: locally

text: maximum

text: rank

text: propagates

text: with

text: unit

text: speed

text: across

text: the

text: graph

text: until

text: it

text: is

text: overcome

text: by

text: a

text: higher

text: ranker.

text: The

text: basic

text: idea

text: of

text: relating

text: a

text: similar

text: process

text: to

text: the

text: FCA

text: dynamics

text: on

text: is

text: the

text: following:

text: we

text: con-

text: struct

text: an

text: accompanying

text: tournament-like

text: system

text: where

text: each

text: site

text: increases

text: its

text: rank

text: if

text: and

text: only

text: if

text: it

text: gets

text: excited.

text: To

text: give

text: a

text: detailed

text: construction,

text: we

text: ﬁrst

text: deﬁne

text: a

text: path

text: integral

text: of

text: the

text: dX

text: A

text: directed

text: t


text: = walk W(cid:126) from a site x to another site y in (cid:90) is a sequence (x )l= of adjacent sites in (cid:90) such that x x = i i 0 0 and x y. Given a κ-color FCA dynamic (X ) ≥ on (cid:90), the path integral of dX over a directed walk = l t t 0 t W(cid:126) (xi)li=0 from x to y is deﬁned by (cid:90) (cid:88) − y = l 1 dX : dX (x ,x + ). (51) t = t i i 1 x i 1 Note that since (cid:90) is simply connected and dX is anti-symmetric, any contour integral of dX is zero. t t Hence the path integral does not depend on the choice of a directed walk between the endpoints. Now let (X ) ≥ be a κ-color FCA trajectory. We deﬁne its tournament expansion by a sequence of t t 0 ≥ rankings (rkt)t t0 deﬁned as follows: set the ra(cid:88)nk of the origin at time t by − = = t 1 rkt(0) net(0) = 1(0 is excited at time s) (52) s 0 ≥ − for all t 0, and then extend to all sites by a path integra(cid:90)l of dX : t − = − x rkt(x) rkt(0) dXt (53) 0 The minus sign in front of the path integral reﬂects the fact that in the (inhibitory) FCA dynamics, nodes tend to adjust their phases toward those lagging the most so we are giving higher ranks to the = = + = = + lagging nodes. For instance, if κ 6 and X (x) 3 and X (x 1) b(6) 2, then x 1 excites x at time + t t t so we give x 1 a higher rank than x at time t, so it is natural to require the following local relation + − = − + > rkt(x 1) rkt(x) dXt(x,x 1) 0, (54) which is incorporated in our construction. Our ﬁrst observation about tournament expansion is that all sites indeed increase their rank by 1 if and only if they get excited: = ≥ Proposition 4.∈1. Let (Xt≥)t≥0 and (rkt)t≥0 be as before, and suppose no particle ﬂips after time t t0 0. For any x, y (cid:90)(cid:90) and t t(cid:90), we have 0 y − y = − + dXt 1 dXt 1(x is excited at time t) 1(y is excited at time t). (55) x x Furthermore, we have − = + rkt 1(x) rkt(x) 1(x is excited at time t). (56) 19

text: walk

font: MMXDPU+Fourier-Math-Letters

text: from

text: a

text: site

text: x

text: to

text: another

text: site

text: y

text: in

font: DEWUCV+Fourier-Math-BlackBoard

text: is

text: a

text: sequence

text: (x

text: )l=

text: of

text: adjacent

text: sites

text: in

text: such

text: that

text: x

text: x

text: i

text: i

text: and

text: x

text: y.

text: Given

text: a

text: κ-color

text: FCA

text: dynamic

text: (X

text: on

text: the

text: path

text: integral

text: of

text: dX

text: over

text: a

text: directed

text: walk

text: l

text: t

text: t

text: t

text: from

text: x

text: to

text: y

text: is

text: deﬁned

text: by

text: y

text: l

text: dX

text: dX

text: (x

text: ,x

text: t

text: t

text: i

text: i

text: x

text: i

text: Note

text: that

text: since

text: is

text: simply

text: connected

text: and

text: dX

text: is

text: anti-symmetric,

text: any

text: contour

text: integral

text: of

text: dX

text: is

text: zero.

text: t

text: t

text: Hence

text: the

text: path

text: integral

text: does

text: not

text: depend

text: on

text: the

text: choice

text: of

text: a

text: directed

text: walk

text: between

text: the

text: endpoints.

text: Now

text: let

text: (X

text: be

text: a

text: κ-color

text: FCA

text: trajectory.

text: We

text: deﬁne

text: its

text: tournament

text: expansion

text: by

text: a

text: sequence

text: of

text: t

text: t

text: rankings

text: (rkt)t

text: deﬁned

text: as

text: follows:

text: set

text: the

text: of

text: the

text: origin

text: at

text: time

text: t

text: by

text: t

text: is

text: excited

text: at

text: time

text: s)

text: s

text: for

text: all

text: t

text: and

text: then

text: extend

text: to

text: all

text: sites

text: by

text: a

text: path

text: of

text: dX

text: t

text: x

text: rkt(x)

text: dXt

text: The

text: minus

text: sign

text: in

text: front

text: of

text: the

text: path

text: integral

text: reﬂects

text: the

text: fact

text: that

text: in

text: the

text: (inhibitory)

text: FCA

text: dynamics,

text: nodes

text: tend

text: to

text: adjust

text: their

text: phases

text: toward

text: those

text: lagging

text: the

text: most

text: so

text: we

text: are

text: giving

text: higher

text: ranks

text: to

text: the

text: lagging

text: nodes.

text: For

text: instance,

text: if

text: κ

text: and

text: X

text: (x)

text: and

text: X

text: (x

text: then

text: x

text: excites

text: x

text: at

text: time

text: t

text: t

text: t

text: so

text: we

text: give

text: x

text: a

text: higher

text: rank

text: than

text: x

text: at

text: time

text: t,

text: so

text: it

text: is

text: natural

text: to

text: require

text: the

text: following

text: local

text: relation

text: rkt(x

text: rkt(x)

text: dXt(x,x

text: which

text: is

text: incorporated

text: in

text: our

text: construction.

text: Our

text: ﬁrst

text: observation

text: about

text: tournament

text: expansion

text: is

text: that

text: all

text: sites

text: indeed

text: increase

text: their

text: rank

text: by

text: if

text: and

text: only

text: if

text: they

text: get

text: excited:

text: Proposition

text: Let

text: and

text: be

text: as

text: before,

text: and

text: suppose

text: no

text: particle

text: ﬂips

text: after

text: time

text: t

text: For

text: any

text: x,

text: y

text: and

text: t

text: we

text: have

text: y

text: y

text: dXt

text: dXt

text: is

text: excited

text: at

text: time

text: t)

text: is

text: excited

text: at

text: time

text: t).

text: x

text: x

text: Furthermore,

text: we

text: have

text: rkt

text: rkt(x)

text: is

text: excited

text: at

text: time

text: t).