text: The evolution of the Sun’s birth cluster and the search for the solar siblings 3 Table 1. Parameters of the Milky Way model potential. Table 2. Parameters of the composite Galaxy model potential. Axisymmetric component Main spiral structure − − Mass of the bulge (Mb) 1.41 × 1010 M(cid:12) Pattern speed (Ωsp1) 26 km s 1 kpc 1− − Scale length bulge (b1) 0.38 kpc Amplitude (Asp1) 650◦–1300 km2 s 2 kpc 1 Disk mass (Md) 8.56 × 1010 M(cid:12) Pitch angle (i1) −7◦ Scale length disk 1 (a2) 5.31 kpc Present-day orientation 20 Scale length disk 2 (b2) 0.25 kpc Initial orientation 171◦ Halo mass (Mh) 1.07 × 1011 M(cid:12) Secondary spiral structure SPSacetamtleeir-nmlesnapjgoetrehdahx(aiΩlsob((aaar)3)) Central431B2.01a–k2r7p0kcpkcm s−1 kpc−1 PAPPairmtetctspeherlnianttnu-sgddplaeeeye((dAoi2r(siΩ)pen2stp)a2ti)on 10−2.5281.08A4◦◦kspm1 s−1 kpc−1 Axis ratio (b/a) 0.37 Initial orientation 158◦ MPInriaetssiaseln(otM-rdibeanyartoa)triioenntation 921.◦08◦−×116079◦–1.4 × 1010 M(cid:12) PSaetmteir-nmsapjoeredax(iΩsb(aar)) Ba43r.01k2mkpsc−1 kpc−1 Spiral arms − − Axis ratio (b/a) 0.37 PLNSapouticmtreuarbslneabrsmepogpefilensidntpuii(ndrΩagels((paARr)msspps))(m) 1326.,5514–023–◦10k1pkc0m0skm12ksp−c2 k1pc−1 MSPIntriraetessinasegln(totMh-rdiobeafnyarttoah)treiioebnnatra(ti(cid:15)obn) 9021..◦083◦× 109 M(cid:12) Pitch angle (i) 12.8 Scale length (RΣ) 2.5◦kpc Present-day orientation 20 The Galactic bar The central bar is modelled with a Ferrers po- ◦ ◦ Initial orientation 103 − 173 tential (Ferrers 1877) which describes the potential associated to an elliptical distribution of mass. In an inertial frame located at the Galactic centre, the bar rotates with a constant pattern speed of ruption time of the cluster which can be a substantial fraction of the − − 40–70 km s 1 kpc 1 (Mart´ınez-Barbosa et al. 2015). This range lifetime of the Sun (Sect. ). of angular velocities places the Outer Lindblad resonance of the Stellar evolution Predicting the observations of the Sun’s birth bar (OLR ) at 10–5 kpc from the Galactic centre. In the same cluster by Gaia requires that we account for the mass-dependent bar inertial frame, the present-day orientation of the bar with respect evolution of the solar siblings, in order to obtain the correct present- ◦ to the negative x-axis is 20 (Pichardo et al. 2004, 2012; Romero- day apparent magnitudes and colours which are used to predict Go´mez et al. 2011, and references therein). In the left panel of Fig. which stars end up in the Gaia catalogue. This prediction also re- 1 we show the present-day orientation of the Galactic bar. In Table quires us to account for interstellar extinction and reddening for 1 we show the parameters used in this study. For further details on which we employ a Galactic extinction model (Sects. , ). the choice of the bar parameters, we refer the reader to Mart´ınez- These elements are described in more detail in the subsequent sub- Barbosa et al. (2015). sections. The spiral arms We model the spiral arms as periodic perturba- tions of the axisymmetric potential (tight winding approximation, 2.1 Galactic model Lin et al. 1969). The spiral arms rotate with a constant pattern − − speed of 15–30 km s 1 kpc 1 (Mart´ınez-Barbosa et al. 2015). This We use an analytical potential to model the Milky Way. This range of values places the co-rotation resonance of these structures potential contains two parts: an axisymmetric component, which (CR ) at 14–7 kpc from the Galactic centre. We assume that the sp corresponds to a bulge, disk and a dark matter halo, and a non- Galaxy has two or four non-transient spiral arms with the same am- axisymmetric component which includes a central bar and spiral plitude. A schematic picture of the present-day conﬁguration of the arms. Bellow we explain these components in more detail. spiral arms is shown in the left and middle panels of Fig. 1. The Axisymmetric component We use the potential of Allen & San- parameters of the spiral arms used in this study are listed in Table tilla´n (1991) to model the axisymmetric component of the Galaxy. 1. For further details on the choice of these parameters, we refer the In this approach, the bulge is modelled with a Plummer (Plum- reader to Mart´ınez-Barbosa et al. (2015). mer 1911) potential; the disk is modelled with a Miyamoto-Nagai (Miyamoto & Nagai 1975) potential and the dark matter halo with Initial orientation of the bar and spiral arms The orientation a logarithmic potential. The parameters used to model the axisym- of the bar and spiral arms at the beginning of the simulations (i.e metric component of the Galaxy are listed in table 1. 4.6 Gyr ago) are deﬁned through the following equations: The model introduced by Allen & Santilla´n (1991) predicts a − rotational velocity of 220 km s 1 at the solar radius, which does not match with the recent observational estimates (see e.g McMil- ϕ = ϕ (0) − Ω t , b b bar lan 2011; Reid et al. 2014). However, J´ılkova´ et al. (2012) did not ϕ = ϕ (0) − Ω t . (1) ﬁnd substantial variations in the orbits of open clusters when us- s s sp ing different models of the axisymmetric structure of the Galaxy. Therefore, we do not expect that the evolution of the Sun’s birth Here ϕ (0) is the present-day orientation of the bar. We as- b cluster and the present-day distribution of solar siblings will be af- sume that the spiral arms start at the tips of the bar, i.e. ϕ (0) = s fected due to the choice of the axisymmetric potential model. ϕ (0) (see Fig. 1). The time, t = 4.6 Gyr corresponds to the age b (cid:13)c 2002 RAS, MNRAS 000, 1–13 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 8 C.A. Mart´ınez-Barbosa et al. G = 20 (Lindegren et al. 2008), where G denotes the apparent d [kpc] d [kpc] magnitude in the white light band of used for the astrometric mea- 700.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 400.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 surements, covering the wavelength range ∼ 350–1050 nm (see 60 a. 35 b. Jordi et al. 2010). Parallaxes ((cid:36)) and proper motions (µ) will be 30 50 measured with accuracies ranging from 10 to 30 micro-arcseconds 25 (µas) for stars brighter than 15 mag, and from 130 to 600 µas for 40 20 sources at G = 20. For ∼ 100 million stars brighter than G = 16 30 15 Gaia will also measure radial velocities (V ), with accuracies rang- 20 − r 10 ing from 1 to 15 km s 1. Gaia will not only revolutionize the cur- 10 5 rent view of the Galaxy but will generate a data set which should 0 0 in principle allow for a search for solar siblings even far away from 5 10 15 20 5 10 15 20 G [mag] G [mag] the Sun. In this section we use our simulations to predict the number d [kpc] d [kpc] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 of solar siblings that will be seen by Gaia, and to study their distri- 40 40 c. d. bution in the space of parallax, proper motion, and radial velocity 35 35 with the aim of establishing efﬁcient ways of selecting solar sibling 30 30 candidates from the Gaia catalogue. 25 25 5.1 The solar siblings in the Gaia catalogue 20 20 15 15 We ﬁrst compare the predicted Gaia survey of the solar sib- 10 10 lings with predictions by Bland-Hawthorn et al. (2010), who con- 5 5 sidered the prospects for a survey like GALAH (De Silva et al. 0 0 2015) to varying limiting magnitudes. Following Bland-Hawthorn 5 10 15 20 5 10 15 20 G [mag] G [mag] et al. (2010) we broadly distinguish the possible present-day phase conﬁgurations for the solar siblings by referring to the cases shown Figure 5. Median number of solar siblings that Gaia is predicted to observe, in the panels of Fig. 3 as model a and model b (compact spatial as a function of their heliocentric distances d (red histograms) and G mag- distribution of solar siblings), model c (spatial distribution of solar nitudes (blue histograms). The letters in the left corner correspond to the siblings obtained with the 2 + 2 composite model) and model d distributions shown in Fig. 3. The vertical dotted black lines in each panel (highly dispersed spatial distribution of solar siblings). represent the limiting magnitude of the GALAH survey, G ∼ 14 mag. In predicting the observed kinematic properties of the solar siblings we want to account for the fact that we do not know which of the stars in our simulated clusters is the Sun. The location of of synthetic spectra (Lejeune et al. 1998), by searching for the stel- the Sun with respect to its siblings will affect the number of sib- lar spectral energy distribution which best matches the mass (M ), s lings that can be observed, especially for clusters that during their radius (R ) and effective temperature (T ) of a given solar sibling, s eﬀ dissolution have not spread all over the Galactic disk in azimuth. where the latter quantities are obtained from the stellar evolution We therefore proceed as follows. All stars in the simulated cluster part of the simulations. FVega(λ) was obtained in the same way by located at Galactocentric distances of R = 8–9 kpc and with stel- using the following parameters (Jordi et al. 2010): Teﬀ −= 9550 K, lar masses around 1 M(cid:12) are considered possible ‘suns’. The Gaia log g = 3.95 dex, [Fe/H] = −0.5 dex and (cid:15)t = 2 km s 1. observables ((cid:36), µ, Vr) of the siblings are then calculated with re- Aλ in Eq. 6 is the extinctio(cid:18)n, which is(cid:19)described by: spect to each of these candidate suns. This results in a set of dis- tributions of siblings over the observables which can be considered b collectively in order to account for the uncertain position of the Sun A = A a + λ , (7) λ V λ R within its dissolved birth cluster. V We used the PYGAIA(cid:63) code to compute the astrometric prop- where A is the extinction in the visual (at λ = 550 nm). The erties of the solar siblings. Since we are interested in solar sib- V value of A within our simulated Galaxy is computed by means lings that can be observed by Gaia, we only include stars for which V (cid:54) of the Drimmel extinction model (Drimmel et al. 2003). R is the G 20. V (cid:82) ratio between the extinction and colour excess in the visual band; The apparent G magnitude is given by the following equation we use R = 3.1. a and b are coefﬁcients calculated trough the (Jordi et al. 2010):  V λ λ (cid:82) Cardelli extinction law (Cardelli et al. 1989).   The function S (λ) in Eq. 6 corresponds to the Gaia pass-   x G = −2.5 log λmλminax F(λ)10−0.4AλSx(λ)dλ + GVega . bqaunadnstu,mwheifcﬁhcideenpcye.nTdoocnotmheputteeletshceospteelltararnmsmagisnsiitoundeainndGth,ewCe CusDe λmax the corresponding pass-band described in Jordi et al. (2010). FVega(λ)S (λ)dλ λmin x (6) Finally, GVega is the magnitude zero point which is ﬁxed through the measurement of the ﬂux of Vega, such that GVega = 0.03 mag. Here F(λ) and FVega(λ) are the ﬂuxes of a solar sibling and In Fig. 5 and Table 4 we show the number of solar siblings Vega, respectively, as measured above the atmosphere of the Earth − − that might be observed by Gaia as a function of their heliocentric (in photons s 1 nm 1). We obtain F(λ) through the BaSeL library distances d and their magnitudes G, where we have averaged over each of the candidate Suns per model. Note that for models a, c and d the largest fraction of solar siblings is located within ∼ 500 pc (cid:63) https://pypi.python.org/pypi/PyGaia/ from the Sun. Yet, the number of solar siblings located at this dis- (cid:13)c 2002 RAS, MNRAS 000, 1–13 sgnilbis ralos fo rebmuN sgnilbis ralos fo rebmuN sgnilbis ralos fo rebmuN sgnilbis ralos fo rebmuN ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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font: default

text: [kpc]

text: d

text: [kpc]

text: magnitude

text: in

text: the

text: white

text: light

text: band

text: of

text: used

text: for

text: the

text: astrometric

text: mea-

text: surements,

text: covering

text: the

text: wavelength

text: range

text: nm

text: (see

text: a.

text: b.

text: Jordi

text: et

text: al.

text: Parallaxes

text: and

text: proper

text: motions

text: (µ)

text: will

text: be

text: measured

text: with

text: accuracies

text: ranging

text: from

text: to

text: micro-arcseconds

text: (µas)

text: for

text: stars

text: brighter

text: than

text: mag,

text: and

text: from

text: to

text: µas

text: for

text: sources

text: at

text: G

text: For

text: million

text: stars

text: brighter

text: than

text: G

text: Gaia

text: will

text: also

text: measure

text: radial

text: velocities

text: (V

text: with

text: accuracies

text: rang-

text: r

text: ing

text: from

text: to

text: km

text: s

text: Gaia

text: will

text: not

text: only

text: revolutionize

text: the

text: cur-

text: rent

text: view

text: of

text: the

text: Galaxy

text: but

text: will

text: generate

text: a

text: data

text: set

text: which

text: should

text: in

text: principle

text: allow

text: for

text: a

text: search

text: for

text: solar

text: siblings

text: even

text: far

text: away

text: from

text: G

text: [mag]

text: G

text: [mag]

text: the

text: Sun.

text: In

text: this

text: section

text: we

text: use

text: our

text: simulations

text: to

text: predict

text: the

text: number

text: d

text: [kpc]

text: d

text: [kpc]

text: of

text: solar

text: siblings

text: that

text: will

text: be

text: seen

text: by

text: Gaia,

text: and

text: to

text: study

text: their

text: distri-

text: c.

text: d.

text: bution

text: in

text: the

text: space

text: of

text: parallax,

text: proper

text: motion,

text: and

text: radial

text: velocity

text: with

text: the

text: aim

text: of

text: establishing

text: efﬁcient

text: ways

text: of

text: selecting

text: solar

text: sibling

text: candidates

text: from

text: the

text: Gaia

text: catalogue.

text: The

text: solar

text: siblings

text: in

text: the

text: Gaia

text: catalogue

text: We

text: ﬁrst

text: compare

text: the

text: predicted

text: Gaia

text: survey

text: of

text: the

text: solar

text: sib-

text: lings

text: with

text: predictions

text: by

text: Bland-Hawthorn

text: et

text: al.

text: who

text: con-

text: sidered

text: the

text: prospects

text: for

text: a

text: survey

text: like

text: GALAH

text: (De

text: Silva

text: et

text: al.

text: to

text: varying

text: limiting

text: magnitudes.

text: Following

text: Bland-Hawthorn

text: G

text: [mag]

text: G

text: [mag]

text: et

text: al.

text: we

text: broadly

text: distinguish

text: the

text: possible

text: present-day

text: phase

text: conﬁgurations

text: for

text: the

text: solar

text: siblings

text: by

text: referring

text: to

text: the

text: cases

text: shown

text: Figure

text: Median

text: number

text: of

text: solar

text: siblings

text: that

text: Gaia

text: is

text: predicted

text: to

text: observe,

text: in

text: the

text: panels

text: of

text: Fig.

text: as

text: model

text: a

text: and

text: model

text: b

text: (compact

text: spatial

text: as

text: a

text: function

text: of

text: their

text: heliocentric

text: distances

text: d

text: (red

text: histograms)

text: and

text: G

text: mag-

text: distribution

text: of

text: solar

text: siblings),

text: model

text: c

text: (spatial

text: distribution

text: of

text: solar

text: nitudes

text: (blue

text: histograms).

text: The

text: letters

text: in

text: the

text: left

text: corner

text: correspond

text: to

text: the

text: siblings

text: obtained

text: with

text: the

text: composite

text: model)

text: and

text: model

text: d

text: distributions

text: shown

text: in

text: Fig.

text: The

text: vertical

text: dotted

text: black

text: lines

text: in

text: each

text: panel

text: (highly

text: dispersed

text: spatial

text: distribution

text: of

text: solar

text: siblings).

text: represent

text: the

text: limiting

text: magnitude

text: of

text: the

text: GALAH

text: survey,

text: G

text: mag.

text: In

text: predicting

text: the

text: observed

text: kinematic

text: properties

text: of

text: the

text: solar

text: siblings

text: we

text: want

text: to

text: account

text: for

text: the

text: fact

text: that

text: we

text: do

text: not

text: know

text: which

text: of

text: the

text: stars

text: in

text: our

text: simulated

text: clusters

text: is

text: the

text: Sun.

text: The

text: location

text: of

text: of

text: synthetic

text: spectra

text: (Lejeune

text: et

text: al.

text: by

text: searching

text: for

text: the

text: stel-

text: the

text: Sun

text: with

text: respect

text: to

text: its

text: siblings

text: will

text: affect

text: the

text: number

text: of

text: sib-

text: lar

text: spectral

text: energy

text: distribution

text: which

text: best

text: matches

text: the

text: mass

text: (M

text: s

text: lings

text: that

text: can

text: be

text: observed,

text: especially

text: for

text: clusters

text: that

text: during

text: their

text: radius

text: (R

text: and

text: effective

text: temperature

text: (T

text: of

text: a

text: given

text: solar

text: sibling,

text: s

text: eﬀ

text: dissolution

text: have

text: not

text: spread

text: all

text: over

text: the

text: Galactic

text: disk

text: in

text: azimuth.

text: where

text: the

text: latter

text: quantities

text: are

text: obtained

text: from

text: the

text: stellar

text: evolution

text: We

text: therefore

text: proceed

text: as

text: follows.

text: All

text: stars

text: in

text: the

text: simulated

text: cluster

text: part

text: of

text: the

text: simulations.

text: FVega(λ)

text: was

text: obtained

text: in

text: the

text: same

text: way

text: by

text: located

text: at

text: Galactocentric

text: distances

text: of

text: R

text: kpc

text: and

text: with

text: stel-

text: using

text: the

text: following

text: parameters

text: (Jordi

text: et

text: al.

text: Teﬀ

text: K,

text: lar

text: masses

text: around

text: are

text: considered

text: possible

text: ‘suns’.

text: The

text: Gaia

text: log

text: g

text: dex,

text: [Fe/H]

text: dex

text: and

text: km

text: s

text: observables

text: µ,

text: Vr)

text: of

text: the

text: siblings

text: are

text: then

text: calculated

text: with

text: re-

text: Aλ

text: in

text: Eq.

text: is

text: the

text: which

text: by:

text: spect

text: to

text: each

text: of

text: these

text: candidate

text: suns.

text: This

text: results

text: in

text: a

text: set

text: of

text: dis-

text: tributions

text: of

text: siblings

text: over

text: the

text: observables

text: which

text: can

text: be

text: considered

text: b

text: collectively

text: in

text: order

text: to

text: account

text: for

text: the

text: uncertain

text: position

text: of

text: the

text: Sun

text: A

text: A

text: a

text: λ

text: λ

text: V

text: λ

text: R

text: within

text: its

text: dissolved

text: birth

text: cluster.

text: V

text: We

text: used

text: the

text: code

text: to

text: compute

text: the

text: astrometric

text: prop-

text: where

text: A

text: is

text: the

text: extinction

text: in

text: the

text: visual

text: (at

text: λ

text: nm).

text: The

text: erties

text: of

text: the

text: solar

text: siblings.

text: Since

text: we

text: are

text: interested

text: in

text: solar

text: sib-

text: V

text: value

text: of

text: A

text: within

text: our

text: simulated

text: Galaxy

text: is

text: computed

text: by

text: means

text: lings

text: that

text: can

text: be

text: observed

text: by

text: Gaia,

text: we

text: only

text: include

text: stars

text: for

text: which

text: V

text: of

text: the

text: Drimmel

text: extinction

text: model

text: (Drimmel

text: et

text: al.

text: R

text: is

text: the

text: G

text: V

text: ratio

text: between

text: the

text: extinction

text: and

text: colour

text: excess

text: in

text: the

text: visual

text: band;

text: The

text: apparent

text: G

text: magnitude

text: is

text: given

text: by

text: the

text: following

text: equation

text: we

text: use

text: R

text: a

text: and

text: b

text: are

text: coefﬁcients

text: calculated

text: trough

text: the

text: (Jordi

text: et

text: al.

text: V

text: λ

text: λ

text: Cardelli

text: extinction

text: law

text: (Cardelli

text: et

text: al.

text: The

text: function

text: S

text: (λ)

text: in

text: Eq.

text: corresponds

text: to

text: the

text: Gaia

text: pass-

text: x

text: G

text: log

text: λmλminax

text: GVega

text: CusDe

text: λmax

text: the

text: corresponding

text: pass-band

text: described

text: in

text: Jordi

text: et

text: al.

text: FVega(λ)S

text: (λ)dλ

text: λmin

text: x

text: Finally,

text: GVega

text: is

text: the

text: magnitude

text: zero

text: point

text: which

text: is

text: ﬁxed

text: through

text: the

text: measurement

text: of

text: the

text: ﬂux

text: of

text: Vega,

text: such

text: that

text: GVega

text: mag.

text: Here

text: F(λ)

text: and

text: FVega(λ)

text: are

text: the

text: ﬂuxes

text: of

text: a

text: solar

text: sibling

text: and

text: In

text: Fig.

text: and

text: Table

text: we

text: show

text: the

text: number

text: of

text: solar

text: siblings

text: Vega,

text: respectively,

text: as

text: measured

text: above

text: the

text: atmosphere

text: of

text: the

text: Earth

text: that

text: might

text: be

text: observed

text: by

text: Gaia

text: as

text: a

text: function

text: of

text: their

text: heliocentric

text: (in

text: photons

text: s

text: nm

text: We

text: obtain

text: F(λ)

text: through

text: the

text: BaSeL

text: library

text: distances

text: d

text: and

text: their

text: magnitudes

text: G,

text: where

text: we

text: have

text: averaged

text: over

text: each

text: of

text: the

text: candidate

text: Suns

text: per

text: model.

text: Note

text: that

text: for

text: models

text: a,

text: c

text: and

text: d

text: the

text: largest

text: fraction

text: of

text: solar

text: siblings

text: is

text: located

text: within

text: pc

text: from

text: the

text: Sun.

text: Yet,

text: the

text: number

text: of

text: solar

text: siblings

text: located

text: at

text: this

text: dis-

text: RAS,

text: MNRAS

text: sgnilbis

text: ralos

text: fo

text: rebmuN

text: sgnilbis

text: ralos

text: fo

text: rebmuN

text: sgnilbis

text: ralos

text: fo

text: rebmuN

text: sgnilbis

text: ralos

text: fo

text: rebmuN

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text: As in the case of the k-support norm, the dual (k, p)-support norm has a simple expression. ∈ R Recall that the dual norm of a vector u d is deﬁned by the optimization problem (cid:8) (cid:9) (cid:107) (cid:107) (cid:104) (cid:105) (cid:107) (cid:107) u ∗ = max u, w : w = 1 . (3.3) (k,p), (k,p) ∈ ∞ (cid:32) (cid:33) Proposition 2. If p (1, ] then the dual (k(cid:88), p)-support norm is given by 1 (cid:107) (cid:107) | | q ∈ R u ∗ = u q , u d, (k,p), i ∈ i I k − ⊂ N where q = p/(p 1) and I is the set of indices of the k largest components of u in absolute value. Furthermore, if p ∈ k(1, ∞)d and u ∈ R(cid:16)d\{0} th(cid:17)en the maximum in (3.3) is attained for   | | −1 ∈ sign(u ) (cid:107) (cid:107)ui p 1 if i I , w = i u ∗ k (3.4) i (k,p), 0 otherwise.  ∞ If p = the maximum is attained for ∈ (cid:54) sign(u ) if i I , u = 0, i k i ∈ − ∈ w = λ [ 1, 1] if i I , u = 0, i i k i 0 otherwise. Note that for p = 2 we recover the dual of the k-support norm in (2.3). 3.1 The Spectral -Support Norm (k, p) From Deﬁnition 1 it is clear that the (k, p)-support norm is a symmetric gauge function. This G follows since contains all groups of cardinality k and the (cid:96) -norms only involve absolute values k p of the components. Hence we can deﬁne the spectral (k, p)-support norm as (cid:107) (cid:107) (cid:107) (cid:107) ∈ R × W = σ(W ) , W d m. (k,p) (k,p) (cid:107) · (cid:107) (cid:107) · (cid:107) Since the dual of any orthogonally invariant norm is given by ∗ = σ( ) ∗, see e.g. [Lewis 1995], we conclude that the dual spectral (k, p)-support norm is given by (cid:107) (cid:107) (cid:107) (cid:107) ∈ R × Z ∗ = σ(Z) ∗, Z d m. (k,p), (k,p), 6 ##LTLine## ##LTLine## ##LTLine##

text: As

text: in

text: the

text: case

text: of

text: the

text: k-support

text: norm,

text: the

text: dual

text: (k,

text: p)-support

text: norm

text: has

text: a

text: simple

text: expression.

text: R

text: Recall

text: that

text: the

text: dual

text: norm

text: of

text: a

text: vector

text: u

text: d

text: is

text: deﬁned

text: by

text: the

text: optimization

text: problem

label: equation

text: u

text: max

text: u,

text: w

text: w

text: (k,p),

text: (k,p)

text: Proposition

text: If

text: p

text: then

text: the

text: dual

text: p)-support

text: norm

text: is

text: given

text: by

text: q

text: R

text: u

text: u

text: q

text: u

text: d,

text: (k,p),

text: i

text: i

text: I

text: k

text: N

text: where

text: q

text: p/(p

text: and

text: I

text: is

text: the

text: set

text: of

text: indices

text: of

text: the

text: k

text: largest

text: components

text: of

text: u

text: in

text: absolute

text: value.

text: Furthermore,

text: if

text: p

text: ∞)d

text: and

text: u

text: the

text: maximum

text: in

text: is

text: attained

text: for

text: −1

text: sign(u

text: p

text: if

text: i

text: I

text: w

text: i

text: u

text: k

text: i

text: (k,p),

text: otherwise.

text: If

text: p

text: the

text: maximum

text: is

text: attained

text: for

text: sign(u

text: if

text: i

text: I

text: u

text: i

text: k

text: i

text: w

text: λ

text: if

text: i

text: I

text: u

text: i

text: i

text: k

text: i

text: otherwise.

text: Note

text: that

text: for

text: p

text: we

text: recover

text: the

text: dual

text: of

text: the

text: k-support

text: norm

text: in

text: The

text: Spectral

text: -Support

text: Norm

text: (k,

text: p)

text: From

text: Deﬁnition

text: it

text: is

text: clear

text: that

text: the

text: (k,

text: p)-support

text: norm

text: is

text: a

text: symmetric

text: gauge

text: function.

text: This

text: G

text: follows

text: since

text: contains

text: all

text: groups

text: of

text: cardinality

text: k

text: and

text: the

text: -norms

text: only

text: involve

text: absolute

text: values

text: k

text: p

text: of

text: the

text: components.

text: Hence

text: we

text: can

text: deﬁne

text: the

text: spectral

text: (k,

text: p)-support

text: norm

text: as

text: R

text: W

text: σ(W

text: W

text: d

text: m.

text: (k,p)

text: (k,p)

text: Since

text: the

text: dual

text: of

text: any

text: orthogonally

text: invariant

text: norm

text: is

text: given

text: by

text: σ(

text: see

text: e.g.

text: [Lewis

text: we

text: conclude

text: that

text: the

text: dual

text: spectral

text: (k,

text: p)-support

text: norm

text: is

text: given

text: by

text: R

text: Z

text: σ(Z)

text: Z

text: d

text: m.

text: (k,p),

text: (k,p),

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The next result characterizes the unit ball of the spectral (k, p)-support norm. Due to the rela- tionship between an orthogonally invariant norm and its corresponding symmetric gauge function, we see that the cardinality constraint for vectors generalizes in a natural manner to the rank opera- tor for matrices. Proposition 3. The unit ball of the spectral (k, p)-support norm is the convex hull of the set of (cid:80) matrices of rank at most k and Schatten p-norm no greater than one. (cid:80) ∞ ∈ R (cid:107) (cid:107) | |↓ In particular, if p = , the dual vector norm is given by u d, by u ∞ ∗ = k u . ∈ R × (cid:107) (cid:107) (k, ), i=1 i Hence, for any Z d m, the dual spectral norm is given by Z ∞ ∗ = k σ (Z), that (k, ), i=1 i is the sum of the k largest singular values, which is also known as the Ky-Fan k-norm, see e.g. [Bhatia 1997]. 4 Computing the Norm In this section we compute the norm, illustrating how it interpolates between the (cid:96) and (cid:96) - 1 p norms. Theorem 4. Let p ∈ (1, ∞). For every w ∈ Rd, and k (cid:54) d, it holds that (cid:34) (cid:32)(cid:80) (cid:33) (cid:35) (cid:88) (cid:107) (cid:107) (cid:96) | |↓ √d |w|↓ p p1 w = ( w )p + i=(cid:96)+−1 i (4.1) (k,p) i q k (cid:96) i=1 { − } where 1 + 1 = 1, and for k = d, we set (cid:96) = d, otherwise (cid:96) is the largest integer in 0, . . . , k 1 p q satisfying (cid:88) − | |↓ (cid:62) d | |↓ (k (cid:96)) w w . (4.2) (cid:96) i i=(cid:96)+1 O Furthermore, the norm can be computed in (d log d) time. Proof. Note ﬁrst that in (4.1) when (cid:96) = 0 we understand the ﬁrst term in the right hand side to be zero, and when (cid:96) = d we understand the second term to be zero. (cid:40) (cid:41) (cid:88) We need to compute (cid:107) (cid:107) d (cid:107) (cid:107) (cid:54) w = max u w : u ∗ 1 (k,p) i i (k,p), i=1 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: next

text: result

text: characterizes

text: the

text: unit

text: ball

text: of

text: the

text: spectral

text: (k,

text: p)-support

text: norm.

text: Due

text: to

text: the

text: rela-

text: tionship

text: between

text: an

text: orthogonally

text: invariant

text: norm

text: and

text: its

text: corresponding

text: symmetric

text: gauge

text: function,

text: we

text: see

text: that

text: the

text: cardinality

text: constraint

text: for

text: vectors

text: generalizes

text: in

text: a

text: natural

text: manner

text: to

text: the

text: rank

text: opera-

text: tor

text: for

text: matrices.

text: Proposition

text: The

text: unit

text: ball

text: of

text: the

text: spectral

text: (k,

text: p)-support

text: norm

text: is

text: the

text: convex

text: hull

text: of

text: the

text: set

text: of

text: matrices

text: of

text: rank

text: at

text: most

text: k

text: and

text: Schatten

text: p-norm

text: no

text: greater

text: than

text: one.

text: R

text: In

text: particular,

text: if

text: p

text: the

text: dual

text: vector

text: norm

text: is

text: given

text: by

text: u

text: d,

text: by

text: u

text: k

text: u

text: R

text: (k,

text: i

text: Hence,

text: for

text: any

text: Z

text: d

text: m,

text: the

text: dual

text: spectral

text: norm

text: is

text: given

text: by

text: Z

text: k

text: σ

text: (Z),

text: that

text: (k,

text: i

text: is

text: the

text: sum

text: of

text: the

text: k

text: largest

text: singular

text: values,

text: which

text: is

text: also

text: known

text: as

text: the

text: Ky-Fan

text: k-norm,

text: see

text: e.g.

text: [Bhatia

text: Computing

text: the

text: Norm

text: In

text: this

text: section

text: we

text: compute

text: the

text: norm,

text: illustrating

text: how

text: it

text: interpolates

text: between

text: the

text: and

text: p

text: norms.

text: Theorem

text: Let

text: p

text: For

text: every

text: w

text: Rd,

text: and

text: k

text: d,

text: it

text: holds

text: that

text: √d

text: |w|↓

text: p

text: w

text: w

text: )p

text: i

text: (k,p)

text: i

text: q

text: k

text: where

text: and

text: for

text: k

text: d,

text: we

text: set

text: d,

text: otherwise

text: is

text: the

text: largest

text: integer

text: in

text: k

text: p

text: q

text: satisfying

text: d

text: (k

text: w

text: w

text: i

text: O

text: Furthermore,

text: the

text: norm

text: can

text: be

text: computed

text: in

text: (d

text: log

text: d)

text: time.

text: Proof.

text: Note

text: ﬁrst

text: that

text: in

text: when

text: we

text: understand

text: the

text: ﬁrst

text: term

text: in

text: the

text: right

text: hand

text: side

text: to

text: be

text: zero,

text: and

text: when

text: d

text: we

text: understand

text: the

text: second

text: term

text: to

text: be

text: zero.

text: We

text: need

text: to

text: compute

text: d

text: w

text: max

text: u

text: w

text: u

text: (k,p)

text: i

text: i

text: (k,p),

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: We use again Ho¨lder’s inequality and keep the result if the ordering constraints are fulﬁlled. Continuing in this way, the generic problem we need to solve is (cid:26) (cid:88) (cid:88) (cid:96) d ma(cid:88)x uizi + u(cid:96)+1 zi : (cid:27) i=1 i=(cid:96)+1 (cid:96) − (cid:54) (cid:62) · · · (cid:62) uq + (k (cid:96))uq 1, u u i (cid:96)+1 1 (cid:96)+1 i=1 ∈ { − } where (cid:96) 0, . . . , k 1 . Without the orde(cid:16)ring(cid:17)constraints the maximum, M , is obtained (cid:55)→ − (cid:96) by the change of variable u(cid:96)+1 (k (cid:96))1q u(cid:96) followed b−y applying Ho¨lder’s inequality. A direct computation provides that the maximizer is u (cid:32)= (cid:80)zi p 1 if i(cid:33)(cid:54) (cid:96), and i M (cid:96) − − d z p 1 (k (cid:96))1q u(cid:96)+1 = (k −i=(cid:96)(cid:96))+1q1Mip . (cid:96) Using the relationship 1 + 1 = 1, we can re(cid:32)wr(cid:80)ite this as (cid:33) p q − d z p 1 u = −i=(cid:96)+1 i . (cid:96)+1 (k (cid:96))Mp (cid:96) (cid:32)(cid:80) (cid:33) Hence, the ordering constraints are satisﬁed if − − (cid:62) d z p 1 zp 1 i=−(cid:96)+1 i , (cid:96) (k (cid:96)) which is equivalent to (4.2). Finally note that M is a nondecreasing function of (cid:96). This is because (cid:96) the problem with a smaller value of (cid:96) is more constrained, namely, it solves (4.3) with the additional · · · ∈ { − constraints u = = u . Moreover, if the constraint (4.2) holds for some value (cid:96) 0, . . . , k } (cid:96)+1 d 1 then it also holds for a smaller value of (cid:96), hence we maximize the objective by choosing the largest (cid:96). The computational complexity stems from using the monotonicity of M with respect to (cid:96), (cid:96) which allows us to identify the critical value of (cid:96) using binary search. Note that for k = d we recover the (cid:96) -norm and for p = 2 we recover the result in p [Argyriou et al. 2012, McDonald et al. 2014], however our proof technique is different from theirs. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: use

text: again

text: Ho¨lder’s

text: inequality

text: and

text: keep

text: the

text: result

text: if

text: the

text: ordering

text: constraints

text: are

text: fulﬁlled.

text: Continuing

text: in

text: this

text: way,

text: the

text: generic

text: problem

text: we

text: need

text: to

text: solve

text: is

text: d

text: uizi

text: zi

text: uq

text: (k

text: u

text: u

text: i

text: where

text: k

text: Without

text: the

text: the

text: maximum,

text: M

text: is

text: obtained

text: by

text: the

text: change

text: of

text: variable

text: (k

text: followed

text: b−y

text: applying

text: Ho¨lder’s

text: inequality.

text: A

text: direct

text: computation

text: provides

text: that

text: the

text: maximizer

text: is

text: u

text: p

text: if

text: and

text: i

text: M

text: d

text: z

text: p

text: (k

text: (k

text: Using

text: the

text: relationship

text: we

text: can

text: this

text: as

text: p

text: q

text: d

text: z

text: p

text: u

text: i

text: (k

text: Hence,

text: the

text: ordering

text: constraints

text: are

text: satisﬁed

text: if

text: d

text: z

text: p

text: zp

text: i

text: (k

text: which

text: is

text: equivalent

text: to

text: Finally

text: note

text: that

text: M

text: is

text: a

text: nondecreasing

text: function

text: of

text: This

text: is

text: because

text: the

text: problem

text: with

text: a

text: smaller

text: value

text: of

text: is

text: more

text: constrained,

text: namely,

text: it

text: solves

text: with

text: the

text: additional

text: constraints

text: u

text: u

text: Moreover,

text: if

text: the

text: constraint

text: holds

text: for

text: some

text: value

text: k

text: d

text: then

text: it

text: also

text: holds

text: for

text: a

text: smaller

text: value

text: of

text: hence

text: we

text: maximize

text: the

text: objective

text: by

text: choosing

text: the

text: largest

text: The

text: computational

text: complexity

text: stems

text: from

text: using

text: the

text: monotonicity

text: of

text: M

text: with

text: respect

text: to

text: which

text: allows

text: us

text: to

text: identify

text: the

text: critical

text: value

text: of

text: using

text: binary

text: search.

text: Note

text: that

text: for

text: k

text: d

text: we

text: recover

text: the

text: -norm

text: and

text: for

text: p

text: we

text: recover

text: the

text: result

text: in

text: p

text: [Argyriou

text: et

text: al.

text: McDonald

text: et

text: al.

text: however

text: our

text: proof

text: technique

text: is

text: different

text: from

text: theirs.

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 +∞ ﬁeld and the trace denotes the integration over all mo- menta as well as the summation over internal indices. This integral is evaluated by choosing R (q2) so that k → 4 Γ approaches the bare action in the limit k Λ and k → the full quantum eﬀective action when k 0 [16]. A detailed study of an extensive class of regulator functions 1 is reported in [17]. In the current case, the optimized − − regulator is chosen R (q2) = (k2 q2) θ(k2 q2) which k provides an analytic result for the momentum integral 0 ≡ [18]. It is convenient to introduce ρ¯ 1φ¯2, which will - 2 be used throughout this paper. Inserting (1) into (2) → ∞ 1 and applying the limit N yields the ﬂow for the 2 - eﬀective potential in the large-N [15]: 4 5 − − (cid:48) 1 - ∂tu = du + (d 2)ρu + 1 + u(cid:48) , (3) -∞ - - - ρ +∞ 1 4 2 1 0 1 1 2 4 3 − where the dimensionless quantities u = Uk d with ρ = − (cid:48) ρ¯k d+2 are introduced and u = ∂ u. ρ An exact solution for the ρ derivative of (3) can be FIG. 1. Each curve corresponds to a critical potential deriva- obtained by using the method of characteristics [15, 19] tive with a particular choice of c in d = 5. The thick black line and the ﬁxed point solutions associated to (3) can be is for the c = 0 solution. The red line segment is considered (cid:48) given as an implicit function ρ = ρ(u∗). u∗ is introduced as the physical branch. The axes are rescaled for clarity. as the dimensionless eﬀective potential at the ﬁxed point. (cid:88) The most compact for∈mZof t(cid:18)hese exact solutions f(cid:19)or d = 2n + 1 and d = 2n (n ) are respectively n u(i)(0) ρ = cu(cid:48)∗d2−1 + (d −1 2) 2F1 2, 1 − d2; 2 − d2; −u(cid:48)∗ (4) u(ρ) = nl→im∞ i=1 i! ρi. (6) (cid:18) (cid:19) The derivatives of the potential are the couplings of and (cid:48) (cid:48)(cid:48) the theory: u (0) = g = m2 (squared mass), u (0) = 1 ρ = c¯u(cid:48)∗d2−1 + (d + 2)(11 + u(cid:48)∗)2 2F1 1, 2; 2 + d2; 1 +1u(cid:48)∗ , gw2as=wλor(kqeudaorutitcfocoruﬁpnldining)g, tehtec.ﬁ..xeAdnpoeﬃintcsieonfttahlegotrhietohrmy for expansions up to order 50, if required. This method (5) is based on the observation that all the couplings can be where c is an arbitrary constant obtained from the inte- − gration, c¯ = c d4π sin(dπ/2) and 2F1 is the hypergeo- eﬁxxperdespsoeidnttsh,rgo∗ug=hgt∗h(ems2∗q)u.ared mass of the system at the metric function. Fig. 1 shows the solutions for the case i i In the case of the ﬁve-dimensional large-N O(N) model, in d = 5. Each curve corresponds to a solution with a (cid:48) ≥ the ﬁxed point structure shows a non-trivial ﬁxed point as particular value of the parameter c. u∗ 0 (4) holds ∈ R well as the non-interacting Gaussian ﬁxed point (GFP). for every c but in order to obtain a continuation (cid:48) ≤ The ﬁxed point position drifts as the order of the Taylor- of the solutions to u∗ 0 the constant c needs to take ≈ expansion increases and converges to m2∗ 0.1392 as imaginary values. There is one exception: the solution (cid:48) shown in Fig. 2. As m2 = u (0), it can be conﬁdently corresponding to c = 0. This is shown in Fig. 1 as the thick black curve that passes smoothly through u∗ = 0 stated that this is the same ﬁxed point solution as found by the analytic study of the ﬂow when c = 0 in (4). and intersects the horizontal and vertical axes at ρ = 1/3 (cid:48) ≈ In fact, this technique naturally singles out a ﬁxed point and u∗(0) 0.1392 on the upper plane, respectively. It solution from all the other solutions that are present only is tempting to consider this ﬁxed point potential as the in the analytical case. In Fig. 1 this corresponds to the physical one since it is analytic at its extremum. On the red line segment on the thick black curve and will be other hand, this curve still has the problem like the other (cid:48) considered as the physical solution. solutions (including their continuation): u∗ can be only ∈ (cid:80) Fig. 3 shows the comparison between the exact critical considered as a multivalued function of ρ [0, 0.6214] potential, computed from (4), and the polynomial expan- [11]. ∗ Focus now shifts to the author’s previous results [12] sion results, u∗(ρ) = n gi (m2∗)ρi, with n = 26. The i=1 i! where a diﬀerent technique, based upon polynomial ex- non-analytic nature of the exact potential is very appar- pansion, was used to ﬁnd the ﬁxed point solutions of the ent as it is restricted to the ﬁnite interval unlike to the ﬂow equation. The potential is assumed to be analytic polynomial potential which was assumed to be analyti- in this case: cal. The most important features of this potential are the * ' u ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ﬁeld

text: and

text: the

text: trace

text: denotes

text: the

text: integration

text: over

text: all

text: mo-

text: menta

text: as

text: well

text: as

text: the

text: summation

text: over

text: internal

text: indices.

text: This

text: integral

text: is

text: evaluated

text: by

text: choosing

text: R

text: so

text: that

text: k

text: Γ

text: approaches

text: the

text: bare

text: action

text: in

text: the

text: limit

text: k

text: Λ

text: and

text: k

text: the

text: full

text: quantum

text: eﬀective

text: action

text: when

text: k

text: A

text: detailed

text: study

text: of

text: an

text: extensive

text: class

text: of

text: regulator

text: functions

text: is

text: reported

text: in

text: In

text: the

text: current

text: case,

text: the

text: optimized

text: regulator

text: is

text: chosen

text: R

text: which

text: k

text: provides

text: an

text: analytic

text: result

text: for

text: the

text: momentum

text: integral

text: It

text: is

text: convenient

text: to

text: introduce

text: ρ¯

text: which

text: will

text: be

text: used

text: throughout

text: this

text: paper.

text: Inserting

text: into

text: and

text: applying

text: the

text: limit

text: N

text: yields

text: the

text: ﬂow

text: for

text: the

text: eﬀective

text: potential

text: in

text: the

text: large-N

text: ∂tu

text: du

text: (d

text: ρ

text: where

text: the

text: dimensionless

text: quantities

text: u

text: Uk

text: d

text: with

text: ρ

text: ρ¯k

text: are

text: introduced

text: and

text: u

text: u.

text: ρ

text: An

text: exact

text: solution

text: for

text: the

text: ρ

text: derivative

text: of

text: can

text: be

text: FIG.

text: Each

text: curve

text: corresponds

text: to

text: a

text: critical

text: potential

text: deriva-

text: obtained

text: by

text: using

text: the

text: method

text: of

text: characteristics

text: tive

text: with

text: a

text: particular

text: choice

text: of

text: c

text: in

text: d

text: The

text: thick

text: black

text: line

text: and

text: the

text: ﬁxed

text: point

text: solutions

text: associated

text: to

text: can

text: be

text: is

text: for

text: the

text: c

text: solution.

text: The

text: red

text: line

text: segment

text: is

text: considered

text: given

text: as

text: an

text: implicit

text: function

text: ρ

text: ρ(u∗).

text: u∗

text: is

text: introduced

text: as

text: the

text: physical

text: branch.

text: The

text: axes

text: are

text: rescaled

text: for

text: clarity.

text: as

text: the

text: dimensionless

text: eﬀective

text: potential

text: at

text: the

text: ﬁxed

text: point.

text: The

text: most

text: compact

text: for∈mZof

text: exact

text: solutions

text: d

text: and

text: d

text: (n

text: are

text: respectively

text: n

text: ρ

text: (d

text: −1

text: u(ρ)

text: nl→im∞

text: i!

text: ρi.

text: The

text: derivatives

text: of

text: the

text: potential

text: are

text: the

text: couplings

text: of

text: and

text: the

text: theory:

text: u

text: g

text: (squared

text: mass),

text: u

text: ρ

text: (d

text: for

text: expansions

text: up

text: to

text: order

text: if

text: required.

text: This

text: method

text: is

text: based

text: on

text: the

text: observation

text: that

text: all

text: the

text: couplings

text: can

text: be

text: where

text: c

text: is

text: an

text: arbitrary

text: constant

text: obtained

text: from

text: the

text: inte-

text: gration,

text: c¯

text: c

text: and

text: is

text: the

text: hypergeo-

text: mass

text: of

text: the

text: system

text: at

text: the

text: metric

text: function.

text: Fig.

text: shows

text: the

text: solutions

text: for

text: the

text: case

text: i

text: i

text: In

text: the

text: case

text: of

text: the

text: ﬁve-dimensional

text: large-N

text: O(N)

text: model,

text: in

text: d

text: Each

text: curve

text: corresponds

text: to

text: a

text: solution

text: with

text: a

text: the

text: ﬁxed

text: point

text: structure

text: shows

text: a

text: non-trivial

text: ﬁxed

text: point

text: as

text: particular

text: value

text: of

text: the

text: parameter

text: c.

text: u∗

text: holds

text: R

text: well

text: as

text: the

text: non-interacting

text: Gaussian

text: ﬁxed

text: point

text: (GFP).

text: for

text: every

text: c

text: but

text: in

text: order

text: to

text: obtain

text: a

text: continuation

text: The

text: ﬁxed

text: point

text: position

text: drifts

text: as

text: the

text: order

text: of

text: the

text: Taylor-

text: of

text: the

text: solutions

text: to

text: u∗

text: the

text: constant

text: c

text: needs

text: to

text: take

text: expansion

text: increases

text: and

text: converges

text: to

text: as

text: imaginary

text: values.

text: There

text: is

text: one

text: exception:

text: the

text: solution

text: shown

text: in

text: Fig.

text: As

text: u

text: it

text: can

text: be

text: conﬁdently

text: corresponding

text: to

text: c

text: This

text: is

text: shown

text: in

text: Fig.

text: as

text: the

text: thick

text: black

text: curve

text: that

text: passes

text: smoothly

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text: Interplay of classical and quantum dynamics in a thermal ensemble of atoms ‡ Arif Warsi Laskar1, Niharika Singh2 , Arunabh Mukherjee3, and Saikat Ghosh4 Department of Physics, Indian Institute of Technology, Kanpur 208016, India E-mail: 4gsaikat@iitk.ac.in Abstract. In a thermal ensemble of atoms driven by coherent ﬁelds, how does evolution of quantum superposition compete with classical dynamics of optical pumping and atomic diﬀusion? Is it optical pumping that ﬁrst prepares a thermal ensemble, with coherent superposition developing subsequently or is it the other way round: coherently superposed atoms driven to steady state via optical pumping? Using a stroboscopic probing technique, here we experimentally explore these questions. A 100 ns pulse is used to probe an experimentally simulated, closed three-level, Λ-like conﬁguration in rubidium atoms, driven by strong coherent (control) and incoherent ﬁelds. Temporal evolution of probe transmission shows an initial overshoot with turn-on of control, resulting in a scenario akin to lasing without inversion (LWI). The corresponding rise time is dictated by coherent dynamics, with a distinct experimental signature of half- cycle Rabi ﬂop in a thermal ensemble of atoms. Our results indicate that, in fact, optical pumping drives the atoms to a steady state in a signiﬁcantly longer time-scale that sustains superposed dark states. Eventual control turn-oﬀ leads to a sudden fall in transmission with an ubiquitous signature for identifying closed and open systems. Numerical simulations and toy-model predictions conﬁrm our claims. These studies reveal new insights into a rich and complex dynamics associated with atoms in thermal ensemble, which are otherwise absent in state-prepared, cold atomic ensembles. PACS numbers: 42.50.-p, 42.50.Ct, 42.50.Gy, 32.80.Qk ‡ 1 , 2These two authors contributed equally

text: Interplay

label: title

text: of

text: classical

text: and

text: quantum

text: dynamics

text: in

text: a

text: thermal

text: ensemble

text: of

text: atoms

label: author

text: Arif

text: Warsi

text: Niharika

text: Arunabh

text: and

text: Saikat

text: Department

text: of

text: Physics,

text: Indian

text: Institute

text: of

text: Technology,

text: Kanpur

text: India

text: E-mail:

label: footer

text: Abstract.

text: In

label: abstract

text: a

text: thermal

text: ensemble

text: of

text: atoms

text: driven

text: by

text: coherent

text: ﬁelds,

text: how

text: does

text: evolution

text: of

text: quantum

text: superposition

text: compete

text: with

text: classical

text: dynamics

text: of

text: optical

text: pumping

text: and

text: atomic

text: diﬀusion?

text: Is

text: it

text: optical

text: pumping

text: that

text: ﬁrst

text: prepares

text: a

text: thermal

text: ensemble,

text: with

text: coherent

text: superposition

text: developing

text: subsequently

text: or

text: is

text: it

text: the

text: other

text: way

text: round:

text: coherently

text: superposed

text: atoms

text: driven

text: to

text: steady

text: state

text: via

text: optical

text: pumping?

text: Using

text: a

text: stroboscopic

text: probing

text: technique,

text: here

text: we

text: experimentally

text: explore

text: these

text: questions.

text: A

text: ns

text: pulse

text: is

text: used

text: to

text: probe

text: an

text: experimentally

text: simulated,

text: closed

text: three-level,

text: Λ-like

text: conﬁguration

text: in

text: rubidium

text: atoms,

text: driven

text: by

text: strong

text: coherent

text: (control)

text: and

text: incoherent

text: ﬁelds.

text: Temporal

text: evolution

text: of

text: probe

text: transmission

text: shows

text: an

text: initial

text: overshoot

text: with

text: turn-on

text: of

text: control,

text: resulting

text: in

text: a

text: scenario

text: akin

text: to

text: lasing

text: without

text: inversion

text: (LWI).

text: The

text: corresponding

text: rise

text: time

text: is

text: dictated

text: by

text: coherent

text: dynamics,

text: with

text: a

text: distinct

text: experimental

text: signature

text: of

text: half-

text: cycle

text: Rabi

text: ﬂop

text: in

text: a

text: thermal

text: ensemble

text: of

text: atoms.

text: Our

text: results

text: indicate

text: that,

text: in

text: fact,

text: optical

text: pumping

text: drives

text: the

text: atoms

text: to

text: a

text: steady

text: state

text: in

text: a

text: signiﬁcantly

text: longer

text: time-scale

text: that

text: sustains

text: superposed

text: dark

text: states.

text: Eventual

text: control

text: turn-oﬀ

text: leads

text: to

text: a

text: sudden

text: fall

text: in

text: transmission

text: with

text: an

text: ubiquitous

text: signature

text: for

text: identifying

text: closed

text: and

text: open

text: systems.

text: Numerical

text: simulations

text: and

text: toy-model

text: predictions

text: conﬁrm

text: our

text: claims.

text: These

text: studies

text: reveal

text: new

text: insights

text: into

text: a

text: rich

text: and

text: complex

text: dynamics

text: associated

text: with

text: atoms

text: in

text: thermal

text: ensemble,

text: which

text: are

text: otherwise

text: absent

text: in

text: state-prepared,

text: cold

text: atomic

text: ensembles.

text: PACS

text: numbers:

text: two

text: authors

text: contributed

text: equally


text: Interplay of classical and quantum dynamics in a thermal ensemble of atoms 7 Figure 3. Probe transmission for case(A) (closed Λ system): (a) Experimentally recorded peak transmission, with I , I being the initial and transmitted probe- o trans peak intensities, respectively. Inset shows energy level conﬁguration of the system. (b) Simulated probe transmission in the absence of re-pumping ﬁeld. Here Ω = 0.01γ , p 3 Ω = 4.5γ , γ = 6 MHz, γ = 12 MHz, Γ = 0.05 MHz, γ = 0.12 MHz, c 3 3 out decoh th where γ = γ + γ = 6 MHz and ρ(0) = ρ(0) = ρeq = ρeq. These parameters are 3 31 32 11 22 11 22 chosen in accordance with the experiment. Inset shows the energy level conﬁguration, | (cid:105) | (cid:105) | (cid:105) populations in bare states 1 and 3 , dark state D and ground state coherence Re(ρ ). 12 transparency and region(IV): a slow rise back to the initial thermal value (IV). These regions are ubiquitous to all the three cases considered. (cid:112) Figure 4 shows simulated probe transmission as a function of initial population | (cid:105) ≡ | (cid:105) − distribution. In an ideal Λ system the dark state is of the form D (Ω 1 | (cid:105) | (cid:105) c Ω 2 )/ Ω2 + Ω2, implying that with all initial population in state 1 (ρ(0) = 0 and p c | (cid:105) p∼ | (cid:105) 22 Ω >> Ω ), D 1 . Thus at control turn on, the system is already in a dark state c p and for large ﬁelds one achieves steady state transparency almost instantaneously. The rise-time τ is then decided by the control ramp time τ when Ω > 1/τ or control Rabi r c c c frequency Ω when Ω < 1/τ . For ideal EIT scenario, one therefore does not expect c c c any overshoot in transmission. On the other hand, when ρ(0) > ρ(0), one observes a Raman gain, due to transfer | (cid:105) 22 11 | (cid:105) | (cid:105) of population to state 3 , which results in population inversion between 1 and 3 . It is intriguing to note that such an overshoot is present only in systems with statistical distribution of initial population in the ground states, and is absent in an otherwise ideal EIT medium with all atoms initially prepared in probe ground state. In simulations, overshoot is not observed for ρ(0)/ρ(0) < 1. 22 11 It can be observed that the overshoot has a corresponding signature of build-up of Raman coherence in the medium. Numerical simulations conﬁrm a peak in two-photon coherence ρ (ﬁgure 4(a)). Accordingly, one expects this peak to bear signature of 12 two-photon Rabi ﬂop. Furthermore, it can be noted that there is a peak in population ##LTFigure## ##LTLine##

text: Interplay

text: of

text: classical

text: and

text: quantum

text: dynamics

text: in

text: a

text: thermal

text: ensemble

text: of

text: atoms

text: Figure

text: Probe

text: transmission

text: for

text: case(A)

text: (closed

text: Λ

text: system):

text: (a)

text: Experimentally

text: recorded

text: peak

text: transmission,

text: with

text: I

text: I

text: being

text: the

text: initial

text: and

text: transmitted

text: probe-

text: o

text: trans

text: peak

text: intensities,

text: respectively.

text: Inset

text: shows

text: energy

text: level

text: conﬁguration

text: of

text: the

text: system.

text: (b)

text: Simulated

text: probe

text: transmission

text: in

text: the

text: absence

text: of

text: re-pumping

text: ﬁeld.

text: Here

text: Ω

text: p

text: Ω

text: γ

text: MHz,

text: γ

text: MHz,

text: Γ

text: MHz,

text: γ

text: MHz,

text: c

text: out

text: decoh

text: th

text: where

text: γ

text: γ

text: γ

text: MHz

text: and

text: ρeq

text: ρeq.

text: These

text: parameters

text: are

text: chosen

text: in

text: accordance

text: with

text: the

text: experiment.

text: Inset

text: shows

text: the

text: energy

text: level

text: conﬁguration,

text: populations

text: in

text: bare

text: states

text: and

text: dark

text: state

text: D

text: and

text: ground

text: state

text: coherence

text: Re(ρ

text: transparency

text: and

text: region(IV):

text: a

text: slow

text: rise

text: back

text: to

text: the

text: initial

text: thermal

text: value

text: (IV).

text: These

text: regions

text: are

text: ubiquitous

text: to

text: all

text: the

text: three

text: cases

text: considered.

text: Figure

text: shows

text: simulated

text: probe

text: transmission

text: as

text: a

text: function

text: of

text: initial

text: population

text: distribution.

text: In

text: an

text: ideal

text: Λ

text: system

text: the

text: dark

text: state

text: is

text: of

text: the

text: form

text: D

text: (Ω

text: c

text: Ω

text: Ω2

text: implying

text: that

text: with

text: all

text: initial

text: population

text: in

text: state

text: and

text: p

text: c

text: p∼

text: Ω

text: Ω

text: D

text: Thus

text: at

text: control

text: turn

text: on,

text: the

text: system

text: is

text: already

text: in

text: a

text: dark

text: state

text: c

text: p

text: and

text: for

text: large

text: ﬁelds

text: one

text: achieves

text: steady

text: state

text: transparency

text: almost

text: instantaneously.

text: The

text: rise-time

text: τ

text: is

text: then

text: decided

text: by

text: the

text: control

text: ramp

text: time

text: τ

text: when

text: Ω

text: or

text: control

text: Rabi

text: r

text: c

text: c

text: c

text: frequency

text: Ω

text: when

text: Ω

text: For

text: ideal

text: EIT

text: scenario,

text: one

text: therefore

text: does

text: not

text: expect

text: c

text: c

text: c

text: any

text: overshoot

text: in

text: transmission.

text: On

text: the

text: other

text: hand,

text: when

text: one

text: observes

text: a

text: Raman

text: gain,

text: due

text: to

text: transfer

text: of

text: population

text: to

text: state

text: which

text: results

text: in

text: population

text: inversion

text: between

text: and

text: It

text: is

text: intriguing

text: to

text: note

text: that

text: such

text: an

text: overshoot

text: is

text: present

text: only

text: in

text: systems

text: with

text: statistical

text: distribution

text: of

text: initial

text: population

text: in

text: the

text: ground

text: states,

text: and

text: is

text: absent

text: in

text: an

text: otherwise

text: ideal

text: EIT

text: medium

text: with

text: all

text: atoms

text: initially

text: prepared

text: in

text: probe

text: ground

text: state.

text: In

text: simulations,

text: overshoot

text: is

text: not

text: observed

text: for

text: It

text: can

text: be

text: observed

text: that

text: the

text: overshoot

text: has

text: a

text: corresponding

text: signature

text: of

text: build-up

text: of

text: Raman

text: coherence

text: in

text: the

text: medium.

text: Numerical

text: simulations

text: conﬁrm

text: a

text: peak

text: in

text: two-photon

text: coherence

text: ρ

text: (ﬁgure

text: Accordingly,

text: one

text: expects

text: this

text: peak

text: to

text: bear

text: signature

text: of

text: two-photon

text: Rabi

text: ﬂop.

text: Furthermore,

text: it

text: can

text: be

text: noted

text: that

text: there

text: is

text: a

text: peak

text: in

text: population

text: ##LTFigure##

text: ##LTLine##


text: Interplay of classical and quantum dynamics in a thermal ensemble of atoms 10 Figure 7. Probe transmission for case(B) (an open Λ system): (a) and (b) depict experimental and simulated time traces, respectively. For simulations, an additional ∼ probe ground state decay term of rate 0.01Ω is used with Ω = 7γ . These c c 3 parameters are chosen in accordance with the experiment. Inset shows energy level | (cid:105) | (cid:105) | (cid:105) | (cid:105) conﬁguration, populations in states 1 , 3 and 4 , dark state D and ground state coherence Re(ρ ). When the control is turned oﬀ, the sharp and partial fall of probe 12 transmission (to point X) is due to adiabatic rotation of dark state atoms to bare states. From X the population relaxes back to initial thermal equilibrium within few transit times. transparency equilibrates back to the initial value (at τ ) in two distinct steps. Initially 0 the transparency falls rapidly to a value much lower than the initial value at τ . This 0 sudden fall in transparency corresponds to a loss of dark state atoms with the control | (cid:105) turn-oﬀ, and an adiabatic rotation of the dark-state to a non-superposed state 1 atoms. In simulations, it can be observed that the fall in transmission mimics the decay time of dark state population (inset of ﬁgure 3(b)). For a closed three level system, since most | (cid:105) of the population ends up in the probe ground state 1 , this results in a population larger than ρ(0) and an enhanced probe absorption in region III. The amplitude of this 11 fall thereby gives a direct and alternative measure of number of atoms in the probe ground state. Region IV shows a straightforward classical dynamics. The atomic ground state population, in absence of pumping ﬁelds, relaxes back to thermal equilibrium. This ∼ happens on a time scale set by transit time of atoms ( 8 µs) through the probe beam diameter. One can thereby conclude that the transient evolution in the closed Λ system is dominated by coherent evolution in regions I and III while incoherent processes dictate the dynamics in regions II and IV. Case B: An open three-level system To compare the observations of a closed system with an open three-level system, we next consider a scenario without the re-pumping ﬁeld. Figure 7 shows typical ##LTFigure##

text: Interplay

text: of

text: classical

text: and

text: quantum

text: dynamics

text: in

text: a

text: thermal

text: ensemble

text: of

text: atoms

text: Figure

text: Probe

text: transmission

text: for

text: case(B)

text: (an

text: open

text: Λ

text: system):

text: (a)

text: and

text: (b)

text: depict

text: experimental

text: and

text: simulated

text: time

text: traces,

text: respectively.

text: For

text: simulations,

text: an

text: additional

text: probe

text: ground

text: state

text: decay

text: term

text: of

text: rate

text: is

text: used

text: with

text: Ω

text: 7γ

text: These

text: c

text: c

text: parameters

text: are

text: chosen

text: in

text: accordance

text: with

text: the

text: experiment.

text: Inset

text: shows

text: energy

text: level

text: conﬁguration,

text: populations

text: in

text: states

text: and

text: dark

text: state

text: D

text: and

text: ground

text: state

text: coherence

text: Re(ρ

text: When

text: the

text: control

text: is

text: turned

text: oﬀ,

text: the

text: sharp

text: and

text: partial

text: fall

text: of

text: probe

text: transmission

text: (to

text: point

text: X)

text: is

text: due

text: to

text: adiabatic

text: rotation

text: of

text: dark

text: state

text: atoms

text: to

text: bare

text: states.

text: From

text: X

text: the

text: population

text: relaxes

text: back

text: to

text: initial

text: thermal

text: equilibrium

text: within

text: few

text: transit

text: times.

text: transparency

text: equilibrates

text: back

text: to

text: the

text: initial

text: value

text: (at

text: τ

text: in

text: two

text: distinct

text: steps.

text: Initially

text: the

text: transparency

text: falls

text: rapidly

text: to

text: a

text: value

text: much

text: lower

text: than

text: the

text: initial

text: value

text: at

text: τ

text: This

text: sudden

text: fall

text: in

text: transparency

text: corresponds

text: to

text: a

text: loss

text: of

text: dark

text: state

text: atoms

text: with

text: the

text: control

text: turn-oﬀ,

text: and

text: an

text: adiabatic

text: rotation

text: of

text: the

text: dark-state

text: to

text: a

text: non-superposed

text: state

text: atoms.

text: In

text: simulations,

text: it

text: can

text: be

text: observed

text: that

text: the

text: fall

text: in

text: transmission

text: mimics

text: the

text: decay

text: time

text: of

text: dark

text: state

text: population

text: (inset

text: of

text: ﬁgure

text: For

text: a

text: closed

text: three

text: level

text: system,

text: since

text: most

text: of

text: the

text: population

text: ends

text: up

text: in

text: the

text: probe

text: ground

text: state

text: this

text: results

text: in

text: a

text: population

text: larger

text: than

text: and

text: an

text: enhanced

text: probe

text: absorption

text: in

text: region

text: III.

text: The

text: amplitude

text: of

text: this

text: fall

text: thereby

text: gives

text: a

text: direct

text: and

text: alternative

text: measure

text: of

text: number

text: of

text: atoms

text: in

text: the

text: probe

text: ground

text: state.

text: Region

text: IV

text: shows

text: a

text: straightforward

text: classical

text: dynamics.

text: The

text: atomic

text: ground

text: state

text: population,

text: in

text: absence

text: of

text: pumping

text: ﬁelds,

text: relaxes

text: back

text: to

text: thermal

text: equilibrium.

text: This

text: happens

text: on

text: a

text: time

text: scale

text: set

text: by

text: transit

text: time

text: of

text: atoms

text: µs)

text: through

text: the

text: probe

text: beam

text: diameter.

text: One

text: can

text: thereby

text: conclude

text: that

text: the

text: transient

text: evolution

text: in

text: the

text: closed

text: Λ

text: system

text: is

text: dominated

text: by

text: coherent

text: evolution

text: in

text: regions

text: I

text: and

text: III

text: while

text: incoherent

text: processes

text: dictate

text: the

text: dynamics

text: in

text: regions

text: II

text: and

text: IV.

text: Case

text: B:

text: An

text: open

text: three-level

text: system

text: To

text: compare

text: the

text: observations

text: of

text: a

text: closed

text: system

text: with

text: an

text: open

text: three-level

text: system,

text: we

text: next

text: consider

text: a

text: scenario

text: without

text: the

text: re-pumping

text: ﬁeld.

text: Figure

text: shows

text: typical

text: ##LTFigure##