text: 2 relaxational equation, it interacts with a fermionic heat bath which expands ﬂuid dynamically [25]. It has been demonstrated in [26–29], that this model is able to de- Veﬀ = Ωqq¯ + ΩA + Uσ + Uχ + Ω0 , (2) scribe critical slowing down as well as spinodal decompo- sition within a dynamical setup. We follow the evolution of the system over various hy- with the quark and(cid:90)gluon co(cid:110)ntr(cid:104)ibutions (cid:105) persurfaces of constant energy density of the coupled sys- tem and compare the ﬂuctuations in the sigma ﬁeld to the − d3p (cid:104) −Eq−(cid:105)µ(cid:111) ﬂuctuations in net-proton numbers which are obtained Ωqq¯ = 2NfNcT (2π)3 ln 1 + e T (3) from a Cooper-Frye particlization prescription. We fur- thermore give a comparison to a mean-ﬁeld evolution in (cid:90) (cid:110) (cid:104) −Eq+µ (cid:105)(cid:111) + ln 1 + e T , order to pin down the ﬂuctuations stemming from the itnraitniasiltsiotant.e versus dynamical ﬂuctuations near the phase ΩA = 2(Nc2 − 1)T (2dπ3p)3 ln 1 − e−ETA , (4) We begin with a description of nonequilibrium chiral ﬂuid dynamics (NχFD) in Sec. , including the equations which depend on temperature T and quark chemical of motion and a brief description of the implementation potential µ = µ /3, Ω in Eq. (2) is an unimportant of initial state and particlization. In Sec. , we investigate B 0 constant to set the total energy to zero in the ground the dynamics of the net-proton and sigma ﬁeld kurtosis (cid:113) (cid:112) state. The quasiparticle energies of constituent quarks for an evolution in the crossover regime near the critical and gluons are generated via their eﬀective masses as point. We conclude and give a brief outlook in Sec. . E = p2 + m2 and E = p2 + m2 . II. NONEQUILIBRIUM CHIRAL FLUID q q A A (cid:104) (cid:105) DYNAMICS The mean-ﬁeld values of the condensate ﬁelds, σ and (cid:104) (cid:105) χ , are obtained by minimizing the eﬀective thermody- We study the ﬂuid dynamical evolution in a heavy-ion namic potential(cid:12)(cid:12)Veﬀ via (cid:12)(cid:12) collision using a quark-meson model with dilaton ﬁeld, (cid:12) (cid:12) (cid:12) (cid:12) ∂V ∂V eﬀ = 0 , eﬀ = 0 . (5) L − 1 1 ∂σ (cid:104) (cid:105) ∂χ (cid:104) (cid:105) = q (iγµ∂ g σ) q + (∂ σ)2 + (∂ χ)2 (1a) σ= σ χ= χ µ q (cid:18)2 (cid:19)µ 2 µ L − − + A Uσ Uχ , In what we will call the mean-ﬁeld evolution, the or- LA = −14(cid:34)AµνAµν +(cid:18)12gA2(cid:19) (cid:35)χχ 2 A(cid:18)µAµ(cid:19), (1b) dn−eerglepcatrianmg eﬂtuecrtuﬁaetldiosnsaraendsetthteoptrhesesirurme eisang-iﬁveelndbvyalpue=s 0 V . The energy and quark number density of the sys- eﬀ − Uσ = λ42 (cid:18)σ2 −(cid:19)fπ2(cid:34) χχ(cid:18) 2(cid:19)2 − h(cid:35) χχ 2 σ , (1c) tneqm=ar∂ep/th∂uµs. eTvahliusatisedeqausiveal=entT∂top/c∂oTnv+enµtinoqnal dpeatenrd- 0 0 ministic ﬂuid dynamical calculations using a chiral equa- 1 χ 4 χ 4 − tion of state (EoS). U = B ln 1 , (1d) χ 4 χ χ From the curvature of the eﬀective potential at the 0 0 equilibrium value, the mass of the sigma ﬁeld and thus the inverse correlation length are o(cid:12)btained as (cid:12) as introduced in [30]. In addition to the breaking and (cid:12) (cid:12) restoration of chiral symmetry it accounts for scale sym- metry via the dilaton or glueball ﬁeld χ that is identiﬁed m2 = 1 = ∂2Veﬀ . (6) with a gluon condensate. In the present version of this σ ξ2 ∂σ2 (cid:104) (cid:105) eq σ= σ model we consider light quarks only, so q = (u, d). The two condensates σ and χ dynamically generate masses for In order to study nonequilibrium eﬀects, we follow our the constituent quarks and gluons, thus allowing us to ﬁx previous works [25, 28, 29, 31], and propagate both order the coupling parameters g = 3.37 and g = 850 MeV q A parameters explicitly. For the chiral condensate we derive from the ground state nucleon and glueball masses. The a stochastic relaxation equation from the two-particle ir- additional parameters of the chiral sigma model are stan- reducible eﬀective action as dard values: the pion decay constant of f = 93 MeV, the π pion mass m = 138 MeV, the explicit symmetry break- π − ing term h = fπm2π and the self-coupling λ2 = m2π2f2m2σ . ∂ ∂µσ + η ∂ σ + δVeﬀ = ξ , (7) For more details, the reader is referred to [29, 30]. π µ σ t δσ In mean-ﬁeld approximation, the eﬀective thermody- namic potential reads which takes into account interactions with the sur- ##LTLine## ##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: relaxational

text: equation,

text: it

text: interacts

text: with

text: a

text: fermionic

text: heat

text: bath

text: which

text: expands

text: ﬂuid

text: dynamically

text: It

text: has

text: been

text: demonstrated

text: in

text: that

text: this

text: model

text: is

text: able

text: to

text: de-

text: Veﬀ

label: equation

text: Ωqq¯

text: ΩA

text: Uσ

text: Uχ

text: Ω0

text: scribe

text: critical

text: slowing

text: down

text: as

text: well

text: as

text: spinodal

text: decompo-

text: sition

text: within

text: a

text: dynamical

text: setup.

text: We

text: follow

text: the

text: evolution

text: of

text: the

text: system

text: over

text: various

text: hy-

text: with

text: the

text: quark

text: persurfaces

text: of

text: constant

text: energy

text: density

text: of

text: the

text: coupled

text: sys-

text: tem

text: and

text: compare

text: the

text: ﬂuctuations

text: in

text: the

text: sigma

text: ﬁeld

text: to

text: the

text: ﬂuctuations

text: in

text: net-proton

text: numbers

text: which

text: are

text: obtained

text: Ωqq¯

text: ln

text: e

text: T

text: from

text: a

text: Cooper-Frye

text: particlization

text: prescription.

text: We

text: fur-

text: thermore

text: give

text: a

text: comparison

text: to

text: a

text: mean-ﬁeld

text: evolution

text: in

text: −Eq+µ

text: ln

text: e

text: T

text: order

text: to

text: pin

text: down

text: the

text: ﬂuctuations

text: stemming

text: from

text: the

text: itnraitniasiltsiotant.e

text: versus

text: dynamical

text: ﬂuctuations

text: near

text: the

text: phase

text: ΩA

text: ln

text: e−ETA

text: We

text: begin

text: with

text: a

text: description

text: of

text: nonequilibrium

text: chiral

text: ﬂuid

text: dynamics

text: (NχFD)

text: in

text: Sec.

text: including

text: the

text: equations

text: which

text: depend

text: on

text: temperature

text: T

text: and

text: quark

text: chemical

text: of

text: motion

text: and

text: a

text: brief

text: description

text: of

text: the

text: implementation

text: potential

text: µ

text: µ

text: Ω

text: in

text: Eq.

text: is

text: an

text: unimportant

text: of

text: initial

text: state

text: and

text: particlization.

text: In

text: Sec.

text: we

text: investigate

text: B

text: constant

text: to

text: set

text: the

text: total

text: energy

text: to

text: zero

text: in

text: the

text: ground

text: the

text: dynamics

text: of

text: the

text: net-proton

text: and

text: sigma

text: ﬁeld

text: kurtosis

text: state.

text: The

text: quasiparticle

text: energies

text: of

text: constituent

text: quarks

text: for

text: an

text: evolution

text: in

text: the

text: crossover

text: regime

text: near

text: the

text: critical

text: and

text: gluons

text: are

text: generated

text: via

text: their

text: eﬀective

text: masses

text: as

text: point.

text: We

text: conclude

text: and

text: give

text: a

text: brief

text: outlook

text: in

text: Sec.

text: E

text: and

text: E

text: II.

text: NONEQUILIBRIUM

text: CHIRAL

text: FLUID

text: q

text: q

text: A

text: A

text: DYNAMICS

text: The

text: mean-ﬁeld

text: values

text: of

text: the

text: condensate

text: ﬁelds,

text: σ

text: and

text: χ

text: are

text: obtained

text: by

text: minimizing

text: the

text: eﬀective

text: thermody-

text: We

text: study

text: the

text: ﬂuid

text: dynamical

text: evolution

text: in

text: a

text: heavy-ion

text: namic

text: via

text: collision

text: using

text: a

text: quark-meson

text: model

text: with

text: dilaton

text: ﬁeld,

text: ∂V

text: ∂V

text: eﬀ

text: eﬀ

text: L

text: ∂σ

text: ∂χ

text: q

text: (iγµ∂

text: g

text: σ)

text: q

text: σ=

text: σ

text: χ=

text: χ

text: µ

text: q

text: µ

text: L

text: A

text: Uσ

text: Uχ

text: In

text: what

text: we

text: will

text: call

text: the

text: mean-ﬁeld

text: evolution,

text: the

text: or-

text: LA

text: dn−eerglepcatrianmg

text: eisang-iﬁveelndbvyalpue=s

text: V

text: The

text: energy

text: and

text: quark

text: number

text: density

text: of

text: the

text: sys-

text: eﬀ

text: Uσ

text: χχ

text: σ

text: tneqm=ar∂ep/th∂uµs.

text: dpeatenrd-

text: ministic

text: ﬂuid

text: dynamical

text: calculations

text: using

text: a

text: chiral

text: equa-

text: χ

text: χ

text: tion

text: of

text: state

text: (EoS).

text: U

text: B

text: ln

text: χ

text: χ

text: χ

text: From

text: the

text: curvature

text: of

text: the

text: eﬀective

text: potential

text: at

text: the

text: equilibrium

text: value,

text: the

text: mass

text: of

text: the

text: sigma

text: ﬁeld

text: and

text: thus

text: the

text: inverse

text: correlation

text: length

text: are

text: as

text: as

text: introduced

text: in

text: In

text: addition

text: to

text: the

text: breaking

text: and

text: restoration

text: of

text: chiral

text: symmetry

text: it

text: accounts

text: for

text: scale

text: sym-

text: metry

text: via

text: the

text: dilaton

text: or

text: glueball

text: ﬁeld

text: χ

text: that

text: is

text: identiﬁed

text: with

text: a

text: gluon

text: condensate.

text: In

text: the

text: present

text: version

text: of

text: this

text: σ

text: ξ2

text: ∂σ2

text: eq

text: σ=

text: σ

text: model

text: we

text: consider

text: light

text: quarks

text: only,

text: so

text: q

text: (u,

text: d).

text: The

text: two

text: condensates

text: σ

text: and

text: χ

text: dynamically

text: generate

text: masses

text: for

text: In

text: order

text: to

text: study

text: nonequilibrium

text: eﬀects,

text: we

text: follow

text: our

text: the

text: constituent

text: quarks

text: and

text: gluons,

text: thus

text: allowing

text: us

text: to

text: ﬁx

text: previous

text: works

text: and

text: propagate

text: both

text: order

text: the

text: coupling

text: parameters

text: g

text: and

text: g

text: MeV

text: q

text: A

text: parameters

text: explicitly.

text: For

text: the

text: chiral

text: condensate

text: we

text: derive

text: from

text: the

text: ground

text: state

text: nucleon

text: and

text: glueball

text: masses.

text: The

text: a

text: stochastic

text: relaxation

text: equation

text: from

text: the

text: two-particle

text: ir-

text: additional

text: parameters

text: of

text: the

text: chiral

text: sigma

text: model

text: are

text: stan-

text: reducible

text: eﬀective

text: action

text: as

text: dard

text: values:

text: the

text: pion

text: decay

text: constant

text: of

text: f

text: MeV,

text: the

text: π

text: pion

text: mass

text: m

text: MeV,

text: the

text: explicit

text: symmetry

text: break-

text: π

text: ing

text: term

text: h

text: and

text: the

text: self-coupling

text: λ2

text: ∂µσ

text: η

text: σ

text: δVeﬀ

text: ξ

text: For

text: more

text: details,

text: the

text: reader

text: is

text: referred

text: to

text: π

text: µ

text: σ

text: t

text: δσ

text: In

text: mean-ﬁeld

text: approximation,

text: the

text: eﬀective

text: thermody-

text: namic

text: potential

text: reads

text: which

text: takes

text: into

text: account

text: interactions

text: with

text: the

text: sur-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 4 6 6 γ =1 γ =0.5 1.2 4 AF 4 AF 0.2 2 2 1 2.0 λ 0 o GPI GPII λ 0 o GPI GPII 10.0 −2 −2 0.8 AF AF −4 −4 0.6 −6 −6 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 α α 0.4 FIG. 1: (Colo−r online) Phase diagram of the Hamiltonian 0.2 (7) in the α λ plane for the isotropic case γ = 1 and an anisotropic case with γ = 0.5. For a ﬁxed α and a 0 large magnitude of the rescaled ﬁeld λ, the spin chain is 0 1 2 3 4 5 anti-ferromagnetic (AF). On the other hand, when λ is re- t duced, the system undergoes quantum phase transitions from 1.8 Gapless-I (GPI) phase characterized by two Fermi points 0.2 to the Gapless-II (GPII) phase characterized by four Fermi 1.6 1.5 points. The vertical line shows the direction of quenching. 1.4 2.0 After [59]. 3.0 1.2 1 0.8 0.6 0.4 0.2 the isotropic case γ = 1. Probing the rate function, we in- deed ﬁnd periodic occurrences of sharp non-analyticities 0 0 1 2 3 4 5 as expected in the case of quenching across an isolated t QCP (see the numerical result presented in the top panel (cid:113) of Fig. (2)); the instants at which these non-analyticities FIG. 2: (Color online) The rate function I(t) as obtained by appea(cid:113)r can be matched with those obtained from numerically integrating Eq. (3) is plotted as a function of Eq. (5) with (cid:15)f = (1/2) λ2 + 1 + γ2 + 2γ cos k∗ = time following a slow quench from a large negative to a large k∗ f positive value of λ. The upper panel corresponds to γ = 1 (1/2) λ2 + 2 + 2 cos k∗ for γ = 1, λ being the ﬁnal where there are periodic occurrences of DPTs for all values f f | of τ’s subject to the condition that p < 0.5, i.e., τ > parameter value reached after the quenching. It is worth | k k=0 τ (γ) → . The lower panel corresponds to the anisotropic mentioning that the dynamics is completely insensitive to 1 γ 1 situation with γ = 0.5 for which the critical value of τ as the fact that the system is driven across gapless phases obtained from Eq. (9) is given by τ (γ) = 1.76. As discussed 2 in the process of quenching and hence no trace of gap- in the text, the ﬁgure shows the presence of DPTs for τ (γ) < 1 less phases is reﬂected in DPTs. However, the oc- τ < τ (γ) while they disappear for τ > τ (γ). The red colored 2 2 currences of DPTs also require the condition that the (solid line) show sharp peaks whereas black colored (dashed minimum value of the non-adiabatic transition probabil- lines) show peaks those are rounded oﬀ. − | | | − ity p = exp( π ∆k 2 k=0τ ) = exp( π(1+γ)2τ ) must be k=0 2 2 less than 1/2 so that a k∗ (for which pk=k∗ = 1/2) exists. − This does not happen if the quenching is too rapid, i.e., in the (γ τ) plane given by the equation: { } (cid:54) τ < τ (γ) = 2 log 2/ π(1 + γ)2 for γ = 1; for γ = 1, |1 τ (γ) → = log 2/(2π). One therefore does not indeed − − 2 log 2 1 γ 1 exp( π(1 γ)2τ/2) = 1/2; τ = τ (γ) = − . observe DPTs even in the isotropic case for too rapid 2 π(1 γ)2 quenching processes. (9) For a ﬁxed γ, if τ exceeds τ (γ), DPTs disappear. This 2 is veriﬁed numerically and shown in the lower panel of We now move to the more interesting situation which (cid:54) Fig. (2) where we evaluate I(t) by numerically calculating a| rise|s in the anisotropic case (γ = 1); in this case, − ∆k 2 = (1 + γ2 + 2γ cos k), assumes the minimum value pks and using the values (cid:15)f1,k = (cid:15)fk and (cid:15)f2,k = (cid:15)fk (cor- at the boundary between AF and the GPI phase for the responding to λ = 10) in Eq. (3); we show that DPTs | | − f mode k = π and is given by ∆ 2 = (1 γ)2, and hence occurring for τ < τ (γ), disappear when τ exceeds τ (γ). k 2 2 the maximum value of the non-adiabatic transition prob- Referring to the situation γ = 1, when maximum value − − | → ∞ ability pmax = exp( π(1 γ)2τ/2). As emphasized be- of p (for k = π) is equal to unity and τ (γ) → , k k 2 γ 1 fore, DPTs can occur only when p = 1/2. If the max- DPTs periodically appear for all values of τ > τ (γ). k (cid:54) 1 imum possible value of pmax is less than 1/2, no DPT On the contrary, for γ = 1, there always exists a critical k can appear even when the system is quenched across the τ (γ) and DPTs appear only when τ (γ) < τ < τ (γ). 2 1 2 QCPs and gapless phases. We therefore ﬁnd a boundary We would like to emphasize that these observations are )t(I )t(I ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: γ

text: γ

text: AF

text: AF

text: λ

text: o

text: GPI

text: GPII

text: λ

text: o

text: GPI

text: GPII

text: −2

text: −2

text: AF

text: AF

text: −4

text: −4

text: −6

text: −6

text: α

text: α

text: FIG.

text: (Colo−r

text: online)

text: Phase

text: diagram

text: of

text: the

text: Hamiltonian

text: in

text: the

text: α

text: λ

text: plane

text: for

text: the

text: isotropic

text: case

text: γ

text: and

text: an

text: anisotropic

text: case

text: with

text: γ

text: For

text: a

text: ﬁxed

text: α

text: and

text: a

text: large

text: magnitude

text: of

text: the

text: rescaled

text: ﬁeld

text: λ,

text: the

text: spin

text: chain

text: is

text: anti-ferromagnetic

text: (AF).

text: On

text: the

text: other

text: hand,

text: when

text: λ

text: is

text: re-

text: t

text: duced,

text: the

text: system

text: undergoes

text: quantum

text: phase

text: transitions

text: from

text: Gapless-I

text: (GPI)

text: phase

text: characterized

text: by

text: two

text: Fermi

text: points

text: to

text: the

text: Gapless-II

text: (GPII)

text: phase

text: characterized

text: by

text: four

text: Fermi

text: points.

text: The

text: vertical

text: line

text: shows

text: the

text: direction

text: of

text: quenching.

text: After

text: the

text: isotropic

text: case

text: γ

text: Probing

text: the

text: rate

text: function,

text: we

text: in-

text: deed

text: ﬁnd

text: periodic

text: occurrences

text: of

text: sharp

text: non-analyticities

text: as

text: expected

text: in

text: the

text: case

text: of

text: quenching

text: across

text: an

text: isolated

text: t

text: QCP

text: (see

text: the

text: numerical

text: result

text: presented

text: in

text: the

text: top

text: panel

text: of

text: Fig.

text: the

text: instants

text: at

text: which

text: these

text: non-analyticities

text: FIG.

text: (Color

text: online)

text: The

text: rate

text: function

text: I(t)

text: as

text: obtained

text: by

text: can

text: be

text: matched

text: with

text: those

text: obtained

text: from

text: numerically

text: integrating

text: Eq.

text: is

text: plotted

text: as

text: a

text: function

text: of

text: Eq.

text: with

text: λ2

text: γ2

text: 2γ

text: cos

text: k∗

text: time

text: following

text: a

text: slow

text: quench

text: from

text: a

text: large

text: negative

text: to

text: a

text: large

text: k∗

text: f

text: positive

text: value

text: of

text: λ.

text: The

text: upper

text: panel

text: corresponds

text: to

text: γ

text: λ2

text: cos

text: k∗

text: for

text: γ

text: λ

text: being

text: the

text: ﬁnal

text: where

text: there

text: are

text: periodic

text: occurrences

text: of

text: DPTs

text: for

text: all

text: values

text: f

text: f

text: of

text: τ’s

text: subject

text: to

text: the

text: condition

text: that

text: p

text: i.e.,

text: τ

text: parameter

text: value

text: reached

text: after

text: the

text: quenching.

text: It

text: is

text: worth

text: k

text: τ

text: (γ)

text: The

text: lower

text: panel

text: corresponds

text: to

text: the

text: anisotropic

text: mentioning

text: that

text: the

text: dynamics

text: is

text: completely

text: insensitive

text: to

text: γ

text: situation

text: with

text: γ

text: for

text: which

text: the

text: critical

text: value

text: of

text: τ

text: as

text: the

text: fact

text: that

text: the

text: system

text: is

text: driven

text: across

text: gapless

text: phases

text: obtained

text: from

text: Eq.

text: is

text: given

text: by

text: τ

text: (γ)

text: As

text: discussed

text: in

text: the

text: process

text: of

text: quenching

text: and

text: hence

text: no

text: trace

text: of

text: gap-

text: in

text: the

text: text,

text: the

text: ﬁgure

text: shows

text: the

text: presence

text: of

text: DPTs

text: for

text: τ

text: (γ)

text: less

text: phases

text: is

text: reﬂected

text: in

text: DPTs.

text: However,

text: the

text: oc-

text: τ

text: τ

text: (γ)

text: while

text: they

text: disappear

text: for

text: τ

text: τ

text: (γ).

text: The

text: red

text: colored

text: currences

text: of

text: DPTs

text: also

text: require

text: the

text: condition

text: that

text: the

text: (solid

text: line)

text: show

text: sharp

text: peaks

text: whereas

text: black

text: colored

text: (dashed

text: minimum

text: value

text: of

text: the

text: non-adiabatic

text: transition

text: probabil-

text: lines)

text: show

text: peaks

text: those

text: are

text: rounded

text: oﬀ.

text: ity

text: p

text: exp(

text: π

text: ∆k

text: exp(

text: must

text: be

text: less

text: than

text: so

text: that

text: a

text: k∗

text: (for

text: which

text: pk=k∗

text: exists.

text: This

text: does

text: not

text: happen

text: if

text: the

text: quenching

text: is

text: too

text: rapid,

text: i.e.,

text: in

text: the

text: (γ

text: τ)

text: plane

text: given

text: by

text: the

text: equation:

text: τ

text: τ

text: (γ)

text: log

text: for

text: γ

text: for

text: γ

text: τ

text: (γ)

text: log

text: One

text: therefore

text: does

text: not

text: indeed

text: log

text: γ

text: exp(

text: τ

text: τ

text: (γ)

text: observe

text: DPTs

text: even

text: in

text: the

text: isotropic

text: case

text: for

text: too

text: rapid

text: quenching

text: processes.

text: For

text: a

text: ﬁxed

text: γ,

text: if

text: τ

text: exceeds

text: τ

text: (γ),

text: DPTs

text: disappear.

text: This

text: is

text: veriﬁed

text: numerically

text: and

text: shown

text: in

text: the

text: lower

text: panel

text: of

text: We

text: now

text: move

text: to

text: the

text: more

text: interesting

text: situation

text: which

text: Fig.

text: where

text: we

text: evaluate

text: I(t)

text: by

text: numerically

text: calculating

text: a|

text: rise|s

text: in

text: the

text: anisotropic

text: case

text: (γ

text: in

text: this

text: case,

text: ∆k

text: γ2

text: 2γ

text: cos

text: k),

text: assumes

text: the

text: minimum

text: value

text: pks

text: and

text: using

text: the

text: values

text: and

text: (cor-

text: at

text: the

text: boundary

text: between

text: AF

text: and

text: the

text: GPI

text: phase

text: for

text: the

text: responding

text: to

text: λ

text: in

text: Eq.

text: we

text: show

text: that

text: DPTs

text: f

text: mode

text: k

text: π

text: and

text: is

text: given

text: by

text: and

text: hence

text: occurring

text: for

text: τ

text: τ

text: (γ),

text: disappear

text: when

text: τ

text: exceeds

text: τ

text: (γ).

text: k

text: the

text: maximum

text: value

text: of

text: the

text: non-adiabatic

text: transition

text: prob-

text: Referring

text: to

text: the

text: situation

text: γ

text: when

text: maximum

text: value

text: ability

text: pmax

text: exp(

text: As

text: emphasized

text: be-

text: of

text: p

text: (for

text: k

text: π)

text: is

text: equal

text: to

text: unity

text: and

text: τ

text: (γ)

text: k

text: k

text: γ

text: fore,

text: DPTs

text: can

text: occur

text: only

text: when

text: p

text: If

text: the

text: max-

text: DPTs

text: periodically

text: appear

text: for

text: all

text: values

text: of

text: τ

text: τ

text: (γ).

text: k

text: imum

text: possible

text: value

text: of

text: pmax

text: is

text: less

text: than

text: no

text: DPT

text: On

text: the

text: contrary,

text: for

text: γ

text: there

text: always

text: exists

text: a

text: critical

text: k

text: can

text: appear

text: even

text: when

text: the

text: system

text: is

text: quenched

text: across

text: the

text: τ

text: (γ)

text: and

text: DPTs

text: appear

text: only

text: when

text: τ

text: (γ)

text: τ

text: τ

text: (γ).

text: QCPs

text: and

text: gapless

text: phases.

text: We

text: therefore

text: ﬁnd

text: a

text: boundary

text: We

text: would

text: like

text: to

text: emphasize

text: that

text: these

text: observations

text: are

text: )t(I

text: )t(I

text: ##LTFigure##

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text: 5 (cid:54) for γ = 1 and = 1. Referring to the Hamiltonian (7), we 2 0 ﬁnd that the ground state and the excited state, i.e., the t 1 adiabatic basis states, for a given λ (say, λi) is given by 1 6 t 2 | (cid:105) θ − θ 1 2 N O - D P T s 1ik = cos 2k (1, 0)T sin 2k (0, 1)T | (cid:105) θ θ 2i = sin k (1, 0)T + cos k (0, 1)T , (10) 8 D P T s k 2 2 −| | where tan θ = ∆ /λ which clearly does not depend k k 4 2 l o g 2 / on α. When the ﬁeld λ is suddenly changed to λ , the p i | | |(cid:104) ||f (cid:105)| l o g 2 / 2 N O - D P T s excitation probability is given by p˜k = u˜k 2 = 1ik 2fk 2. p 0 As discussed before, the ne|cessary condition for the pres- 1 . 0 0 . 8 0 . 6 0 . 4 0 . 2 0 . 0 ence of a DPT requires p˜k k=k∗ = 1/2. Using Eqs. (10), g we immediately ﬁnd (cid:20) (cid:21) | | |(cid:104) || (cid:105)| − p˜ = u˜ 2 =(cid:112)1 (λ ) 2 (λ (cid:113)) 2 = sin2[(θi θf)]/2 k k k i k f | | k k FIG. 3: (Color online) The phase diagram in the γ-τ plane = 1 1 − λfλi + ∆k 2 ; (11) | | | | showing the DPT and the no-DPT regions following a slow 2 (λ2 + ∆ 2) (λ2 + ∆ 2) quench as discussed in Fig. 2. The upper curve corresponds to i k f k − the condition presented in Eq. (9) i.e., τ (γ) = 2 log 2/(π(1 2 γ)2) which diverges in the isotropic case (γ = 1). The lower it should be noted that p˜k depends on λi, λf and γ but curve denoted by τ (γ) = 2 log 2/(π(1+γ)2) is obtained from never on α. 1 | the requirement pk=0 = 1/2 and τ1(γ) γ=1 = log 2/2π. It is to To| pred|ict the|prese|nce of DPTs, it is suﬃcient to ana- be noted that the values of both τ (γ) and τ (γ) are zoomed lyze u˜ 2 and u˜ 2; the necessary condition for DPT 1 2 k=0 | k=|π | | by a factor of 10 for better visibility. The DPTs exist for the γ would then be u˜ 2 > 1/2 and u˜ 2 < 1/2. (We re- | k|=π k=0 and τ values lying in the region bounded by τ2(γ) and τ1(γ). call that the ∆k is minimum for the mode k = π and hence probability of excitation is maximum for that par- ticular mode). If these conditions are satisﬁed, from the all appropriately supported by the behavior of the lines argument of continuity one concludes that there must ex- | | of Fisher zeros, e.g., one can verify that the lines of Fisher ist a k∗ for which u˜k=k∗ 2 = 1/2, ensuring the existence zeros never cross the imaginary axis in the no-DPT re- of DPTs. We note that this is the most generic condition gion in the anisotropic case. All these conditions are for DPTs to occur as long as one can sharply deﬁne a k∗. summarised in Fig. (3). Using (11), one can show that p˜ becomes equal to 1/2 k Interestingly, the relation (9) does not depend on the for a mode k only when parameter α which plays no role in the temporal evo- (cid:112) (cid:113) lution of the system. Let us also addresss the question | | λ λ + ∆ 2 ⇒ | | what happens when α is quenched keeping λ and γ ﬁxed. f i k = 0 = λ λ + ∆ 2 = 0, | | | | f i k In this case, the initial ground state of the Hamiltonian (λ2 + ∆ 2) (λ2 + ∆ 2) i k f k H (which is also the ground state of the LZ part of the k (12) | | Hamiltonian (7)), only evolves through an overall phase where ∆ 2 is evaluated at the corresponding value of k accumulation. Since the LZ part is unaltered, there is no k. To illustrate the main point in a transparent manner, − non-trivial dynamics for any mode k, and hence, no DPT we choose λ = λ = λ, (or the other way round i.e., − f − i is expected; the evolved state is never orthogonal to the | (cid:105) λ = λ = λ) for which Eq. (12) assumes a simpler f i state ψ . It is also noteworthy that in the anisotropic fk for(cid:112)m: (cid:112) case, the defect density shows an exponential decay (with − | | τ) as opposed to the standard power-law KZ scaling59. λ2 + ∆ 2 ⇒ | | | | k | | = 0 = λ2 = ∆ 2. V. SUDDEN QUENCHES: CONDITIONS FOR (λ2 + ∆ 2) (λ2 + ∆ 2) k k k DPTS (13) We shall analyze the condition given in (13) for the (cid:54) In this section, we shall consider a sudden quenching of modes k = 0 and k = π for both γ = 1 and γ = 1. In | | the parameter λ from an initial λ to a ﬁnal value λ . We the former case (γ = 1), u˜ 2 = 1 as the oﬀ-diagonal i f k=π address the questions whether DPTs are always present terms of the LZ part of the Hamiltonian (7) vanish for in the subsequent temporal evolution and how does the k = π so that this mode is temporally frozen. On the | | ≤ situation get altered in the anisotropic case in comparison other hand, the condition that u˜ 2 1/2, demands ≤ | k=0 to the isotropic case. Remarkably, as we shall illustrate λ2 ∆2 = 4. This implies that whenever the ﬁeld k k=0 ≥ − below, in this case also whether DPTs are present or λ is quenched from a value λ 2 to a ﬁnal value ≤ i absent depend on some conditions involving λ and γ both λ +2, DPTs will indeed appear. Otherwise, they are f t ##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##

text: for

text: γ

text: and

text: Referring

text: to

text: the

text: Hamiltonian

text: we

text: ﬁnd

text: that

text: the

text: ground

text: state

text: and

text: the

text: excited

text: state,

text: i.e.,

text: the

text: t

text: adiabatic

text: basis

text: states,

text: for

text: a

text: given

text: λ

text: (say,

text: λi)

text: is

text: given

text: by

text: t

text: θ

text: θ

text: N

text: O

text: D

text: P

text: T

text: s

text: cos

text: sin

text: θ

text: θ

text: sin

text: k

text: cos

text: k

text: D

text: P

text: T

text: s

text: k

text: where

text: tan

text: θ

text: /λ

text: which

text: clearly

text: does

text: not

text: depend

text: k

text: k

text: l

text: o

text: g

text: on

text: α.

text: When

text: the

text: ﬁeld

text: λ

text: is

text: suddenly

text: changed

text: to

text: λ

text: the

text: p

text: i

text: ||f

text: l

text: o

text: g

text: N

text: O

text: D

text: P

text: T

text: s

text: excitation

text: probability

text: is

text: given

text: by

text: p˜k

text: u˜k

text: p

text: As

text: discussed

text: before,

text: the

text: ne|cessary

text: condition

text: for

text: the

text: pres-

text: ence

text: of

text: a

text: DPT

text: requires

text: p˜k

text: k=k∗

text: Using

text: Eqs.

text: g

text: we

text: immediately

text: ﬁnd

text: p˜

text: u˜

text: (λ

text: (λ

text: k

text: k

text: k

text: i

text: k

text: f

text: k

text: k

text: FIG.

text: (Color

text: online)

text: The

text: phase

text: diagram

text: in

text: the

text: γ-τ

text: plane

text: λfλi

text: ∆k

text: showing

text: the

text: DPT

text: and

text: the

text: no-DPT

text: regions

text: following

text: a

text: slow

text: quench

text: as

text: discussed

text: in

text: Fig.

text: The

text: upper

text: curve

text: corresponds

text: to

text: i

text: k

text: f

text: k

text: the

text: condition

text: presented

text: in

text: Eq.

text: i.e.,

text: τ

text: (γ)

text: log

text: which

text: diverges

text: in

text: the

text: isotropic

text: case

text: (γ

text: The

text: lower

text: it

text: should

text: be

text: noted

text: that

text: p˜k

text: depends

text: on

text: λi,

text: λf

text: and

text: γ

text: but

text: curve

text: denoted

text: by

text: τ

text: (γ)

text: log

text: is

text: obtained

text: from

text: never

text: on

text: α.

text: the

text: requirement

text: and

text: log

text: It

text: is

text: to

text: To|

text: pred|ict

text: the|prese|nce

text: of

text: DPTs,

text: it

text: is

text: suﬃcient

text: to

text: ana-

text: be

text: noted

text: that

text: the

text: values

text: of

text: both

text: τ

text: (γ)

text: and

text: τ

text: (γ)

text: are

text: zoomed

text: lyze

text: u˜

text: and

text: u˜

text: the

text: necessary

text: condition

text: for

text: DPT

text: k=|π

text: by

text: a

text: factor

text: of

text: for

text: better

text: visibility.

text: The

text: DPTs

text: exist

text: for

text: the

text: γ

text: would

text: then

text: be

text: u˜

text: and

text: u˜

text: (We

text: re-

text: k|=π

text: and

text: τ

text: values

text: lying

text: in

text: the

text: region

text: bounded

text: by

text: and

text: call

text: that

text: the

text: ∆k

text: is

text: minimum

text: for

text: the

text: mode

text: k

text: π

text: and

text: hence

text: probability

text: of

text: excitation

text: is

text: maximum

text: for

text: that

text: par-

text: ticular

text: mode).

text: If

text: these

text: conditions

text: are

text: satisﬁed,

text: from

text: the

text: all

text: appropriately

text: supported

text: by

text: the

text: behavior

text: of

text: the

text: lines

text: argument

text: of

text: continuity

text: one

text: concludes

text: that

text: there

text: must

text: ex-

text: of

text: Fisher

text: zeros,

text: e.g.,

text: one

text: can

text: verify

text: that

text: the

text: lines

text: of

text: Fisher

text: ist

text: a

text: k∗

text: for

text: which

text: u˜k=k∗

text: ensuring

text: the

text: existence

text: zeros

text: never

text: cross

text: the

text: imaginary

text: axis

text: in

text: the

text: no-DPT

text: re-

text: of

text: DPTs.

text: We

text: note

text: that

text: this

text: is

text: the

text: most

text: generic

text: condition

text: gion

text: in

text: the

text: anisotropic

text: case.

text: All

text: these

text: conditions

text: are

text: for

text: DPTs

text: to

text: occur

text: as

text: long

text: as

text: one

text: can

text: sharply

text: deﬁne

text: a

text: k∗.

text: summarised

text: in

text: Fig.

text: Using

text: one

text: can

text: show

text: that

text: p˜

text: becomes

text: equal

text: to

text: k

text: Interestingly,

text: the

text: relation

text: does

text: not

text: depend

text: on

text: the

text: for

text: a

text: mode

text: k

text: only

text: when

text: parameter

text: α

text: which

text: plays

text: no

text: role

text: in

text: the

text: temporal

text: evo-

text: lution

text: of

text: the

text: system.

text: Let

text: us

text: also

text: addresss

text: the

text: question

text: λ

text: λ

text: what

text: happens

text: when

text: α

text: is

text: quenched

text: keeping

text: λ

text: and

text: γ

text: ﬁxed.

text: f

text: i

text: k

text: λ

text: λ

text: f

text: i

text: k

text: In

text: this

text: case,

text: the

text: initial

text: ground

text: state

text: of

text: the

text: Hamiltonian

text: i

text: k

text: f

text: k

text: H

text: (which

text: is

text: also

text: the

text: ground

text: state

text: of

text: the

text: LZ

text: part

text: of

text: the

text: k

text: Hamiltonian

text: only

text: evolves

text: through

text: an

text: overall

text: phase

text: where

text: is

text: evaluated

text: at

text: the

text: corresponding

text: value

text: of

text: k

text: accumulation.

text: Since

text: the

text: LZ

text: part

text: is

text: unaltered,

text: there

text: is

text: no

text: k.

text: To

text: illustrate

text: the

text: main

text: point

text: in

text: a

text: transparent

text: manner,

text: non-trivial

text: dynamics

text: for

text: any

text: mode

text: k,

text: and

text: hence,

text: no

text: DPT

text: we

text: choose

text: λ

text: λ

text: λ,

text: (or

text: the

text: other

text: way

text: round

text: i.e.,

text: f

text: i

text: is

text: expected;

text: the

text: evolved

text: state

text: is

text: never

text: orthogonal

text: to

text: the

text: λ

text: λ

text: λ)

text: for

text: which

text: Eq.

text: assumes

text: a

text: simpler

text: f

text: i

text: state

text: ψ

text: It

text: is

text: also

text: noteworthy

text: that

text: in

text: the

text: anisotropic

text: fk

text: case,

text: the

text: defect

text: density

text: shows

text: an

text: exponential

text: decay

text: (with

text: τ)

text: as

text: opposed

text: to

text: the

text: standard

text: power-law

text: KZ

text: λ2

text: k

text: λ2

text: V.

text: SUDDEN

text: QUENCHES:

text: CONDITIONS

text: FOR

text: k

text: k

text: k

text: DPTS

text: We

text: shall

text: analyze

text: the

text: condition

text: given

text: in

text: for

text: the

text: In

text: this

text: section,

text: we

text: shall

text: consider

text: a

text: sudden

text: quenching

text: of

text: modes

text: k

text: and

text: k

text: π

text: for

text: both

text: γ

text: and

text: γ

text: In

text: the

text: parameter

text: λ

text: from

text: an

text: initial

text: λ

text: to

text: a

text: ﬁnal

text: value

text: λ

text: We

text: the

text: former

text: case

text: (γ

text: u˜

text: as

text: the

text: oﬀ-diagonal

text: i

text: f

text: k=π

text: address

text: the

text: questions

text: whether

text: DPTs

text: are

text: always

text: present

text: terms

text: of

text: the

text: LZ

text: part

text: of

text: the

text: Hamiltonian

text: vanish

text: for

text: in

text: the

text: subsequent

text: temporal

text: evolution

text: and

text: how

text: does

text: the

text: k

text: π

text: so

text: that

text: this

text: mode

text: is

text: temporally

text: frozen.

text: On

text: the

text: situation

text: get

text: altered

text: in

text: the

text: anisotropic

text: case

text: in

text: comparison

text: other

text: hand,

text: the

text: condition

text: that

text: u˜

text: demands

text: to

text: the

text: isotropic

text: case.

text: Remarkably,

text: as

text: we

text: shall

text: illustrate

text: λ2

text: ∆2

text: This

text: implies

text: that

text: whenever

text: the

text: ﬁeld

text: k

text: below,

text: in

text: this

text: case

text: also

text: whether

text: DPTs

text: are

text: present

text: or

text: λ

text: is

text: quenched

text: from

text: a

text: value

text: λ

text: to

text: a

text: ﬁnal

text: value

text: i

text: absent

text: depend

text: on

text: some

text: conditions

text: involving

text: λ

text: and

text: γ

text: both

text: λ

text: DPTs

text: will

text: indeed

text: appear.

text: Otherwise,

text: they

text: are

text: f

text: t

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text: 7 QCP in the process of quenching; however, it should nec- 4 . 0 ( 1 - ) ( 1 + ) , = - 1 essarily be swept through λ = 0, i.e., either λi or λf 2 2 should be negative for DPTs to appear. We have illus- g £ l f £ g l i 3 . 5 ( 1 - ) 2 ( 1 + ) 2 / 2 , = - 2 trated these diﬀerent scenarios in Fig. 5. This is remark- g / 2 £ l f £ g l i able that DPTs can be made to appear (or disappear) 3 . 0 in the same model by tuning either the anisotropy term 2 . 5 N O - D P T s γ or the inverse quench rate τ for slow quenches, and λ D P T s i and λ for sudden quenches such that the system must f 2 . 0 be driven across λ = 0. 1 . 5 1 . 0 D P T s 0 . 5 Acknowledgments 0 . 0 1 . 0 0 . 8 0 . 6 0 . 4 0 . 2 0 . 0 g − FIG. 5: (Color online) The phase diagram in the λ γ plane f We acknowledge Jun-ichi Inoue for discussions and showing the regions where DPTs will occur following sudden − − quenches with λ = 1 (red triangles) and = 2 (blue circles) Anatoli Polkovnikov and Sei Suzuki for collaboration in i to a ﬁnal value λ . In both the cases, DPTs occur when λ lies related works. Shraddha Sharma acknowledges CSIR, In- f − | | | | f within the range (1 γ)2/ λ and (1 + γ)2/ λ as elaborated dia and also DST, India, and AD and UD acknowledges i i in the text. In the isotropic situation (γ = 1), the condition DST, India, for ﬁnancial support. AD and SS acknowl- ≤ ≤ | | gets simpliﬁed to 0 λf 4/ λi . edge Abdus Salam ICTP for hospitality where the initial part of the work was done. 1 M.E. Fisher, in Boulder Lectures in Theoretical Physics Fazio, Phys. Rev. A 75, 032333 (2007). (University of Colorado, Boulder, 1965), Vol. 7. 19 F. M. Cucchietti, et al, Phys. Rev. A 75, 032337 (2007); 2 C. Yang and T. Lee, Phys. Rev. 87, 404 (1952). C. Cormick and J. P. Paz, Phys. Rev. A 77, 022317 (2008). 3 W. van Saarloos and D. Kurtze, J. Phys. A 17, 1301 20 Lorenzo C Venuti and P. Zanardi, Phys. Rev. A 81, 022113 (1984). (2010); Lorenzo C. Venuti, N. T. Jacobson, S. Santra, and 4 M. Heyl, A. Polkovnikov, and S. Kehrein, Phys. Rev. Lett., P. Zanardi, Phys. Rev. Lett. 107, 010403 (2011). 110, 135704 (2013). 21 S. Sharma, V. Mukherjee, and A. Dutta, Eur. Phys. J. B, 5 S. Sachdev, Quantum Phase Transitions (Cambridge Uni- 85, 143 (2012). versity Press, Cambridge, UK, 1999). 22 V. Mukherjee, S. Sharma, A. Dutta, Phys. Rev. B 86, 6 S. Suzuki, J-i Inoue and Bikas K. Chkarabarti, Quantum 020301 (R) (2012). Ising Phases and Transitions in Transverse Ising Models 23 T. Nag, U. Divakaran and A. Dutta, Phys. Rev. B 86, (Springer, Lecture Notes in Physics, Vol. 862 (2013)). 020401 (R) (2012). 7 C. Karrasch and D. Schuricht, Phys. Rev. B, 87, 195104 24 S. Sharma, A. Russomanno, G. E. Santoro and A. Dutta, (2013). EPL 106, 67003 (2014). 8 N. Kriel, C. Karrasch, and S. Kehrein, Phys. Rev. B 90, 25 B. Dora, F. Pollmann, J. Fortgh, G. Zarand, Phys. Rev. 125106 (2014) Lett. 111, 046402 (2013); R. Sachdeva, T. Nag, A. Agar- 9 F. Andraschko, J. Sirker, Phys. Rev. B 89, 125120 (2014). wal, A. Dutta, Phys. Rev. B 90, 045421 (2014). 10 E. Canovi, P. Werner, and M. Eckstein, Phys. Rev. Lett. 26 S. Suzuki, T. Nag and A. Dutta, arXiv:1509.04649 (to ap- 113, 265702 (2014). pear in Phys. Rev. A (2016)). 11 S. Vajna and B. Dora, Phys. Rev. B 89, 161105(R) (2014). 27 A. Gambassi and A. Silva, arXiv: 1106.2671 (2011); P. 12 S. Sharma, S. Suzuki and A. Dutta, Phys. Rev. B 92, Smacchia and A. Silva, Phys. Rev. E 88, 042109, (2013). 104306 (2015). 28 A. Russomanno, S. Sharma, A. Dutta and G. E. Santoro, 13 S. Vajna and B. Dora, Phys. Rev. B 91, 155127 (2015). J. Stat. Mech., P08030 (2015). 14 M. Schmitt and S. Kehrein, Phys. Rev. B 92, 075114 29 P. Zanardi. H. T. Quan, X. Wang and C. P. Sun, Phys. (2015). Rev. A 75, 032109 (2007). 15 J. C. Budich and M. Heyl, arXiv:1504.05599 (2015). 30 R. Dorner, J. Goold, C. Cormick, M. Paternostro and V. 16 T. Palmai, Phys. Rev. B 92, 235433 (2015). Vedral, Phys. Rev. Lett. 109, 160601 (2012). 17 H.T. Quan, Z. Song, X.F. Liu, P. Zanardi, and C.P. Sun, 31 S. Sharma and A. Dutta, Phys. Rev. E 92, 022108 (2015). Phys.Rev.Lett. 96, 140604 (2006). 32 F. Pollmann, S. Mukerjee, A. G. Green, and J. E. Moore, 18 D. Rossini, T. Calarco, V. Giovannetti, S. Montangero, R. Phys. Rev. E 81, 020101(R) (2010). l f ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: QCP

text: in

text: the

text: process

text: of

text: quenching;

text: however,

text: it

text: should

text: nec-

text: essarily

text: be

text: swept

text: through

text: λ

text: i.e.,

text: either

text: λi

text: or

text: λf

text: should

text: be

text: negative

text: for

text: DPTs

text: to

text: appear.

text: We

text: have

text: illus-

text: g

text: l

text: f

text: g

text: l

text: i

text: trated

text: these

text: diﬀerent

text: scenarios

text: in

text: Fig.

text: This

text: is

text: remark-

text: g

text: l

text: f

text: g

text: l

text: i

text: able

text: that

text: DPTs

text: can

text: be

text: made

text: to

text: appear

text: (or

text: disappear)

text: in

text: the

text: same

text: model

text: by

text: tuning

text: either

text: the

text: anisotropy

text: term

text: N

text: O

text: D

text: P

text: T

text: s

text: γ

text: or

text: the

text: inverse

text: quench

text: rate

text: τ

text: for

text: slow

text: quenches,

text: and

text: λ

text: D

text: P

text: T

text: s

text: i

text: and

text: λ

text: for

text: sudden

text: quenches

text: such

text: that

text: the

text: system

text: must

text: f

text: be

text: driven

text: across

text: λ

text: D

text: P

text: T

text: s

text: Acknowledgments

text: g

text: FIG.

text: (Color

text: online)

text: The

text: phase

text: diagram

text: in

text: the

text: λ

text: γ

text: plane

text: f

text: We

text: acknowledge

text: Jun-ichi

text: Inoue

text: for

text: discussions

text: and

text: showing

text: the

text: regions

text: where

text: DPTs

text: will

text: occur

text: following

text: sudden

text: quenches

text: with

text: λ

text: (red

text: triangles)

text: and

text: (blue

text: circles)

text: Anatoli

text: Polkovnikov

text: and

text: Sei

text: Suzuki

text: for

text: collaboration

text: in

text: i

text: to

text: a

text: ﬁnal

text: value

text: λ

text: In

text: both

text: the

text: cases,

text: DPTs

text: occur

text: when

text: λ

text: lies

text: related

text: works.

text: Shraddha

text: Sharma

text: acknowledges

text: CSIR,

text: In-

text: f

text: f

text: within

text: the

text: range

text: λ

text: and

text: λ

text: as

text: elaborated

text: dia

text: and

text: also

text: DST,

text: India,

text: and

text: AD

text: and

text: UD

text: acknowledges

text: i

text: i

text: in

text: the

text: text.

text: In

text: the

text: isotropic

text: situation

text: (γ

text: the

text: condition

text: DST,

text: India,

text: for

text: ﬁnancial

text: support.

text: AD

text: and

text: SS

text: acknowl-

text: gets

text: simpliﬁed

text: to

text: λf

text: λi

text: edge

text: Abdus

text: Salam

text: ICTP

text: for

text: hospitality

text: where

text: the

text: initial

text: part

text: of

text: the

text: work

text: was

text: done.

text: M.E.

text: Fisher,

text: in

text: Boulder

text: Lectures

text: in

text: Theoretical

text: Physics

text: Fazio,

text: Phys.

text: Rev.

text: A

text: (University

text: of

text: Colorado,

text: Boulder,

text: Vol.

text: F.

text: M.

text: Cucchietti,

text: et

text: al,

text: Phys.

text: Rev.

text: A

text: C.

text: Yang

text: and

text: T.

text: Lee,

text: Phys.

text: Rev.

text: C.

text: Cormick

text: and

text: J.

text: P.

text: Paz,

text: Phys.

text: Rev.

text: A

text: W.

text: van

text: Saarloos

text: and

text: D.

text: Kurtze,

text: J.

text: Phys.

text: A

text: Lorenzo

text: C

text: Venuti

text: and

text: P.

text: Zanardi,

text: Phys.

text: Rev.

text: A

text: Lorenzo

text: C.

text: Venuti,

text: N.

text: T.

text: Jacobson,

text: S.

text: Santra,

text: and

text: M.

text: Heyl,

text: A.

text: Polkovnikov,

text: and

text: S.

text: Kehrein,

text: Phys.

text: Rev.

text: Lett.,

text: P.

text: Zanardi,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Sharma,

text: V.

text: Mukherjee,

text: and

text: A.

text: Dutta,

text: Eur.

text: Phys.

text: J.

text: B,

text: S.

text: Sachdev,

text: Quantum

text: Phase

text: Transitions

text: (Cambridge

text: Uni-

text: versity

text: Press,

text: Cambridge,

text: UK,

text: V.

text: Mukherjee,

text: S.

text: Sharma,

text: A.

text: Dutta,

text: Phys.

text: Rev.

text: B

text: S.

text: Suzuki,

text: J-i

text: Inoue

text: and

text: Bikas

text: K.

text: Chkarabarti,

text: Quantum

text: (R)

text: Ising

text: Phases

text: and

text: Transitions

text: in

text: Transverse

text: Ising

text: Models

text: T.

text: Nag,

text: U.

text: Divakaran

text: and

text: A.

text: Dutta,

text: Phys.

text: Rev.

text: B

text: (Springer,

text: Lecture

text: Notes

text: in

text: Physics,

text: Vol.

text: (R)

text: C.

text: Karrasch

text: and

text: D.

text: Schuricht,

text: Phys.

text: Rev.

text: B,

text: S.

text: Sharma,

text: A.

text: Russomanno,

text: G.

text: E.

text: Santoro

text: and

text: A.

text: Dutta,

text: EPL

text: N.

text: Kriel,

text: C.

text: Karrasch,

text: and

text: S.

text: Kehrein,

text: Phys.

text: Rev.

text: B

text: B.

text: Dora,

text: F.

text: Pollmann,

text: J.

text: Fortgh,

text: G.

text: Zarand,

text: Phys.

text: Rev.

text: Lett.

text: R.

text: Sachdeva,

text: T.

text: Nag,

text: A.

text: Agar-

text: F.

text: Andraschko,

text: J.

text: Sirker,

text: Phys.

text: Rev.

text: B

text: wal,

text: A.

text: Dutta,

text: Phys.

text: Rev.

text: B

text: E.

text: Canovi,

text: P.

text: Werner,

text: and

text: M.

text: Eckstein,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Suzuki,

text: T.

text: Nag

text: and

text: A.

text: Dutta,

text: (to

text: ap-

text: pear

text: in

text: Phys.

text: Rev.

text: A

text: S.

text: Vajna

text: and

text: B.

text: Dora,

text: Phys.

text: Rev.

text: B

text: A.

text: Gambassi

text: and

text: A.

text: Silva,

text: arXiv:

text: P.

text: S.

text: Sharma,

text: S.

text: Suzuki

text: and

text: A.

text: Dutta,

text: Phys.

text: Rev.

text: B

text: Smacchia

text: and

text: A.

text: Silva,

text: Phys.

text: Rev.

text: E

text: A.

text: Russomanno,

text: S.

text: Sharma,

text: A.

text: Dutta

text: and

text: G.

text: E.

text: Santoro,

text: S.

text: Vajna

text: and

text: B.

text: Dora,

text: Phys.

text: Rev.

text: B

text: J.

text: Stat.

text: Mech.,

text: M.

text: Schmitt

text: and

text: S.

text: Kehrein,

text: Phys.

text: Rev.

text: B

text: P.

text: Zanardi.

text: H.

text: T.

text: Quan,

text: X.

text: Wang

text: and

text: C.

text: P.

text: Sun,

text: Phys.

text: Rev.

text: A

text: J.

text: C.

text: Budich

text: and

text: M.

text: Heyl,

text: R.

text: Dorner,

text: J.

text: Goold,

text: C.

text: Cormick,

text: M.

text: Paternostro

text: and

text: V.

text: T.

text: Palmai,

text: Phys.

text: Rev.

text: B

text: Vedral,

text: Phys.

text: Rev.

text: Lett.

text: H.T.

text: Quan,

text: Z.

text: Song,

text: X.F.

text: Liu,

text: P.

text: Zanardi,

text: and

text: C.P.

text: Sun,

text: S.

text: Sharma

text: and

text: A.

text: Dutta,

text: Phys.

text: Rev.

text: E

text: Phys.Rev.Lett.

text: F.

text: Pollmann,

text: S.

text: Mukerjee,

text: A.

text: G.

text: Green,

text: and

text: J.

text: E.

text: Moore,

text: D.

text: Rossini,

text: T.

text: Calarco,

text: V.

text: Giovannetti,

text: S.

text: Montangero,

text: R.

text: Phys.

text: Rev.

text: E

text: l

text: f

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text: 6 | (cid:105) → | (cid:48) (cid:105) pumping transition F = 3 F = 4 to the less ef- ing to a loss rate consistent with losses due to gravity. | (cid:105) → | (cid:48) (cid:105) ﬁcient repumping transition F = 3 F = 3 results From the measured relative absorption and the loading in a higher number of cold atoms loaded into the OM time of 1 s, a mean atom ﬂux from the enhanced LVIS × of CSF2 (Fig. 6). This ﬁnding is probably the result of system of 4 108/s is estimated. two competing eﬀects and depends on the particular re- III. CSF2 OPERATION WITH LVIS LOADING pump laser intensities and geometry of our LVIS setup: | (cid:105) → | (cid:48) (cid:105) While the utilisation of the F = 3 F = 4 repump- The frequency instability of a fountain frequency stan- ing transition is beneﬁcial for the LVIS MOT operation, dard is expressed by the Allan devi(cid:114)ation according to stray light from the repump laser at this transition in- | (cid:105) teracting with the F = 3 atoms in the beam is more | (cid:105) eﬃcient in pumping the atoms to the F = 4 state (re- 1 ∆ν 1 T σ (τ) = c , (1) sulting in the described de|trimen(cid:105)ta→l ac|ce(cid:48)lerat(cid:105)ion eﬀect) y π ν0 SNR τ than stray light from the F = 3 F = 3 repump- ing transition. In our setup, with the given experimental where ∆ν is the full-width-at-half-maximum of the parameters (pump laser power 0.45 mW and a red fre- | (cid:105) → | (cid:48) (cid:105) Ramsey fringe, ν = 9 192 631 770 Hz is the clock tran- quency detuning of 10 Γ from the F = 4 F = 3 0 sition frequency, SNR is the signal-to-noise ratio, T transition), the latter repumping transition is the better c is the cycle time, and τ the measurement time. In of the two choices, resulting in a factor of 40 increase √ the case of quantum projection noise-limited operation, of N compared to operation of CSF2 with OM loading at SNR = N , with N the total detected number of from background gas. at at atoms in the F = 3 and F = 4 hyperﬁne components of D. Velocity and ﬂux of the slow beam the Cs ground state. The SNR was measured directly by operating the foun- To characterize the source of cold atoms, we performed tain CSF2 in a regime where the noise of the local oscilla- ﬂuorescence and absorption measurements in the OM tor does not contribute to the instability [10]√. The mea- zone of the fountain. The ﬁrst step in the atom velocity sured SNR increases linearly as a function of N (mea- at measurement was to allow the atomic beam to reach a sured in relative units) and reaches values larger√than steady state after the LVIS lasers were turned on. The 1700. The linear dependence between SNR and N | (cid:105) at probe laser was also turned on and tuned to the F = 3 allows to calibrate N in terms of the absolute number → | (cid:48) (cid:105) at F = 2 transition. Its intensity was increased to eﬃ- of detected atoms. The expected CSF2 instability, calcu- | (cid:105) ciently heat the cold beam atoms prepared in the F = 3 lated from Eq. 1 for ∆ν = 0.9 Hz, T = 2 s, SNR = 1680 × c× − state by the pump laser, and to eﬀectively prevent them and N = 2.9 106 is σ (1s) = 2.6 10 14. at y from reaching the OM zone. After the atom beam ﬂux To experimentally conﬁrm this value, the fountain was reached a steady state (0.5 s after the LVIS lasers were operated in a regime where the dominant contribution to turned on), as a second step, the probe laser was turned its instability is the quantum projection noise. To reach oﬀ and the OM beams were simultaneously turned on for this regime, an optically-stabilized 9.6 GHz microwave 1.1 s. The measured delay between the turning oﬀ of the signal is generated using a frequency comb as a transfer probe beam (which allows the atoms to reach the OM oscillator, and is used for the frequency synthesis of CSF2 zone) and the observed change in the OM loading rate [11]. The frequency comb is referenced to a laser locked (measured as sudden increase in the OM ﬂuorescence) to an ultra-stable optical cavity, transferring the stabil- as the ﬁrst cold atoms reach the OM zone was used to ity of the laser to a 9.6 GHz dielectric resonator oscillator calculate a mean value for the atom velocity of 8.6 m/s. (DRO). After removing the linear drift of the DRO fre- We estimate the atom ﬂux by performing a measure- quency caused by the drift of the optical cavity, the mea- − ment of the cloud absorption in the OM zone. For this sured Allan deviation σ (τ) shows a τ 1/2 dependence y measurement, the OM and LVIS lasers were operated si- for measurement times up to 100 s (Fig. 7). × − multaneously for 1 s. At the end of this period all lasers The measured fountain instability σ (1s) = 2.7 10 14 y × − were turned oﬀ and the probe beam was turned on, pre- is in good agreement with the value of 2.6 10 14 inferred venting additional arrival of cold atoms in the OM zone. from the SNR measurements. It is also close to the best × − After 10 ms, a weak absorption laser beam, resonant with measured instability of σ (1s) = 1.6 10 14 in a pri- | (cid:105) → | (cid:48) (cid:105) y the F = 4 F = 5 transition was introduced along mary fountain clock, where the atoms were loaded from the direction of one of the OM beams. It propagated a decelerated Cs atomic beam [12]. in the OM ﬁber, and had the same proﬁle and direc- IV. CONCLUSIONS tion as the corresponding OM beam. The absorption beam power was 0.07 µW in a beam diameter of 42 mm Slow atomic beam loading of the PTB fountain CSF2 (1/e2). Its polarization was linear and orthogonal to that is demonstrated. The source of cold atoms is a modiﬁed of the corresponding OM beam, and it was detected with low-velocity intense source setup which includes an ad- a high-gain photodetector after passing the OM zone. ditional pump laser. The pump laser reduces the loss of The absorption beam was kept on for 100 ms. During cold atoms during their ﬂight between the LVIS appa- this measurement time, the absorption changed accord- ratus and the OM loading zone by pumping them into ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: pumping

text: transition

text: F

text: F

text: to

text: the

text: less

text: ef-

text: ing

text: to

text: a

text: loss

text: rate

text: consistent

text: with

text: losses

text: due

text: to

text: gravity.

text: ﬁcient

text: repumping

text: transition

text: F

text: F

text: results

text: From

text: the

text: measured

text: relative

text: absorption

text: and

text: the

text: loading

text: in

text: a

text: higher

text: number

text: of

text: cold

text: atoms

text: loaded

text: into

text: the

text: OM

text: time

text: of

text: s,

text: a

text: mean

text: atom

text: ﬂux

text: from

text: the

text: enhanced

text: LVIS

text: of

text: (Fig.

text: This

text: ﬁnding

text: is

text: probably

text: the

text: result

text: of

text: system

text: of

text: is

text: estimated.

text: two

text: competing

text: eﬀects

text: and

text: depends

text: on

text: the

text: particular

text: re-

text: III.

text: OPERATION

text: WITH

text: LVIS

text: LOADING

text: pump

text: laser

text: intensities

text: and

text: geometry

text: of

text: our

text: LVIS

text: setup:

text: While

text: the

text: utilisation

text: of

text: the

text: F

text: F

text: repump-

text: The

text: frequency

text: instability

text: of

text: a

text: fountain

text: frequency

text: stan-

text: ing

text: transition

text: is

text: beneﬁcial

text: for

text: the

text: LVIS

text: MOT

text: operation,

text: dard

text: is

text: expressed

text: by

text: the

text: Allan

text: according

text: to

text: stray

text: light

text: from

text: the

text: repump

text: laser

text: at

text: this

text: transition

text: in-

text: teracting

text: with

text: the

text: F

text: atoms

text: in

text: the

text: beam

text: is

text: more

text: eﬃcient

text: in

text: pumping

text: the

text: atoms

text: to

text: the

text: F

text: state

text: (re-

text: ∆ν

text: T

text: σ

text: (τ)

text: c

text: sulting

text: in

text: the

text: described

text: eﬀect)

text: y

text: π

text: ν0

text: SNR

text: τ

text: than

text: stray

text: light

text: from

text: the

text: F

text: F

text: repump-

text: ing

text: transition.

text: In

text: our

text: setup,

text: with

text: the

text: given

text: experimental

text: where

text: ∆ν

text: is

text: the

text: full-width-at-half-maximum

text: of

text: the

text: parameters

text: (pump

text: laser

text: power

text: mW

text: and

text: a

text: red

text: fre-

text: Ramsey

text: fringe,

text: ν

text: Hz

text: is

text: the

text: clock

text: tran-

text: quency

text: detuning

text: of

text: Γ

text: from

text: the

text: F

text: F

text: sition

text: frequency,

text: SNR

text: is

text: the

text: signal-to-noise

text: ratio,

text: T

text: transition),

text: the

text: latter

text: repumping

text: transition

text: is

text: the

text: better

text: c

text: is

text: the

text: cycle

text: time,

text: and

text: τ

text: the

text: measurement

text: time.

text: In

text: of

text: the

text: two

text: choices,

text: resulting

text: in

text: a

text: factor

text: of

text: increase

text: the

text: case

text: of

text: quantum

text: projection

text: noise-limited

text: operation,

text: of

text: N

text: compared

text: to

text: operation

text: of

text: with

text: OM

text: loading

text: at

text: SNR

text: N

text: with

text: N

text: the

text: total

text: detected

text: number

text: of

text: from

text: background

text: gas.

text: at

text: at

text: atoms

text: in

text: the

text: F

text: and

text: F

text: hyperﬁne

text: components

text: of

text: D.

text: Velocity

text: and

text: ﬂux

text: of

text: the

text: slow

text: beam

text: the

text: Cs

text: ground

text: state.

text: The

text: SNR

text: was

text: measured

text: directly

text: by

text: operating

text: the

text: foun-

text: To

text: characterize

text: the

text: source

text: of

text: cold

text: atoms,

text: we

text: performed

text: tain

text: in

text: a

text: regime

text: where

text: the

text: noise

text: of

text: the

text: local

text: oscilla-

text: ﬂuorescence

text: and

text: absorption

text: measurements

text: in

text: the

text: OM

text: tor

text: does

text: not

text: contribute

text: to

text: the

text: instability

text: The

text: mea-

text: zone

text: of

text: the

text: fountain.

text: The

text: ﬁrst

text: step

text: in

text: the

text: atom

text: velocity

text: sured

text: SNR

text: increases

text: linearly

text: as

text: a

text: function

text: of

text: N

text: (mea-

text: at

text: measurement

text: was

text: to

text: allow

text: the

text: atomic

text: beam

text: to

text: reach

text: a

text: sured

text: in

text: relative

text: units)

text: and

text: reaches

text: values

text: larger√than

text: steady

text: state

text: after

text: the

text: LVIS

text: lasers

text: were

text: turned

text: on.

text: The

text: The

text: linear

text: dependence

text: between

text: SNR

text: and

text: N

text: at

text: probe

text: laser

text: was

text: also

text: turned

text: on

text: and

text: tuned

text: to

text: the

text: F

text: allows

text: to

text: calibrate

text: N

text: in

text: terms

text: of

text: the

text: absolute

text: number

text: at

text: F

text: transition.

text: Its

text: intensity

text: was

text: increased

text: to

text: eﬃ-

text: of

text: detected

text: atoms.

text: The

text: expected

text: instability,

text: calcu-

text: ciently

text: heat

text: the

text: cold

text: beam

text: atoms

text: prepared

text: in

text: the

text: F

text: lated

text: from

text: Eq.

text: for

text: ∆ν

text: Hz,

text: T

text: s,

text: SNR

text: c×

text: state

text: by

text: the

text: pump

text: laser,

text: and

text: to

text: eﬀectively

text: prevent

text: them

text: and

text: N

text: is

text: σ

text: at

text: y

text: from

text: reaching

text: the

text: OM

text: zone.

text: After

text: the

text: atom

text: beam

text: ﬂux

text: To

text: experimentally

text: conﬁrm

text: this

text: value,

text: the

text: fountain

text: was

text: reached

text: a

text: steady

text: state

text: s

text: after

text: the

text: LVIS

text: lasers

text: were

text: operated

text: in

text: a

text: regime

text: where

text: the

text: dominant

text: contribution

text: to

text: turned

text: on),

text: as

text: a

text: second

text: step,

text: the

text: probe

text: laser

text: was

text: turned

text: its

text: instability

text: is

text: the

text: quantum

text: projection

text: noise.

text: To

text: reach

text: oﬀ

text: and

text: the

text: OM

text: beams

text: were

text: simultaneously

text: turned

text: on

text: for

text: this

text: regime,

text: an

text: optically-stabilized

text: GHz

text: microwave

text: s.

text: The

text: measured

text: delay

text: between

text: the

text: turning

text: oﬀ

text: of

text: the

text: signal

text: is

text: generated

text: using

text: a

text: frequency

text: comb

text: as

text: a

text: transfer

text: probe

text: beam

text: (which

text: allows

text: the

text: atoms

text: to

text: reach

text: the

text: OM

text: oscillator,

text: and

text: is

text: used

text: for

text: the

text: frequency

text: synthesis

text: of

text: zone)

text: and

text: the

text: observed

text: change

text: in

text: the

text: OM

text: loading

text: rate

text: The

text: frequency

text: comb

text: is

text: referenced

text: to

text: a

text: laser

text: locked

text: (measured

text: as

text: sudden

text: increase

text: in

text: the

text: OM

text: ﬂuorescence)

text: to

text: an

text: ultra-stable

text: optical

text: cavity,

text: transferring

text: the

text: stabil-

text: as

text: the

text: ﬁrst

text: cold

text: atoms

text: reach

text: the

text: OM

text: zone

text: was

text: used

text: to

text: ity

text: of

text: the

text: laser

text: to

text: a

text: GHz

text: dielectric

text: resonator

text: oscillator

text: calculate

text: a

text: mean

text: value

text: for

text: the

text: atom

text: velocity

text: of

text: m/s.

text: (DRO).

text: After

text: removing

text: the

text: linear

text: drift

text: of

text: the

text: DRO

text: fre-

text: We

text: estimate

text: the

text: atom

text: ﬂux

text: by

text: performing

text: a

text: measure-

text: quency

text: caused

text: by

text: the

text: drift

text: of

text: the

text: optical

text: cavity,

text: the

text: mea-

text: ment

text: of

text: the

text: cloud

text: absorption

text: in

text: the

text: OM

text: zone.

text: For

text: this

text: sured

text: Allan

text: deviation

text: σ

text: (τ)

text: shows

text: a

text: τ

text: dependence

text: y

text: measurement,

text: the

text: OM

text: and

text: LVIS

text: lasers

text: were

text: operated

text: si-

text: for

text: measurement

text: times

text: up

text: to

text: s

text: (Fig.

text: multaneously

text: for

text: s.

text: At

text: the

text: end

text: of

text: this

text: period

text: all

text: lasers

text: The

text: measured

text: fountain

text: instability

text: σ

text: y

text: were

text: turned

text: oﬀ

text: and

text: the

text: probe

text: beam

text: was

text: turned

text: on,

text: pre-

text: is

text: in

text: good

text: agreement

text: with

text: the

text: value

text: of

text: inferred

text: venting

text: additional

text: arrival

text: of

text: cold

text: atoms

text: in

text: the

text: OM

text: zone.

text: from

text: the

text: SNR

text: measurements.

text: It

text: is

text: also

text: close

text: to

text: the

text: best

text: After

text: ms,

text: a

text: weak

text: absorption

text: laser

text: beam,

text: resonant

text: with

text: measured

text: instability

text: of

text: σ

text: in

text: a

text: pri-

text: y

text: the

text: F

text: F

text: transition

text: was

text: introduced

text: along

text: mary

text: fountain

text: clock,

text: where

text: the

text: atoms

text: were

text: loaded

text: from

text: the

text: direction

text: of

text: one

text: of

text: the

text: OM

text: beams.

text: It

text: propagated

text: a

text: decelerated

text: Cs

text: atomic

text: beam

text: in

text: the

text: OM

text: ﬁber,

text: and

text: had

text: the

text: same

text: proﬁle

text: and

text: direc-

text: IV.

text: CONCLUSIONS

text: tion

text: as

text: the

text: corresponding

text: OM

text: beam.

text: The

text: absorption

text: beam

text: power

text: was

text: µW

text: in

text: a

text: beam

text: diameter

text: of

text: mm

text: Slow

text: atomic

text: beam

text: loading

text: of

text: the

text: PTB

text: fountain

text: Its

text: polarization

text: was

text: linear

text: and

text: orthogonal

text: to

text: that

text: is

text: demonstrated.

text: The

text: source

text: of

text: cold

text: atoms

text: is

text: a

text: modiﬁed

text: of

text: the

text: corresponding

text: OM

text: beam,

text: and

text: it

text: was

text: detected

text: with

text: low-velocity

text: intense

text: source

text: setup

text: which

text: includes

text: an

text: ad-

text: a

text: high-gain

text: photodetector

text: after

text: passing

text: the

text: OM

text: zone.

text: ditional

text: pump

text: laser.

text: The

text: pump

text: laser

text: reduces

text: the

text: loss

text: of

text: The

text: absorption

text: beam

text: was

text: kept

text: on

text: for

text: ms.

text: During

text: cold

text: atoms

text: during

text: their

text: ﬂight

text: between

text: the

text: LVIS

text: appa-

text: this

text: measurement

text: time,

text: the

text: absorption

text: changed

text: accord-

text: ratus

text: and

text: the

text: OM

text: loading

text: zone

text: by

text: pumping

text: them

text: into

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 22 PRZEMYSL(cid:32) AW MAL(cid:32) KIEWICZ Figure 3. Phase space portrait of the semiclassical dynamics in the planes (q, p) and (q, p ) for k = 1, c = 1 and Hˇ = 1 , 1 , 5 , 5 . All the phase space k w true 240 12 24 12 is accessible by the semiclassical dynamics. The trajectories do not diverge but reverse. They represent bouncing universes. (cid:32) (cid:33) c2 − B( k¯ )2 λ (q¯p¯)2 A 1 + λ (q¯p¯)2 (72) Hˇ = w p¯2 cw 2 + 2 D,true 24 q¯2 1 + λ ( k¯ )2 + λ (q¯p¯)2 q¯2 1 + λ ( k¯ )2 + λ (q¯p¯)2 1 cw 2 1 cw 2 that is, it is formally equivalent with the Hamiltonian of Eq. (67). We employ diﬀerent internal clocks to derive semiclassical phase space portraits for the unique quantum dynamics of Bianchi type I model. Next we compare them in a single reduced phase space. We skip the variables α, p and k since they are constants of motion α and no clock eﬀect can occur for them. We focus on the space (q, p, p ). k More precisely, we ﬁx a single reduced phase space parametrised by the barred variables (q¯, p¯, p¯ ) and the remaining reduced phase spaces parametrised by (q, p, p ). In order to k k arrive at the ﬁxed reduced phase space (q¯, p¯, p¯ ), a pseudo-canonical transformation (29) k based on an appropriate delay function D(q, p) is performed on (q, p, p ). k The scheme of computations is following. We use the solution (68,69,70) to the semi- classical Hamilton equations generated by the Hamiltonian (67): (cid:55)→ (73) T (q(T ), p(T ), p (T )). k Next we perform a pseudo-canonical transformation with a suitable delay function D(q, p), described by the eqs (29). Combining it with (73), we obtain: ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: AW

text: KIEWICZ

text: Figure

text: Phase

label: caption

text: space

text: portrait

text: of

text: the

text: semiclassical

text: dynamics

text: in

text: the

text: planes

text: (q,

text: p)

text: and

text: (q,

text: p

text: for

text: k

text: c

text: and

text: Hˇ

text: All

text: the

text: phase

text: space

text: k

text: w

text: true

text: is

text: accessible

text: by

text: the

text: semiclassical

text: dynamics.

text: The

text: trajectories

text: do

text: not

text: diverge

text: but

text: reverse.

text: They

text: represent

text: bouncing

text: universes.

text: B(

text: k¯

text: λ

text: A

text: λ

text: Hˇ

text: w

text: cw

text: D,true

text: λ

text: k¯

text: λ

text: λ

text: k¯

text: λ

text: cw

text: cw

text: that

text: is,

text: it

text: is

text: formally

text: equivalent

text: with

text: the

text: Hamiltonian

text: of

text: Eq.

text: We

text: employ

text: diﬀerent

text: internal

text: clocks

text: to

text: derive

text: semiclassical

text: phase

text: space

text: portraits

text: for

text: the

text: unique

text: quantum

text: dynamics

text: of

text: Bianchi

text: type

text: I

text: model.

text: Next

text: we

text: compare

text: them

text: in

text: a

text: single

text: reduced

text: phase

text: space.

text: We

text: skip

text: the

text: variables

text: α,

text: p

text: and

text: k

text: since

text: they

text: are

text: constants

text: of

text: motion

text: α

text: and

text: no

text: clock

text: eﬀect

text: can

text: occur

text: for

text: them.

text: We

text: focus

text: on

text: the

text: space

text: (q,

text: p,

text: p

text: k

text: More

text: precisely,

text: we

text: ﬁx

text: a

text: single

text: reduced

text: phase

text: space

text: parametrised

text: by

text: the

text: barred

text: variables

text: (q¯,

text: p¯,

text: p¯

text: and

text: the

text: remaining

text: reduced

text: phase

text: spaces

text: parametrised

text: by

text: (q,

text: p,

text: p

text: In

text: order

text: to

text: k

text: k

text: arrive

text: at

text: the

text: ﬁxed

text: reduced

text: phase

text: space

text: (q¯,

text: p¯,

text: p¯

text: a

text: pseudo-canonical

text: transformation

text: k

text: based

text: on

text: an

text: appropriate

text: delay

text: function

text: D(q,

text: p)

text: is

text: performed

text: on

text: (q,

text: p,

text: p

text: k

text: The

text: scheme

text: of

text: computations

text: is

text: following.

text: We

text: use

text: the

text: solution

text: to

text: the

text: semi-

text: classical

text: Hamilton

text: equations

text: generated

text: by

text: the

text: Hamiltonian

text: T

text: (q(T

text: p(T

text: p

text: (T

text: k

text: Next

text: we

text: perform

text: a

text: pseudo-canonical

text: transformation

text: with

text: a

text: suitable

text: delay

text: function

text: D(q,

text: p),

text: described

text: by

text: the

text: eqs

text: Combining

text: it

text: with

text: we

text: obtain:

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: 24 PRZEMYSL(cid:32) AW MAL(cid:32) KIEWICZ Figure 5. From left to right: (i) plot of the delay function D which deﬁnes the 2 transformation between the reduced phase space used to derive the semiclassical dynamics and the reduced phase space used to describe it; (ii) the semiclassical evolution in the (q¯, p¯)-plane for k = 1, c = 1, Hˇ = 1 , 1 , 5 , 5 ; (iii) the w true 240 12 24 12 semiclassical evolution in the (q¯, p¯ )-plane. k Figure 6. From left to right: (i) plot of the delay function D which deﬁnes the 3 transformation between the reduced phase space used to derive the semiclassical dynamics and the reduced phase space used to describe it; (ii) the semiclassical evolution in the (q¯, p¯)-plane for k = 1, c = 1, Hˇ = 1 , 1 , 5 , 5 ; (iii) the w true 240 12 24 12 semiclassical evolution in the (q¯, p¯ )-plane. k The ﬁrst example is − − · q pq 10k − k2 (77) D1(q, p) := 40 η(p + k )η( p2 − k2 )η( p2 + q2 ). q q2 The ﬁgure 4 includes a 3-dimensional plot of the delay function D and the semiclassical 1 phase space portrait in the planes (q¯, p¯) and (q¯, p¯ ) mapped from the planes (q, p) and k (cid:55)→ (q, p ), where the semiclassical motion was derived. The clock transformation T T¯ = k ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: AW

text: KIEWICZ

text: Figure

text: From

text: left

text: to

text: right:

text: (i)

text: plot

text: of

text: the

text: delay

text: function

text: D

text: which

text: deﬁnes

text: the

text: transformation

text: between

text: the

text: reduced

text: phase

text: space

text: used

text: to

text: derive

text: the

text: semiclassical

text: dynamics

text: and

text: the

text: reduced

text: phase

text: space

text: used

text: to

text: describe

text: it;

text: (ii)

text: the

text: semiclassical

text: evolution

text: in

text: the

text: (q¯,

text: p¯)-plane

text: for

text: k

text: c

text: Hˇ

text: (iii)

text: the

text: w

text: true

text: semiclassical

text: evolution

text: in

text: the

text: (q¯,

text: p¯

text: )-plane.

text: k

text: Figure

text: From

text: left

text: to

text: right:

text: (i)

text: plot

text: of

text: the

text: delay

text: function

text: D

text: which

text: deﬁnes

text: the

text: transformation

text: between

text: the

text: reduced

text: phase

text: space

text: used

text: to

text: derive

text: the

text: semiclassical

text: dynamics

text: and

text: the

text: reduced

text: phase

text: space

text: used

text: to

text: describe

text: it;

text: (ii)

text: the

text: semiclassical

text: evolution

text: in

text: the

text: (q¯,

text: p¯)-plane

text: for

text: k

text: c

text: Hˇ

text: (iii)

text: the

text: w

text: true

text: semiclassical

text: evolution

text: in

text: the

text: (q¯,

text: p¯

text: )-plane.

text: k

text: The

text: ﬁrst

text: example

text: is

text: q

text: pq

text: p)

text: η(p

text: k

text: )η(

text: )η(

text: q

text: The

text: ﬁgure

text: includes

text: a

text: plot

text: of

text: the

text: delay

text: function

text: D

text: and

text: the

text: semiclassical

text: phase

text: space

text: portrait

text: in

text: the

text: planes

text: (q¯,

text: p¯)

text: and

text: (q¯,

text: p¯

text: mapped

text: from

text: the

text: planes

text: (q,

text: p)

text: and

text: k

text: (q,

text: p

text: where

text: the

text: semiclassical

text: motion

text: was

text: derived.

text: The

text: clock

text: transformation

text: T

text: T¯

text: k

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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: ##LTLine##


text: 25 Figure 7. From left to right: (i) plot of the delay function D which deﬁnes the 4 transformation between the reduced phase space used to derive the semiclassical dynamics and the reduced phase space used to describe it; (ii) the semiclassical evolution in the (q¯, p¯)-plane for k = 1, c = 1, Hˇ = 1 , 1 , 5 , 5 ; (iii) the w true 240 12 24 12 semiclassical evolution in the (q¯, p¯ )-plane. k T + D (q, p) occurs only in the phase space region where the classical universe contracts. 1 As a result, the semiclassical motion in this region deviates from the one derived with the reference clock T¯. Moreover, the dynamics of the variable p¯ that is coupled to q¯ and p¯ k deviates during the same phase of evolution, i.e. for small and contracting q¯. Finally, we observe that away from the bounce the dynamics coincides with the dynamics derived in T¯ and represented in the ﬁgure 3. The second example is · − k2 (78) D (q, p) := q η(p2 ). 2 q2 The ﬁgure 5 includes the plot of D and the portrait of the quantum motion in the 2 reduced phase space (q¯, p¯, p¯ ) that is diﬀerent by D from the reduced phase space (q, p, p ) k 2 k in which the quantum motion is derived. The respective clock transformation occurs only in the classically forbidden region and as a result, the semiclassical dynamics exhibits some deviations from the dynamics of the ﬁgure 3 precisely in this region. Finally, we observe that away from the bounce the dynamics coincides with the dynamics of the ﬁgure 3. The next example is − − · pq q pq 10k − k2 (79) D3(q, p) := 20 sin(4π p2 − k2 )η(p + k )η( p2 − k2 )η( p2 + q2 ). q2 q q2 The ﬁgure 6 includes the plot of D and the portrait of the quantum motion in the 3 reduced phase space (q¯, p¯, p¯ ) that is diﬀerent by D from the reduced phase space (q, p, p ) k 3 k in which the quantum motion is derived. The respective clock transformation occurs only ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: From

text: left

text: to

text: right:

text: (i)

text: plot

text: of

text: the

text: delay

text: function

text: D

text: which

text: deﬁnes

text: the

text: transformation

text: between

text: the

text: reduced

text: phase

text: space

text: used

text: to

text: derive

text: the

text: semiclassical

text: dynamics

text: and

text: the

text: reduced

text: phase

text: space

text: used

text: to

text: describe

text: it;

text: (ii)

text: the

text: semiclassical

text: evolution

text: in

text: the

text: (q¯,

text: p¯)-plane

text: for

text: k

text: c

text: Hˇ

text: (iii)

text: the

text: w

text: true

text: semiclassical

text: evolution

text: in

text: the

text: (q¯,

text: p¯

text: )-plane.

text: k

text: T

text: D

text: (q,

text: p)

text: occurs

text: only

text: in

text: the

text: phase

text: space

text: region

text: where

text: the

text: classical

text: universe

text: contracts.

text: As

text: a

text: result,

text: the

text: semiclassical

text: motion

text: in

text: this

text: region

text: deviates

text: from

text: the

text: one

text: derived

text: with

text: the

text: reference

text: clock

text: T¯.

text: Moreover,

text: the

text: dynamics

text: of

text: the

text: variable

text: p¯

text: that

text: is

text: coupled

text: to

text: q¯

text: and

text: p¯

text: k

text: deviates

text: during

text: the

text: same

text: phase

text: of

text: evolution,

text: i.e.

text: for

text: small

text: and

text: contracting

text: q¯.

text: Finally,

text: we

text: observe

text: that

text: away

text: from

text: the

text: bounce

text: the

text: dynamics

text: coincides

text: with

text: the

text: dynamics

text: derived

text: in

text: T¯

text: and

text: represented

text: in

text: the

text: ﬁgure

text: The

text: second

text: example

text: is

text: D

text: (q,

text: p)

text: q

text: The

text: ﬁgure

text: includes

text: the

text: plot

text: of

text: D

text: and

text: the

text: portrait

text: of

text: the

text: quantum

text: motion

text: in

text: the

text: reduced

text: phase

text: space

text: (q¯,

text: p¯,

text: p¯

text: that

text: is

text: diﬀerent

text: by

text: D

text: from

text: the

text: reduced

text: phase

text: space

text: (q,

text: p,

text: p

text: k

text: k

text: in

text: which

text: the

text: quantum

text: motion

text: is

text: derived.

text: The

text: respective

text: clock

text: transformation

text: occurs

text: only

text: in

text: the

text: classically

text: forbidden

text: region

text: and

text: as

text: a

text: result,

text: the

text: semiclassical

text: dynamics

text: exhibits

text: some

text: deviations

text: from

text: the

text: dynamics

text: of

text: the

text: ﬁgure

text: precisely

text: in

text: this

text: region.

text: Finally,

text: we

text: observe

text: that

text: away

text: from

text: the

text: bounce

text: the

text: dynamics

text: coincides

text: with

text: the

text: dynamics

text: of

text: the

text: ﬁgure

text: The

text: next

text: example

text: is

text: pq

text: q

text: pq

text: p)

text: )η(p

text: k

text: )η(

text: )η(

text: q

text: The

text: ﬁgure

text: includes

text: the

text: plot

text: of

text: D

text: and

text: the

text: portrait

text: of

text: the

text: quantum

text: motion

text: in

text: the

text: reduced

text: phase

text: space

text: (q¯,

text: p¯,

text: p¯

text: that

text: is

text: diﬀerent

text: by

text: D

text: from

text: the

text: reduced

text: phase

text: space

text: (q,

text: p,

text: p

text: k

text: k

text: in

text: which

text: the

text: quantum

text: motion

text: is

text: derived.

text: The

text: respective

text: clock

text: transformation

text: occurs

text: only

text: ##LTFigure##

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##