text: 3 Practical derivations for the chain rule This section provides speciﬁc applications of the chain rule given in Lemma 1 in which the outer ◦ function of the composition f g assumes a speciﬁc form, commonly encountered in practical deriva- tions. Theorem 4 (Practical derivations for the chain rule). Let X, Y and Z be locally convex topological → → vector spaces, f : Y Z , g : X Y . Assume additionally that g has a chain diﬀerential at some point x in some direction η. Then: ◦ a) if f is a continuous linear function (cid:96), then the composition f g has a chain diﬀerential at point (cid:0) (cid:1) x in the direction η, and it is given by ◦ δ((cid:96) g)(x; η) = (cid:96) δg(x; η) . (10) Leibniz’ rule (cid:55)→ ◦ b) if f is the kth power function y yk, k > 0, then the composition f g has a chain diﬀerential (cid:0) (cid:1) (cid:0) (cid:1) at point x in the direction η, and it is given by (cid:55)→ ◦ − δ (y yk) g (x; η) = k g(x) k 1δg(x; η). (11) ◦ c) if f is the exponential function exp, then the composition f g has a chain diﬀerential at point (cid:0) (cid:1) x in the direction η, and it is given by ◦ δ(exp g)(x; η) = exp g(x) δg(x; η). (12) The proof is given in appendix in Section . Appendix A Proofs A.1 Total chain diﬀerential (Theorem 1) Proof. The result is proved in the case n = 2 from which the general case can be straightforwardly ∈ × ∈ × deduced. Let us ﬁx a point x = (x, y) X X and a direction η = (η, ξ) X X , such that f 1 2 1 2 has partial chain diﬀerentials δ f(x; η) and δ f(x; ξ). Let us then ﬁx arbitrary sequences of directions → ∈ → ∈ 1 2 → ≤ η η X , ξ ξ X , and an arbitrary sequence of real numbers θ 0. For m 0 we can m 1 m 2 m write (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) − − − − θ 1 f(x + θ η) f(x) = θ 1 f(x + θ η , y + θ ξ ) f(x, y) (13a) m m m m m m m − − − − = θ 1 g (y + θ ξ ) g (y) + θ 1 g (x + θ η ) g (x) , (13b) m 1 m m 1 m 2 m m 2 (cid:40) where we deﬁne g (y) and g (x) as follows: 1 2 g (y) = f(x + θ η , y), 1 m m (14) g (x) = f(x, y). 2 (cid:54) R → R Given θ = 0, deﬁne h : as h(t) = g (y+tξ ). From the mean value theorem for real-valued m ∈ 1 m functions, there exists c [0, θ ] such that y m 4

text: Practical

text: derivations

text: for

text: the

text: chain

text: rule

text: This

text: section

text: provides

text: speciﬁc

text: applications

text: of

text: the

text: chain

text: rule

text: given

text: in

text: Lemma

text: in

text: which

text: the

text: outer

text: function

text: of

text: the

text: composition

text: f

text: g

text: assumes

text: a

text: speciﬁc

text: form,

text: commonly

text: encountered

text: in

text: practical

text: deriva-

text: tions.

text: Theorem

text: (Practical

text: derivations

text: for

text: the

text: chain

text: rule).

text: Let

text: X,

text: Y

text: and

text: Z

text: be

text: locally

text: convex

text: topological

text: vector

text: spaces,

text: f

text: Y

text: Z

text: g

text: X

text: Y

text: Assume

text: additionally

text: that

text: g

text: has

text: a

text: chain

text: diﬀerential

text: at

text: some

text: point

text: x

text: in

text: some

text: direction

text: η.

text: Then:

text: a)

text: if

text: f

text: is

text: a

text: continuous

text: linear

text: function

text: then

text: the

text: composition

text: f

text: g

text: has

text: a

text: chain

text: diﬀerential

text: at

text: point

text: x

text: in

text: the

text: direction

text: η,

text: and

text: it

text: is

text: given

text: by

text: g)(x;

text: η)

text: δg(x;

text: η)

text: Leibniz’

text: rule

text: b)

text: if

text: f

text: is

text: the

text: kth

text: power

text: function

text: y

text: yk,

text: k

text: then

text: the

text: composition

text: f

text: g

text: has

text: a

text: chain

text: diﬀerential

text: at

text: point

text: x

text: in

text: the

text: direction

text: η,

text: and

text: it

text: is

text: given

text: by

text: δ

text: (y

text: yk)

text: g

text: (x;

text: η)

text: k

text: g(x)

text: k

text: η).

text: c)

text: if

text: f

text: is

text: the

text: exponential

text: function

text: exp,

text: then

text: the

text: composition

text: f

text: g

text: has

text: a

text: chain

text: diﬀerential

text: at

text: point

text: x

text: in

text: the

text: direction

text: η,

text: and

text: it

text: is

text: given

text: by

text: δ(exp

text: g)(x;

text: η)

text: exp

text: g(x)

text: δg(x;

text: η).

text: The

text: proof

text: is

text: given

text: in

text: appendix

text: in

text: Section

text: Appendix

text: A

text: Proofs

text: Total

text: chain

text: diﬀerential

text: (Theorem

text: Proof.

text: The

text: result

text: is

text: proved

text: in

text: the

text: case

text: n

text: from

text: which

text: the

text: general

text: case

text: can

text: be

text: straightforwardly

text: deduced.

text: Let

text: us

text: ﬁx

text: a

text: point

text: x

text: (x,

text: y)

text: X

text: X

text: and

text: a

text: direction

text: η

text: (η,

text: ξ)

text: X

text: X

text: such

text: that

text: f

text: has

text: partial

text: chain

text: diﬀerentials

text: δ

text: f(x;

text: η)

text: and

text: δ

text: f(x;

text: ξ).

text: Let

text: us

text: then

text: ﬁx

text: arbitrary

text: sequences

text: of

text: directions

text: η

text: η

text: X

text: ξ

text: ξ

text: X

text: and

text: an

text: arbitrary

text: sequence

text: of

text: real

text: numbers

text: θ

text: For

text: m

text: we

text: can

text: m

text: m

text: m

text: write

text: θ

text: f(x

text: θ

text: η)

text: f(x)

text: θ

text: f(x

text: θ

text: η

text: y

text: θ

text: ξ

text: f(x,

text: y)

text: m

text: m

text: m

text: m

text: m

text: m

text: m

text: θ

text: g

text: (y

text: θ

text: ξ

text: g

text: (y)

text: θ

text: g

text: (x

text: θ

text: η

text: g

text: (x)

text: m

text: m

text: m

text: m

text: m

text: m

text: where

text: we

text: deﬁne

text: g

text: (y)

text: and

text: g

text: (x)

text: as

text: follows:

text: g

text: (y)

text: f(x

text: θ

text: η

text: y),

text: m

text: m

text: g

text: (x)

text: f(x,

text: y).

text: R

text: R

text: Given

text: θ

text: deﬁne

text: h

text: as

text: h(t)

text: g

text: (y+tξ

text: From

text: the

text: mean

text: value

text: theorem

text: for

text: real-valued

text: m

text: m

text: functions,

text: there

text: exists

text: c

text: θ

text: such

text: that

text: y

text: m


text: (cid:12) (cid:2) (cid:3) (cid:12) (cid:12) (cid:12) − − dh θ 1 h(θ ) h(0) = (15a) m m dt t=cy = δh(c ; 1), (15b) y which, when replacing h(t)(cid:2)by g1(y + tξm), can b(cid:3)e rew(cid:0)ritten (cid:1) − − θ 1 g (y + θ ξ ) g (y) = δ g (y + c ξ ); 1 (16a) m 1 m m 1 1 y m = δg (y + c ξ ; ξ ), (16b) 1 y m m ∈ where Lemma 1 has been used to obtain the last equality. Similarly for g (x), there exists c 2 x [0, θ ] such that m − − θ 1 [g (x + θ η ) g (x)] = δg (x + c η ; η ). (17) m 2 m m 2 2 x m m (cid:12) (cid:12) (cid:12) (cid:2) (cid:3) (cid:12) Let us now prove tha(cid:12)t the limit of the term (cid:12) − − − − θ 1 f(x + θ η) f(x) δ f(x; η ) δ f(x; ξ ) (18) m m 1 m 2 m (cid:12) → ∞ (cid:12) is equal to 0 w(cid:12)hen r . Substituting (16b) and (17) into (13b), (18) become(cid:12)s − − δg (x + c η ; η ) + δg (y + c ξ ; ξ ) δ f(x; η ) δ f(x; ξ ) . (19) 2 x m m 1 y m m 1 m 2 m (cid:12) (cid:12) (cid:12) (cid:12) By the triang(cid:12)le inequality, (19) is bounded ab(cid:12)ove(cid:12) by the following summation (cid:12) − − δg (x + c η ; η ) δ f(x; η ) + δg (y + c ξ ; ξ ) δ f(x; ξ ) . (20) 2 x m m 1 m 1 y m m 2 m Substitutin(cid:12)g g1 and g2 with f, the bound (20(cid:12)) be(cid:12)comes (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − − δ f(x + c η , y; η ) δ f(x; η ) + δ f(x, y + c ξ ; ξ ) δ f(x; ξ ) , (21) 1 x m m 1 m 2 y m m 2 m → ∞ (cid:55)→ wh(cid:55)→ich tends to 0 when(cid:12)(cid:12) m (cid:2) because of the(cid:3)continuity of the functions(cid:12)(cid:12)(z, ν) δ1f(z, y; ν) and (z, ν) δ2f(x, z; ν). Thu(cid:12)s, it holds that (cid:12) − − − − lim θ 1 f(x + θ η) f(x) δ f(x; η ) δ f(x; ξ ) = 0, (22) →∞ m m 1 m 2 m m that is, f has a total chain diﬀerential in point x in direction η, and it is such that δf(x; η) = δ f(x; η) + δ f(x; ξ), (23) 1 2 which is equivalent to the Proposition 3 in [2]. 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: dh

text: θ

text: h(θ

text: m

text: m

text: dt

text: t=cy

text: δh(c

text: y

text: which,

label: paragraph

text: when

text: replacing

text: tξm),

text: can

text: θ

text: g

text: (y

text: θ

text: ξ

text: g

text: (y)

text: δ

text: g

text: (y

text: c

text: ξ

text: m

text: m

text: m

text: y

text: m

text: δg

text: (y

text: c

text: ξ

text: ξ

text: y

text: m

text: m

text: where

text: Lemma

text: has

text: been

text: used

text: to

text: obtain

text: the

text: last

text: equality.

text: Similarly

text: for

text: g

text: (x),

text: there

text: exists

text: c

text: x

text: θ

text: such

text: that

text: m

text: θ

text: [g

text: (x

text: θ

text: η

text: g

text: (x)]

text: δg

text: (x

text: c

text: η

text: η

text: m

text: m

text: m

text: x

text: m

text: m

text: Let

text: us

text: now

text: prove

text: the

text: limit

text: of

text: the

text: term

text: θ

text: f(x

text: θ

text: η)

text: f(x)

text: δ

text: f(x;

text: η

text: δ

text: f(x;

text: ξ

text: m

text: m

text: m

text: m

text: is

text: equal

text: to

text: r

text: Substituting

text: and

text: into

text: δg

text: (x

text: c

text: η

text: η

text: δg

text: (y

text: c

text: ξ

text: ξ

text: δ

text: f(x;

text: η

text: δ

text: f(x;

text: ξ

text: x

text: m

text: m

text: y

text: m

text: m

text: m

text: m

text: By

text: the

text: inequality,

text: is

text: bounded

text: by

text: the

text: following

text: summation

text: δg

text: (x

text: c

text: η

text: η

text: δ

text: f(x;

text: η

text: δg

text: (y

text: c

text: ξ

text: ξ

text: δ

text: f(x;

text: ξ

text: x

text: m

text: m

text: m

text: y

text: m

text: m

text: m

text: and

text: with

text: f,

text: the

text: bound

text: δ

text: f(x

text: c

text: η

text: y;

text: η

text: δ

text: f(x;

text: η

text: δ

text: f(x,

text: y

text: c

text: ξ

text: ξ

text: δ

text: f(x;

text: ξ

text: x

text: m

text: m

text: m

text: y

text: m

text: m

text: m

text: tends

text: to

text: m

text: because

text: of

text: of

text: the

text: ν)

text: y;

text: ν)

text: and

text: (z,

text: ν)

text: z;

text: ν).

text: it

text: holds

text: that

text: lim

text: θ

text: f(x

text: θ

text: η)

text: f(x)

text: δ

text: f(x;

text: η

text: δ

text: f(x;

text: ξ

text: m

text: m

text: m

text: m

text: m

text: that

text: is,

text: f

text: has

text: a

text: total

text: chain

text: diﬀerential

text: in

text: point

text: x

text: in

text: direction

text: η,

text: and

text: it

text: is

text: such

text: that

text: δf(x;

text: η)

text: δ

text: f(x;

text: η)

text: δ

text: f(x;

text: ξ),

text: which

text: is

text: equivalent

text: to

text: the

text: Proposition

text: in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≥ c) Case n 2. ∈ ∈ Let us ﬁx a point x X and a sequence of directions (η )n Xn such that both f and g have higher i i=1 ⊆ { } order chain diﬀerentials at point x in all sequences of directions (η ) ∈ , I 1, . . . , n . We can then i i I (cid:88) write | | −| | (cid:88)δ π f(x; (η ) ∈ )δn π g(x; (η ) ∈{ }\ ) i i π i i 1,...,n π ⊆{ } π 1,...,n | | − −| | = δ π +1f(x; (η ) ∈ ∪(cid:88){ })δn 1 π g(x; (η ) ∈{ − }\ ) i i π n i i 1,...,n 1 π ⊆{ − } π 1,...,n 1 | | − −| | (cid:88) (cid:0) + (cid:1) δ π f(x; (η ) ∈ )δn 1 π +1g(x; (η ) ∈{ − }\ ∪{ }) (43a) i i π i i 1,...,n 1 π n ⊆{ − } π 1,...,n 1 | | − −| | = δ δ π f(x; (η ) ∈ );(cid:88)η δn 1 π g(x; (η ) ∈{ (cid:0) − }\ ) (cid:1) i i π n i i 1,...,n 1 π ⊆{ − } π 1,...,n 1 | | − −| | (cid:88) (cid:0) + δ π f(x; (η ) ∈ )δ δn 1 π(cid:1)g(x; (η ) ∈{ − }\ ); η (43b) i i π i i 1,...,n 1 π n ⊆{ − } π 1,...,n 1 | | − −| | = (cid:16) (cid:88) δ δ π f(x; (η ) ∈ )δn 1 π g(x; (η ) ∈{ − }\ ); η (cid:17) (43c) i i π i i 1,...,n 1 π n ⊆{ − } π 1,...,n 1 | | − −| | = δ(cid:0) δ π f(x; (η ) ∈(cid:1) )δn 1 π g(x; (η ) ∈{ − }\ ); η (43d) i i π i i 1,...,n 1 π n ⊆{ − } π 1,...,n 1 − · − = δ δn 1(f g)(x; (η )n 1); η , (43e) i i=1 n − · where the last equality was obtained by exploiting case n 1. Thus, f g has a nth-order chain (cid:88) diﬀerential at point x in directions η , . . . , η and it is such that 1 n · | | −| | δn(f g)(x; (η )n ) = δ π f(x; (η ) ∈ )δn π g(x; (η ) ∈{ }\ ). (44) i i=1 i i π i i 1,...,n π ⊆{ } π 1,...,n This ends the proof by induction. A.4 Practical derivations for the chain rule (Theorem 4) ∈ ∈ Proof. Let us ﬁx a point x X and a direction η X, such that g has a chain diﬀerential at point → ∈ → x in direction η. Let us then ﬁx a sequence ν δg(x; η) Y , and a sequence of real numbers θ 0. m m ≥ a) Let us assume(cid:2)that f is a continuous line(cid:3)ar func(cid:2)tion (cid:96). For any m 0 we can(cid:3) write − − − − θ 1 (cid:96)(g(x) + θ ν ) (cid:96)(g(x)) = θ 1 (cid:96)(g(x)) + θ (cid:96)(ν ) (cid:96)(g(x)) (45a) m m m m m m = (cid:96)(ν ). (45b) m Thus, it holds that (cid:2) (cid:3) (cid:0) (cid:1) − − lim θ 1 (cid:96)(g(x) + θ ν ) (cid:96)(g(x)) = lim (cid:96)(ν ) (46a) →∞ m m m →∞ m m m = (cid:96) lim ν (46b) →∞ m m = (cid:96)(δg(x; η)). (46c) 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: c)

text: Case

text: n

text: Let

text: us

text: ﬁx

text: a

text: point

text: x

text: X

text: and

text: a

text: sequence

text: of

text: directions

text: (η

text: )n

text: Xn

text: such

text: that

text: both

text: f

text: and

text: g

text: have

text: higher

text: i

text: order

text: chain

text: diﬀerentials

text: at

text: point

text: x

text: in

text: all

text: sequences

text: of

text: directions

text: (η

text: I

text: n

text: We

text: can

text: then

text: i

text: i

text: I

text: write

text: π

text: f(x;

text: (η

text: )δn

text: π

text: g(x;

text: (η

text: i

text: i

text: π

text: i

text: i

text: π

text: π

text: δ

text: π

text: (η

text: })δn

text: π

text: g(x;

text: (η

text: i

text: i

text: π

text: n

text: i

text: i

text: π

text: π

text: δ

text: π

text: f(x;

text: (η

text: )δn

text: π

text: (η

text: i

text: i

text: π

text: i

text: i

text: π

text: n

text: π

text: δ

text: δ

text: π

text: f(x;

text: (η

text: δn

text: π

text: g(x;

text: (η

text: i

text: i

text: π

text: n

text: i

text: i

text: π

text: π

text: δ

text: π

text: f(x;

text: (η

text: )δ

text: δn

text: (η

text: η

text: i

text: i

text: π

text: i

text: i

text: π

text: n

text: π

text: δ

text: δ

text: π

text: f(x;

text: (η

text: )δn

text: π

text: g(x;

text: (η

text: η

text: i

text: i

text: π

text: i

text: i

text: π

text: n

text: π

text: δ

text: π

text: f(x;

text: (η

text: )δn

text: π

text: g(x;

text: (η

text: η

text: i

text: i

text: π

text: i

text: i

text: π

text: n

text: π

text: δ

text: δn

text: g)(x;

text: (η

text: )n

text: η

text: i

text: n

text: where

text: the

text: last

text: equality

text: was

text: obtained

text: by

text: exploiting

text: case

text: n

text: Thus,

text: f

text: g

text: has

text: a

text: nth-order

text: chain

text: diﬀerential

text: at

text: point

text: x

text: in

text: directions

text: η

text: η

text: and

text: it

text: is

text: such

text: that

text: n

text: δn(f

text: g)(x;

text: (η

text: )n

text: δ

text: π

text: f(x;

text: (η

text: )δn

text: π

text: g(x;

text: (η

text: i

text: i

text: i

text: π

text: i

text: i

text: π

text: π

text: This

text: ends

text: the

text: proof

text: by

text: induction.

text: Practical

text: derivations

text: for

text: the

text: chain

text: rule

text: (Theorem

text: Proof.

text: Let

text: us

text: ﬁx

text: a

text: point

text: x

text: X

text: and

text: a

text: direction

text: η

text: X,

text: such

text: that

text: g

text: has

text: a

text: chain

text: diﬀerential

text: at

text: point

text: x

text: in

text: direction

text: η.

text: Let

text: us

text: then

text: ﬁx

text: a

text: sequence

text: ν

text: δg(x;

text: η)

text: Y

text: and

text: a

text: sequence

text: of

text: real

text: numbers

text: θ

text: m

text: m

text: a)

text: Let

text: us

text: f

text: is

text: a

text: continuous

text: For

text: any

text: m

text: we

text: write

text: θ

text: θ

text: ν

text: θ

text: θ

text: m

text: m

text: m

text: m

text: m

text: m

text: m

text: Thus,

text: it

text: holds

text: that

text: lim

text: θ

text: θ

text: ν

text: lim

text: m

text: m

text: m

text: m

text: m

text: m

text: lim

text: ν

text: m

text: m

text: η)).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 17 − − − (2π) (π) (θ/2) , with φ = arccos( θ/4π). Un- 3φ φ 0 der the BB1 pulse, up to second order of δ is canceled 1 − in the evolution operator (corresponding to sixth order Uk = e iHrot,ktk, (22) preserved in the average gate ﬁdelity). The average gate ﬁdelity of (π/2) with the BB1 pulse is shown in the lower 0 panel of Supplementary Figure 5c. It exhibits a larger re- where t is the duration of the kth pulse. The total k gion of high ﬁdelity in the axis of δ . Thus suppressing evolution under the entire sequence is 1 (cid:89) the δ error by applying the BB1 pulse, in combination 1 with a proper selection of control ﬁeld strength ω to sup- 1 1 press the δ error, can contribute to realization of a high 0 U = U . (23) ﬁdelity (e.g. 0.9999). tot k Supplementary Figure 5d shows a pulse sequence sup- k=N pressing both the δ and δ errors simultaneously. The 0 1 sequence is designed by incorporating the BB1 pulse The time duration of each pulse, tk, is set to be equal within the CORPSE pulse, and is named BB1inC here value τ. The two-qubit evolution operator can be de- for short. There are similar pulse sequences to suppress scribed as (cid:89) the δ and δ error simultaneously [7]. The BB1inC se- 0 1 − − − − quence is depicted as (θ/2) (π) (2π) (π) (θ ) − (θ ) − (θ ) , where0 φ = aφrccos(−θ3/φ4π), θφ = { } P 1 − { } P θ/32 0− arcs2inπ(sin(θ/12)0/2), θ = 2π − 2 arcsin(sin(θ/2)1/2), Useq( ω1,k, φk ) = e iHrot,k( ω1,k,φk )τ , (24) − 2 k=N and θ = 2π arcsin(sin(θ/2)/2). Leading orders of both 3 the δ and δ errors are canceled in the evolution opera- P 0 1 where is the projection operator on the two-qubit tor. The lower panel of Supplementary Figure 5d shows subspace. the average gate ﬁdelity of (π/2) with the BB1inC pulse. 0 The region of high ﬁdelity is much larger, extended in The target gate is the CNOT gate,  both axes of δ and δ . 0 1   Recently the robustness of composite pulse sequences     0 0 1 0 against time-dependent noise is analyzed [8]. It is shown   that composite pulses may also be successfully employed 0 1 0 0 U = . (25) CNOT in the presence of time-dependent noise. The robustness 1 0 0 0 against static as well as time-dependent noise enables 0 0 0 1 composite pulse an eﬀective method to improve single- qubit gate ﬁdelity. The performance function of the GRAPE algorithm is D. Supplementary Note 4 { } the ﬁdelity F , which is a function of ω , φ . The { seq } 1,k k values of ω , φ are initialized with random numbers 1,k k Quantum optimal control method for designing within the experimental conditions. The performance the CNOT gate. GRAPE is a type of quantum opti- function are maximized by iteration. Within each itera- { } mal control method. It can be utilized to design control tion, the value of F ( ω , φ ) as well as its derivative seq 1,k k sequence to realize a target gate with high ﬁdelity. The to ω and φ is calculated, then the value of ω (φ ) 1,k k 1,k k control sequence contains N piece of pulses, with the am- is replaced by the result of its previous value plus the plitude and phase of each piece being diﬀerent. The total derivative multiplied by a proper coeﬃcient. Hamiltonian of the kth pulse in the rotating frame is (see This method works well for designing a sequence with Eqn. 10 and Eqn. 11) high ﬁdelity, if errors due to qubit-environment interac- tion and control ﬁeld ﬂuctuation are not taken into ac- − − H = 2π(ω (S + S2) ω (I I2) + A(S2 + S I )) count. However, our aim is to realize quantum gates, rot,k S z z I z z z z z which are not only of high ﬁdelity, but also being robust + H , C, rot,k to the errors. A method has been presented [3] to design (20) pulse sequence which is robust against the inhomogeneity of ω . Herein, we generalize this method to design pulse with 1 sequence, which is not only robust against the inhomo- √ geneity δ , but also is insensitive to the dephasing noise 1 HC, rot,k = 2πω1,k/ 2(cos φkHx + sin φkHy), (21) δ0. The performance function of the modiﬁed GRAPE is deﬁned as F (ε, U ) (see Eqn. 17). The new perfor- a CNOT mance function is maximized by iteration of the GRAPE where φ is the phase of the kth pulse, ω is the algorithm. In practical implementation of the modiﬁed k 1,k amplitude of the kth microwave pulse and δΩ = 0. algorithm, the integral in Eqn. 17 is replaced by sum of The evolution operator under the kth pulse is written discrete points. We ﬁnd that three points of δ (δ ) are 0 1 as enough. ##LTLine##

text: (π)

text: with

text: φ

text: arccos(

text: Un-

text: 3φ

text: φ

text: der

text: the

text: pulse,

text: up

text: to

text: second

text: order

text: of

text: δ

text: is

text: canceled

text: in

text: the

text: evolution

text: operator

text: (corresponding

text: to

text: sixth

text: order

text: Uk

text: e

text: iHrot,ktk,

text: preserved

text: in

text: the

text: average

text: gate

text: ﬁdelity).

text: The

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text: gate

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text: with

text: the

text: pulse

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text: shown

text: in

text: the

text: lower

text: panel

text: of

text: Supplementary

text: Figure

text: It

text: exhibits

text: a

text: larger

text: re-

text: where

text: t

text: is

text: the

text: duration

text: of

text: the

text: kth

text: pulse.

text: The

text: total

text: k

text: gion

text: of

text: high

text: ﬁdelity

text: in

text: the

text: axis

text: of

text: δ

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text: suppressing

text: evolution

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text: the

text: entire

text: sequence

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text: the

text: δ

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text: by

text: applying

text: the

text: pulse,

text: in

text: combination

text: with

text: a

text: proper

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text: control

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text: strength

text: ω

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text: sup-

text: press

text: the

text: δ

text: error,

text: can

text: contribute

text: to

text: realization

text: of

text: a

text: high

text: U

text: U

text: ﬁdelity

text: (e.g.

text: tot

text: k

text: Supplementary

text: Figure

text: shows

text: a

text: pulse

text: sequence

text: sup-

text: k=N

text: pressing

text: both

text: the

text: δ

text: and

text: δ

text: errors

text: simultaneously.

text: The

text: sequence

text: is

text: designed

text: by

text: incorporating

text: the

text: pulse

text: The

text: time

text: duration

text: of

text: each

text: pulse,

text: tk,

text: is

text: set

text: to

text: be

text: equal

text: within

text: the

text: CORPSE

text: pulse,

text: and

text: is

text: named

text: here

text: value

text: τ.

text: The

text: two-qubit

text: evolution

text: operator

text: can

text: be

text: de-

text: for

text: short.

text: There

text: are

text: similar

text: pulse

text: sequences

text: to

text: suppress

text: scribed

text: as

text: the

text: δ

text: and

text: δ

text: error

text: simultaneously

text: The

text: se-

text: quence

text: is

text: depicted

text: as

text: (π)

text: (π)

text: (θ

text: (θ

text: (θ

text: φ

text: θφ

text: P

text: P

text: 0−

text: θ

text: 2π

text: Useq(

text: φk

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text: iHrot,k(

text: )τ

text: k=N

text: and

text: θ

text: 2π

text: Leading

text: orders

text: of

text: both

text: the

text: δ

text: and

text: δ

text: errors

text: are

text: canceled

text: in

text: the

text: evolution

text: opera-

text: P

text: where

text: is

text: the

text: projection

text: operator

text: on

text: the

text: two-qubit

text: tor.

text: The

text: lower

text: panel

text: of

text: Supplementary

text: Figure

text: shows

text: subspace.

text: the

text: average

text: gate

text: ﬁdelity

text: of

text: with

text: the

text: pulse.

text: The

text: region

text: of

text: high

text: ﬁdelity

text: is

text: much

text: larger,

text: extended

text: in

text: The

text: target

text: gate

text: is

text: the

text: CNOT

text: gate,

text: both

text: axes

text: of

text: δ

text: and

text: δ

text: Recently

text: the

text: robustness

text: of

text: composite

text: pulse

text: sequences

text: against

text: time-dependent

text: noise

text: is

text: analyzed

text: It

text: is

text: shown

text: that

text: composite

text: pulses

text: may

text: also

text: be

text: successfully

text: employed

text: U

text: CNOT

text: in

text: the

text: presence

text: of

text: time-dependent

text: noise.

text: The

text: robustness

text: against

text: static

text: as

text: well

text: as

text: time-dependent

text: noise

text: enables

text: composite

text: pulse

text: an

text: eﬀective

text: method

text: to

text: improve

text: single-

text: qubit

text: gate

text: ﬁdelity.

text: The

text: performance

text: function

text: of

text: the

text: GRAPE

text: algorithm

text: is

text: D.

text: Supplementary

text: Note

text: the

text: ﬁdelity

text: F

text: which

text: is

text: a

text: function

text: of

text: ω

text: φ

text: The

text: seq

text: k

text: values

text: of

text: ω

text: φ

text: are

text: initialized

text: with

text: random

text: numbers

text: k

text: Quantum

text: optimal

text: control

text: method

text: for

text: designing

text: within

text: the

text: experimental

text: conditions.

text: The

text: performance

text: the

text: CNOT

text: gate.

text: GRAPE

text: is

text: a

text: type

text: of

text: quantum

text: opti-

text: function

text: are

text: maximized

text: by

text: iteration.

text: Within

text: each

text: itera-

text: mal

text: control

text: method.

text: It

text: can

text: be

text: utilized

text: to

text: design

text: control

text: tion,

text: the

text: value

text: of

text: F

text: ω

text: φ

text: as

text: well

text: as

text: its

text: derivative

text: seq

text: k

text: sequence

text: to

text: realize

text: a

text: target

text: gate

text: with

text: high

text: ﬁdelity.

text: The

text: to

text: ω

text: and

text: φ

text: is

text: calculated,

text: then

text: the

text: value

text: of

text: ω

text: (φ

text: k

text: k

text: control

text: sequence

text: contains

text: N

text: piece

text: of

text: pulses,

text: with

text: the

text: am-

text: is

text: replaced

text: by

text: the

text: result

text: of

text: its

text: previous

text: value

text: plus

text: the

text: plitude

text: and

text: phase

text: of

text: each

text: piece

text: being

text: diﬀerent.

text: The

text: total

text: derivative

text: multiplied

text: by

text: a

text: proper

text: coeﬃcient.

text: Hamiltonian

text: of

text: the

text: kth

text: pulse

text: in

text: the

text: rotating

text: frame

text: is

text: (see

text: This

text: method

text: works

text: well

text: for

text: designing

text: a

text: sequence

text: with

text: Eqn.

text: and

text: Eqn.

text: high

text: ﬁdelity,

text: if

text: errors

text: due

text: to

text: qubit-environment

text: interac-

text: tion

text: and

text: control

text: ﬁeld

text: ﬂuctuation

text: are

text: not

text: taken

text: into

text: ac-

text: H

text: 2π(ω

text: (S

text: ω

text: (I

text: S

text: I

text: count.

text: However,

text: our

text: aim

text: is

text: to

text: realize

text: quantum

text: gates,

text: rot,k

text: S

text: z

text: z

text: I

text: z

text: z

text: z

text: z

text: z

text: which

text: are

text: not

text: only

text: of

text: high

text: ﬁdelity,

text: but

text: also

text: being

text: robust

text: H

text: C,

text: rot,k

text: to

text: the

text: errors.

text: A

text: method

text: has

text: been

text: presented

text: to

text: design

text: pulse

text: sequence

text: which

text: is

text: robust

text: against

text: the

text: inhomogeneity

text: of

text: ω

text: Herein,

text: we

text: generalize

text: this

text: method

text: to

text: design

text: pulse

text: with

text: sequence,

text: which

text: is

text: not

text: only

text: robust

text: against

text: the

text: inhomo-

text: geneity

text: δ

text: but

text: also

text: is

text: insensitive

text: to

text: the

text: dephasing

text: noise

text: HC,

text: rot,k

text: φkHx

text: sin

text: φkHy),

text: The

text: performance

text: function

text: of

text: the

text: modiﬁed

text: GRAPE

text: is

text: deﬁned

text: as

text: F

text: (ε,

text: U

text: (see

text: Eqn.

text: The

text: new

text: perfor-

text: a

text: CNOT

text: mance

text: function

text: is

text: maximized

text: by

text: iteration

text: of

text: the

text: GRAPE

text: where

text: φ

text: is

text: the

text: phase

text: of

text: the

text: kth

text: pulse,

text: ω

text: is

text: the

text: algorithm.

text: In

text: practical

text: implementation

text: of

text: the

text: modiﬁed

text: k

text: amplitude

text: of

text: the

text: kth

text: microwave

text: pulse

text: and

text: δΩ

text: algorithm,

text: the

text: integral

text: in

text: Eqn.

text: is

text: replaced

text: by

text: sum

text: of

text: The

text: evolution

text: operator

text: under

text: the

text: kth

text: pulse

text: is

text: written

text: discrete

text: points.

text: We

text: ﬁnd

text: that

text: three

text: points

text: of

text: δ

text: (δ

text: are

text: as

text: enough.

text: ##LTLine##


text: 2 Huaiyu Duan oscillation of neutrinos between the electron ﬂavor and other ﬂavors can have im- portant physical consequences in astrophysical environments through the reactions in Eq. (1). Even in the coherent regime (i.e. without neutrino absorption, emission and collision) the transport of neutrino ﬂavors poses a very challenging problem. One of the challenges is that, in an environment with a large neutrino ﬂux, the ﬂavor transport can become nonlinear because of the neutrino-neutrino coupling. In most cases the nonlinear equations of motion (e.o.m.) which govern neutrino oscillations have to be solved numerically. The computational investigations by various groups show that a dense neutrino medium can experience collective oscillations during which neutrinos of diﬀerent momenta oscillate cooperatively.3–16 To obtain theo- retical insights into these numerical results, several toy models such as the bipolar model17–19 have been proposed which can be solved analytically and which bear some of the qualitative features of the more realistic models. Linear stability anal- ysis provides another useful tool to predict the physical regimes where collective neutrino oscillations may or may not occur.20 Another challenge in solving the problem of neutrino transport is simply due to its large dimensionality: the neutrino transport in any real astrophysical envi- ronment even without ﬂavor oscillations involves 1 temporal dimension, 3 spatial dimensions and 3 momentum dimensions. This challenge has been largely bypassed in the previous studies which are limited to the models with imposed symmetries. For example, the isotropic and homogeneous condition is usually assumed for the early universe. However, even though such symmetries can exist in the e.o.m. and may also exist in the physical systems at some point, they may not be preserved during neutrino oscillations.21 Indeed, a series of recent work suggest that these symmetries can be broken spontaneously by neutrino oscillations.22–27 The goals of this paper are to review some of the key concepts of collective neu- trino oscillations through the bipolar neutrino gas model and to explain why spatial and directional symmetries can be broken spontaneously by collective oscillations. We will leave out many important topics such as three-ﬂavor oscillations11,28,29 some of which can be found in an earlier review.30 (cid:126) Throughout this paper we adopt the natural units with = c = 1. We also assume that neutrinos are relativistic with speed v ≈ 1. 2. Collective neutrino oscillations 2.1. Equation of motion Assuming the validity of the mean-ﬁeld theory,31,32 we will use the (Wigner- transformed) neutrino (ﬂavor) density matrices33 ρ (t, x) and ρ¯ (t, x) to describe p p the ﬂavor content of the neutrino and antineutrino of momentum p at time t and position x. For simplicity we will consider neutrino oscillations between the electron ﬂavor and another active ﬂavor, say τ. In the weak-interaction basis (or ﬂavor basis), ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Huaiyu

text: Duan

text: oscillation

text: of

text: neutrinos

text: between

text: the

text: electron

text: ﬂavor

text: and

text: other

text: ﬂavors

text: can

text: have

text: im-

text: portant

text: physical

text: consequences

text: in

text: astrophysical

text: environments

text: through

text: the

text: reactions

text: in

text: Eq.

text: Even

text: in

text: the

text: coherent

text: regime

text: (i.e.

text: without

text: neutrino

text: absorption,

text: emission

text: and

text: collision)

text: the

text: transport

text: of

text: neutrino

text: ﬂavors

text: poses

text: a

text: very

text: challenging

text: problem.

text: One

text: of

text: the

text: challenges

text: is

text: that,

text: in

text: an

text: environment

text: with

text: a

text: large

text: neutrino

text: ﬂux,

text: the

text: ﬂavor

text: transport

text: can

text: become

text: nonlinear

text: because

text: of

text: the

text: neutrino-neutrino

text: coupling.

text: In

text: most

text: cases

text: the

text: nonlinear

text: equations

text: of

text: motion

text: (e.o.m.)

text: which

text: govern

text: neutrino

text: oscillations

text: have

text: to

text: be

text: solved

text: numerically.

text: The

text: computational

text: investigations

text: by

text: various

text: groups

text: show

text: that

text: a

text: dense

text: neutrino

text: medium

text: can

text: experience

text: collective

text: oscillations

text: during

text: which

text: neutrinos

text: of

text: diﬀerent

text: momenta

text: oscillate

text: To

text: obtain

text: theo-

text: retical

text: insights

text: into

text: these

text: numerical

text: results,

text: several

text: toy

text: models

text: such

text: as

text: the

text: bipolar

text: have

text: been

text: proposed

text: which

text: can

text: be

text: solved

text: analytically

text: and

text: which

text: bear

text: some

text: of

text: the

text: qualitative

text: features

text: of

text: the

text: more

text: realistic

text: models.

text: Linear

text: stability

text: anal-

text: ysis

text: provides

text: another

text: useful

text: tool

text: to

text: predict

text: the

text: physical

text: regimes

text: where

text: collective

text: neutrino

text: oscillations

text: may

text: or

text: may

text: not

text: Another

text: challenge

text: in

text: solving

text: the

text: problem

text: of

text: neutrino

text: transport

text: is

text: simply

text: due

text: to

text: its

text: large

text: dimensionality:

text: the

text: neutrino

text: transport

text: in

text: any

text: real

text: astrophysical

text: envi-

text: ronment

text: even

text: without

text: ﬂavor

text: oscillations

text: involves

text: temporal

text: dimension,

text: spatial

text: dimensions

text: and

text: momentum

text: dimensions.

text: This

text: challenge

text: has

text: been

text: largely

text: bypassed

text: in

text: the

text: previous

text: studies

text: which

text: are

text: limited

text: to

text: the

text: models

text: with

text: imposed

text: symmetries.

text: For

text: example,

text: the

text: isotropic

text: and

text: homogeneous

text: condition

text: is

text: usually

text: assumed

text: for

text: the

text: early

text: universe.

text: However,

text: even

text: though

text: such

text: symmetries

text: can

text: exist

text: in

text: the

text: e.o.m.

text: and

text: may

text: also

text: exist

text: in

text: the

text: physical

text: systems

text: at

text: some

text: point,

text: they

text: may

text: not

text: be

text: preserved

text: during

text: neutrino

text: Indeed,

text: a

text: series

text: of

text: recent

text: work

text: suggest

text: that

text: these

text: symmetries

text: can

text: be

text: broken

text: spontaneously

text: by

text: neutrino

text: The

text: goals

text: of

text: this

text: paper

text: are

text: to

text: review

text: some

text: of

text: the

text: key

text: concepts

text: of

text: collective

text: neu-

text: trino

text: oscillations

text: through

text: the

text: bipolar

text: neutrino

text: gas

text: model

text: and

text: to

text: explain

text: why

text: spatial

text: and

text: directional

text: symmetries

text: can

text: be

text: broken

text: spontaneously

text: by

text: collective

text: oscillations.

text: We

text: will

text: leave

text: out

text: many

text: important

text: topics

text: such

text: as

text: three-ﬂavor

text: some

text: of

text: which

text: can

text: be

text: found

text: in

text: an

text: earlier

text: Throughout

text: this

text: paper

text: we

text: adopt

text: the

text: natural

text: units

text: with

text: c

text: We

text: also

text: assume

text: that

text: neutrinos

text: are

text: relativistic

text: with

text: speed

text: v

text: Collective

text: neutrino

text: oscillations

text: Equation

text: of

text: motion

text: Assuming

text: the

text: validity

text: of

text: the

text: mean-ﬁeld

text: we

text: will

text: use

text: the

text: (Wigner-

text: transformed)

text: neutrino

text: (ﬂavor)

text: density

text: ρ

text: (t,

text: x)

text: and

text: ρ¯

text: (t,

text: x)

text: to

text: describe

text: p

text: p

text: the

text: ﬂavor

text: content

text: of

text: the

text: neutrino

text: and

text: antineutrino

text: of

text: momentum

text: p

text: at

text: time

text: t

text: and

text: position

text: x.

text: For

text: simplicity

text: we

text: will

text: consider

text: neutrino

text: oscillations

text: between

text: the

text: electron

text: ﬂavor

text: and

text: another

text: active

text: ﬂavor,

text: say

text: τ.

text: In

text: the

text: weak-interaction

text: basis

text: (or

text: ﬂavor

text: basis),

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Collective neutrino oscillations and spontaneous symmetry breaking 21 38. D. Notzold and G. Raﬀelt, Nucl.Phys. B307 (1988) 924. 39. J. T. Pantaleone, Phys.Rev. D46 (1992) 510. 40. A. Esteban-Pretel, A. Mirizzi, S. Pastor, R. Tomas, G. Raﬀelt et al., Phys.Rev. D78 (2008) 085012, arXiv:0807.0659 [astro-ph]. 41. H. Duan, G. M. Fuller and Y.-Z. Qian, J.Phys. G36 (2009) 105003, arXiv:0808.2046 [astro-ph]. 42. G. G. Raﬀelt and A. Y. Smirnov, Phys.Rev. D76 (2007) 081301, arXiv:0705.1830 [hep-ph]. 43. H. Duan, G. M. Fuller, J. Carlson and Y.-Z. Qian, Phys.Rev. D75 (2007) 125005, arXiv:astro-ph/0703776 [astro-ph]. 44. K. N. Abazajian, J. F. Beacom and N. F. Bell, Phys.Rev. D66 (2002) 013008, arXiv:astro-ph/0203442 [astro-ph]. 45. H. Duan, A. Friedland, G. C. McLaughlin and R. Surman, J. Phys. G38 (2011) 035201, arXiv:1012.0532 [astro-ph.SR]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Collective

text: neutrino

text: oscillations

text: and

text: spontaneous

text: symmetry

text: breaking

label: reference

text: D.

text: Notzold

text: and

text: G.

text: Raﬀelt,

text: Nucl.Phys.

text: J.

text: T.

text: Pantaleone,

text: Phys.Rev.

text: A.

text: Esteban-Pretel,

text: A.

text: Mirizzi,

text: S.

text: Pastor,

text: R.

text: Tomas,

text: G.

text: Raﬀelt

text: et

text: al.,

text: Phys.Rev.

text: [astro-ph].

text: H.

text: Duan,

text: G.

text: M.

text: Fuller

text: and

text: Y.-Z.

text: Qian,

text: J.Phys.

text: [astro-ph].

text: G.

text: G.

text: Raﬀelt

text: and

text: A.

text: Y.

text: Smirnov,

text: Phys.Rev.

text: [hep-ph].

text: H.

text: Duan,

text: G.

text: M.

text: Fuller,

text: J.

text: Carlson

text: and

text: Y.-Z.

text: Qian,

text: Phys.Rev.

text: [astro-ph].

text: K.

text: N.

text: Abazajian,

text: J.

text: F.

text: Beacom

text: and

text: N.

text: F.

text: Bell,

text: Phys.Rev.

text: [astro-ph].

text: H.

text: Duan,

text: A.

text: Friedland,

text: G.

text: C.

text: McLaughlin

text: and

text: R.

text: Surman,

text: J.

text: Phys.

text: [astro-ph.SR].

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Collective neutrino oscillations and spontaneous symmetry breaking 5 2.2. Bipolar model The study of collective neutrino oscillations begins with the bipolar model.17–19 This model describes a homogeneous and isotropic system with ν and ν¯ popula- e e tions of single energy E at time (cid:104)t = 0. The e.o.m. of neu(cid:105)trino oscillations is 0 σ i∂ (cid:37) = (λ − ηω) 3 + H , (cid:37) (14) t ω 2 νν ω with ω = ±ω , and the neutrino potential is 0 H = µ((cid:37) − α(cid:37)− ), (15) νν ω0 ω0 where n α = ν¯ (16) n ν is the ratio of the density of the antineutrino to that of the neutrino, and √ µ = 2G n (17) F ν is a measure of the strength of the neutrino-neutrino coupling. If the neutrino densit(cid:104)y is negligible, Eq(cid:105). (14) has so(cid:104)lution (cid:105) σ σ (cid:37)± (t) ≈ exp −i(λ ∓ ηω )t 3 (cid:37)± (0) exp i(λ ∓ ηω )t 3 . (18) ω0 0 2 ω0 0 2 Because (cid:20) (cid:21) 1 0 (cid:37)± (0) = (19) ω0 0 0 are diagonal and commute with σ , one(cid:20)has(cid:21) 3 1 0 (cid:37)± (t) ≈ . (20) ω 0 0 In other words, neutrino oscillations are suppressed in the presence of dense matter if there is no ambient neutrino. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Collective

text: neutrino

text: oscillations

text: and

text: spontaneous

text: symmetry

text: breaking

label: section

text: Bipolar

text: model

text: The

text: study

text: of

text: collective

text: neutrino

text: oscillations

text: begins

text: with

text: the

text: bipolar

text: This

text: model

text: describes

text: a

text: homogeneous

text: and

text: isotropic

text: system

text: with

text: ν

text: and

text: ν¯

text: popula-

text: e

text: e

text: tions

text: of

text: single

text: energy

text: E

text: at

text: time

text: The

text: e.o.m.

text: of

text: oscillations

text: is

text: σ

text: i∂

text: (λ

text: ηω)

text: H

text: t

text: ω

text: νν

text: ω

text: with

text: ω

text: ±ω

text: and

text: the

text: neutrino

text: potential

text: is

text: H

text: νν

text: ω0

text: ω0

text: where

text: n

text: α

text: ν¯

text: n

text: ν

text: is

text: the

text: ratio

text: of

text: the

text: density

text: of

text: the

text: antineutrino

text: to

text: that

text: of

text: the

text: neutrino,

text: and

text: µ

text: n

text: F

text: ν

text: is

text: a

text: measure

text: of

text: the

text: strength

text: of

text: the

text: neutrino-neutrino

text: coupling.

text: If

text: the

text: neutrino

text: is

text: negligible,

text: has

text: σ

text: σ

text: (t)

text: exp

text: −i(λ

text: ηω

text: )t

text: exp

text: i(λ

text: ηω

text: )t

text: ω0

text: ω0

text: Because

text: ω0

text: are

text: diagonal

text: and

text: commute

text: with

text: σ

text: (t)

text: ω

text: In

text: other

text: words,

text: neutrino

text: oscillations

text: are

text: suppressed

text: in

text: the

text: presence

text: of

text: dense

text: matter

text: if

text: there

text: is

text: no

text: ambient

text: neutrino.

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: 6 Huaiyu Duan However, the presence of dense matter does not necessarily suppress collec- tive oscillations in the dense neutrino medium. This can be seen by performing a “corotating-frame” transformation to the neutrino ﬂavor matrix:18 − (cid:37)˜ (t) = eiλtσ3/2ρ (t)e iλtσ3/2, (21) ω ω which removes the time evolution due to matter. Under this transformation Eq. (14) becomes (cid:104) (cid:105) σ i∂ (cid:37)˜ (t) = −ηω 3 + H˜ , (cid:37)˜ , (22) t ω 2 νν ω where H˜ = µ((cid:37)˜ − α(cid:37)˜− ). Eqs. (14) and (22) are equivalent, and (cid:37) and (cid:37)˜ νν ω0 ω0 have the same diagonal elements. Therefore, if there exists collective ﬂavor trans- formation in the bipolar model, a uniform dense matter will not suppress it.a In the rest of the paper we will always work in the corotating frame by taking λ = 0, but we will drop the tilde for simplicity. To see that a bipolar system can experience collective ﬂavor transformation we note that Eq. (22) is invariant under global phase transformation41 − (cid:37) (t) −→ e iφσ3/2(cid:37) (t)eiφσ3/2, (23) ω ω where φ is a constant. This symmetry leads to the conservation of the “ﬂavor lepton number (density)”42 L = n tr[((cid:37) − α(cid:37)− )σ ] = (n − n ) − (n − n ). (24) ν ω0 ω0 3 νe ντ ν¯e ν¯τ The phase symmetry in Eq. (23) also implies the existence of a collective pre- cession solution: − (cid:37) (t) = e iΩtσ3/2(cid:37) (0)eiΩtσ3/2, (25) ω ω where Ω is a common oscillation frequency independent of the oscillation fre- quency ω of the individual neutrino. The bipolar model can be solved analytically.17 It has also been shown that the bipolar model is equivalent to a gyroscopic pendulum in ﬂavor space,19 and the aHowever, collective neutrino oscillations can be suppressed by a very large matter density in SNe as shown in Ref. 40. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Huaiyu

text: Duan

text: However,

text: the

text: presence

text: of

text: dense

text: matter

text: does

text: not

text: necessarily

text: suppress

text: collec-

text: tive

text: oscillations

text: in

text: the

text: dense

text: neutrino

text: medium.

text: This

text: can

text: be

text: seen

text: by

text: performing

text: a

text: “corotating-frame”

text: transformation

text: to

text: the

text: neutrino

text: ﬂavor

text: (t)

text: (t)e

text: ω

text: ω

text: which

text: removes

text: the

text: time

text: evolution

text: due

text: to

text: matter.

text: Under

text: this

text: transformation

text: Eq.

text: becomes

text: σ

text: i∂

text: (t)

text: −ηω

text: H˜

text: t

text: ω

text: νν

text: ω

text: where

text: H˜

text: Eqs.

text: and

text: are

text: equivalent,

text: and

text: and

text: νν

text: ω0

text: ω0

text: have

text: the

text: same

text: diagonal

text: elements.

text: Therefore,

text: if

text: there

text: exists

text: collective

text: ﬂavor

text: trans-

text: formation

text: in

text: the

text: bipolar

text: model,

text: a

text: uniform

text: dense

text: matter

text: will

text: not

text: suppress

text: it.a

text: In

text: the

text: rest

text: of

text: the

text: paper

text: we

text: will

text: always

text: work

text: in

text: the

text: corotating

text: frame

text: by

text: taking

text: λ

text: but

text: we

text: will

text: drop

text: the

text: tilde

text: for

text: simplicity.

text: To

text: see

text: that

text: a

text: bipolar

text: system

text: can

text: experience

text: collective

text: ﬂavor

text: transformation

text: we

text: note

text: that

text: Eq.

text: is

text: invariant

text: under

text: global

text: phase

text: (t)

text: e

text: ω

text: ω

text: where

text: φ

text: is

text: a

text: constant.

text: This

text: symmetry

text: leads

text: to

text: the

text: conservation

text: of

text: the

text: “ﬂavor

text: lepton

text: number

text: L

text: n

text: )σ

text: (n

text: n

text: (n

text: n

text: ν

text: ω0

text: ω0

text: νe

text: ντ

text: ν¯e

text: ν¯τ

text: The

text: phase

text: symmetry

text: in

text: Eq.

text: also

text: implies

text: the

text: existence

text: of

text: a

text: collective

text: pre-

text: cession

text: solution:

text: (t)

text: e

text: ω

text: ω

text: where

text: Ω

text: is

text: a

text: common

text: oscillation

text: frequency

text: independent

text: of

text: the

text: oscillation

text: fre-

text: quency

text: ω

text: of

text: the

text: individual

text: neutrino.

text: The

text: bipolar

text: model

text: can

text: be

text: solved

text: It

text: has

text: also

text: been

text: shown

text: that

text: the

text: bipolar

text: model

text: is

text: equivalent

text: to

text: a

text: gyroscopic

text: pendulum

text: in

text: ﬂavor

text: and

text: the

text: aHowever,

text: collective

text: neutrino

text: oscillations

text: can

text: be

text: suppressed

text: by

text: a

text: very

text: large

text: matter

text: density

text: in

text: SNe

text: as

text: shown

text: in

text: Ref.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2,1ϑsoc ieνeνPh Collective neutrino oscillations and spontaneous symmetry breaking 7 collective precession solution in Eq. (25) corresponds to the precession motion of this pendulum. However, this collective precession solution can not be achieved exactly for arbitrary initial conditions just as a gyroscopic pendulum usually experiences both nutation and precession. The bipolar model is, of course, very diﬀerent from the neutrino medium in a realistic astrophysical environment. However, it does provide explanations for some of the important results of collective neutrino oscillations in more sophisticated models. For example, one can show that in the collective precession solution neu- trinos will remain in the weak-interaction states if the neutrino density is larger than a critical value19 [see the discussion in Sec. 2.4 and around Eq. (41)]. This corresponds to the “sleeping-top” regime of a gyroscopic pendulum where the fast spinning top or pendulum deﬁes gravity and spins without wobbling. When the neutrino number density decreases below the critical value, the ﬂavor pendulum begins to wobble about the precession solution. Similar phenomenon was indeed observed in numerical simulations of neutrino oscillations in SNe (see Fig. 1). 1 1 0.8 0.5 0.6 0 0.4 -0.5 0.2 -1 0 1 0.8 0.6 0.4 0.2 0 50 100 150 200 nν/nxν r (km) Fig. 1. Left panel: The collective precession solution of a bipolar model with time-varying neutrino density nν (thick lines) and the numerical solution for t−his model (thin lines). The dashed and solid lines represent cosϑ1 = tr((cid:37)ω0σ3) and cosϑ2 = tr((cid:37)−ω0σ3), respectively. Right panel: The energy-averaged neutrino survival probabilities for three representative neutrino trajectories as functions of radius r in a spherical supernova model. The ﬁgures are adapted from Fig. 3 of Ref. 43 and Fig. 2 of Ref. 7, respectively. 2.3. Spectral swap/split The collective precession solution in Eq. (25) also exists in a homogeneous and isotropic neutrino gas with continuous neutrino energy spectra.42 We will again assume that the neutrinos and antineutrinos are in the weak-interaction states at time t = 0. Because Eq. (4) is invariant under ρ → −ρ (without changing H ), we νν can use ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: ieνeνPh

text: Collective

text: neutrino

text: oscillations

text: and

text: spontaneous

text: symmetry

text: breaking

text: collective

text: precession

text: solution

text: in

text: Eq.

text: corresponds

text: to

text: the

text: precession

text: motion

text: of

text: this

text: pendulum.

text: However,

text: this

text: collective

text: precession

text: solution

text: can

text: not

text: be

text: achieved

text: exactly

text: for

text: arbitrary

text: initial

text: conditions

text: just

text: as

text: a

text: gyroscopic

text: pendulum

text: usually

text: experiences

text: both

text: nutation

text: and

text: precession.

text: The

text: bipolar

text: model

text: is,

text: of

text: course,

text: very

text: diﬀerent

text: from

text: the

text: neutrino

text: medium

text: in

text: a

text: realistic

text: astrophysical

text: environment.

text: However,

text: it

text: does

text: provide

text: explanations

text: for

text: some

text: of

text: the

text: important

text: results

text: of

text: collective

text: neutrino

text: oscillations

text: in

text: more

text: sophisticated

text: models.

text: For

text: example,

text: one

text: can

text: show

text: that

text: in

text: the

text: collective

text: precession

text: solution

text: neu-

text: trinos

text: will

text: remain

text: in

text: the

text: weak-interaction

text: states

text: if

text: the

text: neutrino

text: density

text: is

text: larger

text: than

text: a

text: critical

text: [see

text: the

text: discussion

text: in

text: Sec.

text: and

text: around

text: Eq.

text: This

text: corresponds

text: to

text: the

text: “sleeping-top”

text: regime

text: of

text: a

text: gyroscopic

text: pendulum

text: where

text: the

text: fast

text: spinning

text: top

text: or

text: pendulum

text: deﬁes

text: gravity

text: and

text: spins

text: without

text: wobbling.

text: When

text: the

text: neutrino

text: number

text: density

text: decreases

text: below

text: the

text: critical

text: value,

text: the

text: ﬂavor

text: pendulum

text: begins

text: to

text: wobble

text: about

text: the

text: precession

text: solution.

text: Similar

text: phenomenon

text: was

text: indeed

text: observed

text: in

text: numerical

text: simulations

text: of

text: neutrino

text: oscillations

text: in

text: SNe

text: (see

text: Fig.

text: nν/nxν

text: r

text: (km)

text: Fig.

text: Left

text: panel:

text: The

text: collective

text: precession

text: solution

text: of

text: a

text: bipolar

text: model

text: with

text: time-varying

text: neutrino

text: density

text: nν

text: (thick

text: lines)

text: and

text: the

text: numerical

text: solution

text: for

text: t−his

text: model

text: (thin

text: lines).

text: The

text: dashed

text: and

text: solid

text: lines

text: represent

text: and

text: respectively.

text: Right

text: panel:

text: The

text: energy-averaged

text: neutrino

text: survival

text: probabilities

text: for

text: three

text: representative

text: neutrino

text: trajectories

text: as

text: functions

text: of

text: radius

text: r

text: in

text: a

text: spherical

text: supernova

text: model.

text: The

text: ﬁgures

text: are

text: adapted

text: from

text: Fig.

text: of

text: Ref.

text: and

text: Fig.

text: of

text: Ref.

text: respectively.

text: Spectral

text: swap/split

text: The

text: collective

text: precession

text: solution

text: in

text: Eq.

text: also

text: exists

text: in

text: a

text: homogeneous

text: and

text: isotropic

text: neutrino

text: gas

text: with

text: continuous

text: neutrino

text: energy

text: We

text: will

text: again

text: assume

text: that

text: the

text: neutrinos

text: and

text: antineutrinos

text: are

text: in

text: the

text: weak-interaction

text: states

text: at

text: time

text: t

text: Because

text: Eq.

text: is

text: invariant

text: under

text: ρ

text: −ρ

text: (without

text: changing

text: H

text: we

text: νν

text: can

text: use

text: ##LTFigure##

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text: Pattern formation in binary ﬂuid mixtures induced by ∗ short-range competing interactions Cecilia Bores,1 Enrique Lomba,1,1 Aur´elien Perera,1 and No´e G. Almarza1 (Dated: May 27, 2020) Abstract Molecular dynamics simulations and integral equation calculations of a simple equimolar mixture of diatomic molecules and monomers interacting via attractive and repulsive short-range potentials show the existence of pattern formation (microheterogeneity), mostly due to depletion forces away from the demixing region. Eﬀective site-site potentials extracted from the pair correlation func- tions using an inverse Monte Carlo approach and an integral equation inversion procedure exhibit the features characteristic of a short-range attractive and long-range repulsive potential. When charges are incorporated into the model, this becomes a coarse grained representation of a room temperature ionic liquid, and as expected, intermediate range order becomes more pronounced and stable. 1

text: Pattern

label: title

text: formation

text: in

text: binary

text: ﬂuid

text: mixtures

text: induced

text: by

text: short-range

text: competing

text: interactions

text: Cecilia

label: author

text: Enrique

text: Aur´elien

text: and

text: No´e

text: G.

text: (Dated:

text: May

label: date

text: Abstract

text: Molecular

label: abstract

text: dynamics

text: simulations

text: and

text: integral

text: equation

text: calculations

text: of

text: a

text: simple

text: equimolar

text: mixture

text: of

text: diatomic

text: molecules

text: and

text: monomers

text: interacting

text: via

text: attractive

text: and

text: repulsive

text: short-range

text: potentials

text: show

text: the

text: existence

text: of

text: pattern

text: formation

text: (microheterogeneity),

text: mostly

text: due

text: to

text: depletion

text: forces

text: away

text: from

text: the

text: demixing

text: region.

text: Eﬀective

text: site-site

text: potentials

text: extracted

text: from

text: the

text: pair

text: correlation

text: func-

text: tions

text: using

text: an

text: inverse

text: Monte

text: Carlo

text: approach

text: and

text: an

text: integral

text: equation

text: inversion

text: procedure

text: exhibit

text: the

text: features

text: characteristic

text: of

text: a

text: short-range

text: attractive

text: and

text: long-range

text: repulsive

text: potential.

text: When

text: charges

text: are

text: incorporated

text: into

text: the

text: model,

text: this

text: becomes

text: a

text: coarse

text: grained

text: representation

text: of

text: a

text: room

text: temperature

text: ionic

text: liquid,

text: and

text: as

text: expected,

text: intermediate

text: range

text: order

text: becomes

text: more

text: pronounced

text: and

text: stable.


text: TABLE III. captioncolorParameters of tha SALR eﬀective interaction (red10equation.3.10) be- tween B sites ﬁtted to the data extracted from the IMC procedure. Note that the potential is scaled with k T , by which a is dimensionless. B 0 a a a (˚A) a (˚A) 0 1 2 3 System 1 1.788 8.185 3.749 4.297 tablebox System 2 2.066 8.667 0.843 7.282 System 3 3.578 9.927 0.231 14.816 snapshot_toyT09_rho001_B4_N16384.eps snapshot_toyT09_rho001_B6_N16384.eps (a)σ = 8˚A (b)σ = 12˚A BB BB − FIG. 1. captioncolorSnapshots of conﬁgurations for total particle density ρ = 0.00125˚A 3 and temperature T = 226.45K for two B-site diameters. As the size of B-sites grows C monomers cluster in the cavities formed by the B-sites due to excluded volume eﬀects. All other diameters and total density are kept ﬁxed, σ = σ = 4.0 ˚A . AA CC 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: III.

text: captioncolorParameters

text: of

text: tha

text: SALR

text: eﬀective

text: interaction

text: be-

text: tween

text: B

text: sites

text: ﬁtted

text: to

text: the

text: data

text: extracted

text: from

text: the

text: IMC

text: procedure.

text: Note

text: that

text: the

text: potential

text: is

text: scaled

text: with

text: k

text: T

text: by

text: which

text: a

text: is

text: dimensionless.

text: B

text: a

text: a

text: a

text: (˚A)

text: a

text: (˚A)

text: System

text: tablebox

text: System

text: System

text: (a)σ

text: (b)σ

text: BB

text: BB

text: FIG.

text: captioncolorSnapshots

text: of

text: conﬁgurations

text: for

text: total

text: particle

text: density

text: ρ

text: and

text: temperature

text: T

text: for

text: two

text: B-site

text: diameters.

text: As

text: the

text: size

text: of

text: B-sites

text: grows

text: C

text: monomers

text: cluster

text: in

text: the

text: cavities

text: formed

text: by

text: the

text: B-sites

text: due

text: to

text: excluded

text: volume

text: eﬀects.

text: All

text: other

text: diameters

text: and

text: total

text: density

text: are

text: kept

text: ﬁxed,

text: σ

text: σ

text: ˚A

text: AA

text: CC

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: clust_B4_rho0012_T09.eps FIG. 8. captioncolorNormalized cluster distribution function for A, and B sites, and C monomers of System 3. 28 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: captioncolorNormalized

text: cluster

text: distribution

text: function

text: for

text: A,

text: and

text: B

text: sites,

text: and

text: C

text: monomers

text: of

text: System

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: snapshot_toyT09_B40_N16394_q010.eps − (a)q = 0.10e; q = 0.10e A C snapshot_toyT09_B40_N16394_q025.eps − (b)q = 0.25e; q = 0.25e A C FIG. 9. captioncolorSnapshots of the equimolar mixture of AB dimers and C monomers with embedded charges (indicated on the ﬁgures). 29 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)q

text: q

text: A

text: C

text: (b)q

text: q

text: A

text: C

text: FIG.

text: captioncolorSnapshots

text: of

text: the

text: equimolar

text: mixture

text: of

text: AB

text: dimers

text: and

text: C

text: monomers

text: with

text: embedded

text: charges

text: (indicated

text: on

text: the

text: ﬁgures).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: The rest of the paper can be sketched as follows. In the next section we introduce the model in full detail and brieﬂy summarize the methodology. In Section III we introduce our most signiﬁcant results. Conclusions and future prospects are to be found in Section IV. II. MODEL AND METHODS Our model consists in an equimolar ﬂuid mixture of two diﬀerent species, a two-site dimer AB, and a monomer C. The dimers are represented by a two center Lennard-Jones (LJ) site-site potential, in which the sites are separated by a distance l. Our monomers also interact via LJ potentials. In all cases, the interactions are cut and shifted at a distance r , c (cid:34) (cid:35) by which the explicit form(cid:16)of th(cid:17)e site(cid:16)-site(cid:17)poten(cid:18)tials(cid:19)is (cid:18) (cid:19) σ 12 − σ 6 − σ 12 σ 6 ij ij ij ij u (r) = 4(cid:15) + if r < r , (1) ij c r r r r c c and u (r) = 0 otherwise. Our model is to a certain degree inspired by the simple coarse- ij grained model for imidazolium based RTIL of Merlet et al.32. We will see to what extent a simple model, with just two sites and purely short ranged interactions can reproduce the presence of nano-structural order as found in RTILs. To that aim we will however preserve the attractive/repulsive character of the interactions in the RTIL. In our model then, C monomers would correspond to anions, AB dimers to the molecular cations, with the imidazolium ring that contains the positive charge, being represented by site A, and the non-polar tail, by the larger site B. This implies that AA and CC interactions will be repulsive, BB and AC are attractive, ﬁnally BC and AB interactions are also repulsive. For the sizes of A and C particles we have chosen σ = σ = 4 ˚A , the elongation of the AA CC dimer l = 8˚A. The AB distances of the dimers are ﬁxed as constraints of the equations of motion. The LJ well is set to (cid:15) = 2.092 kJ/mol, identical for all interactions. Since the size of the non-polar tail is essential to determine the nanostructural ordering16, we have considered various sizes for σ (with σ > σ always). For the attractive interactions BB BB AA we have truncated and shifted the LJ potential at r = 3σ . For the repulsive interactions, c BB we have simply used r = 21/6σ , thus deﬁning purely repulsive soft spheres following the c ij prescription of Weeks, Chandler and Andersen (WCA)33. The complete set of parameters for all interactions is summarized in Table I. Finally, in order to analyze the eﬀect of charges on the intermediate range order, we have considered explicitly the same model with a positive 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: rest

text: of

text: the

text: paper

text: can

text: be

text: sketched

text: as

text: follows.

text: In

text: the

text: next

text: section

text: we

text: introduce

text: the

text: model

text: in

text: full

text: detail

text: and

text: brieﬂy

text: summarize

text: the

text: methodology.

text: In

text: Section

text: III

text: we

text: introduce

text: our

text: most

text: signiﬁcant

text: results.

text: Conclusions

text: and

text: future

text: prospects

text: are

text: to

text: be

text: found

text: in

text: Section

text: IV.

text: II.

text: MODEL

text: AND

text: METHODS

text: Our

text: model

text: consists

text: in

text: an

text: equimolar

text: ﬂuid

text: mixture

text: of

text: two

text: diﬀerent

text: species,

text: a

text: two-site

text: dimer

text: AB,

text: and

text: a

text: monomer

text: C.

text: The

text: dimers

text: are

text: represented

text: by

text: a

text: two

text: center

text: Lennard-Jones

text: (LJ)

text: site-site

text: potential,

text: in

text: which

text: the

text: sites

text: are

text: separated

text: by

text: a

text: distance

text: l.

text: Our

text: monomers

text: also

text: interact

text: via

text: LJ

text: potentials.

text: In

text: all

text: cases,

text: the

text: interactions

text: are

text: cut

text: and

text: shifted

text: at

text: a

text: distance

text: r

text: c

text: by

text: which

text: the

text: explicit

text: σ

text: σ

text: σ

text: σ

text: ij

text: ij

text: ij

text: ij

text: u

text: (r)

text: if

text: r

text: r

text: ij

text: c

text: r

text: r

text: r

text: r

text: c

text: c

text: and

text: u

text: (r)

text: otherwise.

text: Our

text: model

text: is

text: to

text: a

text: certain

text: degree

text: inspired

text: by

text: the

text: simple

text: coarse-

text: ij

text: grained

text: model

text: for

text: imidazolium

text: based

text: RTIL

text: of

text: Merlet

text: et

text: We

text: will

text: see

text: to

text: what

text: extent

text: a

text: simple

text: model,

text: with

text: just

text: two

text: sites

text: and

text: purely

text: short

text: ranged

text: interactions

text: can

text: reproduce

text: the

text: presence

text: of

text: nano-structural

text: order

text: as

text: found

text: in

text: RTILs.

text: To

text: that

text: aim

text: we

text: will

text: however

text: preserve

text: the

text: attractive/repulsive

text: character

text: of

text: the

text: interactions

text: in

text: the

text: RTIL.

text: In

text: our

text: model

text: then,

text: C

text: monomers

text: would

text: correspond

text: to

text: anions,

text: AB

text: dimers

text: to

text: the

text: molecular

text: cations,

text: with

text: the

text: imidazolium

text: ring

text: that

text: contains

text: the

text: positive

text: charge,

text: being

text: represented

text: by

text: site

text: A,

text: and

text: the

text: non-polar

text: tail,

text: by

text: the

text: larger

text: site

text: B.

text: This

text: implies

text: that

text: AA

text: and

text: CC

text: interactions

text: will

text: be

text: repulsive,

text: BB

text: and

text: AC

text: are

text: attractive,

text: ﬁnally

text: BC

text: and

text: AB

text: interactions

text: are

text: also

text: repulsive.

text: For

text: the

text: sizes

text: of

text: A

text: and

text: C

text: particles

text: we

text: have

text: chosen

text: σ

text: σ

text: ˚A

text: the

text: elongation

text: of

text: the

text: AA

text: CC

text: dimer

text: l

text: The

text: AB

text: distances

text: of

text: the

text: dimers

text: are

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text: Article number, page 18 of 21page.21 A&A proofs: manuscript no. ﬁnal_dif Table 7. Abundances FeI FeII C NaI MgI AlI SiI CaI ScII TiI TiII CrI MnI CoI NiI Zn SrII YII BaII EuII log(cid:15)(X)(cid:12) 7.50 7.50 8.55 6.33 7.58 6.47 7.55 6.36 3.17 5.02 5.02 5.67 5.39 4.92 6.25 4.60 2.97 2.24 2.13 0.51 ET0381 Nb Lines 80 11 1 2 5 1 2 9 5 10 24 6 3 4 3 1 1 2 3 1 − − − − − − − − − − − − − − − − − − − − [X/H] 2.44 2.13 −3.44 −3.03 −2.91 −3.55 −2.81 −2.67 −2.87 −2.97 −2.56 −2.75 −2.63 −2.94 −2.65 < −2.77 −3.30 < −3.31 −3.64 < −2.81 [X/Fe] − 1.00 0.59 0.47 1.11 0.37 0.23 0.43 0.53 0.12 0.31 0.19 0.50 0.21 < 0.33 0.86 < 0.87 1.22 < 0.37 Error 0.08 0.07 0.2 +0.1 0.13 0.10 0.16 0.11 0.07 0.08 0.07 0.09 0.10 0.15 0.16 0.14 0.15 0.06 scl002_06 Nb Lines 82 4 1 2 4 1 2 6 5 4 17 7 4 4 2 1 2 1 1 1 − − − − − − − − − − − − − − − − − − − − [X/H] 3.45 3.28 −4.96 −3.65 3.05 −3.94 3.05 3.06 3.44 −3.54 3.18 −3.75 −3.60 −3.68 3.44 < 2.88 −4.29 < −3.61 < −5.69 < 2.81 [X/Fe] − 1.50 0.20 0.40 0.49 0.40 0.39 0.01 0.09 0.27 0.30 0.15 0.23 0.01 < 0.58 0.84 < 0.15 < 2.23 <0.65 Error 0.07 0.06 1.00 +0.3 0.12 0.09 0.18 0.16 0.05 0.07 0.06 0.10 0.09 0.12 0.14 0.09 0.18 scl_03_059 Nb Lines 97 4 1 2 5 1 2 10 7 10 27 6 3 3 3 1 2 1 4 1 − − − − − − − − − − − − − − − − − − − − [X/H] 3.22 3.08 −4.42 2.82 2.80 −3.88 2.82 2.84 2.96 3.14 2.79 −3.54 −3.99 −3.29 −3.26 < 2.90 −3.51 < −3.41 −3.44 < −3.31 [X/Fe] − 1.20 0.40 0.42 0.66 0.40 0.38 0.26 0.08 0.43 0.32 0.77 0.07 0.04 <0.32 0.29 < 0.20 0.27 < 0.09 Error 0.08 0.08 0.5 +0.3 0.14 0.08 0.15 0.11 0.07 0.07 0.06 0.08 0.08 0.15 0.14 0.12 0.15 0.08 scl031_11 Nb Lines 38 2 1 2 4 2 2 1 2 13 5 3 4 3 1 2 2 1 1 − − − − − − − − − − − − − − − − − − − [X/H] 3.88 3.81 −4.29 −4.20 3.75 −4.70 3.48 3.64 −4.11 3.39 −4.20 −4.42 3.63 3.84 < 2.90 −5.26 < 3.54 < −5.07 < 3.01 [X/Fe] − 0.41 0.32 0.13 0.82 0.40 0.24 0.23 0.49 0.32 0.54 0.25 0.04 < 0.99 1.38 < 0.35 < 1.18 <0.88 Error 0.08 0.08 0.51 +0.22 0.07 0.15 0.14 0.14 0.09 0.13 0.10 0.09 0.18 0.10 0.09 0.12 scl074_02 Nb Lines 61 4 1 2 6 1 2 7 3 4 15 6 3 3 2 1 2 2 2 1 − − − − − − − − − − − − − − − − − − − − [X/H] 3.31 3.13 −4.20 −3.49 3.04 −4.21 2.42 2.80 −3.36 3.31 3.30 −3.55 −3.46 −3.48 3.18 < 2.61 −3.64 < 3.14 < −5.13 < 3.12 [X/Fe] − 0.88 0.18 0.27 0.90 0.89 0.51 0.05 0.00 0.01 0.24 0.15 0.17 0.13 < 0.69 0.33 <0.16 < 1.83 <0.18 Error 0.09 0.10 1.00 +0.3 0.12 0.10 0.26 0.13 0.07 0.27 0.08 0.12 0.12 0.18 0.18 0.08 0.22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Article

text: number,

text: page

text: of

text: A&A

text: proofs:

text: manuscript

text: no.

text: Table

text: Abundances

label: caption

text: FeI

label: table

text: FeII

text: C

text: NaI

text: MgI

text: AlI

text: SiI

text: CaI

text: ScII

text: TiI

text: TiII

text: CrI

text: MnI

text: CoI

text: NiI

text: Zn

text: SrII

text: YII

text: BaII

text: EuII

text: Nb

text: Lines

font: VMYLUC+txsy

text: [X/H]

text: [X/Fe]

text: Error

text: Nb

text: Lines

text: [X/H]

text: [X/Fe]

text: Error

text: Nb

text: Lines

text: [X/H]

text: [X/Fe]

text: Error

text: Nb

text: Lines

text: [X/H]

text: [X/Fe]

text: Error

text: Nb

text: Lines

text: [X/H]

text: [X/Fe]

text: Error

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##