text: 11 A 81, 061801(R) (2010). [28] C. Joshi, F. Nissen, and J. Keeling, Phys. Rev. A 88, [21] I. Carusotto, D. Gerace, H. E. Tu¨reci, S. De Liberato, C. 063835 (2013). Ciuti, and A. Imamogˇlu, Phys. Rev. Lett. 103, 033601 [29] J. Jin, D. Rossini, R. Fazio, M. Leib, and M. J. Hart- (2009). mann, Phys. Rev. Lett. 110, 163605 (2013). [22] S. Ferretti, L. C. Andreani, H. E. Tu¨reci, and Dario Ger- [30] T. Grujic, S. R. Clark, D. Jaksch, and D. G. Angelakis ace, Phys. Rev. A 82, 013841 (2010). Phys. Rev. A 87, 053846 (2013). [23] M. J. Hartmann, Phys. Rev. Lett. 104, 113601 (2010). [31] A. Le Boit´e, G. Orso, and C. Ciuti, Phys. Rev. Lett. 110, [24] M. Leib and M. Hartmann, New J. Phys. 12, 233601 (2013). 093031(2010). [32] P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 [25] M. Hafezi, D. E. Chang, V. Gritsev, E. A. Demler, M.D (1980). Lukin, Europhys. Lett.94, 54006 (2011). [33] I. Vidanovi´c, Daniel Cocks, and Walter Hofstetter, Phys. [26] F. Nissen, S. Schmidt, M. Biondi, G. Blatter, H. E. Rev. A 89, 053614 (2014). Tu¨reci, and J. Keeling, Phys. Rev. Lett. 108, 233603 [34] A. Hu, T. E. Lee, and C. W. Clark, Phys. Rev. A 88, (2012). 053627 (2013). [27] R. O. Umucalilar and I. Carusotto, Phys. Rev. Lett. 108, 206809 (2012).

text: A

text: C.

text: Joshi,

text: F.

text: Nissen,

text: and

text: J.

text: Keeling,

text: Phys.

text: Rev.

text: A

text: I.

text: Carusotto,

text: D.

text: Gerace,

text: H.

text: E.

text: Tu¨reci,

text: S.

text: De

text: Liberato,

text: C.

text: Ciuti,

text: and

text: A.

text: Imamogˇlu,

text: Phys.

text: Rev.

text: Lett.

text: J.

text: Jin,

text: D.

text: Rossini,

text: R.

text: Fazio,

text: M.

text: Leib,

text: and

text: M.

text: J.

text: Hart-

text: mann,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Ferretti,

text: L.

text: C.

text: Andreani,

text: H.

text: E.

text: Tu¨reci,

text: and

text: Dario

text: Ger-

text: T.

text: Grujic,

text: S.

text: R.

text: Clark,

text: D.

text: Jaksch,

text: and

text: D.

text: G.

text: Angelakis

text: ace,

text: Phys.

text: Rev.

text: A

text: Phys.

text: Rev.

text: A

text: M.

text: J.

text: Hartmann,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Le

text: Boit´e,

text: G.

text: Orso,

text: and

text: C.

text: Ciuti,

text: Phys.

text: Rev.

text: Lett.

text: M.

text: Leib

text: and

text: M.

text: Hartmann,

text: New

text: J.

text: Phys.

text: P.

text: D.

text: Drummond

text: and

text: D.

text: F.

text: Walls,

text: J.

text: Phys.

text: A

text: M.

text: Hafezi,

text: D.

text: E.

text: Chang,

text: V.

text: Gritsev,

text: E.

text: A.

text: Demler,

text: M.D

text: Lukin,

text: Europhys.

text: I.

text: Vidanovi´c,

text: Daniel

text: Cocks,

text: and

text: Walter

text: Hofstetter,

text: Phys.

text: F.

text: Nissen,

text: S.

text: Schmidt,

text: M.

text: Biondi,

text: G.

text: Blatter,

text: H.

text: E.

text: Rev.

text: A

text: Tu¨reci,

text: and

text: J.

text: Keeling,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: Hu,

text: T.

text: E.

text: Lee,

text: and

text: C.

text: W.

text: Clark,

text: Phys.

text: Rev.

text: A

text: R.

text: O.

text: Umucalilar

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text: Carusotto,

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text: Lett.


text: 4 1 − 1 1| (cid:105)(cid:104) | 1| (cid:105)(cid:104) | 1| (cid:105)(cid:104) | ∂tρaa = γ[2ρcc ρaa + 2(ρab + ρba)], (17) ρ = 4 0 0 + 2 1 1 + 4 2 2 . (28) 1 − 1 ∂ ρ = γ[ ρ ρ + (ρ + ρ )], (18) t bb 2 cc √bb 2 ab ba The photon density in then equal to one and g(2)(0) = − F2 − γ γ ∂ ρ = ∆ω[( i2 2 )ρ + ], (19) 0.5. We see that in the particular case of multiphotonic t ab −∆ω2 − ∆ω ab 2∆ω resonanc(cid:28)es, one must be careful in discussing the limit ∂tρcc = γ[ρaa + ρbb ρcc (ρab + ρba)]. (20) F , γ 1 as the system behavior varies qualitatively ∆ω ∆ω → depending on the ratio F2 . The state obtained for ξ γ∆ω 0 is highly nonclassical and the closest to a Fock state The stationary value for ρ is then: ab that one can hope for in this context. Besides, this result is not limited to n = 2. In the ρ = 1 1√ , (21) g| e(cid:105)neral c|as(cid:105)e of n-photon resonance, the coupling between ab 2 1 + i2 2F2 0 and n is of order n in F and the energy splitting γ∆ω proportional to (F/∆ω)n. Therefore, a state similar to (cid:29) Eq.(28) may be obtain in the limit Fn− 1. As giving: γ∆ωn 1 shown in the Appendix, the corresponding density matrix is: (cid:18) (cid:19) (cid:88) 1 ρ + ρ = = ξ. (22) ab ba 1 + γ28∆Fω4 2 1 n n | (cid:105)(cid:104) | ρ(n) = k k . (29) 2n k All the other coeﬃcients along with the observables can k=0 be expressed as functions of the parameter ξ. Namely, | (cid:105) | (cid:105) we have: For n > 2, not only the two states 0 and n are degenerate in the absence of driving, but so are all the | (cid:105) | (cid:105) states k , q with k+q = n. This degeneracy is reﬂected (cid:39) 1 − → − ρ ρ = (1 ξ), (23) in the n n k symmetry of Eq.(29). The state of 11 cc 2 Eq.(29) is characterized by: 1 ρ = ρ = (1 + ξ), (24) aa bb 4 (cid:104) † (cid:105) n 1 − 1 1 − b b = , (30) ρ = (ρ + ρ ) (ρ + ρ ) = (1 ξ). (25) 22 2 aa bb 2 ab ba 4 2 − 1 g(2)(0) = 1 . (31) n For this we can compute the mean photon density and g(2)(0): Note that the value of g(2)(0) is the same as in the nth-lobe of the equilibrium model (pure Fock state with (cid:104) † (cid:105) − b b = 1 ξ, (26) n photons). We emphasize that Eq.(29-31) also apply to → → n = 1 (∆ω = 0), in the limit F/U 0 and γ/F 0. 1 g(2)(0) = − . (27) Driving out of multiphotonic resonances. In the case 2(1 ξ) where multiphotonic absorption processes are non reso- nant, the mean photon number is expected to be very A comparison between theses two expressions and the low. By performing an expansion of the master equation exact P-representation formula is shown on Fig.(1). The in powers of F/∆ω and γ/∆ω one can show that the co- mean photon density−and g(2)(0) are plotted as a function eﬃcients of the stationary density matrix∼(in Fock space) of ξ for F/∆ω = 10 2. In order to stay in the domain obey the following hierarchy: Re(ρ ) ( F )n+m and of validity of Eq.(26) and Eq.(27), γ/∆ω is ranging from Im(ρ ) ∼ γ ( F )n+m. F/∆ω anndmγ/∆ω∆aωre assumed F2 to F . In this conditions, the approximations to benbmoth m∆uωch∆smω aller than one, but it is not necessary 10∆ω2 5∆ω (cid:29) underlying the derivation are justiﬁed and the above ex- to impose F γ. The following treatment is still valid pressions are very accurate. if F/∆ω and γ/∆ω are of the same order of magnitude. (cid:28) (cid:29) The two limits F2 1 and F2 1 correspond to Neglecting the probability of having 3 or more photons → → γ∆ω γ∆ω ξ 1 and ξ 0 respectively. In the ﬁrst case, the pho- inside the cavity and keeping the lowest order in F/∆ω ton density goes to zero as expected, and g(2)(0) diverges. and γ/∆ω for the remaining coeﬃcients, we obtain the In the other limit, the ﬁeld is in a statistical mixture of following set of equations (the real part of a complex 3 states and the density matrix in Fock space is: number z is denoted by z, its imaginary part by z˜): ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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

label: equation

text: ∂tρaa

text: ρaa

text: ρba)],

text: ρ

text: ρ

text: γ[

text: ρ

text: ρ

text: (ρ

text: ρ

text: t

text: bb

text: cc

text: √bb

text: ab

text: ba

text: The

text: photon

text: density

text: in

text: then

text: equal

text: to

text: one

text: and

text: γ

text: γ

text: ρ

text: ∆ω[(

text: )ρ

text: We

text: see

text: that

text: in

text: the

text: particular

text: case

text: of

text: multiphotonic

text: t

text: ab

text: −∆ω2

text: ∆ω

text: ab

text: 2∆ω

text: one

text: must

text: be

text: careful

text: in

text: discussing

text: the

text: limit

text: ∂tρcc

text: γ[ρaa

text: ρbb

text: ρcc

text: (ρab

text: ρba)].

text: F

text: γ

text: as

text: the

text: system

text: behavior

text: varies

text: qualitatively

text: ∆ω

text: ∆ω

text: depending

text: on

text: the

text: ratio

text: The

text: state

text: obtained

text: for

text: ξ

text: γ∆ω

text: is

text: highly

text: nonclassical

text: and

text: the

text: closest

text: to

text: a

text: Fock

text: state

text: The

text: stationary

text: value

text: for

text: ρ

text: is

text: then:

text: ab

text: that

text: one

text: can

text: hope

text: for

text: in

text: this

text: context.

text: Besides,

text: this

text: result

text: is

text: not

text: limited

text: to

text: n

text: In

text: the

text: ρ

text: 1√

text: g|

text: of

text: n-photon

text: resonance,

text: the

text: coupling

text: between

text: ab

text: and

text: n

text: is

text: of

text: order

text: n

text: in

text: F

text: and

text: the

text: energy

text: splitting

text: γ∆ω

text: proportional

text: to

text: (F/∆ω)n.

text: Therefore,

text: a

text: state

text: similar

text: to

text: may

text: be

text: obtain

text: in

text: the

text: limit

text: Fn−

text: As

text: giving:

text: γ∆ωn

text: shown

text: in

text: the

text: Appendix,

text: the

text: corresponding

text: density

text: matrix

text: is:

text: ρ

text: ρ

text: ξ.

text: ab

text: ba

text: n

text: n

text: ρ(n)

text: k

text: k

text: k

text: All

text: the

text: other

text: coeﬃcients

text: along

text: with

text: the

text: observables

text: can

text: be

text: expressed

text: as

text: functions

text: of

text: the

text: parameter

text: ξ.

text: Namely,

text: we

text: have:

text: For

text: n

text: not

text: only

text: the

text: two

text: states

text: and

text: n

text: are

text: degenerate

text: in

text: the

text: absence

text: of

text: driving,

text: but

text: so

text: are

text: all

text: the

text: states

text: k

text: q

text: with

text: k+q

text: n.

text: This

text: degeneracy

text: is

text: reﬂected

text: ρ

text: ρ

text: ξ),

text: in

text: the

text: n

text: n

text: k

text: symmetry

text: of

text: The

text: state

text: of

text: cc

text: is

text: characterized

text: by:

text: ρ

text: ρ

text: ξ),

text: aa

text: bb

text: n

text: b

text: b

text: ρ

text: (ρ

text: ρ

text: (ρ

text: ρ

text: ξ).

text: aa

text: bb

text: ab

text: ba

text: n

text: For

text: this

text: we

text: can

text: compute

text: the

text: mean

text: photon

text: density

text: and

text: Note

text: that

text: the

text: value

text: of

text: is

text: the

text: same

text: as

text: in

text: the

text: nth-lobe

text: of

text: the

text: equilibrium

text: model

text: (pure

text: Fock

text: state

text: with

text: b

text: b

text: ξ,

text: n

text: photons).

text: We

text: emphasize

text: that

text: also

text: apply

text: to

text: n

text: (∆ω

text: in

text: the

text: limit

text: F/U

text: and

text: γ/F

text: Driving

text: out

text: of

text: multiphotonic

text: resonances.

text: In

text: the

text: case

text: ξ)

text: where

text: multiphotonic

text: absorption

text: processes

text: are

text: non

text: reso-

text: nant,

text: the

text: mean

text: photon

text: number

text: is

text: expected

text: to

text: be

text: very

text: A

text: comparison

text: between

text: theses

text: two

text: expressions

text: and

text: the

text: low.

text: By

text: performing

text: an

text: expansion

text: of

text: the

text: master

text: equation

text: exact

text: P-representation

text: formula

text: is

text: shown

text: on

text: The

text: in

text: powers

text: of

text: F/∆ω

text: and

text: γ/∆ω

text: one

text: can

text: show

text: that

text: the

text: co-

text: mean

text: photon

text: density−and

text: are

text: plotted

text: as

text: a

text: function

text: eﬃcients

text: of

text: the

text: stationary

text: density

text: matrix∼(in

text: Fock

text: space)

text: of

text: ξ

text: for

text: F/∆ω

text: In

text: order

text: to

text: stay

text: in

text: the

text: domain

text: obey

text: the

text: following

text: hierarchy:

text: Re(ρ

text: F

text: )n+m

text: and

text: of

text: validity

text: of

text: and

text: γ/∆ω

text: is

text: ranging

text: from

text: Im(ρ

text: γ

text: F

text: )n+m.

text: F/∆ω

text: anndmγ/∆ω∆aωre

text: assumed

text: to

text: F

text: In

text: this

text: conditions,

text: the

text: approximations

text: to

text: benbmoth

text: m∆uωch∆smω

text: aller

text: than

text: one,

text: but

text: it

text: is

text: not

text: necessary

text: 5∆ω

text: underlying

text: the

text: derivation

text: are

text: justiﬁed

text: and

text: the

text: above

text: ex-

text: to

text: impose

text: F

text: γ.

text: The

text: following

text: treatment

text: is

text: still

text: valid

text: pressions

text: are

text: very

text: accurate.

text: if

text: F/∆ω

text: and

text: γ/∆ω

text: are

text: of

text: the

text: same

text: order

text: of

text: magnitude.

text: The

text: two

text: limits

text: and

text: correspond

text: to

text: Neglecting

text: the

text: probability

text: of

text: having

text: or

text: more

text: photons

text: γ∆ω

text: γ∆ω

text: ξ

text: and

text: ξ

text: respectively.

text: In

text: the

text: ﬁrst

text: case,

text: the

text: pho-

text: inside

text: the

text: cavity

text: and

text: keeping

text: the

text: lowest

text: order

text: in

text: F/∆ω

text: ton

text: density

text: goes

text: to

text: zero

text: as

text: expected,

text: and

text: diverges.

text: and

text: γ/∆ω

text: for

text: the

text: remaining

text: coeﬃcients,

text: we

text: obtain

text: the

text: In

text: the

text: other

text: limit,

text: the

text: ﬁeld

text: is

text: in

text: a

text: statistical

text: mixture

text: of

text: following

text: set

text: of

text: equations

text: (the

text: real

text: part

text: of

text: a

text: complex

text: states

text: and

text: the

text: density

text: matrix

text: in

text: Fock

text: space

text: is:

text: number

text: z

text: is

text: denoted

text: by

text: z,

text: its

text: imaginary

text: part

text: by

text: z˜):

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 4 B. Bogoliubov Theory 3.5 As shown in our previous work, one can study ﬂuctu- 3 ations of the density matrix around mean-ﬁeld by means of a extended Bogoliubov theory. The ﬂuctuations are 2.5 ω deﬁned as follow: ∆ (cid:79) 2 / U 1.5 ρ = (ρ + δρ ). (60) mf i 1 i (cid:80) 0.5 We also introduce the Fourier transform of the matri- 0 1 J/∆ω 2 3 ces δρi, δρk = √1N Ni=1 e−ik·riδρi In their most general formulations, the equations of FIG. 5: (Color online) Gross-Pitaevski and P-representation evolution that stem from linearization around mean-ﬁeld bistability diagrams. Orange: monostable phase according are: to both approximation schemes ; light blue: bistable accord- ing to Eq.(54) but monostable according to P-representation calculations ; dark blue: bistable phase according to both approximation schemes. L L i∂ δρ = [δρ ] + [δρ ], (61) t k mf k k k high-density phase, the photon density decreases with U. The critical value is then approximately the one for which where the density is maximal. This yields: Uc1 = γ2 (1 + J ). (56) L − iγ † − † − † ∆ω 4F2 ∆ω mf[δρk] = [Hmf, δρk] 2 (2bδρkb b bδρk δρkb b). (62) In fact, this approximate expression corresponds to the ﬁrst term in the expansion in powers of F/∆ω of the This operator is the usual Liouvillian for the eﬀective exact solution. A similar expansion for the other frontier single cavity problem. This term in Eq.(61) is thus inde- in the diagram gives: pendent of k. Propagation eﬀects arise from the second term: U 4∆ω2 c2 = (1 + J/∆ω)3. (57) ∆ω 27F2 L − † † [δρ ] = t (Tr(bδρ )[b , ρ ] + Tr(b δρ )[b, ρ ]), k k k k mf k mf (63) The expansion up to the next term is: with t = J/z(cos k a + cos k a). U γ2 J − γ4 k x y c1 = (1 + ) , (58) The situation is greatly simpliﬁed in the Gross- ∆ω 4F2 ∆ω 64∆ω2F2(1 + J/∆ω) Pitaevski regime where the system is described by clas- Uc2 = 4∆ω2 (1 + J/∆ω)3 + γ2 (1 + J/∆ω). (59) sical complex ﬁeld. Fluctuations around the mean-ﬁeld ∆ω 27F2 12F2 value β then obey the following equation: (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) − − | | − δβ ∆ω t + 2U β 2 iγ/2 Uβ2 δβ i∂ ∗k = k ∗ − | | − ∗k . (64) t δβ− Uβ 2 ∆ω + t 2U β 2 iγ/2 δβ− k k k (cid:112) This leads to a complex Bogoliubov spectrum: ± − − | | − | | −iγ ω±(k) = ( ∆ω t + 2U β 2)2 U2 β 4 . (65) k 2 ##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##

font: default

text: B.

label: section

text: Bogoliubov

text: Theory

text: As

text: shown

text: in

text: our

text: previous

text: work,

text: one

text: can

text: study

text: ﬂuctu-

text: ations

text: of

text: the

text: density

text: matrix

text: around

text: mean-ﬁeld

text: by

text: means

text: of

text: a

text: extended

text: Bogoliubov

text: theory.

text: The

text: ﬂuctuations

text: are

text: ω

text: deﬁned

text: as

text: follow:

text: U

text: ρ

text: (ρ

text: δρ

text: mf

text: i

text: i

text: We

text: also

text: introduce

text: the

text: Fourier

text: transform

text: of

text: the

text: matri-

text: J/∆ω

text: ces

text: δρi,

text: δρk

text: e−ik·riδρi

text: In

text: their

text: most

text: general

text: formulations,

text: the

text: equations

text: of

text: FIG.

text: (Color

text: online)

text: Gross-Pitaevski

text: and

text: P-representation

text: evolution

text: that

text: stem

text: from

text: linearization

text: around

text: mean-ﬁeld

text: bistability

text: diagrams.

text: Orange:

text: monostable

text: phase

text: according

text: are:

text: to

text: both

text: approximation

text: schemes

text: light

text: blue:

text: bistable

text: accord-

text: ing

text: to

text: but

text: monostable

text: according

text: to

text: P-representation

text: calculations

text: dark

text: blue:

text: bistable

text: phase

text: according

text: to

text: both

text: approximation

text: schemes.

text: L

text: L

text: i∂

text: δρ

text: [δρ

text: [δρ

text: t

text: k

text: mf

text: k

text: k

text: k

text: high-density

text: phase,

text: the

text: photon

text: density

text: decreases

text: with

text: U.

text: The

text: critical

text: value

text: is

text: then

text: approximately

text: the

text: one

text: for

text: which

text: where

text: the

text: density

text: is

text: maximal.

text: This

text: yields:

text: γ2

text: J

text: L

text: iγ

text: ∆ω

text: ∆ω

text: mf[δρk]

text: [Hmf,

text: δρk]

text: b

text: bδρk

text: δρkb

text: b).

text: In

text: fact,

text: this

text: approximate

text: expression

text: corresponds

text: to

text: the

text: ﬁrst

text: term

text: in

text: the

text: expansion

text: in

text: powers

text: of

text: F/∆ω

text: of

text: the

text: This

text: operator

text: is

text: the

text: usual

text: Liouvillian

text: for

text: the

text: eﬀective

text: exact

text: solution.

text: A

text: similar

text: expansion

text: for

text: the

text: other

text: frontier

text: single

text: cavity

text: problem.

text: This

text: term

text: in

text: is

text: thus

text: inde-

text: in

text: the

text: diagram

text: gives:

text: pendent

text: of

text: k.

text: Propagation

text: eﬀects

text: arise

text: from

text: the

text: second

text: term:

text: U

text: 4∆ω2

text: ∆ω

text: L

text: [δρ

text: t

text: (Tr(bδρ

text: )[b

text: ρ

text: Tr(b

text: δρ

text: )[b,

text: ρ

text: k

text: k

text: k

text: k

text: mf

text: k

text: mf

text: The

text: expansion

text: up

text: to

text: the

text: next

text: term

text: is:

text: with

text: t

text: J/z(cos

text: k

text: a

text: cos

text: k

text: a).

text: U

text: γ2

text: J

text: γ4

text: k

text: x

text: y

text: The

text: situation

text: is

text: greatly

text: simpliﬁed

text: in

text: the

text: Gross-

text: ∆ω

text: ∆ω

text: J/∆ω)

text: Pitaevski

text: regime

text: where

text: the

text: system

text: is

text: described

text: by

text: clas-

text: 4∆ω2

text: γ2

text: J/∆ω).

text: sical

text: complex

text: ﬁeld.

text: Fluctuations

text: around

text: the

text: mean-ﬁeld

text: ∆ω

text: value

text: β

text: then

text: obey

text: the

text: following

text: equation:

text: δβ

text: ∆ω

text: t

text: β

text: δβ

text: i∂

text: ∗k

text: k

text: ∗k

text: t

text: δβ−

text: Uβ

text: ∆ω

text: t

text: β

text: δβ−

text: k

text: k

text: k

text: This

text: leads

text: to

text: a

text: complex

text: Bogoliubov

text: spectrum:

text: −iγ

text: ω±(k)

text: ∆ω

text: t

text: β

text: β

text: k

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: EPJ manuscript No. (will be inserted by the editor) De Haas-van Alphen oscillations in the compensated organic · metal α-’pseudo-κ’-(ET) H O[Fe(C O ) ] (C H Br ) 4 3 2 4 3 6 4 2 Alain Audouard1a, Jean-Yves Fortin2b, Vladimir N. Laukhin3,4, David Vignolles1, Tatyana G. Prokhorova5, Eduard B. Yagubskii5, and Enric Canadell4 1 Laboratoire National des Champs Magn´etiques Intenses (UPR 3228 CNRS, INSA, UJF, UPS) 143 avenue de Rangueil, F-31400 Toulouse, France. 2 Institut Jean Lamour, D´epartement de Physique de la Mati`ere et des Mat´eriaux, CNRS-UMR 7198, Vandoeuvre-les-Nancy, F-54506, France. 3 Institucio´ Catalana de Recerca i Estudis Avanc¸ats (ICREA), 08010 Barcelona, Spain. 4 Institut de Ci`encia de Materials de Barcelona, CSIC, Campus de la UAB, 08193, Bellaterra, Spain. 5 Institute of Problems of Chemical Physics, Russian Academy of Sciences, 142432 Chernogolovka, MD, Russia Received: date / Revised version: date Abstract. Field-, temperature- and angle-dependent Fourier amplitude of de Haas-van Alphen (dHvA) oscillations are calculated for compensated two-dimensional (2D) metals with textbook Fermi surface (FS) composed of one hole and two electron orbits connected by magnetic breakdown. It is demonstrated that, taking into account the opposite sign of electron and hole orbits, a given Fourier component involves combination of several orbits, the contribution of which must be included in the calculations. Such FS is · observed in the strongly 2D organic metal α-’pseudo-κ’-(ET) H O[Fe(C O ) ] (C H Br ), dHvA oscilla- 4 3 2 4 3 6 4 2 tions of which have been studied up to 55 T for various directions of the magnetic ﬁeld with respect to the conducting plane. Calculations are in good quantitative agreement with the data. PACS. 71.18.+y Fermi surface: calculations and measurements; eﬀective mass, g factor – 71.20.Rv Poly- mers and organic compounds ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EPJ

text: manuscript

text: No.

text: (will

text: be

text: inserted

text: by

text: the

text: editor)

text: De

label: title

text: Haas-van

text: Alphen

text: oscillations

text: in

text: the

text: compensated

text: organic

text: metal

text: α-’pseudo-κ’-(ET)

text: H

text: O[Fe(C

text: O

text: (C

text: H

text: Br

text: Alain

label: author

text: Jean-Yves

text: Vladimir

text: N.

text: David

text: Tatyana

text: G.

text: Eduard

text: B.

text: and

text: Enric

text: Laboratoire

text: National

text: des

text: Champs

text: Magn´etiques

text: Intenses

text: (UPR

text: CNRS,

text: INSA,

text: UJF,

text: UPS)

text: avenue

text: de

text: Rangueil,

text: Toulouse,

text: France.

text: Institut

text: Jean

text: Lamour,

text: D´epartement

text: de

text: Physique

text: de

text: la

text: Mati`ere

text: et

text: des

text: Mat´eriaux,

text: CNRS-UMR

text: Vandoeuvre-les-Nancy,

text: France.

text: Institucio´

text: Catalana

text: de

text: Recerca

text: i

text: Estudis

text: Avanc¸ats

text: (ICREA),

text: Barcelona,

text: Spain.

text: Institut

text: de

text: Ci`encia

text: de

text: Materials

text: de

text: Barcelona,

text: CSIC,

text: Campus

text: de

text: la

text: UAB,

text: Bellaterra,

text: Spain.

text: Institute

text: of

text: Problems

text: of

text: Chemical

text: Physics,

text: Russian

text: Academy

text: of

text: Sciences,

text: Chernogolovka,

text: MD,

text: Russia

text: Received:

text: date

text: Revised

text: version:

text: date

text: Abstract.

text: Field-,

text: temperature-

text: and

text: angle-dependent

text: Fourier

text: amplitude

text: of

text: de

text: Haas-van

text: Alphen

text: (dHvA)

text: oscillations

text: are

text: calculated

text: for

text: compensated

text: two-dimensional

text: metals

text: with

text: textbook

text: Fermi

text: surface

text: (FS)

text: composed

text: of

text: one

text: hole

text: and

text: two

text: electron

text: orbits

text: connected

text: by

text: magnetic

text: breakdown.

text: It

text: is

text: demonstrated

text: that,

text: taking

text: into

text: account

text: the

text: opposite

text: sign

text: of

text: electron

text: and

text: hole

text: orbits,

text: a

text: given

text: Fourier

text: component

text: involves

text: combination

text: of

text: several

text: orbits,

text: the

text: contribution

text: of

text: which

text: must

text: be

text: included

text: in

text: the

text: calculations.

text: Such

text: FS

text: is

text: observed

text: in

text: the

text: strongly

text: organic

text: metal

text: α-’pseudo-κ’-(ET)

text: H

text: O[Fe(C

text: O

text: (C

text: H

text: Br

text: dHvA

text: oscilla-

text: tions

text: of

text: which

text: have

text: been

text: studied

text: up

text: to

text: T

text: for

text: various

text: directions

text: of

text: the

text: magnetic

text: ﬁeld

text: with

text: respect

text: to

text: the

text: conducting

text: plane.

text: Calculations

text: are

text: in

text: good

text: quantitative

text: agreement

text: with

text: the

text: data.

text: PACS.

text: Fermi

text: surface:

text: calculations

text: and

text: measurements;

text: eﬀective

text: mass,

text: g

text: factor

text: Poly-

text: mers

text: and

text: organic

text: compounds

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 F. Mottez Figure 10. Schematic explanation of the parallel electric ﬁeld E parallel to the average X magnetic ﬁeld B(cid:126) . 0 the ion distribution function has a wave number m = 4 for initial Alfv´en wave numbers m = m = 2. This is consistent with the multiplication by two of the initial wave vector 1 2 seen, for instance, in Eq. (3.4). This is not compatible with the slightly more complex wave number ratios involved with parametric instabilities. Therefore, we are tempted to conclude that ion acceleration is directly related to APAWI, and not to a parametric instability. Therefore we can conclude that APAWI does not involve ponderomotive force, inertial or kinetic Alfv´en waves, resonant acceleration or propagation eﬀects on a non-uniform plasma, or parametric instabilities. Of course, APAWI can act simultaneously with one of these and the consequences of APAWI can be similar to those of other acceleration mechanisms. For instance, like ponderomotive forces, APAWI causes deep plasma density depletions; like the parametric instabilities, it can cause ion beams. In conclusion of this paper, acceleration by parallel Alfv´en waves interaction (APAWI) is a new method of accelerating plasma particles with Alfv´en waves. The existence of APAWI does not rule out the relevance of other Alfv´enic processes, but it enlarges the range of plasma parameters and waves polarisations that make possible plasma accel- eration by Alfv´en waves. More speciﬁcally, it is the only acceleration process known up to now that involves only parallel Alfv´en waves and an initially homogeneous plasma. APAWI may be relevant in a very large number of space plasma contexts, including planetary magnetospheres and magnetic loops in the Sun corona. It is a very eﬃcient way of creating plasma cavities such as those observed in the Earth auroral zone. Appendix A. Parallel electric ﬁeld We deﬁne the two angles θ and θ by the relations Y Z B B Y = tan θ and Z = tan θ . (A 1) B Y B Z 0 0 Because we consider that the MHD waves amplitude is low relative to the ambient magnetic ﬁeld B , we consider that these angles are small. 0 The ideal Ohm’s law, used for MHD waves, implies that the electric ﬁeld and the magnetic ﬁeld are orthogonal. This holds when the Ohm’s law includes the Hall eﬀect, ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F.

text: Mottez

text: Figure

text: Schematic

label: caption

text: explanation

text: of

text: the

text: parallel

text: electric

text: ﬁeld

text: E

text: parallel

text: to

text: the

text: average

text: X

text: magnetic

text: ﬁeld

text: the

text: ion

text: distribution

text: function

text: has

text: a

text: wave

text: number

text: m

text: for

text: initial

text: Alfv´en

text: wave

text: numbers

text: m

text: m

text: This

text: is

text: consistent

text: with

text: the

text: multiplication

text: by

text: two

text: of

text: the

text: initial

text: wave

text: vector

text: seen,

text: for

text: instance,

text: in

text: Eq.

text: This

text: is

text: not

text: compatible

text: with

text: the

text: slightly

text: more

text: complex

text: wave

text: number

text: ratios

text: involved

text: with

text: parametric

text: instabilities.

text: Therefore,

text: we

text: are

text: tempted

text: to

text: conclude

text: that

text: ion

text: acceleration

text: is

text: directly

text: related

text: to

text: APAWI,

text: and

text: not

text: to

text: a

text: parametric

text: instability.

text: Therefore

text: we

text: can

text: conclude

text: that

text: APAWI

text: does

text: not

text: involve

text: ponderomotive

text: force,

text: inertial

text: or

text: kinetic

text: Alfv´en

text: waves,

text: resonant

text: acceleration

text: or

text: propagation

text: eﬀects

text: on

text: a

text: non-uniform

text: plasma,

text: or

text: parametric

text: instabilities.

text: Of

text: course,

text: APAWI

text: can

text: act

text: simultaneously

text: with

text: one

text: of

text: these

text: and

text: the

text: consequences

text: of

text: APAWI

text: can

text: be

text: similar

text: to

text: those

text: of

text: other

text: acceleration

text: mechanisms.

text: For

text: instance,

text: like

text: ponderomotive

text: forces,

text: APAWI

text: causes

text: deep

text: plasma

text: density

text: depletions;

text: like

text: the

text: parametric

text: instabilities,

text: it

text: can

text: cause

text: ion

text: beams.

text: In

text: conclusion

text: of

text: this

text: paper,

text: acceleration

text: by

text: parallel

text: Alfv´en

text: waves

text: interaction

text: (APAWI)

text: is

text: a

text: new

text: method

text: of

text: accelerating

text: plasma

text: particles

text: with

text: Alfv´en

text: waves.

text: The

text: existence

text: of

text: APAWI

text: does

text: not

text: rule

text: out

text: the

text: relevance

text: of

text: other

text: Alfv´enic

text: processes,

text: but

text: it

text: enlarges

text: the

text: range

text: of

text: plasma

text: parameters

text: and

text: waves

text: polarisations

text: that

text: make

text: possible

text: plasma

text: accel-

text: eration

text: by

text: Alfv´en

text: waves.

text: More

text: speciﬁcally,

text: it

text: is

text: the

text: only

text: acceleration

text: process

text: known

text: up

text: to

text: now

text: that

text: involves

text: only

text: parallel

text: Alfv´en

text: waves

text: and

text: an

text: initially

text: homogeneous

text: plasma.

text: APAWI

text: may

text: be

text: relevant

text: in

text: a

text: very

text: large

text: number

text: of

text: space

text: plasma

text: contexts,

text: including

text: planetary

text: magnetospheres

text: and

text: magnetic

text: loops

text: in

text: the

text: Sun

text: corona.

text: It

text: is

text: a

text: very

text: eﬃcient

text: way

text: of

text: creating

text: plasma

text: cavities

text: such

text: as

text: those

text: observed

text: in

text: the

text: Earth

text: auroral

text: zone.

text: Appendix

text: A.

text: Parallel

text: electric

text: ﬁeld

text: We

text: deﬁne

text: the

text: two

text: angles

text: θ

text: and

text: θ

text: by

text: the

text: relations

text: Y

text: Z

text: B

text: B

text: Y

text: tan

text: θ

text: and

text: Z

text: tan

text: θ

text: (A

text: B

text: Y

text: B

text: Z

text: Because

text: we

text: consider

text: that

text: the

text: MHD

text: waves

text: amplitude

text: is

text: low

text: relative

text: to

text: the

text: ambient

text: magnetic

text: ﬁeld

text: B

text: we

text: consider

text: that

text: these

text: angles

text: are

text: small.

text: The

text: ideal

text: Ohm’s

text: law,

text: used

text: for

text: MHD

text: waves,

text: implies

text: that

text: the

text: electric

text: ﬁeld

text: and

text: the

text: magnetic

text: ﬁeld

text: are

text: orthogonal.

text: This

text: holds

text: when

text: the

text: Ohm’s

text: law

text: includes

text: the

text: Hall

text: eﬀect,

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Acceleration by parallel Alfv´en waves 3 (e(cid:126) , e(cid:126) , e(cid:126) ) where e(cid:126) is parallel to B(cid:126) . The unit vectors e(cid:126) , e(cid:126) are set arbitrarily, as X Y Z X 0 Y Z long as they contribute with e(cid:126) to form an orthonormal basis. In the approximation of X a uniform magnetic ﬁeld B(cid:126) applied in the computation, this coordinate system is time 0 invariant and uniform. We can attach a local frame of reference to the local magnetic ﬁeld B(cid:126)(X, t). The e(cid:126) axis of this local frame of reference of coordinates (x, y, z) is parallel x to the local magnetic ﬁeld. For convenience, the basis (e(cid:126) , e(cid:126) , e(cid:126) ) can be deﬁned as the x y z result of the rotation R of axis e(cid:126) and angle θ of the set (e(cid:126) , e(cid:126) , e(cid:126) ), followed by a Z Z − Y X Y Z rotation R of axis e(cid:126) = R (e(cid:126) ) and angle θ . Y 1 Y 1 Z y Z This paper is based on a series of numerical simulations, whose naming convention is AWC (for Alfv´en wave crossing) followed by a number, or a number and a letter. These simulation share most of their characteristics with AWC009, that is therefore used as a reference simulation. In AWC009, two wave packets propagate in a uniform plasma. There are 1638400 particles of each species in the simulation; it corresponds to 100 particles per cell. There are 65536 time steps, deﬁned by ∆t = 0.1, corresponding to a time lapse t = 6553.6. × max The size of the whole simulation domain is 4096∆x 4∆y. where ∆x = ∆y = 0.10 is the size of the grid cells. The electron thermal velocity is v = 0.1; the ion and electron te temperatures are the same. The ion to electron mass ratio is reduced to m /m = 400.. i e Each wave packet is initialized as the sum of 8 sinusoidal waves with their maximum at X = 300 for the ﬁrst wave packet, and X = 100 for the second wave packet. The equations deﬁning the waves polarisations are given in the appendix in (Mottez 2008). The ﬁrst wave packet propagates downward (in the direction of decreasing values of X). The sinusoidal waves that compose the wave packet have the same amplitude, given × by the wave magnetic ﬁeld δB = 0.05 B . The waves have a right handed circular 0 polarization. Their wavelengths are λ = λ /m where λ = 409.6 and m varies from 1 to 0 0 8. The phase velocities vary from 0.109 for the shortest wave to 0.046 for the longest. (The Alfven velocity is V = 0.04.) The second wave packet propagates upward. Apart A from its direction of propagation and location, its characteristics are the same. The B component of the magnetic ﬁeld is a proxy of the wave packets propagation. z It is plotted as a function of X and time on Fig. 1.We can see that the two packets cross ∼ ∼ ∼ ∼ each other at time t 104 and X 200, and latter a times t 2.104, t 3.104 etc. For the other simulations, a few parameters are varied. When a physical parameter is varied (for instance, a wave amplitude), the number is diﬀerent. The parameters of these simulations are summarized in Table 1. We can notice that these simulations all involve the right-hand circularly polarized waves propagating in opposite directions, as in AWC009. We made the deliberate choice to concentrate on this subset of parameters because, as shown in (Mottez 2012a), it represent a favourable case for parallel to B 0 electric ﬁeld, and because the RH waves are less dispersive than the LH ones. This make the physical interpretation of the simulation easier. Concerning dimensions, a 2D simulation was presented in (Mottez 2012a) with the same initial conﬁguration as the 1D simulations. The plasma behaved exactly as in 1D simulations. This is why 2D simulations are not repeated in the present study. The simulation AWC009 is based on a magnetic ﬁeld B = 0.8 that is ﬁve times 0 smaller than those in (Mottez 2012a). This is a deliberate choice in order to check the robustness of APAWI relative to the ambient magnetic ﬁeld amplitude. Nevertheless, we can notice that the plasma β (see Table 1) is typical of a highly magnetized plasma − (β << 1). In the simulations with B = 4, β < m /m = 10 2 is typical of the largest 0 e i part of acceleration region of the auroral zone. To illustrate more precisely the physical context, let us notice that the ”classical” theories of acceleration by oblique Alfv´en waves (therefore not APAWI) involve inertial AW in this range of altitudes (Thompson & Lysak

text: Acceleration

text: by

text: parallel

text: Alfv´en

text: waves

text: where

text: is

text: parallel

text: to

text: The

text: unit

text: vectors

text: are

text: set

text: arbitrarily,

text: as

text: X

text: Y

text: Z

text: X

text: Y

text: Z

text: long

text: as

text: they

text: contribute

text: with

text: to

text: form

text: an

text: orthonormal

text: basis.

text: In

text: the

text: approximation

text: of

text: X

text: a

text: uniform

text: magnetic

text: ﬁeld

text: applied

text: in

text: the

text: computation,

text: this

text: coordinate

text: system

text: is

text: time

text: invariant

text: and

text: uniform.

text: We

text: can

text: attach

text: a

text: local

text: frame

text: of

text: reference

text: to

text: the

text: local

text: magnetic

text: ﬁeld

text: t).

text: The

text: axis

text: of

text: this

text: local

text: frame

text: of

text: reference

text: of

text: coordinates

text: (x,

text: y,

text: z)

text: is

text: parallel

text: x

text: to

text: the

text: local

text: magnetic

text: ﬁeld.

text: For

text: convenience,

text: the

text: basis

text: can

text: be

text: deﬁned

text: as

text: the

text: x

text: y

text: z

text: result

text: of

text: the

text: rotation

text: R

text: of

text: axis

text: and

text: angle

text: θ

text: of

text: the

text: set

text: followed

text: by

text: a

text: Z

text: Z

text: Y

text: X

text: Y

text: Z

text: rotation

text: R

text: of

text: axis

text: R

text: and

text: angle

text: θ

text: Y

text: Y

text: Z

text: y

text: Z

text: This

text: paper

text: is

text: based

text: on

text: a

text: series

text: of

text: numerical

text: simulations,

text: whose

text: naming

text: convention

text: is

text: AWC

text: (for

text: Alfv´en

text: wave

text: crossing)

text: followed

text: by

text: a

text: number,

text: or

text: a

text: number

text: and

text: a

text: letter.

text: These

text: simulation

text: share

text: most

text: of

text: their

text: characteristics

text: with

text: that

text: is

text: therefore

text: used

text: as

text: a

text: reference

text: simulation.

text: In

text: two

text: wave

text: packets

text: propagate

text: in

text: a

text: uniform

text: plasma.

text: There

text: are

text: particles

text: of

text: each

text: species

text: in

text: the

text: simulation;

text: it

text: corresponds

text: to

text: particles

text: per

text: cell.

text: There

text: are

text: time

text: steps,

text: deﬁned

text: by

text: ∆t

text: corresponding

text: to

text: a

text: time

text: lapse

text: t

text: max

text: The

text: size

text: of

text: the

text: whole

text: simulation

text: domain

text: is

text: where

text: ∆x

text: ∆y

text: is

text: the

text: size

text: of

text: the

text: grid

text: cells.

text: The

text: electron

text: thermal

text: velocity

text: is

text: v

text: the

text: ion

text: and

text: electron

text: te

text: temperatures

text: are

text: the

text: same.

text: The

text: ion

text: to

text: electron

text: mass

text: ratio

text: is

text: reduced

text: to

text: m

text: /m

text: i

text: e

text: Each

text: wave

text: packet

text: is

text: initialized

text: as

text: the

text: sum

text: of

text: sinusoidal

text: waves

text: with

text: their

text: maximum

text: at

text: X

text: for

text: the

text: ﬁrst

text: wave

text: packet,

text: and

text: X

text: for

text: the

text: second

text: wave

text: packet.

text: The

text: equations

text: deﬁning

text: the

text: waves

text: polarisations

text: are

text: given

text: in

text: the

text: appendix

text: in

text: (Mottez

text: The

text: ﬁrst

text: wave

text: packet

text: propagates

text: downward

text: (in

text: the

text: direction

text: of

text: decreasing

text: values

text: of

text: X).

text: The

text: sinusoidal

text: waves

text: that

text: compose

text: the

text: wave

text: packet

text: have

text: the

text: same

text: amplitude,

text: given

text: by

text: the

text: wave

text: magnetic

text: ﬁeld

text: δB

text: B

text: The

text: waves

text: have

text: a

text: right

text: handed

text: circular

text: polarization.

text: Their

text: wavelengths

text: are

text: λ

text: λ

text: /m

text: where

text: λ

text: and

text: m

text: varies

text: from

text: to

text: The

text: phase

text: velocities

text: vary

text: from

text: for

text: the

text: shortest

text: wave

text: to

text: for

text: the

text: longest.

text: (The

text: Alfven

text: velocity

text: is

text: V

text: The

text: second

text: wave

text: packet

text: propagates

text: upward.

text: Apart

text: A

text: from

text: its

text: direction

text: of

text: propagation

text: and

text: location,

text: its

text: characteristics

text: are

text: the

text: same.

text: The

text: B

text: component

text: of

text: the

text: magnetic

text: ﬁeld

text: is

text: a

text: proxy

text: of

text: the

text: wave

text: packets

text: propagation.

text: z

text: It

text: is

text: plotted

text: as

text: a

text: function

text: of

text: X

text: and

text: time

text: on

text: Fig.

text: can

text: see

text: that

text: the

text: two

text: packets

text: cross

text: each

text: other

text: at

text: time

text: t

text: and

text: X

text: and

text: latter

text: a

text: times

text: t

text: t

text: etc.

text: For

text: the

text: other

text: simulations,

text: a

text: few

text: parameters

text: are

text: varied.

text: When

text: a

text: physical

text: parameter

text: is

text: varied

text: (for

text: instance,

text: a

text: wave

text: amplitude),

text: the

text: number

text: is

text: diﬀerent.

text: The

text: parameters

text: of

text: these

text: simulations

text: are

text: summarized

text: in

text: Table

text: We

text: can

text: notice

text: that

text: these

text: simulations

text: all

text: involve

text: the

text: right-hand

text: circularly

text: polarized

text: waves

text: propagating

text: in

text: opposite

text: directions,

text: as

text: in

text: We

text: made

text: the

text: deliberate

text: choice

text: to

text: concentrate

text: on

text: this

text: subset

text: of

text: parameters

text: because,

text: as

text: shown

text: in

text: (Mottez

text: it

text: represent

text: a

text: favourable

text: case

text: for

text: parallel

text: to

text: B

text: electric

text: ﬁeld,

text: and

text: because

text: the

text: RH

text: waves

text: are

text: less

text: dispersive

text: than

text: the

text: LH

text: ones.

text: This

text: make

text: the

text: physical

text: interpretation

text: of

text: the

text: simulation

text: easier.

text: Concerning

text: dimensions,

text: a

text: simulation

text: was

text: presented

text: in

text: (Mottez

text: with

text: the

text: same

text: initial

text: conﬁguration

text: as

text: the

text: simulations.

text: The

text: plasma

text: behaved

text: exactly

text: as

text: in

text: simulations.

text: This

text: is

text: why

text: simulations

text: are

text: not

text: repeated

text: in

text: the

text: present

text: study.

text: The

text: simulation

text: is

text: based

text: on

text: a

text: magnetic

text: ﬁeld

text: B

text: that

text: is

text: ﬁve

text: times

text: smaller

text: than

text: those

text: in

text: (Mottez

text: This

text: is

text: a

text: deliberate

text: choice

text: in

text: order

text: to

text: check

text: the

text: robustness

text: of

text: APAWI

text: relative

text: to

text: the

text: ambient

text: magnetic

text: ﬁeld

text: amplitude.

text: Nevertheless,

text: we

text: can

text: notice

text: that

text: the

text: plasma

text: β

text: (see

text: Table

text: is

text: typical

text: of

text: a

text: highly

text: magnetized

text: plasma

text: (β

text: In

text: the

text: simulations

text: with

text: B

text: β

text: m

text: /m

text: is

text: typical

text: of

text: the

text: largest

text: e

text: i

text: part

text: of

text: acceleration

text: region

text: of

text: the

text: auroral

text: zone.

text: To

text: illustrate

text: more

text: precisely

text: the

text: physical

text: context,

text: let

text: us

text: notice

text: that

text: the

text: ”classical”

text: theories

text: of

text: acceleration

text: by

text: oblique

text: Alfv´en

text: waves

text: (therefore

text: not

text: APAWI)

text: involve

text: inertial

text: AW

text: in

text: this

text: range

text: of

text: altitudes

text: (Thompson

text: Lysak


text: – 15 – − F = F ν αδ(3+α) , (9) 0 − − (see Dermer & Menon 2009) where δ = [Γ (1 β cos Φ] 1 is the jet’s Doppler factor, j − − β = (1 Γ 2)1/2 its global velocity in units of speed of light, Φ is the viewing angle, and α is the j spectral index in the optical bands. The observed degree of polarization p depends on the rest–frame angle between the line of sight and the compression axis Ψ, the spectral index α, and the shock compression factor η, which is the ratio of densities of a plasma of relativistic electrons from the shocked to the unshocked region η = η /η (Hughes & Miller 1991): shock unshock − − ≈ α + 1 (1 η 2) sin2 Ψ p − − − , (10) α + 5/3 2 (1 η 2) sin2 Ψ and      (cid:113)  − sin Φ Ψ = tan 1 . (11) − − − Γ (cos Φ 1 Γ 2) j j Following Acciari et al. (2009), we assumed a bulk Lorentz factor Γ = 20 for W Comae. We j also used the value of α = 0.87, given by Tosti et al. (1998). From equation (9), we can estimate the Doppler factor as a function of time. The value of F is determined by F = F να/δ(3+α), where F is the maximum observed ﬂux and δ is 0 0 max D max D ≈ obtained from Φ , which is calculated from equations (10) and (11) for p pmax this being the 0 var maximum value of the polarization degree of the variable component (see Table 4). From Hughes ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: YQUCVP+txsy

text: F

font: HNPYWK+rtxr

text: F

text: ν

font: OJSRFV+rtxmi

text: (see

text: Dermer

text: Menon

text: where

text: δ

text: [Γ

text: β

text: cos

text: Φ]

text: is

text: the

text: jet’s

text: Doppler

text: factor,

text: j

text: β

text: Γ

text: its

text: global

text: velocity

text: in

text: units

text: of

text: speed

text: of

text: light,

text: Φ

text: is

text: the

text: viewing

text: angle,

text: and

text: α

text: is

text: the

text: j

text: spectral

text: index

text: in

text: the

text: optical

text: bands.

text: The

text: observed

text: degree

text: of

text: polarization

text: p

text: depends

text: on

text: the

text: rest–frame

text: angle

text: between

text: the

text: line

text: of

text: sight

text: and

text: the

text: compression

text: axis

text: Ψ,

text: the

text: spectral

text: index

text: α,

text: and

text: the

text: shock

text: compression

text: factor

text: η,

text: which

text: is

text: the

text: ratio

text: of

text: densities

text: of

text: a

text: plasma

text: of

text: relativistic

text: electrons

text: from

text: the

text: shocked

text: to

text: the

text: unshocked

text: region

text: η

text: η

text: /η

text: (Hughes

text: Miller

text: shock

text: unshock

text: α

text: η

text: Ψ

text: p

text: α

text: η

text: Ψ

text: and

text: sin

text: Φ

text: Ψ

text: tan

text: Γ

text: (cos

text: Φ

text: Γ

text: j

text: j

text: Following

text: Acciari

text: et

text: al.

text: we

text: assumed

text: a

text: bulk

text: Lorentz

text: factor

text: Γ

text: for

text: W

text: Comae.

text: We

text: j

text: also

text: used

text: the

text: value

text: of

text: α

text: given

text: by

text: Tosti

text: et

text: al.

text: From

text: equation

text: we

text: can

text: estimate

text: the

text: Doppler

text: factor

text: as

text: a

text: function

text: of

text: time.

text: The

text: value

text: of

text: F

text: is

text: determined

text: by

text: F

text: F

text: where

text: F

text: is

text: the

text: maximum

text: observed

text: ﬂux

text: and

text: δ

text: is

text: max

text: D

text: max

text: D

text: obtained

text: from

text: Φ

text: which

text: is

text: calculated

text: from

text: equations

text: and

text: for

text: p

text: pmax

text: this

text: being

text: the

text: var

text: maximum

text: value

text: of

text: the

text: polarization

text: degree

text: of

text: the

text: variable

text: component

text: (see

text: Table

text: From

text: Hughes

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: – 18 – ∼ ◦ the polarization angle tends to a preferential orientation of 65 . The large rotation associated with the ﬂare in γ-rays suggests that both optical / γ-ray emissions could be produced in the same jet’s region. From the analysis of the Stokes parameters, we infer the presence of two optically thin synchrotron components with diﬀerent polarimetric characteristics: one is a variable component ≈ and the other one is stable with a constant degree of polarization of p 11%, and a constant cons ≈ ◦ position angle of θ 65 . Assuming that the 2008 June optical ﬂare has originated in a cons transversal shock propagating down a twisted jet, and that the source is a spherical blob of radius r , moving with a Lorentz’s factor of Γ = 20, from our polarimetric data we estimated a Doppler b ∼ ∼ ◦ factor of δ 27 when the ﬂux was maximum, and a visual angle of the jet Φ 2 .0. We also D ∼ obtained a magnetic ﬁeld intensity B 0.12 G. Finally, an upper limit for the size of the emission ≤ × region of r 2 1017 cm was estimated. b The variability timescales displayed by W Comae show two main characteristics: (1) There are two components in the light-curve, one contributing to the long-term brightening with timescales going from 2 to 4 months, and the other that contributes to the short timescale ∼ variations ( 3 days). This result is in agreement with Tosti et al. (2002). (2) The Doppler factor ≈ − changes (δ(t) 16 27) could be due to changes in the viewing angle of the jet, implying ﬂux ∼ variations lasting 0.9 yr. Based on the the anticorrelation found between the polarization percentage and the ﬂux, we propose that the observed long-term ﬂux behavior can be explained with a spiral jet and a transversal shock-wave models. This anticorrelation depends on the Doppler factor time-variations δ(t) for a range of values of the viewing angle θ(t). ∼ ◦ From our observations, we found that the EPVA in the optical has a value 110 in 2008 June 7, a day before the gamma-ray ﬂare. This value is identical to the projected angle of the radio jet found by Gabuzda et al. (1994) and Massaro et al. (2001). Later, after the strong gamma-ray

text: the

text: polarization

text: angle

text: tends

text: to

text: a

text: preferential

text: orientation

text: of

text: The

text: large

text: rotation

text: associated

text: with

text: the

text: ﬂare

text: in

text: γ-rays

text: suggests

text: that

text: both

text: optical

text: γ-ray

text: emissions

text: could

text: be

text: produced

text: in

text: the

text: same

text: jet’s

text: region.

text: From

text: the

text: analysis

text: of

text: the

text: Stokes

text: parameters,

text: we

text: infer

text: the

text: presence

text: of

text: two

text: optically

text: thin

text: synchrotron

text: components

text: with

text: diﬀerent

text: polarimetric

text: characteristics:

text: one

text: is

text: a

text: variable

text: component

text: and

text: the

text: other

text: one

text: is

text: stable

text: with

text: a

text: constant

text: degree

text: of

text: polarization

text: of

text: p

text: and

text: a

text: constant

text: cons

text: position

text: angle

text: of

text: θ

text: Assuming

text: that

text: the

text: June

text: optical

text: ﬂare

text: has

text: originated

text: in

text: a

text: cons

text: transversal

text: shock

text: propagating

text: down

text: a

text: twisted

text: jet,

text: and

text: that

text: the

text: source

text: is

text: a

text: spherical

text: blob

text: of

text: radius

text: r

text: moving

text: with

text: a

text: Lorentz’s

text: factor

text: of

text: Γ

text: from

text: our

text: polarimetric

text: data

text: we

text: estimated

text: a

text: Doppler

text: b

text: factor

text: of

text: δ

text: when

text: the

text: ﬂux

text: was

text: maximum,

text: and

text: a

text: visual

text: angle

text: of

text: the

text: jet

text: Φ

text: We

text: also

text: D

text: obtained

text: a

text: magnetic

text: ﬁeld

text: intensity

text: B

text: G.

text: Finally,

text: an

text: upper

text: limit

text: for

text: the

text: size

text: of

text: the

text: emission

text: region

text: of

text: r

text: cm

text: was

text: estimated.

text: b

text: The

text: variability

text: timescales

text: displayed

text: by

text: W

text: Comae

text: show

text: two

text: main

text: characteristics:

text: There

text: are

text: two

text: components

text: in

text: the

text: light-curve,

text: one

text: contributing

text: to

text: the

text: long-term

text: brightening

text: with

text: timescales

text: going

text: from

text: to

text: months,

text: and

text: the

text: other

text: that

text: contributes

text: to

text: the

text: short

text: timescale

text: variations

text: days).

text: This

text: result

text: is

text: in

text: agreement

text: with

text: Tosti

text: et

text: al.

text: The

text: Doppler

text: factor

text: changes

text: (δ(t)

text: could

text: be

text: due

text: to

text: changes

text: in

text: the

text: viewing

text: angle

text: of

text: the

text: jet,

text: implying

text: ﬂux

text: variations

text: lasting

text: yr.

text: Based

text: on

text: the

text: the

text: anticorrelation

text: found

text: between

text: the

text: polarization

text: percentage

text: and

text: the

text: ﬂux,

text: we

text: propose

text: that

text: the

text: observed

text: long-term

text: ﬂux

text: behavior

text: can

text: be

text: explained

text: with

text: a

text: spiral

text: jet

text: and

text: a

text: transversal

text: shock-wave

text: models.

text: This

text: anticorrelation

text: depends

text: on

text: the

text: Doppler

text: factor

text: time-variations

text: δ(t)

text: for

text: a

text: range

text: of

text: values

text: of

text: the

text: viewing

text: angle

text: θ(t).

text: From

text: our

text: observations,

text: we

text: found

text: that

text: the

text: EPVA

text: in

text: the

text: optical

text: has

text: a

text: value

text: in

text: June

text: a

text: day

text: before

text: the

text: gamma-ray

text: ﬂare.

text: This

text: value

text: is

text: identical

text: to

text: the

text: projected

text: angle

text: of

text: the

text: radio

text: jet

text: found

text: by

text: Gabuzda

text: et

text: al.

text: and

text: Massaro

text: et

text: al.

text: Later,

text: after

text: the

text: strong

text: gamma-ray


text: – 22 – Nilsson, K., Pasanen, M., Takalo, L. O., et al. 2007, A&A, 475, 199 Nilsson, K., Pursimo, T., Heidt, J., et al. 2003, A&A, 400, 95 Padovani, P., & Giommi, P. 1995, ApJ, 444, 567 Sasada, M., Uemura, M., Fukazawa, Y., et al. 2011, PASJ, 63, 489 Schmidt, G. D., Elston, R., & Lupie, O. L. 1992, AJ, 104, 1563 Sorcia, M., Ben´ıtez, E., Hiriart, D., et al. 2013, ApJS, 206, 11 Tagliaferri, G., Ghisellini, G., Giommi, P., et al. 2000, A&A, 354, 431 Tosti, G., Fiorucci, M., Luciani, M., et al. 1998, A&AS, 130, 109 Tosti, G., Massaro, E., Nesci, R., et al. 2002, A&A, 395, 11 Zhang, X., Zheng, Y. G., Zhang, L., et al. 2008, PASJ, 60, 145 This manuscript was prepared with the AAS LATEX macros v5.2. ##LTLine##

text: Nilsson,

label: reference

text: K.,

text: Pasanen,

text: M.,

text: Takalo,

text: L.

text: O.,

text: et

text: al.

text: A&A,

text: Nilsson,

text: K.,

text: Pursimo,

text: T.,

text: Heidt,

text: J.,

text: et

text: al.

text: A&A,

text: Padovani,

text: P.,

text: Giommi,

text: P.

text: ApJ,

text: Sasada,

text: M.,

text: Uemura,

text: M.,

text: Fukazawa,

text: Y.,

text: et

text: al.

text: PASJ,

text: Schmidt,

text: G.

text: D.,

text: Elston,

text: R.,

text: Lupie,

text: O.

text: L.

text: AJ,

text: Sorcia,

text: M.,

text: Ben´ıtez,

text: E.,

text: Hiriart,

text: D.,

text: et

text: al.

text: ApJS,

text: Tagliaferri,

text: G.,

text: Ghisellini,

text: G.,

text: Giommi,

text: P.,

text: et

text: al.

text: A&A,

text: Tosti,

text: G.,

text: Fiorucci,

text: M.,

text: Luciani,

text: M.,

text: et

text: al.

text: A&AS,

text: Tosti,

text: G.,

text: Massaro,

text: E.,

text: Nesci,

text: R.,

text: et

text: al.

text: A&A,

text: Zhang,

text: X.,

text: Zheng,

text: Y.

text: G.,

text: Zhang,

text: L.,

text: et

text: al.

text: PASJ,

text: This

text: manuscript

text: was

text: prepared

text: with

text: the

text: AAS

text: LATEX

text: macros

text: ##LTLine##


text: – 4 – Massaro et al. 1999). Also, it has shown rapid variations on scales of hours (Babadzhanyants & Belokon’ 2002). Tosti et al. (1998) observed the highest brightness value ever observed for W Comae since 1940, reaching a maximum of B = 14.2 mag in 1997 January. Later, Massaro et al. (1999) reported a very strong ﬂare of W Comae when the object reached a historical maximum of ∼ R 12.2 mag in 1998 April 23. Optical polarization of W Comae was also reported in Massaro et al. (1999). Their multi-band optical observations were done just before and during its brightest phase (1998 April 17–25). During the brightest state, the polarization was low in the UBV ﬁlters ∼ ( 2% to 4%) with less than 0.4% in the R and I ﬁlters. c c The γ–ray emission of W Comae has been detected by the Energetic Gamma Ray Experiment Telescope (EGRET) on board of the Compton Gamma Ray Observatory (CGRO) in the 100 MeV to 10 GeV band (Hartman et al. 1999). BeppoSAX data analysis of W Comae given by Tagliaferri et al. (2000) demonstrates that this source is an IBL source. This blazar was considered as a very interesting target for the very high energy (VHE) observatories due to the possibility of being a γ-ray source that could be detected by Cerenkov telescopes such as the High Energy Stereoscopic System (HESS), the MAGIC telescopes, and the Very Energetic Radiation Imaging Telescope Array System (VERITAS) (see, e.g. Bo¨ttcher et al. 2002). This prediction was conﬁrmed later, when W Comae was discovered to be a γ–ray emitter at VHE by VERITAS in 2008 March 15 (see Acciari et al. 2008). Thus, W Comae is the ﬁrst IBL detected at VHE. A subsequent multiwavelength campaign on this object was coordinated during a major γ-ray ﬂare in 2008 June (Acciari et al. 2009). A very high γ-ray signal was detected by VERITAS in 2008 June 8 that was brighter, by a factor of three, than the previous emission detected in 2008 March. In this paper we report the results of the photopolarimetric monitoring of the TeV–blazar W Comae carried out from 2008 February to 2013 May. Our main goal is to establish the long–term optical variability properties of the polarized emission in the R–band. The variability of the Stokes parameters obtained from our observations is analyzed in terms of a two–component

text: Massaro

text: et

text: al.

text: Also,

text: it

text: has

text: shown

text: rapid

text: variations

text: on

text: scales

text: of

text: hours

text: (Babadzhanyants

text: Belokon’

text: Tosti

text: et

text: al.

text: observed

text: the

text: highest

text: brightness

text: value

text: ever

text: observed

text: for

text: W

text: Comae

text: since

text: reaching

text: a

text: maximum

text: of

text: B

text: mag

text: in

text: January.

text: Later,

text: Massaro

text: et

text: al.

text: reported

text: a

text: very

text: strong

text: ﬂare

text: of

text: W

text: Comae

text: when

text: the

text: object

text: reached

text: a

text: historical

text: maximum

text: of

text: R

text: mag

text: in

text: April

text: Optical

text: polarization

text: of

text: W

text: Comae

text: was

text: also

text: reported

text: in

text: Massaro

text: et

text: al.

text: Their

text: multi-band

text: optical

text: observations

text: were

text: done

text: just

text: before

text: and

text: during

text: its

text: brightest

text: phase

text: April

text: During

text: the

text: brightest

text: state,

text: the

text: polarization

text: was

text: low

text: in

text: the

text: UBV

text: ﬁlters

text: to

text: with

text: less

text: than

text: in

text: the

text: R

text: and

text: I

text: ﬁlters.

text: c

text: c

text: The

text: γ–ray

text: emission

text: of

text: W

text: Comae

text: has

text: been

text: detected

text: by

text: the

text: Energetic

text: Gamma

text: Ray

text: Experiment

text: Telescope

text: (EGRET)

text: on

text: board

text: of

text: the

text: Compton

text: Gamma

text: Ray

text: Observatory

text: (CGRO)

text: in

text: the

text: MeV

text: to

text: GeV

text: band

text: (Hartman

text: et

text: al.

text: BeppoSAX

text: data

text: analysis

text: of

text: W

text: Comae

text: given

text: by

text: Tagliaferri

text: et

text: al.

text: demonstrates

text: that

text: this

text: source

text: is

text: an

text: IBL

text: source.

text: This

text: blazar

text: was

text: considered

text: as

text: a

text: very

text: interesting

text: target

text: for

text: the

text: very

text: high

text: energy

text: (VHE)

text: observatories

text: due

text: to

text: the

text: possibility

text: of

text: being

text: a

text: γ-ray

text: source

text: that

text: could

text: be

text: detected

text: by

text: Cerenkov

text: telescopes

text: such

text: as

text: the

text: High

text: Energy

text: Stereoscopic

text: System

text: (HESS),

text: the

text: MAGIC

text: telescopes,

text: and

text: the

text: Very

text: Energetic

text: Radiation

text: Imaging

text: Telescope

text: Array

text: System

text: (VERITAS)

text: (see,

text: e.g.

text: Bo¨ttcher

text: et

text: al.

text: This

text: prediction

text: was

text: conﬁrmed

text: later,

text: when

text: W

text: Comae

text: was

text: discovered

text: to

text: be

text: a

text: γ–ray

text: emitter

text: at

text: VHE

text: by

text: VERITAS

text: in

text: March

text: (see

text: Acciari

text: et

text: al.

text: Thus,

text: W

text: Comae

text: is

text: the

text: ﬁrst

text: IBL

text: detected

text: at

text: VHE.

text: A

text: subsequent

text: multiwavelength

text: campaign

text: on

text: this

text: object

text: was

text: coordinated

text: during

text: a

text: major

text: γ-ray

text: ﬂare

text: in

text: June

text: (Acciari

text: et

text: al.

text: A

text: very

text: high

text: γ-ray

text: signal

text: was

text: detected

text: by

text: VERITAS

text: in

text: June

text: that

text: was

text: brighter,

text: by

text: a

text: factor

text: of

text: three,

text: than

text: the

text: previous

text: emission

text: detected

text: in

text: March.

text: In

text: this

text: paper

text: we

text: report

text: the

text: results

text: of

text: the

text: photopolarimetric

text: monitoring

text: of

text: the

text: TeV–blazar

text: W

text: Comae

text: carried

text: out

text: from

text: February

text: to

text: May.

text: Our

text: main

text: goal

text: is

text: to

text: establish

text: the

text: long–term

text: optical

text: variability

text: properties

text: of

text: the

text: polarized

text: emission

text: in

text: the

text: R–band.

text: The

text: variability

text: of

text: the

text: Stokes

text: parameters

text: obtained

text: from

text: our

text: observations

text: is

text: analyzed

text: in

text: terms

text: of

text: a

text: two–component


text: 13 the system is given by Eq. (13). Here, we parameterize (cid:126)σ as (sin θ cos θ , sin θ sin θ , cos θ ) and we choose m(cid:126) 1 2 1 2 1 as m(cid:126) = (m sin φ cos φ , m sin φ sin φ , m cos φ ). There- 2 1 2 1 1 fore, the mea(cid:20)n(cid:82)ﬁeld equation reads (cid:21) (cid:82) (cid:126)σ exp(βJm(cid:126) .(cid:126)σ + β(cid:15)ησ ) sin θ dθ dθ z 2 2 1 m(cid:126) = Av . η exp(βJm(cid:126) .(cid:126)σ + β(cid:15)ησ ) sin θ dθ dθ z 2 2 1 (68) A. The pure system: (cid:15) = 0 Consider ﬁrst the case, when (cid:15) = 0. After some alge- bra, it can be shown that the three components of the mean ﬁeld equation (Eq. (68)) reduce to the single equa- tion − − 1 m + coth(Jβm) = 0. (69) Jβm Numerical simulations provide the following picture. The pluses in Fig. 11(a) show two real solutions of the Eq. (69). A nontrivial solution appears only when β is ≈ greater than a certain critical temperature β0,3 3 and c approaches unity at very low temperature. The system behaves uniformly in all possible directions of the 3D space, i.e., the solutions of Eq. (68) form a sphere of radius m for a given magnetization m(cid:126) . 0 0 To ﬁnd β0,3 analytically, we perform the Taylor series c expansion of Eq. (69) in m around m = 0. We obtain FIG. 10. (Color online.) Plot of the functions S(x, j) (top) and R(x, j) (bottom) with respect to x and j. Both of them are again negative for the entire range of x and j. These are in (−1 + Jβ )m − 1 (J3β3)m3 + o(m4) = 0. (70) agreement with the numerical results in Figs. 6(a) and 6(b). x 3 45 is measured in radians. All other quantities are dimensionless. The nontrivial solutions of Eq. (70) are given by √ As seen in Fig. 10 (bottom), R is always negative, showing that the length of the magnetization decreases ± 15 − − m0,3 = β 3/2(Jβ 3)1/2. (71) in the presence of disorder. Note that the behavior of the J3/2 length and phase obtained perturbatively, matches with what is shown schematically in Fig. 7. ≥ m0,3 vanishes if Jβ = 3 and is nonzero iﬀ Jβ 3, V. CLASSICAL HEISENBERG MODEL IN A which implies that the critical temperature is given by RANDOM FIELD 3 Till now we have explicitly considered only the situa- β0,3 = . (72) c J tion when the spins were two-dimensional. It is natural to ask analogous questions for three-dimensional spins with continuous symmetry. In this section we argue that for Note that in the case of the XY model, we also found the canonical system of this kind - the classical Heisen- a similar behavior of the magnetization near its critical berg model - the behavior is similar to that of the XY temperature. (cid:54) model. B. The system with disorder: (cid:15) = 0 We study the behavior of magnetization in the pres- ence of disorder in the lattice Heisenberg model where Breaking down m(cid:126) in Eq. (68) into its components along the spins are three-dimensional with continuous symme- the X, Y and Z axes, we have three diﬀerent equations. try. We assume that at all sites, random ﬁelds are di- It turns out that the equations along the X and Y axes, rected in the Z-direction. The mean-ﬁeld Hamiltonian of both of which are in a direction transverse to the applied ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: the

text: system

text: is

text: given

text: by

text: Eq.

text: Here,

text: we

text: parameterize

text: as

text: (sin

text: θ

text: cos

text: θ

text: sin

text: θ

text: sin

text: θ

text: cos

text: θ

text: and

text: we

text: choose

text: as

text: (m

text: sin

text: φ

text: cos

text: φ

text: m

text: sin

text: φ

text: sin

text: φ

text: m

text: cos

text: φ

text: There-

text: fore,

text: the

text: equation

text: reads

text: sin

text: θ

text: dθ

text: dθ

text: z

text: Av

text: η

text: sin

text: θ

text: dθ

text: dθ

text: z

text: A.

text: The

text: pure

text: system:

text: Consider

text: ﬁrst

text: the

text: case,

text: when

text: After

text: some

text: alge-

text: bra,

text: it

text: can

text: be

text: shown

text: that

text: the

text: three

text: components

text: of

text: the

text: mean

text: ﬁeld

text: equation

text: (Eq.

text: reduce

text: to

text: the

text: single

text: equa-

text: tion

text: m

text: coth(Jβm)

text: Jβm

text: Numerical

text: simulations

text: provide

text: the

text: following

text: picture.

text: The

text: pluses

text: in

text: Fig.

text: show

text: two

text: real

text: solutions

text: of

text: the

text: Eq.

text: A

text: nontrivial

text: solution

text: appears

text: only

text: when

text: β

text: is

text: greater

text: than

text: a

text: certain

text: critical

text: temperature

text: and

text: c

text: approaches

text: unity

text: at

text: very

text: low

text: temperature.

text: The

text: system

text: behaves

text: uniformly

text: in

text: all

text: possible

text: directions

text: of

text: the

text: space,

text: i.e.,

text: the

text: solutions

text: of

text: Eq.

text: form

text: a

text: sphere

text: of

text: radius

text: m

text: for

text: a

text: given

text: magnetization

text: To

text: ﬁnd

text: analytically,

text: we

text: perform

text: the

text: Taylor

text: series

text: c

text: expansion

text: of

text: Eq.

text: in

text: m

text: around

text: m

text: We

text: obtain

text: FIG.

text: (Color

text: online.)

text: Plot

text: of

text: the

text: functions

text: S(x,

text: j)

text: (top)

text: and

text: R(x,

text: j)

text: (bottom)

text: with

text: respect

text: to

text: x

text: and

text: j.

text: Both

text: of

text: them

text: are

text: again

text: negative

text: for

text: the

text: entire

text: range

text: of

text: x

text: and

text: j.

text: These

text: are

text: in

text: Jβ

text: )m

text: agreement

text: with

text: the

text: numerical

text: results

text: in

text: Figs.

text: and

text: x

text: is

text: measured

text: in

text: radians.

text: All

text: other

text: quantities

text: are

text: dimensionless.

text: The

text: nontrivial

text: solutions

text: of

text: Eq.

text: are

text: given

text: by

text: As

text: seen

text: in

text: Fig.

text: (bottom),

text: R

text: is

text: always

text: negative,

text: showing

text: that

text: the

text: length

text: of

text: the

text: magnetization

text: decreases

text: β

text: in

text: the

text: presence

text: of

text: disorder.

text: Note

text: that

text: the

text: behavior

text: of

text: the

text: length

text: and

text: phase

text: obtained

text: perturbatively,

text: matches

text: with

text: what

text: is

text: shown

text: schematically

text: in

text: Fig.

text: vanishes

text: if

text: Jβ

text: and

text: is

text: nonzero

text: iﬀ

text: Jβ

text: V.

text: CLASSICAL

text: HEISENBERG

text: MODEL

text: IN

text: A

text: which

text: implies

text: that

text: the

text: critical

text: temperature

text: is

text: given

text: by

text: RANDOM

text: FIELD

text: Till

text: now

text: we

text: have

text: explicitly

text: considered

text: only

text: the

text: situa-

text: c

text: J

text: tion

text: when

text: the

text: spins

text: were

text: two-dimensional.

text: It

text: is

text: natural

text: to

text: ask

text: analogous

text: questions

text: for

text: three-dimensional

text: spins

text: with

text: continuous

text: symmetry.

text: In

text: this

text: section

text: we

text: argue

text: that

text: for

text: Note

text: that

text: in

text: the

text: case

text: of

text: the

text: XY

text: model,

text: we

text: also

text: found

text: the

text: canonical

text: system

text: of

text: this

text: kind

text: the

text: classical

text: Heisen-

text: a

text: similar

text: behavior

text: of

text: the

text: magnetization

text: near

text: its

text: critical

text: berg

text: model

text: the

text: behavior

text: is

text: similar

text: to

text: that

text: of

text: the

text: XY

text: temperature.

text: model.

text: B.

text: The

text: system

text: with

text: disorder:

text: We

text: study

text: the

text: behavior

text: of

text: magnetization

text: in

text: the

text: pres-

text: ence

text: of

text: disorder

text: in

text: the

text: lattice

text: Heisenberg

text: model

text: where

text: Breaking

text: down

text: in

text: Eq.

text: into

text: its

text: components

text: along

text: the

text: spins

text: are

text: three-dimensional

text: with

text: continuous

text: symme-

text: the

text: X,

text: Y

text: and

text: Z

text: axes,

text: we

text: have

text: three

text: diﬀerent

text: equations.

text: try.

text: We

text: assume

text: that

text: at

text: all

text: sites,

text: random

text: ﬁelds

text: are

text: di-

text: It

text: turns

text: out

text: that

text: the

text: equations

text: along

text: the

text: X

text: and

text: Y

text: axes,

text: rected

text: in

text: the

text: Z-direction.

text: The

text: mean-ﬁeld

text: Hamiltonian

text: of

text: both

text: of

text: which

text: are

text: in

text: a

text: direction

text: transverse

text: to

text: the

text: applied

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 We now want to see the eﬀect of a random ﬁeld that does not have the rotational symmetry of the XY model 2 β0,2 = . (10) interaction, by considering the case when c J (cid:98) (cid:126)h = η .e , (12) III. FERROMAGNETIC XY MODEL IN A i i y RANDOM FIELD (cid:98) where η are scalar random variables with a distribu- i We now consider the eﬀect of additional quenched ran- tion symmetric about 0 and e denoting the unit vector y dom ﬁelds in the system. Let us begin by the notions of in the y direction. The main result of [12] is that on the quenched disorder and quenched averaging. two-dimensional lattice such a random ﬁeld will break A. Quenched averaging the continuous symmetry and the system will magnetize, even in two dimensions, thus destroying the Mermin- The disorder considered in this paper is “quenched”, Wagner-Hohenberg eﬀect. Above two dimensions, the i.e., its conﬁguration remains unchanged for a time that pure XY model magnetizes at low temperatures and it is much larger than the duration of the dynamics con- has been suggested in [12] that the uniaxial random ﬁeld sidered. In the systems that we study, it is the local as described above may enhance this magnetization. In magnetic ﬁelds that are disordered. They are random the present paper we want to study related eﬀects at the variables with a certain probability distributions. Since level of a simpler, mean-ﬁeld model, which allows for a the disorder is quenched, a particular realization of all more detailed analysis and more accurate simulations. the random variables remains ﬁxed for the whole time The mean-ﬁeld Hamiltonian in this case is given by necessary for the system to equilibrate. An average of a physical quantity, say A, is thus to be carried out in the − − following order: H = Jm(cid:126) .(cid:126)σ (cid:15)(cid:126)η.(cid:126)σ, (13) (a) Compute the value of the physical quantity A, with the ﬁxed conﬁguration of the disorder. where (cid:126)η is the quenched random ﬁeld in the y-direction, (b) Average over the disordered parameters. (cid:126)η = η.eˆ . The random variable here is assumed to be This mode of averaging is called quenched. It may be y Gaussian with zero mean and unit variance. (cid:15)(> 0) is mentioned that an averaging in which items (a) and (b) typically a small parameter that quantiﬁes the strength are interchanged in order, is called annealed averaging. of randomness. In the mean ﬁeld equation, the magneti- Physically it corresponds to a situation when the disor- zation, which is obtained by averaging over the disorder, der ﬂuctuates on time scales comparable to the system’s is given by (cid:34)(cid:82) (cid:35) thermal ﬂuctuations. B. The model and the mean ﬁeld equation for (cid:82) magnetization 2π (cid:126)σ exp(βJm(cid:126) .(cid:126)σ + β(cid:15)ησ )d(cid:126)σ m(cid:126) = Av 0 y , (14) η 2π exp(βJm(cid:126) .(cid:126)σ + β(cid:15)ησ )d(cid:126)σ The XY model with an inhomogeneous magnetic ﬁeld 0 y has the interaction (cid:88) (cid:88) · where Av ( ) denotes the average over the disorder, η i.e., the integral over η with the appropriate distri- − − H = J (cid:126)σ .(cid:126)σ (cid:15) (cid:126)h .(cid:126)σ , (11) bution (here assumed to be unit normal). Set m(cid:126) = i j i i | − | (m cos φ , m sin φ ). As demonstrated in Appendix A, i j =1 i 1 1 we obtain a coupled set of the fo(cid:20)llowing two (cid:21)equations: where the two-dimensional vectors (cid:126)h are the exter- i nal magnetic ﬁelds, up to a coeﬃcient (cid:15). In the sequel, ≡ I [βr] (cid:126)hi, which are random variables of order one, model the m(cid:15)⊥,2 m cos φ1 = Avη cos αI10[βr] , (15) disorder in the system and thus (cid:15) measures the disorder strength. More precisely, let (cid:126)hi be independent and iden- and (cid:20) (cid:21) tically distributed random variables (vector-valued). We want to study the eﬀect of including such a random ﬁeld term in the XY Hamiltonian at small values of (cid:15). As m(cid:15)(cid:107),2 ≡ m sin φ = Av sin αI1[βr] , (16) argued in [12], in lattice XY models, this eﬀect depends 1 η I [βr] 0 critically on the properties of the probability distribution of the random ﬁelds. where (cid:112) If the distribution of the (cid:126)h is invariant under rotations, i there is no spontaneous magnetization at any nonzero ≤ temperature in any dimension d 4 [9–11]. r = β J2m2 + (cid:15)2η2 + 2Jm(cid:15)η sin φ , (17) 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: now

text: want

text: to

text: see

text: the

text: eﬀect

text: of

text: a

text: random

text: ﬁeld

text: that

text: does

text: not

text: have

text: the

text: rotational

text: symmetry

text: of

text: the

text: XY

text: model

text: interaction,

text: by

text: considering

text: the

text: case

text: when

text: c

text: J

text: η

text: .e

text: III.

text: FERROMAGNETIC

text: XY

text: MODEL

text: IN

text: A

text: i

text: i

text: y

text: RANDOM

text: FIELD

text: where

text: η

text: are

text: scalar

text: random

text: variables

text: with

text: a

text: distribu-

text: i

text: We

text: now

text: consider

text: the

text: eﬀect

text: of

text: additional

text: quenched

text: ran-

text: tion

text: symmetric

text: about

text: and

text: e

text: denoting

text: the

text: unit

text: vector

text: y

text: dom

text: ﬁelds

text: in

text: the

text: system.

text: Let

text: us

text: begin

text: by

text: the

text: notions

text: of

text: in

text: the

text: y

text: direction.

text: The

text: main

text: result

text: of

text: is

text: that

text: on

text: the

text: quenched

text: disorder

text: and

text: quenched

text: averaging.

text: two-dimensional

text: lattice

text: such

text: a

text: random

text: ﬁeld

text: will

text: break

text: A.

text: Quenched

text: averaging

text: the

text: continuous

text: symmetry

text: and

text: the

text: system

text: will

text: magnetize,

text: even

text: in

text: two

text: dimensions,

text: thus

text: destroying

text: the

text: Mermin-

text: The

text: disorder

text: considered

text: in

text: this

text: paper

text: is

text: “quenched”,

text: Wagner-Hohenberg

text: eﬀect.

text: Above

text: two

text: dimensions,

text: the

text: i.e.,

text: its

text: conﬁguration

text: remains

text: unchanged

text: for

text: a

text: time

text: that

text: pure

text: XY

text: model

text: magnetizes

text: at

text: low

text: temperatures

text: and

text: it

text: is

text: much

text: larger

text: than

text: the

text: duration

text: of

text: the

text: dynamics

text: con-

text: has

text: been

text: suggested

text: in

text: that

text: the

text: uniaxial

text: random

text: ﬁeld

text: sidered.

text: In

text: the

text: systems

text: that

text: we

text: study,

text: it

text: is

text: the

text: local

text: as

text: described

text: above

text: may

text: enhance

text: this

text: magnetization.

text: In

text: magnetic

text: ﬁelds

text: that

text: are

text: disordered.

text: They

text: are

text: random

text: the

text: present

text: paper

text: we

text: want

text: to

text: study

text: related

text: eﬀects

text: at

text: the

text: variables

text: with

text: a

text: certain

text: probability

text: distributions.

text: Since

text: level

text: of

text: a

text: simpler,

text: mean-ﬁeld

text: model,

text: which

text: allows

text: for

text: a

text: the

text: disorder

text: is

text: quenched,

text: a

text: particular

text: realization

text: of

text: all

text: more

text: detailed

text: analysis

text: and

text: more

text: accurate

text: simulations.

text: the

text: random

text: variables

text: remains

text: ﬁxed

text: for

text: the

text: whole

text: time

text: The

text: mean-ﬁeld

text: Hamiltonian

text: in

text: this

text: case

text: is

text: given

text: by

text: necessary

text: for

text: the

text: system

text: to

text: equilibrate.

text: An

text: average

text: of

text: a

text: physical

text: quantity,

text: say

text: A,

text: is

text: thus

text: to

text: be

text: carried

text: out

text: in

text: the

text: following

text: order:

text: H

text: (a)

text: Compute

text: the

text: value

text: of

text: the

text: physical

text: quantity

text: A,

text: with

text: the

text: ﬁxed

text: conﬁguration

text: of

text: the

text: disorder.

text: where

text: is

text: the

text: quenched

text: random

text: ﬁeld

text: in

text: the

text: y-direction,

text: (b)

text: Average

text: over

text: the

text: disordered

text: parameters.

text: η.eˆ

text: The

text: random

text: variable

text: here

text: is

text: assumed

text: to

text: be

text: This

text: mode

text: of

text: averaging

text: is

text: called

text: quenched.

text: It

text: may

text: be

text: y

text: Gaussian

text: with

text: zero

text: mean

text: and

text: unit

text: variance.

text: is

text: mentioned

text: that

text: an

text: averaging

text: in

text: which

text: items

text: (a)

text: and

text: (b)

text: typically

text: a

text: small

text: parameter

text: that

text: quantiﬁes

text: the

text: strength

text: are

text: interchanged

text: in

text: order,

text: is

text: called

text: annealed

text: averaging.

text: of

text: randomness.

text: In

text: the

text: mean

text: ﬁeld

text: equation,

text: the

text: magneti-

text: Physically

text: it

text: corresponds

text: to

text: a

text: situation

text: when

text: the

text: disor-

text: zation,

text: which

text: is

text: obtained

text: by

text: averaging

text: over

text: the

text: disorder,

text: der

text: ﬂuctuates

text: on

text: time

text: scales

text: comparable

text: to

text: the

text: system’s

text: is

text: given

text: by

text: thermal

text: ﬂuctuations.

text: B.

text: The

text: model

text: and

text: the

text: mean

text: ﬁeld

text: equation

text: for

text: magnetization

text: 2π

text: Av

text: y

text: η

text: 2π

text: The

text: XY

text: model

text: with

text: an

text: inhomogeneous

text: magnetic

text: ﬁeld

text: y

text: has

text: the

text: interaction

text: where

text: Av

text: denotes

text: the

text: average

text: over

text: the

text: disorder,

text: η

text: i.e.,

text: the

text: integral

text: over

text: η

text: with

text: the

text: appropriate

text: distri-

text: H

text: J

text: bution

text: (here

text: assumed

text: to

text: be

text: unit

text: normal).

text: Set

text: i

text: j

text: i

text: i

text: (m

text: cos

text: φ

text: m

text: sin

text: φ

text: As

text: demonstrated

text: in

text: Appendix

text: A,

text: i

text: j

text: i

text: we

text: obtain

text: a

text: coupled

text: set

text: of

text: the

text: two

text: where

text: the

text: two-dimensional

text: vectors

text: are

text: the

text: exter-

text: i

text: nal

text: magnetic

text: ﬁelds,

text: up

text: to

text: a

text: coeﬃcient

text: In

text: the

text: sequel,

text: I

text: [βr]

text: which

text: are

text: random

text: variables

text: of

text: order

text: one,

text: model

text: the

text: m

text: cos

text: φ1

text: Avη

text: cos

text: disorder

text: in

text: the

text: system

text: and

text: thus

text: measures

text: the

text: disorder

text: strength.

text: More

text: precisely,

text: let

text: be

text: independent

text: and

text: iden-

text: and

text: tically

text: distributed

text: random

text: variables

text: (vector-valued).

text: We

text: want

text: to

text: study

text: the

text: eﬀect

text: of

text: including

text: such

text: a

text: random

text: ﬁeld

text: term

text: in

text: the

text: XY

text: Hamiltonian

text: at

text: small

text: values

text: of

text: As

text: m

text: sin

text: φ

text: Av

text: sin

text: argued

text: in

text: in

text: lattice

text: XY

text: models,

text: this

text: eﬀect

text: depends

text: η

text: I

text: [βr]

text: critically

text: on

text: the

text: properties

text: of

text: the

text: probability

text: distribution

text: of

text: the

text: random

text: ﬁelds.

text: where

text: If

text: the

text: distribution

text: of

text: the

text: is

text: invariant

text: under

text: rotations,

text: i

text: there

text: is

text: no

text: spontaneous

text: magnetization

text: at

text: any

text: nonzero

text: temperature

text: in

text: any

text: dimension

text: d

text: r

text: β

text: sin

text: φ

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##