text: 2 temperatures [Fig. 1(b)], although our theory could read- sponse coeﬃcients satisfy Onsager-like relations and are ily be applied to situations which combine ac potentials thus not independent of one another. In Sec. , we gen- with both chemical-potential and temperature gradients. eralize the thermodynamical framework to include the The physics of these setups can be understood by analogy time-averaged work per unit time performed by the ac to the Archimedes device, a pipe with a rotating screw, forces as a third ﬂux, along with the heat and particle (cid:126) which can be used to pump water against gravity. This is ﬂuxes. We also identify the scaled frequency ω/T of the a classical analog of an adiabatic quantum pump, where driving as the appropriate third aﬃnity, complementing ac driving pumps a certain amount of electric charge per the temperature and chemical-potential diﬀerences. In cycle. Speciﬁcally, this charge can be pumped against an Sec. , we deﬁne and analyze eﬃciency and ﬁgure of merit applied dc bias voltage,21 in which case quantum pump for the various quantum machines sketched in Fig. 1. We realizes a generator. ﬁnd that the deﬁnition of the appropriate ﬁgure of merit The Archimedes screw can also be operated in reverse, analogous to the ZT value diﬀers in characteristic ways, with water ﬂowing between the reservoirs by gravity and reﬂecting the fact that the usual oﬀ-diagonal thermoelec- setting the screw into rotational motion. An analogous tric response coeﬃcients, the oﬀ-diagonal coeﬃcients in- eﬀect can be used to turn an adiabatic quantum pump volving the third ﬂux or aﬃnity do not enter into the en- into an adiabatic quantum motor. This is most easily tropy production. To illustrate these concepts, we apply understood when imagining that the time dependence of our theory to an example device in Sec. which is based the ac potentials derives from the (classical) dynamics of, on a simple model for a quantum pump. We summarize say, one or more mechanical degrees of freedom.14 Then, in Sec. . a charge current pushed through the coherent conductor II. ADIABATIC RESPONSE AND ONSAGER will set the mechanical degrees of freedom into motion. RELATIONS Generator and quantum motor are driven by a bias voltage and correspond to the setup sketched in Fig. 1(a). We begin by evaluating the forces and currents induced Alternatively, we can also consider devices involving tem- by a set of time-periodic parameters in the adiabatic ap- perature gradients instead of bias voltages, which realize proximation. We will see that this can be done in close heat pumps and heat engines. Such a device is depicted in analogy to linear-response theory, allowing us to derive Fig. 1(b). The devices in Fig. 1 are examples of nanomo- Onsager-like relations. H tors and nanoengines, which have received much atten- We collect the parameterTs Vi(t) of the Hamiltonian ˆ tion recently.14–19 We note that the eﬀect of ac potentials iHnto aHvector V(t) = VT(t+ ) = (V1(t), V2(t), . . .) so that on the conventional thermoelectric eﬀects has been stud- ˆ = ˆ(V(t)), where = 2π/ω is the driving period.24 ied in a number of recent papers.20–23 Quite generally the Hamiltonian of the system can be The aim of the present work is to extend the linear- expressed as (cid:88) response theory of thermoelectrics to such nanomotors and nanoengines, to understand their eﬃciencies, and to Hˆ(V(t)) = Hˆ − Fˆ V (t), (2) identify appropriate ﬁgures of merit. This program poses 0 j j j several conceptual questions: (i) We need to identify the current that complements the charge and heat currents H where is the time-independent part of the Hamil- and accounts for the eﬀects of the ac potentials. Simi- 0 tonian and F are hermitian operators that play the role larly, we need to identify the aﬃnity that complements j of generalized forces the (scaled) temperature diﬀerence and bias voltage. (ii) We need to develop the generalized linear response theory H which includes these additional quantities. While this is −∂ ˆ(t) Fˆ(t) = . (3) a conventional linear-response theory for traditional ther- ∂V(t) moelectrics, the ac potentials are not actually weak but only slowly varying. (iii) We ﬁnally need to identify ap- The quantum expectation value tr{ρˆFˆ} in terms of the propriate eﬃciencies and ﬁgures of merit. We will see electronic density matrix ρˆ are just conventional forces that the latter also diﬀer in essential ways from those when the V denote regular cartesian coordinates of a j deﬁned in conventional thermoelectrics. classical system obeying Newtonian dynamics. In Sec. , we generalize linear-response theory to include At lowest order in the adiabatic approximation, the the response to the adiabatically varying ac potentials in system is described by the frozen density matrix ρˆ for H t addition to the applied bias voltage. We do this by work- the Hamiltonian ˆ with t treated as a parameter. Ac- t ing to linear order in the rate of change (or velocity) of the counting for the temporal variation of V(t) to lowest or- ac potentials. We ﬁnd that this can be done in a manner der, we can approximate the time(cid:90)evolution operator as which closely resembles the derivation of Kubo formulas in linear response theory. Consequently, we derive gen- eral Kubo-like expressions for the response of both the Uˆ(t, t ) (cid:39) T exp{−iHˆ (t − t ) − i t dt(cid:48)(t − t(cid:48))Fˆ · V˙ (t)}. 0 t 0 charge current and the generalized forces conjugate to t0 the ac potentials. These expressions imply that the re- (4) ##LTLine##

text: temperatures

text: [Fig.

text: although

text: our

text: theory

text: could

text: read-

text: sponse

text: coeﬃcients

text: satisfy

text: Onsager-like

text: relations

text: and

text: are

text: ily

text: be

text: applied

text: to

text: situations

text: which

text: combine

text: ac

text: potentials

text: thus

text: not

text: independent

text: of

text: one

text: another.

text: In

text: Sec.

text: we

text: gen-

text: with

text: both

text: chemical-potential

text: and

text: temperature

text: gradients.

text: eralize

text: the

text: thermodynamical

text: framework

text: to

text: include

text: the

text: The

text: physics

text: of

text: these

text: setups

text: can

text: be

text: understood

text: by

text: analogy

text: time-averaged

text: work

text: per

text: unit

text: time

text: performed

text: by

text: the

text: ac

text: to

text: the

text: Archimedes

text: device,

text: a

text: pipe

text: with

text: a

text: rotating

text: screw,

text: forces

text: as

text: a

text: third

text: ﬂux,

text: along

text: with

text: the

text: heat

text: and

text: particle

text: which

text: can

text: be

text: used

text: to

text: pump

text: water

text: against

text: gravity.

text: This

text: is

text: ﬂuxes.

text: We

text: also

text: identify

text: the

text: scaled

text: frequency

text: ω/T

text: of

text: the

text: a

text: classical

text: analog

text: of

text: an

text: adiabatic

text: quantum

text: pump,

text: where

text: driving

text: as

text: the

text: appropriate

text: third

text: aﬃnity,

text: complementing

text: ac

text: driving

text: pumps

text: a

text: certain

text: amount

text: of

text: electric

text: charge

text: per

text: the

text: temperature

text: and

text: chemical-potential

text: diﬀerences.

text: In

text: cycle.

text: Speciﬁcally,

text: this

text: charge

text: can

text: be

text: pumped

text: against

text: an

text: Sec.

text: we

text: deﬁne

text: and

text: analyze

text: eﬃciency

text: and

text: ﬁgure

text: of

text: merit

text: applied

text: dc

text: bias

text: in

text: which

text: case

text: quantum

text: pump

text: for

text: the

text: various

text: quantum

text: machines

text: sketched

text: in

text: Fig.

text: We

text: realizes

text: a

text: generator.

text: ﬁnd

text: that

text: the

text: deﬁnition

text: of

text: the

text: appropriate

text: ﬁgure

text: of

text: merit

text: The

text: Archimedes

text: screw

text: can

text: also

text: be

text: operated

text: in

text: reverse,

text: analogous

text: to

text: the

text: ZT

text: value

text: diﬀers

text: in

text: characteristic

text: ways,

text: with

text: water

text: ﬂowing

text: between

text: the

text: reservoirs

text: by

text: gravity

text: and

text: reﬂecting

text: the

text: fact

text: that

text: the

text: usual

text: oﬀ-diagonal

text: thermoelec-

text: setting

text: the

text: screw

text: into

text: rotational

text: motion.

text: An

text: analogous

text: tric

text: response

text: coeﬃcients,

text: the

text: oﬀ-diagonal

text: coeﬃcients

text: in-

text: eﬀect

text: can

text: be

text: used

text: to

text: turn

text: an

text: adiabatic

text: quantum

text: pump

text: volving

text: the

text: third

text: ﬂux

text: or

text: aﬃnity

text: do

text: not

text: enter

text: into

text: the

text: en-

text: into

text: an

text: adiabatic

text: quantum

text: motor.

text: This

text: is

text: most

text: easily

text: tropy

text: production.

text: To

text: illustrate

text: these

text: concepts,

text: we

text: apply

text: understood

text: when

text: imagining

text: that

text: the

text: time

text: dependence

text: of

text: our

text: theory

text: to

text: an

text: example

text: device

text: in

text: Sec.

text: which

text: is

text: based

text: the

text: ac

text: potentials

text: derives

text: from

text: the

text: (classical)

text: dynamics

text: of,

text: on

text: a

text: simple

text: model

text: for

text: a

text: quantum

text: pump.

text: We

text: summarize

text: say,

text: one

text: or

text: more

text: mechanical

text: degrees

text: of

text: Then,

text: in

text: Sec.

text: a

text: charge

text: current

text: pushed

text: through

text: the

text: coherent

text: conductor

text: II.

text: ADIABATIC

text: RESPONSE

text: AND

text: ONSAGER

text: will

text: set

text: the

text: mechanical

text: degrees

text: of

text: freedom

text: into

text: motion.

text: RELATIONS

text: Generator

text: and

text: quantum

text: motor

text: are

text: driven

text: by

text: a

text: bias

text: voltage

text: and

text: correspond

text: to

text: the

text: setup

text: sketched

text: in

text: Fig.

text: We

text: begin

text: by

text: evaluating

text: the

text: forces

text: and

text: currents

text: induced

text: Alternatively,

text: we

text: can

text: also

text: consider

text: devices

text: involving

text: tem-

text: by

text: a

text: set

text: of

text: time-periodic

text: parameters

text: in

text: the

text: adiabatic

text: ap-

text: perature

text: gradients

text: instead

text: of

text: bias

text: voltages,

text: which

text: realize

text: proximation.

text: We

text: will

text: see

text: that

text: this

text: can

text: be

text: done

text: in

text: close

text: heat

text: pumps

text: and

text: heat

text: engines.

text: Such

text: a

text: device

text: is

text: depicted

text: in

text: analogy

text: to

text: linear-response

text: theory,

text: allowing

text: us

text: to

text: derive

text: Fig.

text: The

text: devices

text: in

text: Fig.

text: are

text: examples

text: of

text: nanomo-

text: Onsager-like

text: relations.

text: H

text: tors

text: and

text: nanoengines,

text: which

text: have

text: received

text: much

text: atten-

text: We

text: collect

text: the

text: parameterTs

text: Vi(t)

text: of

text: the

text: Hamiltonian

text: ˆ

text: tion

text: We

text: note

text: that

text: the

text: eﬀect

text: of

text: ac

text: potentials

text: iHnto

text: aHvector

text: V(t)

text: VT(t+

text: so

text: that

text: on

text: the

text: conventional

text: thermoelectric

text: eﬀects

text: has

text: been

text: stud-

text: ˆ

text: ˆ(V(t)),

text: where

text: 2π/ω

text: is

text: the

text: driving

text: ied

text: in

text: a

text: number

text: of

text: recent

text: Quite

text: generally

text: the

text: Hamiltonian

text: of

text: the

text: system

text: can

text: be

text: The

text: aim

text: of

text: the

text: present

text: work

text: is

text: to

text: extend

text: the

text: linear-

text: expressed

text: as

text: response

text: theory

text: of

text: thermoelectrics

text: to

text: such

text: nanomotors

text: and

text: nanoengines,

text: to

text: understand

text: their

text: eﬃciencies,

text: and

text: to

text: Hˆ(V(t))

text: Hˆ

text: Fˆ

text: V

text: (t),

text: identify

text: appropriate

text: ﬁgures

text: of

text: merit.

text: This

text: program

text: poses

text: j

text: j

text: j

text: several

text: conceptual

text: questions:

text: (i)

text: We

text: need

text: to

text: identify

text: the

text: current

text: that

text: complements

text: the

text: charge

text: and

text: heat

text: currents

text: H

text: where

text: is

text: the

text: time-independent

text: part

text: of

text: the

text: Hamil-

text: and

text: accounts

text: for

text: the

text: eﬀects

text: of

text: the

text: ac

text: potentials.

text: Simi-

text: tonian

text: and

text: F

text: are

text: hermitian

text: operators

text: that

text: play

text: the

text: role

text: larly,

text: we

text: need

text: to

text: identify

text: the

text: aﬃnity

text: that

text: complements

text: j

text: of

text: generalized

text: forces

text: the

text: (scaled)

text: temperature

text: diﬀerence

text: and

text: bias

text: voltage.

text: (ii)

text: We

text: need

text: to

text: develop

text: the

text: generalized

text: linear

text: response

text: theory

text: H

text: which

text: includes

text: these

text: additional

text: quantities.

text: While

text: this

text: is

text: ˆ(t)

text: Fˆ(t)

text: a

text: conventional

text: linear-response

text: theory

text: for

text: traditional

text: ther-

text: ∂V(t)

text: moelectrics,

text: the

text: ac

text: potentials

text: are

text: not

text: actually

text: weak

text: but

text: only

text: slowly

text: varying.

text: (iii)

text: We

text: ﬁnally

text: need

text: to

text: identify

text: ap-

text: The

text: quantum

text: expectation

text: value

text: tr{ρˆFˆ}

text: in

text: terms

text: of

text: the

text: propriate

text: eﬃciencies

text: and

text: ﬁgures

text: of

text: merit.

text: We

text: will

text: see

text: electronic

text: density

text: matrix

text: ρˆ

text: are

text: just

text: conventional

text: forces

text: that

text: the

text: latter

text: also

text: diﬀer

text: in

text: essential

text: ways

text: from

text: those

text: when

text: the

text: V

text: denote

text: regular

text: cartesian

text: coordinates

text: of

text: a

text: j

text: deﬁned

text: in

text: conventional

text: thermoelectrics.

text: classical

text: system

text: obeying

text: Newtonian

text: dynamics.

text: In

text: Sec.

text: we

text: generalize

text: linear-response

text: theory

text: to

text: include

text: At

text: lowest

text: order

text: in

text: the

text: adiabatic

text: approximation,

text: the

text: the

text: response

text: to

text: the

text: adiabatically

text: varying

text: ac

text: potentials

text: in

text: system

text: is

text: described

text: by

text: the

text: frozen

text: density

text: matrix

text: ρˆ

text: for

text: H

text: t

text: addition

text: to

text: the

text: applied

text: bias

text: voltage.

text: We

text: do

text: this

text: by

text: work-

text: the

text: Hamiltonian

text: ˆ

text: with

text: t

text: treated

text: as

text: a

text: parameter.

text: Ac-

text: t

text: ing

text: to

text: linear

text: order

text: in

text: the

text: rate

text: of

text: change

text: (or

text: velocity)

text: of

text: the

text: counting

text: for

text: the

text: temporal

text: variation

text: of

text: V(t)

text: to

text: lowest

text: or-

text: ac

text: potentials.

text: We

text: ﬁnd

text: that

text: this

text: can

text: be

text: done

text: in

text: a

text: manner

text: der,

text: we

text: can

text: approximate

text: the

text: operator

text: as

text: which

text: closely

text: resembles

text: the

text: derivation

text: of

text: Kubo

text: formulas

text: in

text: linear

text: response

text: theory.

text: Consequently,

text: we

text: derive

text: gen-

text: eral

text: Kubo-like

text: expressions

text: for

text: the

text: response

text: of

text: both

text: the

text: Uˆ(t,

text: t

text: T

text: exp{−iHˆ

text: (t

text: t

text: i

text: t

text: V˙

text: (t)}.

text: t

text: charge

text: current

text: and

text: the

text: generalized

text: forces

text: conjugate

text: to

text: the

text: ac

text: potentials.

text: These

text: expressions

text: imply

text: that

text: the

text: re-

text: ##LTLine##


text: 2 microscopic Hamiltonian, representing the system shown in (cid:126) Fig. 1, reads (hereafter = 1 and n = 1, 2): (cid:88) (cid:2) (cid:3) H = H + H + H + H , (1) res q JC int † Hres = (cid:88) ωr(ens)anan + fn(cid:2) an exp(iωd(nrv)t) + H.c.(cid:3), (2) n H = (cid:88) ωq(n) σz + Ωn σ− exp(iω(n)t) + H.c. , (3) q 2 n 2 n drv n H = (cid:2)g (a σ+ + a† σ−), (cid:3) (4) JC n n n n n n (cid:2) (cid:3) H = J a exp(iω(1) t) + H.c. int 12 1 drv × a exp(iω(2) t) + H.c. , (5) 2 drv † FIG. 1: (Color online) Schematic diagram for the couplings in the where a and a are, respectively, the phonon annihilation n n dissipative hybrid system described by the Hamiltonian given in and creation operators for the nth NAMR, Eq. (1). The system consists of two linear nanomechanical resonators (NAMRs), with frequencies ω(n), and a pair of two-level systems res − (TLSs), with frequencies ωq(n), driven by classical ﬁelds, with fre- an = (2mnωr(ens)) 1/2(mnωr(ens)xn + ipn), quencies ωd(nrv) and ω¯d(nrv), respectively. Moreover, Ωn, fn, gn, and a† = (2m ω(n))−1/2(m ω(n)x − ip ), (6) J denote the coupling strengths of the depicted subsystems; γ are n n res n res n n 12 † n the NAMR decay rates; a (a ) is the phonon annihilation (creation) n n operator for the nth NAMR, and σz is the Pauli operator for the nth TLS. n which are given in terms of the position operator xn, mo- mentum operator p , and frequency ω(n) of the NAMR. | n(cid:105)(cid:104) | − | (cid:105)(cid:104) | res Moreover, σz = e e g g is the Pauli Z opera- n n n − n | n(cid:105)(cid:104) | | (cid:105)(cid:104) | in an analogous way to how Coulomb blockade can lead to the tor for the nth TLS, while σ = g e (σ+ = e g ) n n n n n n observation of single-electron tunneling. is the qubit lowering (raising) operator given in terms of the | (cid:105) | (cid:105) We discuss here how the form of the multiphonon block- ground ( g ) and excited ( e ) states of the nth TLS; ω(n) n n q ade can be tuned via driving, and verify the resultant highly is the TLS frequency and ω(n) (ω(n)) is the NAMR (TLS) nonclassical states with a variety of measures. drv drv driving-ﬁeld frequency. The coupling strengths of the subsys- This paper is organized as follows: In Sec. II, we describe tems, as shown in Fig. 1, are denoted by Ω , f , g , and J . a model for a hybrid system of coupled linear NAMRs and n n n 12 The symbol H.c. denotes the Hermitian conjugated term. TLSs. In this section and in Appendix A, we also show how a This system, described by Eq. (1), can be realized in vari- Kerr-type nonlinearity can be induced via NAMR-TLS inter- ous ways as a combination of two types of implementations, actions, and derive an effective Hamiltonian for the coupled e.g.: (i) the proposal of Ref. [43] for observing single-mode nonlinear NAMRs. The possibility of observing multiphonon phonon blockade in a driven single NAMR coupled to a su- blockades in this system is described in Sec. III. We summa- perconducting quantum two-level system and (ii) the system rize several methods to assess nonclassicality, which we then of two nonlinearly-coupled NAMRs, which was experimen- apply in our analysis of phonon blockade in Secs. IV and V. tally realized in the NTT experiments (see, e.g., Ref [54] and We conclude in Sec. VI. references therein). It is worth noting that we assumed that II. MODEL the interaction H is additionally driven at frequencies ω(1) int drv A. Time-dependent Hamiltonian and ωd(2r)v, as given in Eq. (5), which slightly generalizes the model applied in Ref. [54]. We also note that the interaction described by Eq. (4) conserves the number of excitations, in We consider a hybrid system, as schematically depicted in contrast to that described by Eq. (5). Fig. 1, consisting of two interacting driven linear nanome- B. Time-independent Hamiltonian in a rotated dressed-qubit chanical resonators (NAMRs), described by the Hamiltonian basis H , coupled to two driven two-level systems (TLSs, qubits), res given by the Hamiltonian H . The interaction H between q JC In the following, we assume the following large detunings: the nth NAMR and nth TLS (for n = 1, 2) is described by a Jaynes-Cummings-like model under the rotating wave- approximation. The interaction Hint between the two NAMRs ∆(n) ≡ ω(n) − ω(n) = ω(n) − ω(n) (cid:29) g > 0, (7) can be interpreted as a combined driven process of frequency rq res q drv drv n ≡ ≡ − − (cid:29) conversion and parametric ampliﬁcation [54]. Thus, the total ∆ ∆(n) ω(n) ω(n) = ω(n) ω(n) Ω > 0. (8) n rd res drv q drv n ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: microscopic

text: Hamiltonian,

text: representing

text: the

text: system

text: shown

text: in

text: Fig.

text: reads

text: (hereafter

text: and

text: n

label: equation

text: H

text: H

text: H

text: H

text: H

text: res

text: q

text: JC

text: int

text: Hres

text: ωr(ens)anan

text: an

text: exp(iωd(nrv)t)

text: n

text: H

text: ωq(n)

text: σz

text: Ωn

text: σ−

text: exp(iω(n)t)

text: H.c.

text: q

text: n

text: n

text: drv

text: n

text: H

text: (a

text: σ+

text: a†

text: σ−),

text: JC

text: n

text: n

text: n

text: n

text: n

text: n

text: H

text: J

text: a

text: t)

text: H.c.

text: int

text: drv

text: a

text: t)

text: H.c.

text: drv

text: FIG.

text: (Color

text: online)

text: Schematic

text: diagram

text: for

text: the

text: couplings

text: in

text: the

text: where

text: a

text: and

text: a

text: are,

text: respectively,

text: the

text: phonon

text: annihilation

text: n

text: n

text: dissipative

text: hybrid

text: system

text: described

text: by

text: the

text: Hamiltonian

text: given

text: in

text: and

text: creation

text: operators

text: for

text: the

text: nth

text: NAMR,

text: Eq.

text: The

text: system

text: consists

text: of

text: two

text: linear

text: nanomechanical

text: resonators

text: (NAMRs),

text: with

text: frequencies

text: ω(n),

text: and

text: a

text: pair

text: of

text: two-level

text: systems

text: res

text: (TLSs),

text: with

text: frequencies

text: ωq(n),

text: driven

text: by

text: classical

text: ﬁelds,

text: with

text: fre-

text: an

text: ipn),

text: quencies

text: ωd(nrv)

text: and

text: ω¯d(nrv),

text: respectively.

text: Moreover,

text: Ωn,

text: fn,

text: gn,

text: and

text: a†

text: ω(n)x

text: ip

text: J

text: denote

text: the

text: coupling

text: strengths

text: of

text: the

text: depicted

text: subsystems;

text: γ

text: are

text: n

text: n

text: res

text: n

text: res

text: n

text: n

text: n

text: the

text: NAMR

text: decay

text: rates;

text: a

text: (a

text: is

text: the

text: phonon

text: annihilation

text: (creation)

text: n

text: n

text: operator

text: for

text: the

text: nth

text: NAMR,

text: and

text: σz

text: is

text: the

text: Pauli

text: operator

text: for

text: the

text: nth

text: TLS.

text: n

text: which

text: are

text: given

text: in

text: terms

text: of

text: the

text: position

text: operator

text: xn,

text: mo-

text: mentum

text: operator

text: p

text: and

text: frequency

text: ω(n)

text: of

text: the

text: NAMR.

text: res

text: Moreover,

text: σz

text: e

text: e

text: g

text: g

text: is

text: the

text: Pauli

text: Z

text: opera-

text: n

text: n

text: n

text: n

text: in

text: an

text: analogous

text: way

text: to

text: how

text: Coulomb

text: blockade

text: can

text: lead

text: to

text: the

text: tor

text: for

text: the

text: nth

text: TLS,

text: while

text: σ

text: g

text: e

text: (σ+

text: e

text: g

text: n

text: n

text: n

text: n

text: n

text: n

text: observation

text: of

text: single-electron

text: tunneling.

text: is

text: the

text: qubit

text: lowering

text: (raising)

text: operator

text: given

text: in

text: terms

text: of

text: the

text: We

text: discuss

text: here

text: how

text: the

text: form

text: of

text: the

text: multiphonon

text: block-

text: ground

text: g

text: and

text: excited

text: e

text: states

text: of

text: the

text: nth

text: TLS;

text: ω(n)

text: n

text: n

text: q

text: ade

text: can

text: be

text: tuned

text: via

text: driving,

text: and

text: verify

text: the

text: resultant

text: highly

text: is

text: the

text: TLS

text: frequency

text: and

text: ω(n)

text: (ω(n))

text: is

text: the

text: NAMR

text: (TLS)

text: nonclassical

text: states

text: with

text: a

text: variety

text: of

text: measures.

text: drv

text: drv

text: driving-ﬁeld

text: frequency.

text: The

text: coupling

text: strengths

text: of

text: the

text: subsys-

text: This

text: paper

text: is

text: organized

text: as

text: follows:

text: In

text: Sec.

text: II,

text: we

text: describe

text: tems,

text: as

text: shown

text: in

text: Fig.

text: are

text: denoted

text: by

text: Ω

text: f

text: g

text: and

text: J

text: a

text: model

text: for

text: a

text: hybrid

text: system

text: of

text: coupled

text: linear

text: NAMRs

text: and

text: n

text: n

text: n

text: The

text: symbol

text: H.c.

text: denotes

text: the

text: Hermitian

text: conjugated

text: term.

text: TLSs.

text: In

text: this

text: section

text: and

text: in

text: Appendix

text: A,

text: we

text: also

text: show

text: how

text: a

text: This

text: system,

text: described

text: by

text: Eq.

text: can

text: be

text: realized

text: in

text: vari-

text: Kerr-type

text: nonlinearity

text: can

text: be

text: induced

text: via

text: NAMR-TLS

text: inter-

text: ous

text: ways

text: as

text: a

text: combination

text: of

text: two

text: types

text: of

text: implementations,

text: actions,

text: and

text: derive

text: an

text: effective

text: Hamiltonian

text: for

text: the

text: coupled

text: e.g.:

text: (i)

text: the

text: proposal

text: of

text: Ref.

text: for

text: observing

text: single-mode

text: nonlinear

text: NAMRs.

text: The

text: possibility

text: of

text: observing

text: multiphonon

text: phonon

text: blockade

text: in

text: a

text: driven

text: single

text: NAMR

text: coupled

text: to

text: a

text: su-

text: blockades

text: in

text: this

text: system

text: is

text: described

text: in

text: Sec.

text: III.

text: We

text: summa-

text: perconducting

text: quantum

text: two-level

text: system

text: and

text: (ii)

text: the

text: system

text: rize

text: several

text: methods

text: to

text: assess

text: nonclassicality,

text: which

text: we

text: then

text: of

text: two

text: nonlinearly-coupled

text: NAMRs,

text: which

text: was

text: experimen-

text: apply

text: in

text: our

text: analysis

text: of

text: phonon

text: blockade

text: in

text: Secs.

text: IV

text: and

text: V.

text: tally

text: realized

text: in

text: the

text: NTT

text: experiments

text: (see,

text: e.g.,

text: Ref

text: and

text: We

text: conclude

text: in

text: Sec.

text: VI.

text: references

text: therein).

text: It

text: is

text: worth

text: noting

text: that

text: we

text: assumed

text: that

text: II.

text: MODEL

text: the

text: interaction

text: H

text: is

text: additionally

text: driven

text: at

text: frequencies

text: int

text: drv

text: A.

text: Time-dependent

text: Hamiltonian

text: and

text: as

text: given

text: in

text: Eq.

text: which

text: slightly

text: generalizes

text: the

text: model

text: applied

text: in

text: Ref.

text: We

text: also

text: note

text: that

text: the

text: interaction

text: described

text: by

text: Eq.

text: conserves

text: the

text: number

text: of

text: excitations,

text: in

text: We

text: consider

text: a

text: hybrid

text: system,

text: as

text: schematically

text: depicted

text: in

text: contrast

text: to

text: that

text: described

text: by

text: Eq.

text: Fig.

text: consisting

text: of

text: two

text: interacting

text: driven

text: linear

text: nanome-

text: B.

text: Time-independent

text: Hamiltonian

text: in

text: a

text: rotated

text: dressed-qubit

text: chanical

text: resonators

text: (NAMRs),

text: described

text: by

text: the

text: Hamiltonian

text: basis

text: H

text: coupled

text: to

text: two

text: driven

text: two-level

text: systems

text: (TLSs,

text: qubits),

text: res

text: given

text: by

text: the

text: Hamiltonian

text: H

text: The

text: interaction

text: H

text: between

text: q

text: JC

text: In

text: the

text: following,

text: we

text: assume

text: the

text: following

text: large

text: detunings:

text: the

text: nth

text: NAMR

text: and

text: nth

text: TLS

text: (for

text: n

text: is

text: described

text: by

text: a

text: Jaynes-Cummings-like

text: model

text: under

text: the

text: rotating

text: wave-

text: approximation.

text: The

text: interaction

text: Hint

text: between

text: the

text: two

text: NAMRs

text: ∆(n)

text: ω(n)

text: ω(n)

text: ω(n)

text: ω(n)

text: g

text: can

text: be

text: interpreted

text: as

text: a

text: combined

text: driven

text: process

text: of

text: frequency

text: rq

text: res

text: q

text: drv

text: drv

text: n

text: conversion

text: and

text: parametric

text: ampliﬁcation

text: Thus,

text: the

text: total

text: ∆(n)

text: ω(n)

text: ω(n)

text: ω(n)

text: ω(n)

text: Ω

text: n

text: rd

text: res

text: drv

text: q

text: drv

text: n

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 FIG. 13: (Color online) Non-stationary solutions for model 1: (a) Entanglement, measured by the PPT entanglement cost E (ρ), cost (b) dimensionality of entanglement D (ρ), and (c) nonclassical- dim ity, measured by the entanglement potential EP(ρ(1)), of the ﬁrst (and, equivalently, second) NAMR for the states ρ generated in the non-dissipative (blue solid upper curves, γ = 0) and dissipative (red dashed lower curves, γ = J/3) systems described by the effective (cid:48) Hamiltonian H . Parameters are same as in Fig. 7. eﬀ nonlinear systems, instead of a single system, one can more FIG. 14: (Color online) Non-stationary solutions for model 2: Same (cid:48)(cid:48) easily tune various types of multi-phonon two-mode block- as in Fig. 13 but for the effective Hamiltonian H . Additionally, eﬀ ades, as studied in detail in this paper. panel (d) shows the nonclassicality, measured by the entanglement It is important to clarify the main differences between potential EP(ρ(2)), of the second NAMR. our model of coupled oscillators and that studied in, e.g., Refs. [44, 64, 65]: (1) Here we assumed that the oscilla- tors are nonlinearly coupled as described by the Hamiltonian classicality of a total system, as measured by, e.g., the entan- ∼ † † H (a + a )(a + a ) in contrast to the linear cou- glement potential. It is also worth noting that, by applying int 1 1 ∼2 † 2 † pling given by H (a a + a a ), which was applied in our precise numerical calculations, we found the steady-states int 1 2 1 2 Refs. [44, 64, 65]. Note that this linear coupler (which is for- of the NAMRs to be only partially coherent (partially mixed), mally equivalent to a beam splitter or a frequency converter), while the steady states calculated in, e.g., Ref. [65] were as- does not change the nonclassicality of a total system [73]. In sumed to be completely coherent (perfectly pure). (2) We contrast to this, the nonlinear coupler can increase the non- have derived an effective Kerr-type Hamiltonian from a mi- ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

label: caption

text: online)

text: Non-stationary

text: solutions

text: for

text: model

text: (a)

text: Entanglement,

text: measured

text: by

text: the

text: PPT

text: entanglement

text: cost

text: E

text: (ρ),

text: cost

text: (b)

text: dimensionality

text: of

text: entanglement

text: D

text: (ρ),

text: and

text: (c)

text: nonclassical-

text: dim

text: ity,

text: measured

text: by

text: the

text: entanglement

text: potential

text: of

text: the

text: ﬁrst

text: (and,

text: equivalently,

text: second)

text: NAMR

text: for

text: the

text: states

text: ρ

text: generated

text: in

text: the

text: non-dissipative

text: (blue

text: solid

text: upper

text: curves,

text: γ

text: and

text: dissipative

text: (red

text: dashed

text: lower

text: curves,

text: γ

text: systems

text: described

text: by

text: the

text: effective

text: Hamiltonian

text: H

text: Parameters

text: are

text: same

text: as

text: in

text: Fig.

text: eﬀ

text: nonlinear

text: systems,

text: instead

text: of

text: a

text: single

text: system,

text: one

text: can

text: more

text: FIG.

text: (Color

text: online)

text: Non-stationary

text: solutions

text: for

text: model

text: Same

text: easily

text: tune

text: various

text: types

text: of

text: multi-phonon

text: two-mode

text: block-

text: as

text: in

text: Fig.

text: but

text: for

text: the

text: effective

text: Hamiltonian

text: H

text: Additionally,

text: eﬀ

text: ades,

text: as

text: studied

text: in

text: detail

text: in

text: this

text: paper.

text: panel

text: (d)

text: shows

text: the

text: nonclassicality,

text: measured

text: by

text: the

text: entanglement

text: It

text: is

text: important

text: to

text: clarify

text: the

text: main

text: differences

text: between

text: potential

text: of

text: the

text: second

text: NAMR.

text: our

text: model

text: of

text: coupled

text: oscillators

text: and

text: that

text: studied

text: in,

text: e.g.,

text: Refs.

text: Here

text: we

text: assumed

text: that

text: the

text: oscilla-

text: tors

text: are

text: nonlinearly

text: coupled

text: as

text: described

text: by

text: the

text: Hamiltonian

text: classicality

text: of

text: a

text: total

text: system,

text: as

text: measured

text: by,

text: e.g.,

text: the

text: entan-

text: H

text: (a

text: a

text: )(a

text: a

text: in

text: contrast

text: to

text: the

text: linear

text: cou-

text: glement

text: potential.

text: It

text: is

text: also

text: worth

text: noting

text: that,

text: by

text: applying

text: int

text: ∼2

text: pling

text: given

text: by

text: H

text: (a

text: a

text: a

text: a

text: which

text: was

text: applied

text: in

text: our

text: precise

text: numerical

text: calculations,

text: we

text: found

text: the

text: steady-states

text: int

text: Refs.

text: Note

text: that

text: this

text: linear

text: coupler

text: (which

text: is

text: for-

text: of

text: the

text: NAMRs

text: to

text: be

text: only

text: partially

text: coherent

text: (partially

text: mixed),

text: mally

text: equivalent

text: to

text: a

text: beam

text: splitter

text: or

text: a

text: frequency

text: converter),

text: while

text: the

text: steady

text: states

text: calculated

text: in,

text: e.g.,

text: Ref.

text: were

text: as-

text: does

text: not

text: change

text: the

text: nonclassicality

text: of

text: a

text: total

text: system

text: In

text: sumed

text: to

text: be

text: completely

text: coherent

text: (perfectly

text: pure).

text: We

text: contrast

text: to

text: this,

text: the

text: nonlinear

text: coupler

text: can

text: increase

text: the

text: non-

text: have

text: derived

text: an

text: effective

text: Kerr-type

text: Hamiltonian

text: from

text: a

text: mi-

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 12 croscopic one, given in Eq. (1), while an analogous Kerr-type Hamiltonian in Refs. [44, 64, 65] was assumed without deriva- (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) tion. Moreover, we studied here the Kerr interaction under H = H + H + H , (A1) sys drv int two different resonance conditions, as described by Eqs. (29) and (33). References [44, 64, 65] discussed only the Kerr in- where (cid:88) teraction given by Eq. (29). (3) We analyzed the blockade of mechanical phonons, contrary to photon blockade studied in Refs. [64, 65]. Moreover, the interplay between single- (cid:48)(cid:48)(cid:48) (cid:88)(cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) † (cid:48)(cid:48)(cid:48) H = H + H = ∆ a a + H , (A2) phonon blockade in one oscillator and single-photon block- sys 0 JC n n n JC ade in another oscillator was studied in Ref. [44]. Here we n (cid:48)(cid:48)(cid:48) † predicted tunable k-phonon blockades (with k = 1, 2, 3) in Hdrv = fn(a + a ). (A3) each oscillator, where the k-phonon Fock state impedes the n excitation of more phonons. To our knowledge, multiphonon (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48) blockade has not been studied before. Here H is given by Eq. (22) and H = H is given by JC int int Eq. (15). In conclusion, we showed here a rich tapestry of phonon Our derivation is based on the method described in Ref. [81] blockade effects in two coupled nonlinear nanomechanical for the exact diagonalization of the Jaynes-Cummings model resonators. Different types of phonon blockade could be with the following unitary transformation “picked out” of this tapestry by controlling the nonlinearity via ancilla TLS, and by changing the driving frequency of − † − − the resonators themselves. Within these different types of U = exp[ Λ(λ )(a ρ a ρ+)], (A4) n n n n n n phonon blockade, the coupled NAMRs can be made to be- have like two coupled qubits, a qutrit coupled to a quartit, or even two coupled qudits. We veriﬁed this picture by looking where √the √operator Λ(λ ) = − n at the nonclassical properties of these states including their arctan(2λ N )/(2 N ) is given in terms of the n n n † | (cid:105)(cid:104) | single-NAMR nonclassicality, and two-NAMR entanglement total number of excitations N = a a + E E in the n n n n n and entanglement dimensionality. The nonclassical properties nth NAMR and TLS. Thus, the expansions of the annihilation † − † − of these states were also analyzed in phase space by applying operators, a¯ = U aU and ρ¯ = U ρ U of Ref. [81] can n n n n n n n the s-parametrized Cahill-Glauber quasiprobability distribu- be rewritten for our dressed qubit states as follows: tions and, in particular, the Wigner function. We expect that, if realized in experiment, the ability to operate in these dif- − O ferent regimes will have a range of applications in quantum a¯n = an(1 + 12λ2nρzn) + λnh(n3)ρn + λ3na2nρ+n + (λ4n),(A5) − − − O information and quantum technologies. ρ¯ = h(1)ρ + λ a ρz λ2a2ρ+ + (λ3), (A6) n n n n n n n n n n Acknowledgement − † O where h(k) = 1 kλ2(a a +1/2) and (λk) denotes the n n∼ n n n A.M. was supported by the Polish National Science Cen- omitted terms of order λk and higher. Now one can easily n (cid:48)(cid:48)(cid:48) tre under the grants No. DEC-2011/02/A/ST2/00305 and No. transform the Hamiltonian H into DEC-2011/03/B/ST2/01903. A.M. acknowledges a long-term fellowship from the Japan Society for the Promotion of Sci- † † (cid:48)(cid:48)(cid:48) ence (JSPS). Y.X.L. was supported by the National Natural H¯ = U1U2H U2U1. (A7) Science Foundation of China under Grant No. 61328502, the National Basic Research Program of China (973 Program) (cid:48)(cid:48)(cid:48) In particular, by applying(cid:88)Eq. (A5), H transforms into under Grant No. 2014CB921401, the Tsinghua University drv Initiative Scientiﬁc Research Program, and the Tsinghua Na- tional Laboratory for Information Science and Technology † H¯ = f [a¯ + a¯ ]. (A8) (TNList) Cross-discipline Foundation. J.B. was supported drv n n n by the Palacky´ University under the Project No. IGA-PˇrF- 2015-002. F.N. was partially supported by the RIKEN iTHES | (cid:105) If the qubits remain in the excited dressed-qubit states E , Project, the MURI Center for Dynamic Magneto-Optics via n given in Eq. (17), then (cid:88) the AFOSR award number FA9550-14-1-0040, the IMPACT program of JST, and a Grant-in-Aid for Scientiﬁc Research (cid:104) | | (cid:105) † O (A). E E H¯ E E = f (a +a )(1+λ2/2)+ (λ4). 1 2 drv 1 2 n n n n n Appendix A: Derivation of the effective Hamiltonian n (A9) Here we show how to derive the effective Hamiltonian The assumption of a “frozen” state of both qubits is physi- (cid:48)(cid:48)(cid:48) (cid:29) H , given by Eq. (28), from the Hamiltonian H , given in cally justiﬁed for the large detuning ∆(n) g , as speciﬁed eﬀ (cid:48)(cid:48)(cid:48) rq n Eq. (23). The latter can be divided into the following parts: in Eq. (7). Moreover, H transforms into int ##LTLine## ##LTLine## ##LTLine##

text: croscopic

text: one,

text: given

text: in

text: Eq.

text: while

text: an

text: analogous

text: Kerr-type

text: Hamiltonian

text: in

text: Refs.

text: was

text: assumed

text: without

text: deriva-

text: tion.

text: Moreover,

text: we

text: studied

text: here

text: the

text: Kerr

text: interaction

text: under

text: H

text: H

text: H

text: H

text: sys

text: drv

text: int

text: two

text: different

text: resonance

text: conditions,

text: as

text: described

text: by

text: Eqs.

text: and

text: References

text: discussed

text: only

text: the

text: Kerr

text: in-

text: where

text: teraction

text: given

text: by

text: Eq.

text: We

text: analyzed

text: the

text: blockade

text: of

text: mechanical

text: phonons,

text: contrary

text: to

text: photon

text: blockade

text: studied

text: in

text: Refs.

text: Moreover,

text: the

text: interplay

text: between

text: single-

text: H

text: H

text: H

text: a

text: a

text: H

text: phonon

text: blockade

text: in

text: one

text: oscillator

text: and

text: single-photon

text: block-

text: sys

text: JC

text: n

text: n

text: n

text: JC

text: ade

text: in

text: another

text: oscillator

text: was

text: studied

text: in

text: Ref.

text: Here

text: we

text: n

text: predicted

text: tunable

text: k-phonon

text: blockades

text: (with

text: k

text: in

text: Hdrv

text: fn(a

text: a

text: each

text: oscillator,

text: where

text: the

text: k-phonon

text: Fock

text: state

text: impedes

text: the

text: n

text: excitation

text: of

text: more

text: phonons.

text: To

text: our

text: knowledge,

text: multiphonon

text: blockade

text: has

text: not

text: been

text: studied

text: before.

text: Here

text: H

text: is

text: given

text: by

text: Eq.

text: and

text: H

text: H

text: is

text: given

text: by

text: JC

text: int

text: int

text: Eq.

text: In

text: conclusion,

text: we

text: showed

text: here

text: a

text: rich

text: tapestry

text: of

text: phonon

text: Our

text: derivation

text: is

text: based

text: on

text: the

text: method

text: described

text: in

text: Ref.

text: blockade

text: effects

text: in

text: two

text: coupled

text: nonlinear

text: nanomechanical

text: for

text: the

text: exact

text: diagonalization

text: of

text: the

text: Jaynes-Cummings

text: model

text: resonators.

text: Different

text: types

text: of

text: phonon

text: blockade

text: could

text: be

text: with

text: the

text: following

text: unitary

text: transformation

text: “picked

text: out”

text: of

text: this

text: tapestry

text: by

text: controlling

text: the

text: nonlinearity

text: via

text: ancilla

text: TLS,

text: and

text: by

text: changing

text: the

text: driving

text: frequency

text: of

text: the

text: resonators

text: themselves.

text: Within

text: these

text: different

text: types

text: of

text: U

text: exp[

text: Λ(λ

text: )(a

text: ρ

text: a

text: ρ+)],

text: n

text: n

text: n

text: n

text: n

text: n

text: phonon

text: blockade,

text: the

text: coupled

text: NAMRs

text: can

text: be

text: made

text: to

text: be-

text: have

text: like

text: two

text: coupled

text: qubits,

text: a

text: qutrit

text: coupled

text: to

text: a

text: quartit,

text: or

text: even

text: two

text: coupled

text: qudits.

text: We

text: veriﬁed

text: this

text: picture

text: by

text: looking

text: where

text: √the

text: √operator

text: Λ(λ

text: n

text: at

text: the

text: nonclassical

text: properties

text: of

text: these

text: states

text: including

text: their

text: N

text: N

text: is

text: given

text: in

text: terms

text: of

text: the

text: n

text: n

text: n

text: single-NAMR

text: nonclassicality,

text: and

text: two-NAMR

text: entanglement

text: total

text: number

text: of

text: excitations

text: N

text: a

text: a

text: E

text: E

text: in

text: the

text: n

text: n

text: n

text: n

text: n

text: and

text: entanglement

text: dimensionality.

text: The

text: nonclassical

text: properties

text: nth

text: NAMR

text: and

text: TLS.

text: Thus,

text: the

text: expansions

text: of

text: the

text: annihilation

text: of

text: these

text: states

text: were

text: also

text: analyzed

text: in

text: phase

text: space

text: by

text: applying

text: operators,

text: a¯

text: U

text: aU

text: and

text: ρ¯

text: U

text: ρ

text: U

text: of

text: Ref.

text: can

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: the

text: s-parametrized

text: Cahill-Glauber

text: quasiprobability

text: distribu-

text: be

text: rewritten

text: for

text: our

text: dressed

text: qubit

text: states

text: as

text: follows:

text: tions

text: and,

text: in

text: particular,

text: the

text: Wigner

text: function.

text: We

text: expect

text: that,

text: if

text: realized

text: in

text: experiment,

text: the

text: ability

text: to

text: operate

text: in

text: these

text: dif-

text: O

text: ferent

text: regimes

text: will

text: have

text: a

text: range

text: of

text: applications

text: in

text: quantum

text: a¯n

text: O

text: information

text: and

text: quantum

text: technologies.

text: ρ¯

text: λ

text: a

text: ρz

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: Acknowledgement

text: O

text: where

text: h(k)

text: a

text: and

text: (λk)

text: denotes

text: the

text: n

text: n∼

text: n

text: n

text: n

text: A.M.

text: was

text: supported

text: by

text: the

text: Polish

text: National

text: Science

text: Cen-

text: omitted

text: terms

text: of

text: order

text: λk

text: and

text: higher.

text: Now

text: one

text: can

text: easily

text: n

text: tre

text: under

text: the

text: grants

text: No.

text: and

text: No.

text: transform

text: the

text: Hamiltonian

text: H

text: into

text: A.M.

text: acknowledges

text: a

text: long-term

text: fellowship

text: from

text: the

text: Japan

text: Society

text: for

text: the

text: Promotion

text: of

text: Sci-

text: ence

text: (JSPS).

text: Y.X.L.

text: was

text: supported

text: by

text: the

text: National

text: Natural

text: H¯

text: Science

text: Foundation

text: of

text: China

text: under

text: Grant

text: No.

text: the

text: National

text: Basic

text: Research

text: Program

text: of

text: China

text: Program)

text: In

text: particular,

text: by

text: H

text: transforms

text: into

text: under

text: Grant

text: No.

text: the

text: Tsinghua

text: University

text: drv

text: Initiative

text: Scientiﬁc

text: Research

text: Program,

text: and

text: the

text: Tsinghua

text: Na-

text: tional

text: Laboratory

text: for

text: Information

text: Science

text: and

text: Technology

text: H¯

text: f

text: [a¯

text: a¯

text: (TNList)

text: Cross-discipline

text: Foundation.

text: J.B.

text: was

text: supported

text: drv

text: n

text: n

text: n

text: by

text: the

text: Palacky´

text: University

text: under

text: the

text: Project

text: No.

text: IGA-PˇrF-

text: F.N.

text: was

text: partially

text: supported

text: by

text: the

text: RIKEN

text: iTHES

text: If

text: the

text: qubits

text: remain

text: in

text: the

text: excited

text: dressed-qubit

text: states

text: E

text: Project,

text: the

text: MURI

text: Center

text: for

text: Dynamic

text: Magneto-Optics

text: via

text: n

text: given

text: in

text: Eq.

text: then

text: the

text: AFOSR

text: award

text: number

text: the

text: IMPACT

text: program

text: of

text: JST,

text: and

text: a

text: Grant-in-Aid

text: for

text: Scientiﬁc

text: Research

text: O

text: (A).

text: E

text: E

text: H¯

text: E

text: E

text: f

text: (a

text: +a

text: drv

text: n

text: n

text: n

text: n

text: n

text: Appendix

text: A:

text: Derivation

text: of

text: the

text: effective

text: Hamiltonian

text: n

text: Here

text: we

text: show

text: how

text: to

text: derive

text: the

text: effective

text: Hamiltonian

text: The

text: assumption

text: of

text: a

text: “frozen”

text: state

text: of

text: both

text: qubits

text: is

text: physi-

text: H

text: given

text: by

text: Eq.

text: from

text: the

text: Hamiltonian

text: H

text: given

text: in

text: cally

text: justiﬁed

text: for

text: the

text: large

text: detuning

text: ∆(n)

text: g

text: as

text: speciﬁed

text: eﬀ

text: rq

text: n

text: Eq.

text: The

text: latter

text: can

text: be

text: divided

text: into

text: the

text: following

text: parts:

text: in

text: Eq.

text: Moreover,

text: H

text: transforms

text: into

text: int

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 13 (cid:88) † † H¯int = J12(a¯1 + a¯1)(a¯2 + a¯2), (A10) c (t) = exp(−iE(cid:48) t)(cid:104)E(cid:48) |00(cid:105)(cid:104)xy|E(cid:48) (cid:105), (B1) xy n n n n and, thus, (cid:48)(cid:48) | (cid:48)(cid:48)(cid:105) (cid:48)(cid:48)| (cid:48)(cid:48)(cid:105) (cid:104)E E |H¯ |E E (cid:105) = J(a +a†)(a +a†)+O(λ4), (A11) theasnedpraonballeomgos,ulselyt ufsolrimHiteﬀtheEdnime=nsiEonnoEfnthe. HTilobesritmspplaicfye 1 2 int 1 2 1 1 2 2 n to that of two qutrits and assume K = K = 10J and 1 2 J = F = F (as in Figs. 7 and 8). Then the Hamiltonians, 1 2 where J = J (1 + λ2/2)(1 + λ2/2). Analogously, by given in Eqs. (34) and (35), reduce to 12 1 2 generalizing the results of Ref. [81] for the dressed-qubit op-   erators ρz , one can ﬁnd that n    √ √   0 1 0 1 1 0 0 0 0  H¯ = U†H(cid:48)(cid:48)(cid:48) U(cid:88)− iU(cid:0)† ∂ U(cid:112) (cid:1) (A12)  1 √0 2 1 √1 2 0 0 0  sys sys ∂t  0 2 20 0 2 1 √0 √0 0  = H(cid:48)(cid:48)(cid:48) − (cid:88)1 δ 1 − 1 + 4N λ2 ρz (cid:48)  1 1 √0 0 1 √0 √2 √2 0  ≈ H0(cid:48)(cid:48)(cid:48) + 2 n2Kna† a + h ρz +n Kn (an† )2a2ρz , HJeﬀ =  10 √12 12 √01 √√02 202 02 22 √22  , 0 n n n n n n n n n √ √ √ 0 0 0 2 2 0 20 1 0 n √ √ 0 0 0 2 2 2 1 20 2 (cid:48)(cid:48)(cid:48) † where H is deﬁned in Eq. (A2), h = χ (a a + 1), 0 0 0 0 2 2 0 2 40 with χ = g0(cid:48) λ (1 − λ2), and the effectnive Kernr nonnlninear2ity (B2) n n−n(cid:48) n − (cid:48) reads K = g λ3 = (g )4/δ3. Moreover, U = U U n n n n(cid:88) n 1 2 and ∂ U = 0. Thus, one can write   ∂t (cid:104) | | (cid:105) †   E1E2 H¯sys E1E2 = Cn + Dnanan  20 1 √0 1 1 √0 0 0 0  n † O  1 √0 2 1 √1 2 0 0 0  +Kn(an)2a2n + (λ4n), (A13)  0 2 0 0 2 1 √0 √0 0    (cid:48)(cid:48)  1 1 √0 20 1 √0 √2 √2 0  H   terwmh)earnedCDn =≡χEn/−2 (ωw(hni)ch=c∆an b+e 2nKegle+cteχd a.sFaincaollnys,tathnet Jeﬀ =  10 √12 12 √01 √√02 02 02 22 √22  , effective Hamn iltonnian drv n n n 0 0 0 √2 √2 0 40 1 √0 √ √ 0 0 0 2 2 2 1 20 2 0 0 0 0 2 2 0 2 20 ≡ (cid:104) | | (cid:105) H E E (H¯ + H¯ + H¯ ) E E , (A14) (B3) eﬀ 1 2 sys drv int 1 2 ∼ O respectively. It is seen that the matrices, given by Eqs. (B2) is given explicitly by Eq. (28), where the terms (λ4n) and (B3), differ only in their diagonal terms. Unfortunately, are omitted. even in these special cases, it is very unlikely that exact analyt- Appendix B: Probability amplitudes in Eqs. (36) and (38) ical compact-form solutions of these eigenvalue problems can be found, as they require ﬁnding the roots of sixth and ninth (cid:104) | (cid:105) The probability amplitudes cxy(t) = xy ψ(t) (for x, y = order equations, respectively. Especially, highly-irregular os- 0ei,g.e..n,v3a)l,ugeivdeencoinmEpqoss.it(i3o6n)saHnd(cid:48) (3|E8)(cid:48),(cid:105)ca=nEbe(cid:48) o|Ebt(cid:48)a(cid:105)i,naesd using the cthililsactioonncsluosfiocnxy. (t) for model 2, as shown in Fig. 8, conﬁrm eﬀ n n n [1] X. M. H. Huang, C. A. Zorman, M. Mehregany, and 291 (2003). M. L. Roukes, Nanoelectromechanical systems: Nanodevice [3] M. P. Blencowe, Nanomechanical Quantum Limits, Science motion at microwave frequencies, Nature (London) 421, 496 304, 56 (2004). (2003). [4] M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, Ap- [2] R. G. Knobel and A. N. Cleland, Nanometre-scale displacement proaching the quantum limit of a nanomechanical resonator, sensing using a single electron transistor, Nature (London) 424, Science 304, 74 (2004). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: H¯int

text: c

text: (t)

text: xy

text: n

text: n

text: n

text: n

text: and,

text: thus,

text: E

text: |H¯

text: |E

text: E

text: J(a

text: +a†)(a

text: theasnedpraonballeomgos,ulselyt

text: ufsolrimHiteﬀtheEdnime=nsiEonnoEfnthe.

text: HTilobesritmspplaicfye

text: int

text: n

text: to

text: that

text: of

text: two

text: qutrits

text: and

text: assume

text: K

text: K

text: and

text: J

text: F

text: F

text: (as

text: in

text: Figs.

text: and

text: Then

text: the

text: Hamiltonians,

text: where

text: J

text: J

text: Analogously,

text: by

text: given

text: in

text: Eqs.

text: and

text: reduce

text: to

text: generalizing

text: the

text: results

text: of

text: Ref.

text: for

text: the

text: dressed-qubit

text: op-

text: erators

text: ρz

text: one

text: can

text: ﬁnd

text: that

text: n

text: H¯

text: √0

text: √1

text: sys

text: sys

text: ∂t

text: √0

text: √0

text: δ

text: λ2

text: ρz

text: √0

text: √0

text: √2

text: √2

text: a

text: h

text: ρz

text: +n

text: Kn

text: (an†

text: HJeﬀ

text: √√02

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: where

text: H

text: is

text: deﬁned

text: in

text: Eq.

text: h

text: χ

text: (a

text: a

text: with

text: χ

text: λ

text: and

text: the

text: effectnive

text: Kernr

text: n

text: n

text: reads

text: K

text: g

text: λ3

text: (g

text: Moreover,

text: U

text: U

text: U

text: n

text: n

text: n

text: n

text: and

text: U

text: Thus,

text: one

text: can

text: write

text: ∂t

text: H¯sys

text: Cn

text: Dnanan

text: √0

text: √0

text: n

text: O

text: √0

text: √1

text: √0

text: √0

text: √0

text: √0

text: √2

text: √2

text: H

text: terwmh)earnedCDn

text: (ωw(hni)ch=c∆an

text: b+e

text: a.sFaincaollnys,tathnet

text: Jeﬀ

text: √√02

text: effective

text: Hamn

text: iltonnian

text: drv

text: n

text: n

text: n

text: √2

text: √2

text: √0

text: H

text: E

text: E

text: (H¯

text: H¯

text: H¯

text: E

text: E

text: eﬀ

text: sys

text: drv

text: int

text: O

text: respectively.

text: It

text: is

text: seen

text: that

text: the

text: matrices,

text: given

text: by

text: Eqs.

text: is

text: given

text: explicitly

text: by

text: Eq.

text: where

text: the

text: terms

text: and

text: differ

text: only

text: in

text: their

text: diagonal

text: terms.

text: Unfortunately,

text: are

text: omitted.

text: even

text: in

text: these

text: special

text: cases,

text: it

text: is

text: very

text: unlikely

text: that

text: exact

text: analyt-

text: Appendix

text: B:

text: Probability

text: amplitudes

text: in

text: Eqs.

text: and

text: ical

text: compact-form

text: solutions

text: of

text: these

text: eigenvalue

text: problems

text: can

text: be

text: found,

text: as

text: they

text: require

text: ﬁnding

text: the

text: roots

text: of

text: sixth

text: and

text: ninth

text: The

text: probability

text: amplitudes

text: cxy(t)

text: xy

text: ψ(t)

text: (for

text: x,

text: y

text: order

text: equations,

text: respectively.

text: Especially,

text: highly-irregular

text: os-

text: using

text: the

text: cthililsactioonncsluosfiocnxy.

text: (t)

text: for

text: model

text: as

text: shown

text: in

text: Fig.

text: conﬁrm

text: eﬀ

text: n

text: n

text: n

text: X.

text: M.

text: H.

text: Huang,

text: C.

text: A.

text: Zorman,

text: M.

text: Mehregany,

text: and

text: M.

text: L.

text: Roukes,

text: Nanoelectromechanical

text: systems:

text: Nanodevice

text: M.

text: P.

text: Blencowe,

text: Nanomechanical

text: Quantum

text: Limits,

text: Science

text: motion

text: at

text: microwave

text: frequencies,

text: Nature

text: (London)

text: M.

text: D.

text: LaHaye,

text: O.

text: Buu,

text: B.

text: Camarota,

text: and

text: K.

text: C.

text: Schwab,

text: Ap-

text: R.

text: G.

text: Knobel

text: and

text: A.

text: N.

text: Cleland,

text: Nanometre-scale

text: displacement

text: proaching

text: the

text: quantum

text: limit

text: of

text: a

text: nanomechanical

text: resonator,

text: sensing

text: using

text: a

text: single

text: electron

text: transistor,

text: Nature

text: (London)

text: Science

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: 3 (cid:48) To transform the Hamiltonian H , given by Eq. (14), into (cid:29) JC The large detuning ∆(n) g implies that, e.g., the qubit the dressed-qubit basis, ﬁrst we note that rq n states cannot be ﬂipped by the interaction with the NAMR (see (cid:29) Appendix A). While the large detuning ∆n Ωn enables us − − to omit, in particular, the terms which do not conserve the aσ+ = cos2(x )aρ+ sin2(x )aρ + 1 sin(2x )aρz , (18) n n n n n 2 n n number of excitations in the TLS-NAMR interaction Hamil- tonian, as will be explained below. Note that the assumption − | (cid:105)(cid:104) | | (cid:105)(cid:104) | that ω(n) − ω(n) = ω(n) − ω(n) is not essential in our deriva- where ρn = Gn En , and ρ+n = En Gn . By res drv q drv recalling the assumption, given in Eq. (8), we can write tion, and is applied only for simplicity. ≈ aσ+ cos2(x )aρ+. For example, if Ω /∆ = 0.1 then First, we transform the Hamiltonian H, given in Eq. (1), n n n n n ≈ cos2(x ) = 0.9975, sin2(x ) = 0.0025, and sin(2x )/2 into a rotating re(cid:89)ference(cid:16)frame by the unitary transfor(cid:17)mation n n n 0.05. Thus, we can omit the second and third terms in Eq. (18), and their Hermitian conjugates, which do not con- UR(t) = exp −iωd(nrv)a†nant − 21iω¯d(nrv)σnzt , (9) scearnvaeptphreonxuimmabtee(cid:88)rlyobfeextrcaintastfioornms.edTihnetno, the Hamiltonian HJ(cid:48)C n which results in the following effective Hamiltonian (cid:48)(cid:48) ≈ (cid:48) − H g [a ρ+ exp( i∆(n)t) + H.c.], (19) JC n n n rq n (cid:48) † − † ∂ H = URHUR iUR ∂tUR, (10) where g(cid:48) = g cos2(x ). n n n To obtain a time-independent total Hamiltonian, we trans- which can be given explicitly as form it into a qubit rotating frame by applying the standard unitary transformation (cid:88) (cid:0) (cid:1) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) H = H + H + H + H , (11) res q JC int − (cid:48)(cid:48) H(cid:48) = (cid:88) ∆ a† a + f a + a† , (12) Uq = exp( iHq t). (20) res n n n n n n n H(cid:48) = (cid:88) ∆n σz + Ωn σx, (13) This resu(cid:88)lts in (cid:110) (cid:111) q 2 n 2 n n (cid:48) − HJC = gn[anσn+ exp( i∆(rnq)t) + H.c.], (14) H(cid:48)(cid:48)(cid:48) = g(cid:48) a ρ+ exp[−i(∆(n) − ∆¯ )t] + H.c. . JC n n n rq n n (cid:48) † † n H = J (a + a )(a + a ), (15) (21) int 12 1 1 2 2 − By assuming ∆(rnq) = ∆¯ n (for n = 1, 2), one obtains where σx = σ+ + σ . Note that this Hamiltonian is still the time-independent Jaynes-Cummings Hamiltonian in the n n n time dependent. dressed-qubit basis, (cid:88) Now we diagonalize the qubit Hamiltonian, given by Eq. (13), by transforming it into a dressed-qubit basis follow- (cid:48)(cid:48)(cid:48) (cid:48) † − ing the method described in, e.g., Refs. [61, 62]. Thus, one H = g (a ρ+ + a ρ ). (22) JC n n n n ﬁnds (cid:88) n Thus, after these transformations, the total Hamiltonian (cid:48)(cid:48) ∆¯ (cid:112)H = n ρz , (16) reads q 2 n n (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48) where ∆¯ = ∆2 + Ω2 and the dressed-qubit operator H = H + H + H , (23) | (cid:105)(cid:104)n | − | n (cid:105)(cid:104) n| res JC int ρz = E E G G can be deﬁned by the dressed- n n n n n qubit basis states [62]: (cid:48)(cid:48)(cid:48) (cid:48) (cid:48)(cid:48)(cid:48) (cid:48) (cid:48)(cid:48)(cid:48) where H = H and H = H . Note that H = 0. int int res res q C. Effective Hamiltonian with a qubit-induced nonlinearity | (cid:105) | (cid:105) | (cid:105) E = cos x e + sin x g , n n n n n | (cid:105) − | (cid:105) | (cid:105) G = sin x e + cos x g , (17) We assume another large detuning n n n n n − ≡ − (cid:29) (cid:48) where xn = (1/2) tan 1(Ωn/∆n). It is seen that the δn ∆¯ n ∆n gn, (24) dressed nth qubit refers to the nth TLS dressed with the nth NAMR phononic ﬁeld (for a general discussion see Ref. [61]). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: To

text: transform

text: the

text: Hamiltonian

text: H

text: given

text: by

text: Eq.

text: into

text: JC

text: The

text: large

text: detuning

text: ∆(n)

text: g

text: implies

text: that,

text: e.g.,

text: the

text: qubit

text: the

text: dressed-qubit

text: basis,

text: ﬁrst

text: we

text: note

text: that

text: rq

text: n

text: states

text: cannot

text: be

text: ﬂipped

text: by

text: the

text: interaction

text: with

text: the

text: NAMR

text: (see

text: Appendix

text: A).

text: While

text: the

text: large

text: detuning

text: ∆n

text: Ωn

text: enables

text: us

text: to

text: omit,

text: in

text: particular,

text: the

text: terms

text: which

text: do

text: not

text: conserve

text: the

text: aσ+

text: )aρ+

text: )aρ

text: )aρz

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: number

text: of

text: excitations

text: in

text: the

text: TLS-NAMR

text: interaction

text: Hamil-

text: tonian,

text: as

text: will

text: be

text: explained

text: below.

text: Note

text: that

text: the

text: assumption

text: that

text: ω(n)

text: ω(n)

text: ω(n)

text: ω(n)

text: is

text: not

text: essential

text: in

text: our

text: deriva-

text: where

text: ρn

text: Gn

text: En

text: and

text: ρ+n

text: En

text: Gn

text: By

text: res

text: drv

text: q

text: drv

text: recalling

text: the

text: assumption,

text: given

text: in

text: Eq.

text: we

text: can

text: write

text: tion,

text: and

text: is

text: applied

text: only

text: for

text: simplicity.

text: aσ+

text: )aρ+.

text: For

text: example,

text: if

text: Ω

text: then

text: First,

text: we

text: transform

text: the

text: Hamiltonian

text: H,

text: given

text: in

text: Eq.

text: n

text: n

text: n

text: n

text: n

text: and

text: into

text: a

text: rotating

text: by

text: the

text: unitary

text: n

text: n

text: n

text: Thus,

text: we

text: can

text: omit

text: the

text: second

text: and

text: third

text: terms

text: in

text: Eq.

text: and

text: their

text: Hermitian

text: conjugates,

text: which

text: do

text: not

text: con-

text: UR(t)

text: exp

text: −iωd(nrv)a†nant

text: the

text: Hamiltonian

text: n

text: which

text: results

text: in

text: the

text: following

text: effective

text: Hamiltonian

text: H

text: g

text: [a

text: ρ+

text: exp(

text: i∆(n)t)

text: H.c.],

text: JC

text: n

text: n

text: n

text: rq

text: n

text: H

text: URHUR

text: iUR

text: ∂tUR,

text: where

text: g

text: n

text: n

text: n

text: To

text: obtain

text: a

text: time-independent

text: total

text: Hamiltonian,

text: we

text: trans-

text: which

text: can

text: be

text: given

text: explicitly

text: as

text: form

text: it

text: into

text: a

text: qubit

text: rotating

text: frame

text: by

text: applying

text: the

text: standard

text: unitary

text: transformation

text: H

text: H

text: H

text: H

text: H

text: res

text: q

text: JC

text: int

text: a†

text: a

text: f

text: a

text: a†

text: Uq

text: exp(

text: iHq

text: t).

text: res

text: n

text: n

text: n

text: n

text: n

text: n

text: n

text: ∆n

text: σz

text: Ωn

text: σx,

text: This

text: in

text: q

text: n

text: n

text: n

text: HJC

text: gn[anσn+

text: exp(

text: i∆(rnq)t)

text: H.c.],

text: a

text: ρ+

text: exp[−i(∆(n)

text: )t]

text: H.c.

text: JC

text: n

text: n

text: n

text: rq

text: n

text: n

text: n

text: H

text: J

text: (a

text: a

text: )(a

text: a

text: int

text: By

text: assuming

text: ∆(rnq)

text: n

text: (for

text: n

text: one

text: obtains

text: where

text: σx

text: σ+

text: σ

text: Note

text: that

text: this

text: Hamiltonian

text: is

text: still

text: the

text: time-independent

text: Jaynes-Cummings

text: Hamiltonian

text: in

text: the

text: n

text: n

text: n

text: time

text: dependent.

text: dressed-qubit

text: basis,

text: Now

text: we

text: diagonalize

text: the

text: qubit

text: Hamiltonian,

text: given

text: by

text: Eq.

text: by

text: transforming

text: it

text: into

text: a

text: dressed-qubit

text: basis

text: follow-

text: ing

text: the

text: method

text: described

text: in,

text: e.g.,

text: Refs.

text: Thus,

text: one

text: H

text: g

text: (a

text: ρ+

text: a

text: ρ

text: JC

text: n

text: n

text: n

text: n

text: ﬁnds

text: n

text: Thus,

text: after

text: these

text: transformations,

text: the

text: total

text: Hamiltonian

text: n

text: ρz

text: reads

text: q

text: n

text: n

text: where

text: ∆2

text: Ω2

text: and

text: the

text: dressed-qubit

text: operator

text: H

text: H

text: H

text: H

text: n

text: n|

text: res

text: JC

text: int

text: ρz

text: E

text: E

text: G

text: G

text: can

text: be

text: deﬁned

text: by

text: the

text: dressed-

text: n

text: n

text: n

text: n

text: n

text: qubit

text: basis

text: states

text: where

text: H

text: H

text: and

text: H

text: H

text: Note

text: that

text: H

text: int

text: int

text: res

text: res

text: q

text: C.

text: Effective

text: Hamiltonian

text: with

text: a

text: qubit-induced

text: nonlinearity

text: E

text: cos

text: x

text: e

text: sin

text: x

text: g

text: n

text: n

text: n

text: n

text: n

text: G

text: sin

text: x

text: e

text: cos

text: x

text: g

text: We

text: assume

text: another

text: large

text: detuning

text: n

text: n

text: n

text: n

text: n

text: where

text: xn

text: tan

text: It

text: is

text: seen

text: that

text: the

text: δn

text: n

text: ∆n

text: gn,

text: dressed

text: nth

text: qubit

text: refers

text: to

text: the

text: nth

text: TLS

text: dressed

text: with

text: the

text: nth

text: NAMR

text: phononic

text: ﬁeld

text: (for

text: a

text: general

text: discussion

text: see

text: Ref.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 FIG. 4: (Color online) Energy levels E , given by FIG. 3: (Color online) Energy levels for the Kerr-type Hamiltonians H¯|m , m (cid:105) =(cid:80)E |m , m (cid:105), for a simpliﬁedm1mmo2del 2, i.e., the gEive≡n Ein (a) Eq. (29) (with n = 1) and (b) Eq. (33) (with n = 2 and system1 of2two uncmou1pmle2d no1nlin2ear NAMRs, described by the Hamil- 12). E ≡ E E 2 2 tonian H¯ = H(n) , given in Eq. (29) assuming = . n Kerr 1 2 Here, m and m are the Fock states of the two NAMRs. It is seen 1 2 (cid:54) E that the levels E and E are off-resonance if K , K = /2 and, than the Kerr nonlinearities K , strong single-time photon (or 20 02 1 2 n thus, they are much less occupied than the other shown levels. This phonon) antibunching can still be observed as an indicator of explains the occurrence of phonon blockade in this coupled system. the photon (phonon) blockade [64, 65]. We will analyze free and dissipative evolutions of the NAMR systems described by the Hamiltonian given in Eq. (28), under two different resonance conditions, as spec- iﬁed later in Eqs. (34) and (35). III. PHONON BLOCKADE A. Phonon blockade in two models In this section we will analyze phonon blockade in two models, which are special cases of the general model de- scribed by H in Eq. (28) under different resonance condi- eﬀ tions. FIG. 5: (Color online) Energy levels for a single NAMR, as a sim- In one model, we assume that the frequency ω(n) of the pliﬁed version of model 2: How to induce a three-phonon resonant drv driving ﬁeld of the nth NAMR is tuned precisely to the effec- transition in a single nonlinear NAMR described by the Hamiltonian E tive free energy (n = 1, 2). Then, the effective Hamilto- given in Eq. (33). If the driving ﬁeld frequency is resonant with the n | (cid:105) | (cid:105) − E nian H ,(cid:88)given in Eq. (28), simpliﬁes to transition between the energy levels 1 and 2 , E E = , then eﬀ 2 1 one can also induce the three-phonon transition between the energy | (cid:105) | (cid:105) − E H(cid:48) = (cid:88)K a†(cid:0)a (a† a (cid:1)− 1) lreevaell(svi0rtuaanl)den3er,gsyinlceevEels3. WEe0n=ote3th.aStoolnidly(dsainshgeled-)plhinoensondetnraonte- eﬀ n n n n n sitions can be observed if the driving ﬁeld frequency is tuned with n | (cid:105) | (cid:105) (cid:54) † † † the transition between other levels n and n + 1 (n = 1). More- + F a + a + J(a + a )(a + a ), (34) n n n 1 1 2 2 over, we cannot observe such multiphonon transitions if the system n is described by the Hamiltonian given in Eq. (29) for any n. which is referred here to as model 1. The occurrence of single-phonon blockade in this model is explained in Fig. 4. In another model, we set the frequency ω(n) of the driving Some lowest energy levels for the Kerr-type Hamiltonian of ﬁeld of the ﬁrst (second) NAMR to be tuneddwrvith the effective Eq. (35) for n = 1 are shown in Fig. 3(a) in comparison with fHreaemeilntoerngiaynEH1(cid:48)(cid:48) =rEed1u(cEe2(cid:48)s(cid:48) t=o: E2 + 2K2). Thus, the effective tFhiogs.e3(fbo)r snho=ws2thsahtoEw3n−inEF0ig=. 33(bE)2. ≡A 3cElo,swerhiacnhaliymsipslioesf eﬀ that a three-phonon resonant transition can be observed, as (cid:88) (cid:0) (cid:1) shown in Fig. 5. The occurrence of multiphonon blockade in (cid:48)(cid:48) † † − † − † − H = K a a (a a 1) + K (a a 1)(a a 2) this coupled system can clearly be understood by analyzing eﬀ 1 1 1 1 1 2 2 2 2 2 † † † Fig. 6. + F a + a + J(a + a )(a + a ), (35) n n n 1 1 2 2 B. Phonon blockade in non-dissipative systems n which is referred here to as model 2. Note that we ignored In order to show phonon blockade in the non-dissipative in Eq. (35) the irrelevant constant terms Ckl (for n = 1, 2). model 1, we start from the analysis of the system, described n ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

text: online)

text: Energy

text: levels

text: E

text: given

text: by

text: FIG.

text: (Color

text: online)

text: Energy

text: levels

text: for

text: the

text: Kerr-type

text: Hamiltonians

text: H¯|m

text: m

text: |m

text: m

text: for

text: a

text: i.e.,

text: the

text: gEive≡n

text: Ein

text: (a)

text: Eq.

text: (with

text: n

text: and

text: (b)

text: Eq.

text: (with

text: n

text: and

text: NAMRs,

text: described

text: by

text: the

text: Hamil-

text: E

text: E

text: E

text: tonian

text: H¯

text: H(n)

text: given

text: in

text: Eq.

text: assuming

text: n

text: Kerr

text: Here,

text: m

text: and

text: m

text: are

text: the

text: Fock

text: states

text: of

text: the

text: two

text: NAMRs.

text: It

text: is

text: seen

text: E

text: that

text: the

text: levels

text: E

text: and

text: E

text: are

text: off-resonance

text: if

text: K

text: K

text: and,

text: than

text: the

text: Kerr

text: nonlinearities

text: K

text: strong

text: single-time

text: photon

text: (or

text: n

text: thus,

text: they

text: are

text: much

text: less

text: occupied

text: than

text: the

text: other

text: shown

text: levels.

text: This

text: phonon)

text: antibunching

text: can

text: still

text: be

text: observed

text: as

text: an

text: indicator

text: of

text: explains

text: the

text: occurrence

text: of

text: phonon

text: blockade

text: in

text: this

text: coupled

text: system.

text: the

text: photon

text: (phonon)

text: blockade

text: We

text: will

text: analyze

text: free

text: and

text: dissipative

text: evolutions

text: of

text: the

text: NAMR

text: systems

text: described

text: by

text: the

text: Hamiltonian

text: given

text: in

text: Eq.

text: under

text: two

text: different

text: resonance

text: conditions,

text: as

text: spec-

text: iﬁed

text: later

text: in

text: Eqs.

text: and

text: III.

text: PHONON

text: BLOCKADE

text: A.

text: Phonon

text: blockade

text: in

text: two

text: models

text: In

text: this

text: section

text: we

text: will

text: analyze

text: phonon

text: blockade

text: in

text: two

text: models,

text: which

text: are

text: special

text: cases

text: of

text: the

text: general

text: model

text: de-

text: scribed

text: by

text: H

text: in

text: Eq.

text: under

text: different

text: resonance

text: condi-

text: eﬀ

text: tions.

text: FIG.

text: (Color

text: online)

text: Energy

text: levels

text: for

text: a

text: single

text: NAMR,

text: as

text: a

text: sim-

text: In

text: one

text: model,

text: we

text: assume

text: that

text: the

text: frequency

text: ω(n)

text: of

text: the

text: pliﬁed

text: version

text: of

text: model

text: How

text: to

text: induce

text: a

text: three-phonon

text: resonant

text: drv

text: driving

text: ﬁeld

text: of

text: the

text: nth

text: NAMR

text: is

text: tuned

text: precisely

text: to

text: the

text: effec-

text: transition

text: in

text: a

text: single

text: nonlinear

text: NAMR

text: described

text: by

text: the

text: Hamiltonian

text: E

text: tive

text: free

text: energy

text: (n

text: Then,

text: the

text: effective

text: Hamilto-

text: given

text: in

text: Eq.

text: If

text: the

text: driving

text: ﬁeld

text: frequency

text: is

text: resonant

text: with

text: the

text: n

text: E

text: nian

text: H

text: in

text: Eq.

text: simpliﬁes

text: to

text: transition

text: between

text: the

text: energy

text: levels

text: and

text: E

text: E

text: then

text: eﬀ

text: one

text: can

text: also

text: induce

text: the

text: three-phonon

text: transition

text: between

text: the

text: energy

text: E

text: (a†

text: a

text: eﬀ

text: n

text: n

text: n

text: n

text: n

text: sitions

text: can

text: be

text: observed

text: if

text: the

text: driving

text: ﬁeld

text: frequency

text: is

text: tuned

text: with

text: n

text: the

text: transition

text: between

text: other

text: levels

text: n

text: and

text: n

text: (n

text: More-

text: F

text: a

text: a

text: J(a

text: a

text: )(a

text: a

text: n

text: n

text: n

text: over,

text: we

text: cannot

text: observe

text: such

text: multiphonon

text: transitions

text: if

text: the

text: system

text: n

text: is

text: described

text: by

text: the

text: Hamiltonian

text: given

text: in

text: Eq.

text: for

text: any

text: n.

text: which

text: is

text: referred

text: here

text: to

text: as

text: model

text: The

text: occurrence

text: of

text: single-phonon

text: blockade

text: in

text: this

text: model

text: is

text: explained

text: in

text: Fig.

text: In

text: another

text: model,

text: we

text: set

text: the

text: frequency

text: ω(n)

text: of

text: the

text: driving

text: Some

text: lowest

text: energy

text: levels

text: for

text: the

text: Kerr-type

text: Hamiltonian

text: of

text: ﬁeld

text: of

text: the

text: ﬁrst

text: (second)

text: NAMR

text: to

text: be

text: tuneddwrvith

text: the

text: effective

text: Eq.

text: for

text: n

text: are

text: shown

text: in

text: Fig.

text: in

text: comparison

text: with

text: t=o:

text: Thus,

text: the

text: effective

text: ≡A

text: eﬀ

text: that

text: a

text: three-phonon

text: resonant

text: transition

text: can

text: be

text: observed,

text: as

text: shown

text: in

text: Fig.

text: The

text: occurrence

text: of

text: multiphonon

text: blockade

text: in

text: H

text: K

text: a

text: a

text: (a

text: a

text: K

text: (a

text: a

text: a

text: this

text: coupled

text: system

text: can

text: clearly

text: be

text: understood

text: by

text: analyzing

text: eﬀ

text: Fig.

text: F

text: a

text: a

text: J(a

text: a

text: )(a

text: a

text: n

text: n

text: n

text: B.

text: Phonon

text: blockade

text: in

text: non-dissipative

text: systems

text: n

text: which

text: is

text: referred

text: here

text: to

text: as

text: model

text: Note

text: that

text: we

text: ignored

text: In

text: order

text: to

text: show

text: phonon

text: blockade

text: in

text: the

text: non-dissipative

text: in

text: Eq.

text: the

text: irrelevant

text: constant

text: terms

text: Ckl

text: (for

text: n

text: model

text: we

text: start

text: from

text: the

text: analysis

text: of

text: the

text: system,

text: described

text: n

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: Electromagnetic form factors in the Hypercentral Constituent Quark Model M. De Sanctis Istituto Nazionale di Fisica Nucleare, Sezione di Roma, P.le A. Moro 2, I-00185 Roma, Italy M.M. Giannini, E. Santopinto, A. Vassallo Dipartimento di Fisica dell’Universit`a di Genova and Istituto Nazionale di Fisica Nucleare, Sezione di Genova, via Dodecaneso 33, I-16146 Genova, Italy Abstract We discuss the recent experimental results on the ratio bertween the electric and magnetic proton form factors and how they can be described by the hypercentral Constituent Quark Model. 1 INTRODUCTION The interest in the electromagnetic form factors of the nucleon has been increased by the recent results of the Jeﬀerson Laboratory on the ratio between the electric and magnetic form factors of the proton [1, 2]. At variance with the ≥ expectations, the ratio deviates strongly from 1 and, for Q2 1 (GeV/c)2, the ratio decreases with an almost linear behaviour, pointing towards the possible ≈ existence of a zero at Q2 8 (GeV/c)2. These unexpected results pose some problems. The ﬁrst one is the compat- ibility of the new data with the traditional ones obtained from a Rosenbluth plot. In this respect much attention is been devoted to the two-photon exchange mechanisms twoph, which however seem to be too small for reconciling the two sets of data. A critical re-analysis of the Rosenbluth procedure is also being performed [4], with promising results. The main further problem is the physical picture emerging from the data, that is the origin of the decrease of the ratio and the eventual presence of a zero in the electric form factor forces. The planned experiments at higher Q2 will provide the answer about the occurrence of a zero in the electric form factor of the proton. From the theo- retical point of view such zero is a challenge for most theoretical models for the internal proton structure. 1

text: Electromagnetic

label: title

text: form

text: factors

text: in

text: the

text: Hypercentral

text: Constituent

text: Quark

text: Model

text: M.

label: author

text: De

text: Sanctis

text: Istituto

text: Nazionale

text: di

text: Fisica

text: Nucleare,

text: Sezione

text: di

text: Roma,

text: P.le

text: A.

text: Moro

text: Roma,

text: Italy

text: M.M.

text: Giannini,

text: E.

text: Santopinto,

text: A.

text: Vassallo

text: Dipartimento

text: di

text: Fisica

text: dell’Universit`a

text: di

text: Genova

text: and

text: Istituto

text: Nazionale

text: di

text: Fisica

text: Nucleare,

text: Sezione

text: di

text: Genova,

text: via

text: Dodecaneso

text: Genova,

text: Italy

text: Abstract

text: We

label: abstract

text: discuss

text: the

text: recent

text: experimental

text: results

text: on

text: the

text: ratio

text: bertween

text: the

text: electric

text: and

text: magnetic

text: proton

text: form

text: factors

text: and

text: how

text: they

text: can

text: be

text: described

text: by

text: the

text: hypercentral

text: Constituent

text: Quark

text: Model.

label: section

text: INTRODUCTION

text: The

text: interest

text: in

text: the

text: electromagnetic

text: form

text: factors

text: of

text: the

text: nucleon

text: has

text: been

text: increased

text: by

text: the

text: recent

text: results

text: of

text: the

text: Jeﬀerson

text: Laboratory

text: on

text: the

text: ratio

text: between

text: the

text: electric

text: and

text: magnetic

text: form

text: factors

text: of

text: the

text: proton

text: At

text: variance

text: with

text: the

text: expectations,

text: the

text: ratio

text: deviates

text: strongly

text: from

text: and,

text: for

text: the

text: ratio

text: decreases

text: with

text: an

text: almost

text: linear

text: behaviour,

text: pointing

text: towards

text: the

text: possible

text: existence

text: of

text: a

text: zero

text: at

text: These

text: unexpected

text: results

text: pose

text: some

text: problems.

text: The

text: ﬁrst

text: one

text: is

text: the

text: compat-

text: ibility

text: of

text: the

text: new

text: data

text: with

text: the

text: traditional

text: ones

text: obtained

text: from

text: a

text: Rosenbluth

text: plot.

text: In

text: this

text: respect

text: much

text: attention

text: is

text: been

text: devoted

text: to

text: the

text: two-photon

text: exchange

text: mechanisms

text: twoph,

text: which

text: however

text: seem

text: to

text: be

text: too

text: small

text: for

text: reconciling

text: the

text: two

text: sets

text: of

text: data.

text: A

text: critical

text: re-analysis

text: of

text: the

text: Rosenbluth

text: procedure

text: is

text: also

text: being

text: performed

text: with

text: promising

text: results.

text: The

text: main

text: further

text: problem

text: is

text: the

text: physical

text: picture

text: emerging

text: from

text: the

text: data,

text: that

text: is

text: the

text: origin

text: of

text: the

text: decrease

text: of

text: the

text: ratio

text: and

text: the

text: eventual

text: presence

text: of

text: a

text: zero

text: in

text: the

text: electric

text: form

text: factor

text: forces.

text: The

text: planned

text: experiments

text: at

text: higher

text: will

text: provide

text: the

text: answer

text: about

text: the

text: occurrence

text: of

text: a

text: zero

text: in

text: the

text: electric

text: form

text: factor

text: of

text: the

text: proton.

text: From

text: the

text: theo-

text: retical

text: point

text: of

text: view

text: such

text: zero

text: is

text: a

text: challenge

text: for

text: most

text: theoretical

text: models

text: for

text: the

text: internal

text: proton

text: structure.


text: In this contribution we report the results of recent relativistic calculations of the elstic nucleon form factors within the hypercentral Constituent Quark Model (hCQM). 2 THE HYPERCENTRAL CONSTITUENT QUARK MODEL In the hCQM the SU(6) invariant quark potential is assumed to be [5] − V (x) = frac1x + αx (1) (cid:112) where x is the hyperradius. x = ρ2 + λ2, (2) Interactions of the type linear plus Coulomb-like have been used since long time for the meson sector, e.g. the Cornell potential. This form has been obtained in recent Lattice QCD calculations [6, 7] for SU(3) invariant static quark sources. The three-quark potential 1, depending on the hyperradius x only, is hyper- central. Then it can be considered as the lattice two-body interaction within the so called hypercentral approximation, that is averaged over the hyperan- gle ξ = arctg(ρ) and the angles Ω , Ω ; this approximation has been shown λ ρ λ to be valid, specially for the lower energy states [8]. On the other hand, the hyperradius x depends on the coordinates of all the three quarks, therefore the interaction V (x) is in general a three-body potential. − The ’hypercoulomb’ part of τ of the potential 1 has interesting properties x [8, 9]. It leads to a power-law behaviour of the proton form factor and of all the transition form factors and it has a perfect degeneracy between the ﬁrst 0+ − excitated state and the ﬁrst 1 states. The SU(6) violation is taking into account by adding a standard hyperﬁne interaction H [10], treated as a perturbation. The three quark hamiltonian hyp for the hCQM is then [5] p2 p2 − τ H = ρ + λ + αx + H (3) 2m 2m x hyp − The spectrum is described with τ = 4.59 and α = 1.61 fm 2 and − the standard strength of the hyperﬁne interaction needed for the N ∆ mass diﬀerence [10]. 3 THE ELECTROMAGNETIC FORM FACTORS The model has been used for the prediction of various physical quantities of interest, namely the photocouplings [11], the electromagnetic transition am- plitudes [12], the elastic nucleon form factors [13] and the ratio between the electric and magnetic proton form factors [14]. 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: this

text: contribution

text: we

text: report

text: the

text: results

text: of

text: recent

text: relativistic

text: calculations

text: of

text: the

text: elstic

text: nucleon

text: form

text: factors

text: within

text: the

text: hypercentral

text: Constituent

text: Quark

text: Model

text: (hCQM).

text: THE

text: HYPERCENTRAL

text: CONSTITUENT

text: QUARK

text: MODEL

text: In

text: the

text: hCQM

text: the

text: invariant

text: quark

text: potential

text: is

text: assumed

text: to

text: be

text: V

text: (x)

text: αx

text: where

text: x

text: is

text: the

text: hyperradius.

text: x

text: ρ2

text: Interactions

text: of

text: the

text: type

text: linear

text: plus

text: Coulomb-like

text: have

text: been

text: used

text: since

text: long

text: time

text: for

text: the

text: meson

text: sector,

text: e.g.

text: the

text: Cornell

text: potential.

text: This

text: form

text: has

text: been

text: obtained

text: in

text: recent

text: Lattice

text: QCD

text: calculations

text: for

text: invariant

text: static

text: quark

text: sources.

text: The

text: three-quark

text: potential

text: depending

text: on

text: the

text: hyperradius

text: x

text: only,

text: is

text: hyper-

text: central.

text: Then

text: it

text: can

text: be

text: considered

text: as

text: the

text: lattice

text: two-body

text: interaction

text: within

text: the

text: so

text: called

text: hypercentral

text: approximation,

text: that

text: is

text: averaged

text: over

text: the

text: hyperan-

text: gle

text: ξ

text: arctg(ρ)

text: and

text: the

text: angles

text: Ω

text: Ω

text: this

text: approximation

text: has

text: been

text: shown

text: λ

text: ρ

text: λ

text: to

text: be

text: valid,

text: specially

text: for

text: the

text: lower

text: energy

text: states

text: On

text: the

text: other

text: hand,

text: the

text: hyperradius

text: x

text: depends

text: on

text: the

text: coordinates

text: of

text: all

text: the

text: three

text: quarks,

text: therefore

text: the

text: interaction

text: V

text: (x)

text: is

text: in

text: general

text: a

text: three-body

text: potential.

text: The

text: ’hypercoulomb’

text: part

text: of

text: τ

text: of

text: the

text: potential

text: has

text: interesting

text: properties

text: x

text: It

text: leads

text: to

text: a

text: power-law

text: behaviour

text: of

text: the

text: proton

text: form

text: factor

text: and

text: of

text: all

text: the

text: transition

text: form

text: factors

text: and

text: it

text: has

text: a

text: perfect

text: degeneracy

text: between

text: the

text: ﬁrst

text: excitated

text: state

text: and

text: the

text: ﬁrst

text: states.

text: The

text: violation

text: is

text: taking

text: into

text: account

text: by

text: adding

text: a

text: standard

text: hyperﬁne

text: interaction

text: H

text: treated

text: as

text: a

text: perturbation.

text: The

text: three

text: quark

text: hamiltonian

text: hyp

text: for

text: the

text: hCQM

text: is

text: then

text: τ

text: H

text: ρ

text: λ

text: αx

text: H

text: x

text: hyp

text: The

text: spectrum

text: is

text: described

text: with

text: τ

text: and

text: α

text: fm

text: and

text: the

text: standard

text: strength

text: of

text: the

text: hyperﬁne

text: interaction

text: needed

text: for

text: the

text: N

text: mass

text: diﬀerence

text: THE

text: ELECTROMAGNETIC

text: FORM

text: FACTORS

text: The

text: model

text: has

text: been

text: used

text: for

text: the

text: prediction

text: of

text: various

text: physical

text: quantities

text: of

text: interest,

text: namely

text: the

text: photocouplings

text: the

text: electromagnetic

text: transition

text: am-

text: plitudes

text: the

text: elastic

text: nucleon

text: form

text: factors

text: and

text: the

text: ratio

text: between

text: the

text: electric

text: and

text: magnetic

text: proton

text: form

text: factors

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:0) (cid:1) (cid:0) (cid:1) Thus, using the deﬁnition of the chain diﬀerential (2), we have δ exp g(x); δg(x; η) = exp g(x) δg(x; η). (53) Using the chain rule (6) ends the proof. References [1] J.-P. Aubin and H. Frankowska. Set Valued Analysis. Birkhauser, 1990. [2] P. Bernhard. Chain diﬀerentials with an application to the mathematical fear operator. Nonlinear Analysis, 62:1225–1233, 2005. [3] D. E. Clark and J. Houssineau. Fa`a di Bruno’s formula for chain diﬀerentials. arXiv:1310.2833, 2013. [4] F. Fa`a di Bruno. Note sur un nouvelle formule de calcul diﬀ´erentiel. Quarterly Journal of Pure and Applied Mathematics, 1:359–360, 1857. [5] R. Gˆateaux. Fonctions d’une inﬁnit´e de variables ind´ependantes. Bulletin de la Soci´et´e Math´ematique de France, 47:70–96, 1919. [6] M. Hardy. Combinatorics of partial derivatives. Electronic Journal of Combinatorics, 13:R1, 2006. [7] H.-N. Huang, S. A. M. Marcantognini, and N. J. Young. Chain Rules for Higher Derivatives. The Mathematical Intelligencer, 28(2):61–69, 2006. [8] T.W. Ma. Higher Chain Formula proved by Combinatorics. Electronic Journal of Combinatorics, 16:N21, 2009. [9] R. Stanley. Enumerative Combinatorics, volume 1. Cambridge University Press, 2 edition, 2012. 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Thus,

text: using

text: the

text: deﬁnition

text: of

text: the

text: chain

text: diﬀerential

text: we

text: have

text: δ

text: exp

text: g(x);

text: δg(x;

text: η)

text: exp

text: g(x)

text: δg(x;

text: η).

text: Using

text: the

text: chain

text: rule

text: ends

text: the

text: proof.

text: References

label: reference

text: J.-P.

text: Aubin

text: and

text: H.

text: Frankowska.

text: Set

text: Valued

text: Analysis.

text: Birkhauser,

text: P.

text: Bernhard.

text: Chain

text: diﬀerentials

text: with

text: an

text: application

text: to

text: the

text: mathematical

text: fear

text: operator.

text: Nonlinear

text: Analysis,

text: D.

text: E.

text: Clark

text: and

text: J.

text: Houssineau.

text: Fa`a

text: di

text: Bruno’s

text: formula

text: for

text: chain

text: diﬀerentials.

text: F.

text: Fa`a

text: di

text: Bruno.

text: Note

text: sur

text: un

text: nouvelle

text: formule

text: de

text: calcul

text: diﬀ´erentiel.

text: Quarterly

text: Journal

text: of

text: Pure

text: and

text: Applied

text: Mathematics,

text: R.

text: Gˆateaux.

text: Fonctions

text: d’une

text: inﬁnit´e

text: de

text: variables

text: ind´ependantes.

text: Bulletin

text: de

text: la

text: Soci´et´e

text: Math´ematique

text: de

text: France,

text: M.

text: Hardy.

text: Combinatorics

text: of

text: partial

text: derivatives.

text: Electronic

text: Journal

text: of

text: Combinatorics,

text: H.-N.

text: Huang,

text: S.

text: A.

text: M.

text: Marcantognini,

text: and

text: N.

text: J.

text: Young.

text: Chain

text: Rules

text: for

text: Higher

text: Derivatives.

text: The

text: Mathematical

text: Intelligencer,

text: T.W.

text: Ma.

text: Higher

text: Chain

text: Formula

text: proved

text: by

text: Combinatorics.

text: Electronic

text: Journal

text: of

text: Combinatorics,

text: R.

text: Stanley.

text: Enumerative

text: Combinatorics,

text: volume

text: Cambridge

text: University

text: Press,

text: edition,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##