text: decompose the resulting evolution into discrete steps. With an ion trap, a digital simulation of this system can be implemented efﬁciently using steps of the form (cid:16) (cid:17) (cid:16) (cid:17) π π UM† S 2 UσzmUMS 2 . (1) Here, UMS(π2 ) is an MS gate and Uσzm is a local rotation on the mth ion in the trap. The MS gate couples every pair of ions in the trap evenly, UMS(θ) = e−iθ4 ∑Lk,l=1σxkσxl, (2) where σk is the Pauli x operator on ion k, and L the total number of ions. Although MS gates can be implemented with high x ﬁdelity in ion traps22,23, they are much slower than the trap period due to their reliance on the Lamb-Dicke regime and the vibrational rotating-wave approximation. Furthermore, as the number of addressed ions increases to scale up the simulation, the motional sideband spectral density increases and the gate time increases accordingly. It would thus be difﬁcult to use MS gates, or alternative sideband-resolving gates, for a digital quantum simulation of a sufﬁcient size to outperform current classical computers, with dozens of interacting spins or fermions. It was also shown in the simulation proposal17 that each MS gate can be replaced by an ‘ultrafast multiqubit’ (UMQ) gate in the simulation step, UUMQ = e−iπ4 σxm∑Lk=1,k(cid:54)=mσxk, (3) coupling only pairs involving the locally-rotated ion. It was suggested that these UMQ gates could be constructed using fast two-qubit phase gates24, which can be performed much faster than the trap period on ion crystals of arbitrary length, with high ﬁdelity25–28. In light of recent progress towards the implementation of the ﬁrst fast entangling gate using ultrafast laser pulses29–33, we explore this approach to validate the pursuit of fast gates as a scaling tool as well as to benchmark the required laser control and environmental coupling rates. In this paper, we perform numerical simulations of fast gates in a full 40-qubit crystal to calculate the ﬁdelity of a digital quantum simulation scheme under typical noise parameters. We benchmark the laser repetition rates which will allow us to overcome the time barrier imposed by the MS gate and signiﬁcantly reduce the operation time of digital quantum simulations, as well as greatly increasing the ﬁnal ﬁdelity. The latter implies that large-scale quantum simulations could be performed without error correction, bringing nearer the solutions for problems in condensed matter, quantum chemistry, and high-energy physics, which are infeasible to classical computers. Results Fast gate model Fast two-qubit gates can be performed beyond the trap-period limit and implement the ideal unitary for a geometric phase gate, UI = eiπ4 σz1σz2, (4) where σk is the Pauli z operator on the kth ion. These gates are implemented using counter-propagating laser π-pulse pairs, z which act as state-dependent momentum kicks on the motional modes while preserving the internal states of the ions. This behaviour is described in the optical rotating-wave approximation by the kick unitary Ukick = e−2iζk(x1σz1+x2σz2). (5) 2/10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: decompose

text: the

text: resulting

text: evolution

text: into

text: discrete

text: steps.

text: With

text: an

text: ion

text: trap,

text: a

text: digital

text: simulation

text: of

text: this

text: system

text: can

text: be

text: implemented

text: efﬁciently

text: using

text: steps

text: of

text: the

text: form

text: π

text: π

text: UM†

text: S

text: UσzmUMS

text: Here,

text: is

text: an

text: MS

text: gate

text: and

text: Uσzm

text: is

text: a

text: local

text: rotation

text: on

text: the

text: mth

text: ion

text: in

text: the

text: trap.

text: The

text: MS

text: gate

text: couples

text: every

text: pair

text: of

text: ions

text: in

text: the

text: trap

text: evenly,

text: UMS(θ)

text: where

text: σk

text: is

text: the

text: Pauli

text: x

text: operator

text: on

text: ion

text: k,

text: and

text: L

text: the

text: total

text: number

text: of

text: ions.

text: Although

text: MS

text: gates

text: can

text: be

text: implemented

text: with

text: high

text: x

text: ﬁdelity

text: in

text: ion

text: they

text: are

text: much

text: slower

text: than

text: the

text: trap

text: period

text: due

text: to

text: their

text: reliance

text: on

text: the

text: Lamb-Dicke

text: regime

text: and

text: the

text: vibrational

text: rotating-wave

text: approximation.

text: Furthermore,

text: as

text: the

text: number

text: of

text: addressed

text: ions

text: increases

text: to

text: scale

text: up

text: the

text: simulation,

text: the

text: motional

text: sideband

text: spectral

text: density

text: increases

text: and

text: the

text: gate

text: time

text: increases

text: accordingly.

text: It

text: would

text: thus

text: be

text: difﬁcult

text: to

text: use

text: MS

text: gates,

text: or

text: alternative

text: sideband-resolving

text: gates,

text: for

text: a

text: digital

text: quantum

text: simulation

text: of

text: a

text: sufﬁcient

text: size

text: to

text: outperform

text: current

text: classical

text: computers,

text: with

text: dozens

text: of

text: interacting

text: spins

text: or

text: fermions.

text: It

text: was

text: also

text: shown

text: in

text: the

text: simulation

text: that

text: each

text: MS

text: gate

text: can

text: be

text: replaced

text: by

text: an

text: ‘ultrafast

text: multiqubit’

text: (UMQ)

text: gate

text: in

text: the

text: simulation

text: step,

text: UUMQ

text: coupling

text: only

text: pairs

text: involving

text: the

text: locally-rotated

text: ion.

text: It

text: was

text: suggested

text: that

text: these

text: UMQ

text: gates

text: could

text: be

text: constructed

text: using

text: fast

text: two-qubit

text: phase

text: which

text: can

text: be

text: performed

text: much

text: faster

text: than

text: the

text: trap

text: period

text: on

text: ion

text: crystals

text: of

text: arbitrary

text: length,

text: with

text: high

text: In

text: light

text: of

text: recent

text: progress

text: towards

text: the

text: implementation

text: of

text: the

text: ﬁrst

text: fast

text: entangling

text: gate

text: using

text: ultrafast

text: laser

text: we

text: explore

text: this

text: approach

text: to

text: validate

text: the

text: pursuit

text: of

text: fast

text: gates

text: as

text: a

text: scaling

text: tool

text: as

text: well

text: as

text: to

text: benchmark

text: the

text: required

text: laser

text: control

text: and

text: environmental

text: coupling

text: rates.

text: In

text: this

text: paper,

text: we

text: perform

text: numerical

text: simulations

text: of

text: fast

text: gates

text: in

text: a

text: full

text: crystal

text: to

text: calculate

text: the

text: ﬁdelity

text: of

text: a

text: digital

text: quantum

text: simulation

text: scheme

text: under

text: typical

text: noise

text: parameters.

text: We

text: benchmark

text: the

text: laser

text: repetition

text: rates

text: which

text: will

text: allow

text: us

text: to

text: overcome

text: the

text: time

text: barrier

text: imposed

text: by

text: the

text: MS

text: gate

text: and

text: signiﬁcantly

text: reduce

text: the

text: operation

text: time

text: of

text: digital

text: quantum

text: simulations,

text: as

text: well

text: as

text: greatly

text: increasing

text: the

text: ﬁnal

text: ﬁdelity.

text: The

text: latter

text: implies

text: that

text: large-scale

text: quantum

text: simulations

text: could

text: be

text: performed

text: without

text: error

text: correction,

text: bringing

text: nearer

text: the

text: solutions

text: for

text: problems

text: in

text: condensed

text: matter,

text: quantum

text: chemistry,

text: and

text: high-energy

text: physics,

text: which

text: are

text: infeasible

text: to

text: classical

text: computers.

text: Results

text: Fast

text: gate

text: model

text: Fast

text: two-qubit

text: gates

text: can

text: be

text: performed

text: beyond

text: the

text: trap-period

text: limit

text: and

text: implement

text: the

text: ideal

text: unitary

text: for

text: a

text: geometric

text: phase

text: gate,

text: UI

text: where

text: σk

text: is

text: the

text: Pauli

text: z

text: operator

text: on

text: the

text: kth

text: ion.

text: These

text: gates

text: are

text: implemented

text: using

text: counter-propagating

text: laser

text: π-pulse

text: pairs,

text: z

text: which

text: act

text: as

text: state-dependent

text: momentum

text: kicks

text: on

text: the

text: motional

text: modes

text: while

text: preserving

text: the

text: internal

text: states

text: of

text: the

text: ions.

text: This

text: behaviour

text: is

text: described

text: in

text: the

text: optical

text: rotating-wave

text: approximation

text: by

text: the

text: kick

text: unitary

text: Ukick

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Here, x is the position operator on ion c, ζ is the number of π-pulse pairs used for the momentum kick and k is the laser c wavenumber. These kicks are nearly instantaneous compared to the trap period, so the evolution of a real gate operation can be expressed as the(cid:32)product of the ki(cid:33)ck unitaries interspersed with free motional evolution, Ure = ∏N ∏L e−iνpa†papδtc Ukcick, (6) c=1 p=1 where Ukcick is the cth kick unitary, δtc is the time between kicks c and c + 1 (0 if c = N) and N is the total number of kicks. An analytic solution for this evolution to implement the ideal unitary in Eq. (4) cannot generally be found, so numerical methods are used to optimise the ﬁdelity of the resulting gate. The high-ﬁdelity solutions found by this optimisation approach will restore the initial motional state of the trap, while the state-dependent phase-space trajectories for the motional modes will enclose areas corresponding to the π phase required by the ideal unitary. This results in a geometric phase gate that produces 4 minimal trap heating. The structure of the kicks is chosen to be antisymmetric, which approximately restores the motional state - the dominant motional error term is cubic with respect to gate time25. This reduces the difﬁculty of the numerical optimisation and produces generally higher-ﬁdelity solutions. In particular, the Fast Robust Antisymmetric Gate (FRAG) scheme30 was found to give the shortest gate times for a target ﬁdelity using realistic experimental parameters. This scheme uses the kick sequence − − − ζ = ( n,2n, 2n,2n, 2n,n), − − − t = ( τ1, τ2, τ3,τ3,τ2,τ1), (7) where z is the set of kick magnitudes associated with the times t and n is an integer. Negative and positive kick amplitudes represent kicks in opposite directions. The originally proposed FRAG scheme restricts the order of the kicks, so τ1 > τ2 > τ3 > 0. Removing this restriction preserves the antisymmetric structure of the sequence whilst allowing numerical optimisations to search a larger parameter space. This often results in higher-ﬁdelity gate solutions, so we ignore the kick ordering restriction in this work. We address the axial phonon modes of the trap - it is possible to perform quantum gates using transverse phonon modes34, but for a typical linear trap the electrode geometry will limit their speed due to micromotion. Addressing the axial modes requires that ground-state population in the ions not addressed by each fast gate be ‘shelved’43 into a third state such that they are not affected by the fast gate pulses. In the following analysis we neglect the inﬁdelity of the shelving technique, as each ion can be shelved with a single optical π pulse (albeit at a different wavelength) and thus errors will contribute negligibly compared with the inﬁdelity of the fast gates, which use hundreds of π pulses. We can use fast geometric phase gates to implement the UMQ gates required in digital quantum simulations. If we label the ions in a Paul trap from 1 to L starting at one end, this is done by performing a fast gate on ions 1 and 2, a SWAP gate on these − ions, then a fast gate on ions 2 and 3 and so on until we reach ions L 1 and L. The SWAP gate is given in the computational basis by       1 0 0 0 0 0 1 0 USWAP = 0 1 0 0 , (8) 0 0 0 1 and is equivalent to ‘swapping’ the internal states of the two ions. This process gives the UMQ unitary required up to local rotations, not preserving the position of the internal states in the trap. The reversed UMQ gate in the simulation step restores the position of the internal states. It should be noted that this method of constructing a UMQ gate requires fast SWAP gates, which can be implemented with three CNOT gates35. Each CNOT requires one fast geometric phase gate and four local rotations. − − This allows us to implement a non-position-preserving UMQ gate using L 2 SWAP gates and an additional L 1 fast − gates, for a total of 4L 7 fast gates. Due to the relative timescales and ﬁdelities of local rotations, which require only single laser pulses that take negligible time and can be performed with very high ﬁdelity36, we consider only inﬁdelity and time cost of the fast two-qubit gates in our results. This approximation is valid because the fast two-qubit gates require hundreds of laser 3/10 ##LTLine##

text: Here,

label: paragraph

text: x

text: is

text: the

text: position

text: operator

text: on

text: ion

text: c,

text: ζ

text: is

text: the

text: number

text: of

text: π-pulse

text: pairs

text: used

text: for

text: the

text: momentum

text: kick

text: and

text: k

text: is

text: the

text: laser

text: c

text: wavenumber.

text: These

text: kicks

text: are

text: nearly

text: instantaneous

text: compared

text: to

text: the

text: trap

text: period,

text: so

text: the

text: evolution

text: of

text: a

text: real

text: gate

text: operation

text: can

text: be

text: expressed

text: as

label: equation

text: of

text: the

text: unitaries

text: interspersed

text: with

text: free

text: motional

text: evolution,

text: Ure

text: ∏N

text: ∏L

text: e−iνpa†papδtc

text: Ukcick,

text: where

text: Ukcick

text: is

text: the

text: cth

text: kick

text: unitary,

text: δtc

text: is

text: the

text: time

text: between

text: kicks

text: c

text: and

text: c

text: if

text: c

text: N)

text: and

text: N

text: is

text: the

text: total

text: number

text: of

text: kicks.

text: An

text: analytic

text: solution

text: for

text: this

text: evolution

text: to

text: implement

text: the

text: ideal

text: unitary

text: in

text: Eq.

text: cannot

text: generally

text: be

text: found,

text: so

text: numerical

text: methods

text: are

text: used

text: to

text: optimise

text: the

text: ﬁdelity

text: of

text: the

text: resulting

text: gate.

text: The

text: high-ﬁdelity

text: solutions

text: found

text: by

text: this

text: optimisation

text: approach

text: will

text: restore

text: the

text: initial

text: motional

text: state

text: of

text: the

text: trap,

text: while

text: the

text: state-dependent

text: phase-space

text: trajectories

text: for

text: the

text: motional

text: modes

text: will

text: enclose

text: areas

text: corresponding

text: to

text: the

text: π

text: phase

text: required

text: by

text: the

text: ideal

text: unitary.

text: This

text: results

text: in

text: a

text: geometric

text: phase

text: gate

text: that

text: produces

text: minimal

text: trap

text: heating.

text: The

text: structure

text: of

text: the

text: kicks

text: is

text: chosen

text: to

text: be

text: antisymmetric,

text: which

text: approximately

text: restores

text: the

text: motional

text: state

text: the

text: dominant

text: motional

text: error

text: term

text: is

text: cubic

text: with

text: respect

text: to

text: gate

text: This

text: reduces

text: the

text: difﬁculty

text: of

text: the

text: numerical

text: optimisation

text: and

text: produces

text: generally

text: higher-ﬁdelity

text: solutions.

text: In

text: particular,

text: the

text: Fast

text: Robust

text: Antisymmetric

text: Gate

text: (FRAG)

text: was

text: found

text: to

text: give

text: the

text: shortest

text: gate

text: times

text: for

text: a

text: target

text: ﬁdelity

text: using

text: realistic

text: experimental

text: parameters.

text: This

text: scheme

text: uses

text: the

text: kick

text: sequence

text: ζ

text: t

text: where

text: z

text: is

text: the

text: set

text: of

text: kick

text: magnitudes

text: associated

text: with

text: the

text: times

text: t

text: and

text: n

text: is

text: an

text: integer.

text: Negative

text: and

text: positive

text: kick

text: amplitudes

text: represent

text: kicks

text: in

text: opposite

text: directions.

text: The

text: originally

text: proposed

text: FRAG

text: scheme

text: restricts

text: the

text: order

text: of

text: the

text: kicks,

text: so

text: τ1

text: τ2

text: τ3

text: Removing

text: this

text: restriction

text: preserves

text: the

text: antisymmetric

text: structure

text: of

text: the

text: sequence

text: whilst

text: allowing

text: numerical

text: optimisations

text: to

text: search

text: a

text: larger

text: parameter

text: space.

text: This

text: often

text: results

text: in

text: higher-ﬁdelity

text: gate

text: solutions,

text: so

text: we

text: ignore

text: the

text: kick

text: ordering

text: restriction

text: in

text: this

text: work.

text: We

text: address

text: the

text: axial

text: phonon

text: modes

text: of

text: the

text: trap

text: it

text: is

text: possible

text: to

text: perform

text: quantum

text: gates

text: using

text: transverse

text: phonon

text: but

text: for

text: a

text: typical

text: linear

text: trap

text: the

text: electrode

text: geometry

text: will

text: limit

text: their

text: speed

text: due

text: to

text: micromotion.

text: Addressing

text: the

text: axial

text: modes

text: requires

text: that

text: ground-state

text: population

text: in

text: the

text: ions

text: not

text: addressed

text: by

text: each

text: fast

text: gate

text: be

text: into

text: a

text: third

text: state

text: such

text: that

text: they

text: are

text: not

text: affected

text: by

text: the

text: fast

text: gate

text: pulses.

text: In

text: the

text: following

text: analysis

text: we

text: neglect

text: the

text: inﬁdelity

text: of

text: the

text: shelving

text: technique,

text: as

text: each

text: ion

text: can

text: be

text: shelved

text: with

text: a

text: single

text: optical

text: π

text: pulse

text: (albeit

text: at

text: a

text: different

text: wavelength)

text: and

text: thus

text: errors

text: will

text: contribute

text: negligibly

text: compared

text: with

text: the

text: inﬁdelity

text: of

text: the

text: fast

text: gates,

text: which

text: use

text: hundreds

text: of

text: π

text: pulses.

text: We

text: can

text: use

text: fast

text: geometric

text: phase

text: gates

text: to

text: implement

text: the

text: UMQ

text: gates

text: required

text: in

text: digital

text: quantum

text: simulations.

text: If

text: we

text: label

text: the

text: ions

text: in

text: a

text: Paul

text: trap

text: from

text: to

text: L

text: starting

text: at

text: one

text: end,

text: this

text: is

text: done

text: by

text: performing

text: a

text: fast

text: gate

text: on

text: ions

text: and

text: a

text: SWAP

text: gate

text: on

text: these

text: ions,

text: then

text: a

text: fast

text: gate

text: on

text: ions

text: and

text: and

text: so

text: on

text: until

text: we

text: reach

text: ions

text: L

text: and

text: L.

text: The

text: SWAP

text: gate

text: is

text: given

text: in

text: the

text: computational

text: basis

text: by

text: 1

text: 0

text: USWAP

text: and

text: is

text: equivalent

text: to

text: ‘swapping’

text: the

text: internal

text: states

text: of

text: the

text: two

text: ions.

text: This

text: process

text: gives

text: the

text: UMQ

text: unitary

text: required

text: up

text: to

text: local

text: rotations,

text: not

text: preserving

text: the

text: position

text: of

text: the

text: internal

text: states

text: in

text: the

text: trap.

text: The

text: reversed

text: UMQ

text: gate

text: in

text: the

text: simulation

text: step

text: restores

text: the

text: position

text: of

text: the

text: internal

text: states.

text: It

text: should

text: be

text: noted

text: that

text: this

text: method

text: of

text: constructing

text: a

text: UMQ

text: gate

text: requires

text: fast

text: SWAP

text: gates,

text: which

text: can

text: be

text: implemented

text: with

text: three

text: CNOT

text: Each

text: CNOT

text: requires

text: one

text: fast

text: geometric

text: phase

text: gate

text: and

text: four

text: local

text: rotations.

text: This

text: allows

text: us

text: to

text: implement

text: a

text: non-position-preserving

text: UMQ

text: gate

text: using

text: L

text: SWAP

text: gates

text: and

text: an

text: additional

text: L

text: fast

text: gates,

text: for

text: a

text: total

text: of

text: fast

text: gates.

text: Due

text: to

text: the

text: relative

text: timescales

text: and

text: ﬁdelities

text: of

text: local

text: rotations,

text: which

text: require

text: only

text: single

text: laser

text: pulses

text: that

text: take

text: negligible

text: time

text: and

text: can

text: be

text: performed

text: with

text: very

text: high

text: we

text: consider

text: only

text: inﬁdelity

text: and

text: time

text: cost

text: of

text: the

text: fast

text: two-qubit

text: gates

text: in

text: our

text: results.

text: This

text: approximation

text: is

text: valid

text: because

text: the

text: fast

text: two-qubit

text: gates

text: require

text: hundreds

text: of

text: laser

text: ##LTLine##


text: correspond to ﬁdelity of around 95%, without including error correction, for 10 Trotter steps of a 20-site fermionic lattice simulation, requiring 2680 UMQ gates. UMQ gates with n = 50 or more are capable, as shown in Fig. (1a), of exceeding the error threshold required for a 20-site simulation without error correction with 70% ﬁdelity. Current repetition rates for applicable pulsed laser systems are up to 300 MHz, with small modiﬁcations required to increase repetition rates to approximately 1 GHz33,42. This is still well below the 10 GHz required for n = 50 fast gates, but work is in progress on a pulsed laser system for 40Ca+ control with a 5 GHz repetition rate44. In Fig. (1b), we analyze the dependence on time of our fast gate scheme. Again focusing on the n = 50 case, the time required for a 40-ion UMQ gate is 17.9 µs, comparable with the trap period of 10 µs for this trap. Furthermore, the trap period does not place a√ny fundamental limit on the speed of fast gates as it does with MS gates. In general, an MS gate for N ions will take a time Nt , t being the time it takes for 1 ion. This can be seen from the expression of the gate’s phase in terms 1 1 of experimental parameters, θ ∝ (Ωη )2. Here Ω is the Rabi frequency, which does not depen√d on the number of ions and is δ limited by the laser power, η is the Lamb-Dicke parameter, which for N ions is given by η1/ N, where η1 is the single-ion Lamb-Dicke parameter, when the COM vibrational mode is considered. The detuning δ sets the gate time to t = 1/δ.√To → compensate for the 1/N decay rate of θ with the√number of ions, the detuning has to be decreased accordingly, δ δ/ N, → which increases the time of the MS gate as t Nt. A two-qubit MS gate is typically around 50 µs long22, thus a 40-ion gate takes an order of magnitude more time than our UMQ gate implementation with n = 50, as shown in the ﬁgure. Higher available repetition rates would lead to even more signiﬁcant gains, as seen in the n = 100 case. The lowest heating rates in − current traps are on the order of 10 s 1, which limits the total simulation time to around 100 ms. This places a corresponding maximal threshold on gate time, shown in Fig. (1b). It can be seen that fast gates are below this threshold, and therefore that they would allow for a simulation outperforming classical computers within the trap coherence time. A large scale quantum simulation with MS gates sits on the verge of current motional coherence times,√and should be possible with coherence times expected in the near future. However, from the time scaling of MS (scaling as N) and UMQ (linear in N) gates, it can be seen that UMQ gates would perform faster for a number of ions on the order of 1000 or below. This is far beyond the expected capacity of linear traps, and for this number of ions more creative ion arrays will be required, most likely composed of linear traps where the UMQ gates prove to be the most practical. Finally, to achieve a better understanding of the interplay between laser repetition rate, momentum (i.e. fast gate n) and gate time, we numerically optimised fast gates using the FRAG scheme with n = 1 to n = 100 for our two-ion trap. The resulting gate times are shown in Figure (2). (cid:1) 1000 (cid:1) 500 (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) 120000 (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) 1 5 10 50 100 Fast gate n (cid:1) Calculated gate times Fit line Figure 2. Gate times for FRAG fast gates on a two-ion trap with varying n. The points represent the numerical solutions for × − each n, and the dashed line is a least-squares ﬁt to the points with the equation TG = 1.149 µs n 0.6704, where TG is gate time. − A simple least-squares ﬁt to this data reveals that gate time scales with n (and thus momentum) approximately as n 2/3. This agrees very well with theoretical results for the GZC pulse scheme24 and as our UMQ gates are constructed using fast gates, we expect the UMQ gate time to follow the same scaling. To understand the relationship between laser repetition rate fr and gate time, w−e can use the approximation n = TG fr, which works very well for fast gates in this regime25. We can combine this with the n 2/3 scaling of gate time with the number of 5/10 )sn( emit etaG ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: correspond

text: to

text: ﬁdelity

text: of

text: around

text: without

text: including

text: error

text: correction,

text: for

text: Trotter

text: steps

text: of

text: a

text: fermionic

text: lattice

text: simulation,

text: requiring

text: UMQ

text: gates.

text: UMQ

text: gates

text: with

text: n

text: or

text: more

text: are

text: capable,

text: as

text: shown

text: in

text: Fig.

text: of

text: exceeding

text: the

text: error

text: threshold

text: required

text: for

text: a

text: simulation

text: without

text: error

text: correction

text: with

text: ﬁdelity.

text: Current

text: repetition

text: rates

text: for

text: applicable

text: pulsed

text: laser

text: systems

text: are

text: up

text: to

text: MHz,

text: with

text: small

text: modiﬁcations

text: required

text: to

text: increase

text: repetition

text: rates

text: to

text: approximately

text: This

text: is

text: still

text: well

text: below

text: the

text: GHz

text: required

text: for

text: n

text: fast

text: gates,

text: but

text: work

text: is

text: in

text: progress

text: on

text: a

text: pulsed

text: laser

text: system

text: for

text: control

text: with

text: a

text: GHz

text: repetition

text: In

text: Fig.

text: we

text: analyze

text: the

text: dependence

text: on

text: time

text: of

text: our

text: fast

text: gate

text: scheme.

text: Again

text: focusing

text: on

text: the

text: n

text: case,

text: the

text: time

text: required

text: for

text: a

text: UMQ

text: gate

text: is

text: µs,

text: comparable

text: with

text: the

text: trap

text: period

text: of

text: µs

text: for

text: this

text: trap.

text: Furthermore,

text: the

text: trap

text: period

text: does

text: not

text: place

text: a√ny

text: fundamental

text: limit

text: on

text: the

text: speed

text: of

text: fast

text: gates

text: as

text: it

text: does

text: with

text: MS

text: gates.

text: In

text: general,

text: an

text: MS

text: gate

text: for

text: N

text: ions

text: will

text: take

text: a

text: time

text: Nt

text: t

text: being

text: the

text: time

text: it

text: takes

text: for

text: ion.

text: This

text: can

text: be

text: seen

text: from

text: the

text: expression

text: of

text: the

text: gate’s

text: phase

text: in

text: terms

text: of

text: experimental

text: parameters,

text: θ

text: (Ωη

text: Here

text: Ω

text: is

text: the

text: Rabi

text: frequency,

text: which

text: does

text: not

text: depen√d

text: on

text: the

text: number

text: of

text: ions

text: and

text: is

text: δ

text: limited

text: by

text: the

text: laser

text: power,

text: η

text: is

text: the

text: Lamb-Dicke

text: parameter,

text: which

text: for

text: N

text: ions

text: is

text: given

text: by

text: N,

text: where

text: η1

text: is

text: the

text: single-ion

text: Lamb-Dicke

text: parameter,

text: when

text: the

text: COM

text: vibrational

text: mode

text: is

text: considered.

text: The

text: detuning

text: δ

text: sets

text: the

text: gate

text: time

text: to

text: t

text: compensate

text: for

text: the

text: decay

text: rate

text: of

text: θ

text: with

text: the√number

text: of

text: ions,

text: the

text: detuning

text: has

text: to

text: be

text: decreased

text: accordingly,

text: δ

text: δ/

text: N,

text: which

text: increases

text: the

text: time

text: of

text: the

text: MS

text: gate

text: as

text: t

text: Nt.

text: A

text: two-qubit

text: MS

text: gate

text: is

text: typically

text: around

text: µs

text: thus

text: a

text: gate

text: takes

text: an

text: order

text: of

text: magnitude

text: more

text: time

text: than

text: our

text: UMQ

text: gate

text: implementation

text: with

text: n

text: as

text: shown

text: in

text: the

text: ﬁgure.

text: Higher

text: available

text: repetition

text: rates

text: would

text: lead

text: to

text: even

text: more

text: signiﬁcant

text: gains,

text: as

text: seen

text: in

text: the

text: n

text: case.

text: The

text: lowest

text: heating

text: rates

text: in

text: current

text: traps

text: are

text: on

text: the

text: order

text: of

text: s

text: which

text: limits

text: the

text: total

text: simulation

text: time

text: to

text: around

text: ms.

text: This

text: places

text: a

text: corresponding

text: maximal

text: threshold

text: on

text: gate

text: time,

text: shown

text: in

text: Fig.

text: It

text: can

text: be

text: seen

text: that

text: fast

text: gates

text: are

text: below

text: this

text: threshold,

text: and

text: therefore

text: that

text: they

text: would

text: allow

text: for

text: a

text: simulation

text: outperforming

text: classical

text: computers

text: within

text: the

text: trap

text: coherence

text: time.

text: A

text: large

text: scale

text: quantum

text: simulation

text: with

text: MS

text: gates

text: sits

text: on

text: the

text: verge

text: of

text: current

text: motional

text: coherence

text: times,√and

text: should

text: be

text: possible

text: with

text: coherence

text: times

text: expected

text: in

text: the

text: near

text: future.

text: However,

text: from

text: the

text: time

text: scaling

text: of

text: MS

text: (scaling

text: as

text: N)

text: and

text: UMQ

text: (linear

text: in

text: N)

text: gates,

text: it

text: can

text: be

text: seen

text: that

text: UMQ

text: gates

text: would

text: perform

text: faster

text: for

text: a

text: number

text: of

text: ions

text: on

text: the

text: order

text: of

text: or

text: below.

text: This

text: is

text: far

text: beyond

text: the

text: expected

text: capacity

text: of

text: linear

text: traps,

text: and

text: for

text: this

text: number

text: of

text: ions

text: more

text: creative

text: ion

text: arrays

text: will

text: be

text: required,

text: most

text: likely

text: composed

text: of

text: linear

text: traps

text: where

text: the

text: UMQ

text: gates

text: prove

text: to

text: be

text: the

text: most

text: practical.

text: Finally,

text: to

text: achieve

text: a

text: better

text: understanding

text: of

text: the

text: interplay

text: between

text: laser

text: repetition

text: rate,

text: momentum

text: (i.e.

text: fast

text: gate

text: n)

text: and

text: gate

text: time,

text: we

text: numerically

text: optimised

text: fast

text: gates

text: using

text: the

text: FRAG

text: scheme

text: with

text: n

text: to

text: n

text: for

text: our

text: two-ion

text: trap.

text: The

text: resulting

text: gate

text: times

text: are

text: shown

text: in

text: Figure

text: Fast

text: gate

text: n

text: Calculated

text: gate

text: times

text: Fit

text: line

text: Figure

text: Gate

text: times

text: for

text: FRAG

text: fast

text: gates

text: on

text: a

text: two-ion

text: trap

text: with

text: varying

text: n.

text: The

text: points

text: represent

text: the

text: numerical

text: solutions

text: for

text: each

text: n,

text: and

text: the

text: dashed

text: line

text: is

text: a

text: least-squares

text: ﬁt

text: to

text: the

text: points

text: with

text: the

text: equation

text: TG

text: µs

text: n

text: where

text: TG

text: is

text: gate

text: time.

text: A

text: simple

text: least-squares

text: ﬁt

text: to

text: this

text: data

text: reveals

text: that

text: gate

text: time

text: scales

text: with

text: n

text: (and

text: thus

text: momentum)

text: approximately

text: as

text: n

text: This

text: agrees

text: very

text: well

text: with

text: theoretical

text: results

text: for

text: the

text: GZC

text: pulse

text: and

text: as

text: our

text: UMQ

text: gates

text: are

text: constructed

text: using

text: fast

text: gates,

text: we

text: expect

text: the

text: UMQ

text: gate

text: time

text: to

text: follow

text: the

text: same

text: scaling.

text: To

text: understand

text: the

text: relationship

text: between

text: laser

text: repetition

text: rate

text: fr

text: and

text: gate

text: time,

text: w−e

text: can

text: use

text: the

text: approximation

text: n

text: TG

text: fr,

text: which

text: works

text: very

text: well

text: for

text: fast

text: gates

text: in

text: this

text: We

text: can

text: combine

text: this

text: with

text: the

text: n

text: scaling

text: of

text: gate

text: time

text: with

text: the

text: number

text: of

text: )sn(

text: emit

text: etaG

text: ##LTFigure##

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text: 2 Publications of the Astronomical Society of Japan, (2014), Vol. 00, No. 0 be of the order of 0.02 for dark cloud cores (?). ? suggested that angular resolution toward the Orion A GMC (?). N H+ and − 2 § the J/M R relation can be explained for virial equilibrium NH are thought to trace similar volumes of dense gas ( 3.5). 3 ± cores if β is constant against the core size. β and the ratio α of In this work we adopt a distance of 418 6 pc for the Orion the thermal to gravitational energy is thought to affect the core A GMC, based on the work of ?, as the best estimate rather than fragmentation process, which will be related to the frequency of 450 pc used in ?. At this distance, 1(cid:48) corresponds to 0.122 pc. binary and multiple star formation (e.g., ?; ?; see also ?). We correct the core radius (in pc) and core mass of ? decreasing by 6.5% and 12.5%, respectively. Galactic molecular clouds can be divided into two cate- Table ?? lists the N H+ cores in the Orion A GMC from gories: giant molecular clouds (GMCs) and cold dark clouds 2 (excluding infrared dark clouds here) (e.g., ?; ?; ?). These two ?. The kinetic temperature Tk is calculated from the rotation temperature from NH observations of ? and by using the con- categories of clouds show different ranges of cloud mass, and 3 version given in ?. In general, N H+ core positions do not star-formation modes: GMCs are most likely associated with 2 match NH observed positions, so we take the value of the near- cluster star formation including massive stars, while cold dark 3 clouds show preferentially isolated low-mass star formation. est NH3 position. The total linewidth ∆vTOT, including both Furthermore, molecular cloud cores inside these clouds show thermal and non-thermal contribution, is calculated by using Tk and by correcting for the difference of the mean molecular mass different characteristics: Warm-temperature, turbulent cores in (2.33 u) and the mass of the observed molecule (N H+, 29 u) GMCs and cold thermal cores in cold dark clouds. It was sug- 2 gested that the coefﬁcient A (or intercept in the log-log form) (?). The core mass M is taken from ?, which was obtained by assuming local thermodynamic equilibrium (LTE) and cor- of the linewidth-size relation differs between cores in these two categories. ? have studied molecular cloud cores in the Orion A rected for the distance revision. The virial mass Mvir is deﬁned GMC in CS J = 1 → 0, and suggested that this coefﬁcient A (or as 210 R (pc) ∆vT2OT for a uniform density sphere (?). The intercept) is larger in Orion cores compared with that for cores virial parameter αvir is deﬁned as the ratio of the virial mass in cold dark clouds. They argued that a larger coefﬁcient means Mvir to±the core mass M, and ±its average for Orion N2H+ cores that Orion cores have higher external pressure and/or stronger is 0.79 0.61 (log αvir = -0.21 0.31). They are not far deviated magnetic ﬁelds. ? have also suggested different intercepts (and from virial equilibrium. The blank in the Tk column means that we do not have a good nearest NH counterpart. The blank different slopes) between GMC cores and cold dark cloud cores 3 using the Orion A GMC data of ? and other complementary in the M column (and then ∆vTOT, Mvir, and αvir columns) data. Originally, ? pointed out that the linewidth is relatively means either no Tk value or failure in the N2H+ hyperﬁne ﬁt- ting to restrict the optical depth. small in the Taurus cold dark cloud, larger in the ρ Ophiuchus complex, and even larger in the Orion A GMC (meaning dif- ferent intercepts). ? compared the linewidth-size relation be- tween “well-deﬁned” Orion A GMC cores and cold dark cloud cores (see their Figure 7) and concluded that the intercept is clearly different, but the power-law index is not so different. Further evidence of variation of intercept was obtained toward the Galactic Center, where the intercept is even larger than that for Orion cores (?). ?, ?, and ? have also shown that the co- efﬁcient (or intercept) of the linewidth-size relation is highly related to the surface density (higher external pressure and/or stronger magnetic ﬁelds lead to higher surface density), and it has deviations among clouds. It seems that different coefﬁcients (or intercepts) of the linewidth-size relation in different cate- gories of Galactic clouds are well established. Given that intercept of the linewidth-size relation differs − among Galactic clouds, we wonder whether the J/M R re- − lation also shows any difference. ? compared the J/M R re- lation of the Orion CS cores with that for cold dark cloud cores derived in NH by ?, and found that the Orion cores may have 3 slightly larger J/M. However, their results were not conclusive because of large data scattering and insufﬁcient linear-scale res- olution in pc. In this paper, we re-investigate the difference in − the J/M R relation by using data obtained in N H+ at higher 2 ##LTLine##

font: WBTOMN+NimbusSanL-Bold

text: Publications

font: ZDDFZZ+NimbusSanL-ReguItal

text: of

text: the

text: Astronomical

text: Society

text: of

text: Japan,

font: NYFNKI+NimbusSanL-Regu

text: Vol.

text: No.

text: be

text: of

text: the

text: order

text: of

text: for

text: dark

text: cloud

text: cores

text: suggested

text: that

text: angular

text: resolution

text: toward

text: the

text: Orion

text: A

text: GMC

text: N

text: H+

text: and

text: the

text: J/M

text: R

text: relation

text: can

text: be

text: explained

text: for

text: virial

text: equilibrium

text: NH

text: are

text: thought

text: to

text: trace

text: similar

text: volumes

text: of

text: dense

text: gas

text: cores

text: if

text: β

text: is

text: constant

text: against

text: the

text: core

text: size.

text: β

text: and

text: the

text: ratio

text: α

text: of

text: In

text: this

text: work

text: we

text: adopt

text: a

text: distance

text: of

text: pc

text: for

text: the

text: Orion

text: the

text: thermal

text: to

text: gravitational

text: energy

text: is

text: thought

text: to

text: affect

text: the

text: core

text: A

text: GMC,

text: based

text: on

text: the

text: work

text: of

text: as

text: the

text: best

text: estimate

text: rather

text: than

text: fragmentation

text: process,

text: which

text: will

text: be

text: related

text: to

text: the

text: frequency

text: of

text: pc

text: used

text: in

text: At

text: this

text: distance,

text: corresponds

text: to

text: pc.

text: binary

text: and

text: multiple

text: star

text: formation

text: (e.g.,

text: see

text: also

text: We

text: correct

text: the

text: core

text: radius

text: (in

text: pc)

text: and

text: core

text: mass

text: of

text: decreasing

text: by

text: and

text: respectively.

text: Galactic

text: molecular

text: clouds

text: can

text: be

text: divided

text: into

text: two

text: cate-

text: Table

text: lists

text: the

text: N

text: H+

text: cores

text: in

text: the

text: Orion

text: A

text: GMC

text: from

text: gories:

text: giant

text: molecular

text: clouds

text: (GMCs)

text: and

text: cold

text: dark

text: clouds

text: (excluding

text: infrared

text: dark

text: clouds

text: here)

text: (e.g.,

text: These

text: two

text: The

text: kinetic

text: temperature

text: Tk

text: is

text: calculated

text: from

text: the

text: rotation

text: temperature

text: from

text: NH

text: observations

text: of

text: and

text: by

text: using

text: the

text: con-

text: categories

text: of

text: clouds

text: show

text: different

text: ranges

text: of

text: cloud

text: mass,

text: and

text: version

text: given

text: in

text: In

text: general,

text: N

text: H+

text: core

text: positions

text: do

text: not

text: star-formation

text: modes:

text: GMCs

text: are

text: most

text: likely

text: associated

text: with

text: match

text: NH

text: observed

text: positions,

text: so

text: we

text: take

text: the

text: value

text: of

text: the

text: near-

text: cluster

text: star

text: formation

text: including

text: massive

text: stars,

text: while

text: cold

text: dark

text: clouds

text: show

text: preferentially

text: isolated

text: low-mass

text: star

text: formation.

text: est

text: position.

text: The

text: total

text: linewidth

text: ∆vTOT,

text: including

text: both

text: Furthermore,

text: molecular

text: cloud

text: cores

text: inside

text: these

text: clouds

text: show

text: thermal

text: and

text: non-thermal

text: contribution,

text: is

text: calculated

text: by

text: using

text: Tk

text: and

text: by

text: correcting

text: for

text: the

text: difference

text: of

text: the

text: mean

text: molecular

text: mass

text: different

text: characteristics:

text: Warm-temperature,

text: turbulent

text: cores

text: in

text: u)

text: and

text: the

text: mass

text: of

text: the

text: observed

text: molecule

text: (N

text: H+,

text: u)

text: GMCs

text: and

text: cold

text: thermal

text: cores

text: in

text: cold

text: dark

text: clouds.

text: It

text: was

text: sug-

text: gested

text: that

text: the

text: coefﬁcient

text: A

text: (or

text: intercept

text: in

text: the

text: log-log

text: form)

text: The

text: core

text: mass

text: M

text: is

text: taken

text: from

text: which

text: was

text: obtained

text: by

text: assuming

text: local

text: thermodynamic

text: equilibrium

text: (LTE)

text: and

text: cor-

text: of

text: the

text: linewidth-size

text: relation

text: differs

text: between

text: cores

text: in

text: these

text: two

text: categories.

text: have

text: studied

text: molecular

text: cloud

text: cores

text: in

text: the

text: Orion

text: A

text: rected

text: for

text: the

text: distance

text: revision.

text: The

text: virial

text: mass

text: Mvir

text: is

text: deﬁned

text: GMC

text: in

text: CS

text: J

text: and

text: suggested

text: that

text: this

text: coefﬁcient

text: A

text: (or

text: as

text: R

text: (pc)

text: for

text: a

text: uniform

text: density

text: sphere

text: The

text: intercept)

text: is

text: larger

text: in

text: Orion

text: cores

text: compared

text: with

text: that

text: for

text: cores

text: virial

text: parameter

text: αvir

text: is

text: deﬁned

text: as

text: the

text: ratio

text: of

text: the

text: virial

text: mass

text: in

text: cold

text: dark

text: clouds.

text: They

text: argued

text: that

text: a

text: larger

text: coefﬁcient

text: means

text: Mvir

text: to±the

text: core

text: mass

text: M,

text: and

text: ±its

text: average

text: for

text: Orion

text: cores

text: that

text: Orion

text: cores

text: have

text: higher

text: external

text: pressure

text: and/or

text: stronger

text: is

text: (log

text: αvir

text: They

text: are

text: not

text: far

text: deviated

text: magnetic

text: ﬁelds.

text: have

text: also

text: suggested

text: different

text: intercepts

text: (and

text: from

text: virial

text: equilibrium.

text: The

text: blank

text: in

text: the

text: Tk

text: column

text: means

text: that

text: we

text: do

text: not

text: have

text: a

text: good

text: nearest

text: NH

text: counterpart.

text: The

text: blank

text: different

text: slopes)

text: between

text: GMC

text: cores

text: and

text: cold

text: dark

text: cloud

text: cores

text: using

text: the

text: Orion

text: A

text: GMC

text: data

text: of

text: and

text: other

text: complementary

text: in

text: the

text: M

text: column

text: (and

text: then

text: ∆vTOT,

text: Mvir,

text: and

text: αvir

text: columns)

text: data.

text: Originally,

text: pointed

text: out

text: that

text: the

text: linewidth

text: is

text: relatively

text: means

text: either

text: no

text: Tk

text: value

text: or

text: failure

text: in

text: the

text: hyperﬁne

text: ﬁt-

text: ting

text: to

text: restrict

text: the

text: optical

text: depth.

text: small

text: in

text: the

text: Taurus

text: cold

text: dark

text: cloud,

text: larger

text: in

text: the

text: ρ

text: Ophiuchus

text: complex,

text: and

text: even

text: larger

text: in

text: the

text: Orion

text: A

text: GMC

text: (meaning

text: dif-

text: ferent

text: intercepts).

text: compared

text: the

text: linewidth-size

text: relation

text: be-

text: tween

text: “well-deﬁned”

text: Orion

text: A

text: GMC

text: cores

text: and

text: cold

text: dark

text: cloud

text: cores

text: (see

text: their

text: Figure

text: and

text: concluded

text: that

text: the

text: intercept

text: is

text: clearly

text: different,

text: but

text: the

text: power-law

text: index

text: is

text: not

text: so

text: different.

text: Further

text: evidence

text: of

text: variation

text: of

text: intercept

text: was

text: obtained

text: toward

text: the

text: Galactic

text: Center,

text: where

text: the

text: intercept

text: is

text: even

text: larger

text: than

text: that

text: for

text: Orion

text: cores

text: and

text: have

text: also

text: shown

text: that

text: the

text: co-

text: efﬁcient

text: (or

text: intercept)

text: of

text: the

text: linewidth-size

text: relation

text: is

text: highly

text: related

text: to

text: the

text: surface

text: density

text: (higher

text: external

text: pressure

text: and/or

text: stronger

text: magnetic

text: ﬁelds

text: lead

text: to

text: higher

text: surface

text: density),

text: and

text: it

text: has

text: deviations

text: among

text: clouds.

text: It

text: seems

text: that

text: different

text: coefﬁcients

text: (or

text: intercepts)

text: of

text: the

text: linewidth-size

text: relation

text: in

text: different

text: cate-

text: gories

text: of

text: Galactic

text: clouds

text: are

text: well

text: established.

text: Given

text: that

text: intercept

text: of

text: the

text: linewidth-size

text: relation

text: differs

text: among

text: Galactic

text: clouds,

text: we

text: wonder

text: whether

text: the

text: J/M

text: R

text: re-

text: lation

text: also

text: shows

text: any

text: difference.

text: compared

text: the

text: J/M

text: R

text: re-

text: lation

text: of

text: the

text: Orion

text: CS

text: cores

text: with

text: that

text: for

text: cold

text: dark

text: cloud

text: cores

text: derived

text: in

text: NH

text: by

text: and

text: found

text: that

text: the

text: Orion

text: cores

text: may

text: have

text: slightly

text: larger

text: J/M.

text: However,

text: their

text: results

text: were

text: not

text: conclusive

text: because

text: of

text: large

text: data

text: scattering

text: and

text: insufﬁcient

text: linear-scale

text: res-

text: olution

text: in

text: pc.

text: In

text: this

text: paper,

text: we

text: re-investigate

text: the

text: difference

text: in

text: the

text: J/M

text: R

text: relation

text: by

text: using

text: data

text: obtained

text: in

text: N

text: H+

text: at

text: higher

text: ##LTLine##


text: 10 Publications of the Astronomical Society of Japan, (2014), Vol. 00, No. 0 1 2 3 -05 00 5 4 6 7 8 9 10 10 11 12 13 14 15 20 16 17 18 19 20 21 22 30 23 24 40 25 26 27 50 28 30 29 31 -06 00 32 10 33 34 053630 15 00 3545 30 15 00 3445 RIGHT ASCENSION (J2000) Fig. 10. The orientation of the velocity gradient is plotted on the N H+ taken from ?. The contour map represents the velocity-integrated intensity of the N H+ → → 2 − × 2 J = 1 0 F1 = 2 1 hyperﬁne group. The contour levels are 0.749 K km s 1 (1, 2, 4, 8). The intensity maxima of the N2H+ cores and the orientations of the velocity gradient are shown as pluses associated with core numbers and arrows, respectively. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: Publications

text: of

text: the

text: Astronomical

text: Society

text: of

text: Japan,

text: Vol.

text: No.

font: default

text: RIGHT

text: ASCENSION

text: Fig.

text: The

text: orientation

text: of

text: the

text: velocity

text: gradient

text: is

text: plotted

text: on

text: the

text: N

text: H+

text: taken

text: from

text: The

text: contour

text: map

text: represents

text: the

text: velocity-integrated

text: intensity

text: of

text: the

text: N

text: H+

text: J

text: hyperﬁne

text: group.

text: The

text: contour

text: levels

text: are

text: K

text: km

text: s

text: The

text: intensity

text: maxima

text: of

text: the

text: cores

text: and

text: the

text: orientations

text: of

text: the

text: velocity

text: gradient

text: are

text: shown

text: as

text: pluses

text: associated

text: with

text: core

text: numbers

text: and

text: arrows,

text: respectively.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: The geometry of the triple junction between three ﬂuids in equilibrium Blank, Elcrat, and Treinen May 28, 2020 Abstract We conduct an analysis of the blow up at the triple junction of three ﬂuids in equilibrium. Energy minimizers have been shown to exist in the class of functions of bounded variations, and the classical theory implies that an interface between two ﬂuids is an analytic surface. We prove two monotonicity formulas at the triple junction for the three-ﬂuid conﬁguration, and show that blow up limits exist and are always cones. We discuss some of the geometric consequences of our results. Keywords: Floating Drops, Capillarity, Regularity, and Blow up. AMS subject classiﬁcation: 76B45, 35R35, and 35B65 1 Introduction ⊂ Let Ω IRn be a bounded domain with boundary smooth enough that the interior sphere condition holds. Then consider a partition of Ω into three sets E , j = 0, 1, 2. Each E will j j represent a ﬂuid, and we assum(cid:32)e that the three ﬂuids are immiscible and are in(cid:33)equilibrium (cid:90) (cid:90) (cid:90) with respect to the energy f(cid:88)unctional 2 F { } | | H − ( E ) := α Dχ + β χ d n 1 + ρ g z dV SWP j j Ej j Ej j j=0 Ω ∂Ω Ej where g is determined by the force of gravity, and where the constants α , β , and ρ are j j j determined by constitutive properties of our ﬂuids. It will make the most sense to consider sets with ﬁnite perimeter, as this functional is inﬁnite otherwise, and accordingly, we will work within the framework aﬀorded to us by functions of bounded variation. We will deﬁne this functional more carefully and state some assumptions that we will make on the constitutive constants in Section below. Two common physical situations where this mathematical model arise include ﬁrst, if there is a double sessile drop of two distinct immiscible ﬂuids resting on a surface with air above, and second, if a drop of a light ﬂuid is ﬂoating on the top of a heavier ﬂuid and below a lighter ﬂuid as would be the case when oil ﬂoats on water and below air. See Figure 1 for an example of the ﬁrst situation, and Figure 2 (found within Section ) for an example of the second situation. The terms in the energy functional given above arise from (in 1

text: The

label: title

text: geometry

text: of

text: the

text: triple

text: junction

text: between

text: three

text: ﬂuids

text: in

text: equilibrium

text: Blank,

label: author

text: Elcrat,

text: and

text: Treinen

text: May

text: Abstract

text: We

label: abstract

text: conduct

text: an

text: analysis

text: of

text: the

text: blow

text: up

text: at

text: the

text: triple

text: junction

text: of

text: three

text: ﬂuids

text: in

text: equilibrium.

text: Energy

text: minimizers

text: have

text: been

text: shown

text: to

text: exist

text: in

text: the

text: class

text: of

text: functions

text: of

text: bounded

text: variations,

text: and

text: the

text: classical

text: theory

text: implies

text: that

text: an

text: interface

text: between

text: two

text: ﬂuids

text: is

text: an

text: analytic

text: surface.

text: We

text: prove

text: two

text: monotonicity

text: formulas

text: at

text: the

text: triple

text: junction

text: for

text: the

text: three-ﬂuid

text: conﬁguration,

text: and

text: show

text: that

text: blow

text: up

text: limits

text: exist

text: and

text: are

text: always

text: cones.

text: We

text: discuss

text: some

text: of

text: the

text: geometric

text: consequences

text: of

text: our

text: results.

text: Keywords:

text: Floating

text: Drops,

text: Capillarity,

text: Regularity,

text: and

text: Blow

text: up.

text: AMS

text: subject

text: classiﬁcation:

text: and

text: Introduction

text: Let

text: Ω

text: IRn

text: be

text: a

text: bounded

text: domain

text: with

text: boundary

text: smooth

text: enough

text: that

text: the

text: interior

text: sphere

text: condition

text: holds.

text: Then

text: consider

text: a

text: partition

text: of

text: Ω

text: into

text: three

text: sets

text: E

text: j

text: Each

text: E

text: will

text: j

text: j

text: represent

text: a

text: ﬂuid,

text: and

text: we

text: that

text: the

text: three

text: ﬂuids

text: are

text: immiscible

text: and

text: are

text: with

text: respect

text: to

text: the

text: energy

text: F

text: H

text: E

text: α

text: Dχ

text: β

text: χ

text: d

text: n

text: ρ

text: g

text: z

text: dV

text: SWP

text: j

text: j

text: Ej

text: j

text: Ej

text: j

text: Ω

text: ∂Ω

text: Ej

text: where

text: g

text: is

text: determined

text: by

text: the

text: force

text: of

text: gravity,

text: and

text: where

text: the

text: constants

text: α

text: β

text: and

text: ρ

text: are

text: j

text: j

text: j

text: determined

text: by

text: constitutive

text: properties

text: of

text: our

text: ﬂuids.

text: It

text: will

text: make

text: the

text: most

text: sense

text: to

text: consider

text: sets

text: with

text: ﬁnite

text: perimeter,

text: as

text: this

text: functional

text: is

text: inﬁnite

text: otherwise,

text: and

text: accordingly,

text: we

text: will

text: work

text: within

text: the

text: framework

text: aﬀorded

text: to

text: us

text: by

text: functions

text: of

text: bounded

text: variation.

text: We

text: will

text: deﬁne

text: this

text: functional

text: more

text: carefully

text: and

text: state

text: some

text: assumptions

text: that

text: we

text: will

text: make

text: on

text: the

text: constitutive

text: constants

text: in

text: Section

text: below.

text: Two

text: common

text: physical

text: situations

text: where

text: this

text: mathematical

text: model

text: arise

text: include

text: ﬁrst,

text: if

text: there

text: is

text: a

text: double

text: sessile

text: drop

text: of

text: two

text: distinct

text: immiscible

text: ﬂuids

text: resting

text: on

text: a

text: surface

text: with

text: air

text: above,

text: and

text: second,

text: if

text: a

text: drop

text: of

text: a

text: light

text: ﬂuid

text: is

text: ﬂoating

text: on

text: the

text: top

text: of

text: a

text: heavier

text: ﬂuid

text: and

text: below

text: a

text: lighter

text: ﬂuid

text: as

text: would

text: be

text: the

text: case

text: when

text: oil

text: ﬂoats

text: on

text: water

text: and

text: below

text: air.

text: See

text: Figure

text: for

text: an

text: example

text: of

text: the

text: ﬁrst

text: situation,

text: and

text: Figure

text: (found

text: within

text: Section

text: for

text: an

text: example

text: of

text: the

text: second

text: situation.

text: The

text: terms

text: in

text: the

text: energy

text: functional

text: given

text: above

text: arise

text: from

text: (in


text: and deﬁne Φ (r) and Ψ (r) by replacing ϕ with ϕ in the deﬁnitions of Φ(r) and Ψ(r) (cid:15) (cid:15) (cid:15) respectively. Then, by approximation using Theorem 2, p.172 of [EG] or something similar, for ∈ (cid:15) (0, 1) and ϕ = ϕ in (6.2) and (6.3) we obtain (cid:15) (cid:48) (cid:48) Φ (r) − − Φ (r) Ψ (r) ∈ (cid:15)− (n 1) (cid:15) = (cid:15)− a.e. r (0, d). (6.18) rn 1 rn rn 1 F { } (cid:90) Deﬁne Φ (r) = ( E(cid:88), B ) and 0 S j r · 2 (ν (x) x)2 H − ∈ γ(r) = α E|j | d n 1(x), r (0, d). (6.19) j ∩ ∗ x n+1 j=0 Br ∂ Ej ∈ (cid:90) For r (0, d), the Lebesg(cid:88)ue Dominated Convergence Theorem implies 2 → H − → Φ α d n 1(x) = Φ , as (cid:15) 0. (6.20) (cid:15) j ∗ ∩ 0 j=0 ∂ Ej Br ∈ Claim 6.4 For a.e. r (0, d) (cid:48) → (cid:48) (cid:48) → − (cid:48) Φ (r) Φ (r), Ψ (r) rn 1γ (r), (6.21) (cid:15) 0 (cid:15) → ∈ as (cid:15) 0. In particular, this holds for every r (0, d) where Φ and γ are diﬀerentiable. 0 Proof. (of claim)(cid:32)Upon examining, we write (cid:33) (cid:90) (cid:90) (cid:18) (cid:19) (cid:88) | | 2 H − 1 − x H − Φ (r) = α d n 1(x) + d n 1(x) . (6.22) (cid:15) j ∗ ∩ ∗ ∩ \ (cid:15) (cid:15)r j=0 ∂ Ej Br(1−(cid:15)) ∂ Ej (Br Br(1−(cid:15))) We wish to diﬀe(cid:90)rentiate the (cid:18)term in the p(cid:19)arentheses abo(cid:90)ve. We c(cid:18)an expres(cid:19)s that term as: | | | | − 1 x H − 1 − x H − I (r) + I (r) := 1 + d n 1(x) + d n 1(x) . 1 2 ∗ ∩ (cid:15) (cid:15)r ∗ ∩ (cid:15) (cid:15)r ∂ Ej Br(1−(cid:15)) ∂ Ej Br (6.23) Then (cid:90) (cid:90) (cid:18) (cid:19) | | (cid:48) (cid:90) (cid:18)H −(cid:19) − x H − I (r) = 0 d n 1(x) d n 1(x) 1 ∗ ∩ ∗ ∩ (cid:15)r2 ∂ Ej ∂Br(1−(cid:15)) | | ∂ Ej Br(1−(cid:15)) − x H − = d n 1(x) , (6.24) ∗ ∩ (cid:15)r2 ∂ Ej Br(1−(cid:15)) 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: deﬁne

text: Φ

text: (r)

text: and

text: Ψ

text: (r)

text: by

text: replacing

text: ϕ

text: with

text: ϕ

text: in

text: the

text: deﬁnitions

text: of

text: Φ(r)

text: and

text: Ψ(r)

text: respectively.

text: Then,

text: by

text: approximation

text: using

text: Theorem

text: of

text: [EG]

text: or

text: something

text: similar,

text: for

text: and

text: ϕ

text: ϕ

text: in

text: and

text: we

text: obtain

text: Φ

text: (r)

text: Φ

text: (r)

text: Ψ

text: (r)

text: (n

text: a.e.

text: r

text: d).

text: rn

text: rn

text: rn

text: F

text: Deﬁne

text: Φ

text: (r)

text: B

text: and

text: S

text: j

text: r

text: (ν

text: (x)

text: H

text: γ(r)

text: α

text: E|j

text: d

text: n

text: r

text: d).

text: j

text: x

text: Br

text: Ej

text: For

text: r

text: d),

text: the

text: Dominated

text: Convergence

text: Theorem

text: implies

text: H

text: Φ

text: α

text: d

text: n

text: Φ

text: as

text: j

text: Ej

text: Br

text: Claim

text: For

text: a.e.

text: r

text: d)

text: Φ

text: (r)

text: Φ

text: (r),

text: Ψ

text: (r)

text: rn

text: 1γ

text: (r),

text: as

text: In

text: particular,

text: this

text: holds

text: for

text: every

text: r

text: d)

text: where

text: Φ

text: and

text: γ

text: are

text: diﬀerentiable.

text: Proof.

text: (of

text: examining,

text: we

text: write

text: H

text: x

text: H

text: Φ

text: (r)

text: α

text: d

text: n

text: d

text: n

text: j

text: Ej

text: Ej

text: (Br

text: We

text: wish

text: to

text: the

text: in

text: the

text: We

text: that

text: term

text: as:

text: x

text: H

text: x

text: H

text: I

text: (r)

text: I

text: (r)

text: d

text: n

text: d

text: n

text: Ej

text: Ej

text: Br

text: Then

text: x

text: H

text: I

text: (r)

text: d

text: n

text: d

text: n

text: Ej

text: Ej

text: x

text: H

text: d

text: n

text: Ej

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: { } ∀ ⊂⊂ Ψ ( E , A) = 0 , A Ω (7.3) V j,k (with potentially diﬀerent Dirichlet data and volume constraints for each k). Suppose there { } exists a triple E such that j → χ χ in L1 (Ω), j = 0, 1, 2 (7.4) Ej,k Ej loc { } F Then E is a DV-minimizer of over Ω: j S { } ∀ ⊂⊂ Ψ ( E , A) = 0 , A Ω. (7.5) V j ⊂⊂ (cid:90) Moreover, if L Ω is any open set such that | | Dχ = 0, j = 0, 1, 2, (7.6) Ej ∂L then we have F { } F { } lim ( E , L) = ( E , L). (7.7) →∞ S j,k S j k Remark 7.3 (Weakness of some of the hypotheses) Equation (7.4) can be guaranteed by Helly’s Selection Theorem as long as all of the conﬁgurations have uniformly bounded energy. ⊂⊂ (cid:32) (cid:18) (cid:19) (cid:33) Proof. Let A Ω. We may suppose that ∂A is smooth, so that for every(cid:88)k: − F { } ≤ H − 1 max diam of A n 1 2 ( E , A) n 1(∂A) + ω − α , (7.8) S j,k n 1 j 2 2 j=0 { } which follows by covering ∂A with all three values, and bounding the minimal energy of E j by a (standard) competitor on a possibly larger domain. Then lower semicontinuity implies the { } { } same inequality holds with E replaced with E . j,k j For t > 0, let { ∈ } A = x Ω : dist(x, A) < t . (7.9) t (cid:90) We have | − | lim χ χ dx = 0, j = 0, 1, 2 (7.10) k→∞ At Ej,k Ej 22 ##LTLine## ##LTLine##

text: Ψ

text: E

text: A)

text: A

text: Ω

text: V

text: j,k

text: (with

text: potentially

text: diﬀerent

text: Dirichlet

text: data

text: and

text: volume

text: constraints

text: for

text: each

text: k).

text: Suppose

text: there

text: exists

text: a

text: triple

text: E

text: such

text: that

text: j

text: χ

text: χ

text: in

text: (Ω),

text: j

text: Ej,k

text: Ej

text: loc

text: F

text: Then

text: E

text: is

text: a

text: DV-minimizer

text: of

text: over

text: Ω:

text: j

text: S

text: Ψ

text: E

text: A)

text: A

text: Ω.

text: V

text: j

text: Moreover,

text: if

text: L

text: Ω

text: is

text: any

text: open

text: set

text: such

text: that

text: Dχ

text: j

text: Ej

text: ∂L

text: then

text: we

text: have

text: F

text: F

text: lim

text: E

text: L)

text: E

text: L).

text: S

text: j,k

text: S

text: j

text: k

text: Remark

text: (Weakness

text: of

text: some

text: of

text: the

text: hypotheses)

text: Equation

text: can

text: be

text: guaranteed

text: by

text: Helly’s

text: Selection

text: Theorem

text: as

text: long

text: as

text: all

text: of

text: the

text: conﬁgurations

text: have

text: uniformly

text: bounded

text: energy.

text: Proof.

text: Let

text: A

text: Ω.

text: We

text: may

text: suppose

text: that

text: ∂A

text: is

text: smooth,

text: so

text: that

text: for

text: F

text: H

text: max

text: diam

text: of

text: A

text: n

text: E

text: A)

text: n

text: ω

text: α

text: S

text: j,k

text: n

text: j

text: which

text: follows

text: by

text: covering

text: ∂A

text: with

text: all

text: three

text: values,

text: and

text: bounding

text: the

text: minimal

text: energy

text: of

text: E

text: j

text: by

text: a

text: (standard)

text: competitor

text: on

text: a

text: possibly

text: larger

text: domain.

text: Then

text: lower

text: semicontinuity

text: implies

text: the

text: same

text: inequality

text: holds

text: with

text: E

text: replaced

text: with

text: E

text: j,k

text: j

text: For

text: t

text: let

text: A

text: x

text: Ω

text: dist(x,

text: A)

text: t

text: t

text: We

text: have

text: lim

text: χ

text: χ

text: dx

text: j

text: k→∞

text: At

text: Ej,k

text: Ej

text: ##LTLine##

text: ##LTLine##


text: (cid:90) { } and therefore there exists a subsequence E such that for almost every t close to 0 j,ki | − | H − li(cid:80)m χ χ d n 1 = 0, j = 0, 1, 2. (7.11) k→∞ ∂At Ej,ki Ej | | → From Lemma 7.1, as 2 ∆V 0, we have for those t j=0 j,k { { lim Υ ( E , A ) = Υ ( E , A ) (7.12) k→∞ V j,ki t V j t and by lower semicontinuity { } Ψ ( E , A ) = 0(cid:82). (7.13) V j t ⊂⊂ | | Thus (7.5) holds. Now let L Ω be such that Dχ = 0 for j = 0, 1, 2, and let ⊂⊂ ⊂⊂ ∂{L }Ej { } A be a smooth open set such that L A Ω. Let F be any subsequence of E . j,k { } j,k Repeating the same argument as above, there is a set A and a subsequence F such that t j,ki { } { } lim Υ ( F , A ) = Υ ( F , A ). (7.14) k→∞ V j,ki t V j t { } { } Since Ψ ( F , A ) = Ψ ( E , A ) = 0 we have V j,ki t V j t F { } F { } lim ( F , A ) = ( F , A ), (7.15) k→∞ S j,ki t S j t thus from Theorem 2.3 (with A playing the role of Ω) t F { } F { } lim ( F , L) = ( E , L), (7.16) →∞ S j,k S j k completing the proof. Figure 4: Limiting conﬁguration as cones. 23 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: and

text: therefore

text: there

text: exists

text: a

text: subsequence

text: E

text: such

text: that

text: for

text: almost

text: every

text: t

text: close

text: to

text: j,ki

text: H

text: χ

text: χ

text: d

text: n

text: j

text: k→∞

text: ∂At

text: Ej,ki

text: Ej

text: From

text: Lemma

text: as

text: ∆V

text: we

text: have

text: for

text: those

text: t

text: j,k

text: lim

text: Υ

text: E

text: A

text: Υ

text: E

text: A

text: k→∞

text: V

text: j,ki

text: t

text: V

text: j

text: t

text: and

text: by

text: lower

text: semicontinuity

text: Ψ

text: E

text: A

text: V

text: j

text: t

text: Thus

text: holds.

text: Now

text: let

text: L

text: Ω

text: be

text: such

text: that

text: Dχ

text: for

text: j

text: and

text: let

text: ∂{L

text: }Ej

text: A

text: be

text: a

text: smooth

text: open

text: set

text: such

text: that

text: L

text: A

text: Ω.

text: Let

text: F

text: be

text: any

text: subsequence

text: of

text: E

text: j,k

text: j,k

text: Repeating

text: the

text: same

text: argument

text: as

text: above,

text: there

text: is

text: a

text: set

text: A

text: and

text: a

text: subsequence

text: F

text: such

text: that

text: t

text: j,ki

text: lim

text: Υ

text: F

text: A

text: Υ

text: F

text: A

text: k→∞

text: V

text: j,ki

text: t

text: V

text: j

text: t

text: Since

text: Ψ

text: F

text: A

text: Ψ

text: E

text: A

text: we

text: have

text: V

text: j,ki

text: t

text: V

text: j

text: t

text: F

text: F

text: lim

text: F

text: A

text: F

text: A

text: k→∞

text: S

text: j,ki

text: t

text: S

text: j

text: t

text: thus

text: from

text: Theorem

text: (with

text: A

text: playing

text: the

text: role

text: of

text: Ω)

text: t

text: F

text: F

text: lim

text: F

text: L)

text: E

text: L),

text: S

text: j,k

text: S

text: j

text: k

text: completing

text: the

text: proof.

text: Figure

text: Limiting

text: conﬁguration

text: as

text: cones.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: and so F { } − F { } − F { } ( C , B(x , ρ)) = t1 n ( C , B(x , ρt)) = ρn 1 ( C , B(x , 1)) (8.3) S j,t 0 S j 0 S j 0 { } The argument in the proof of Theorem 7.4 implies the existence of a sequence t converging { } { } i to 0 such that C converges to cones Q , each with a vertex at x , and that V-minimize F j,i j 0 over IRn. S { } It remains to prove that Q are cylinders with axes through 0 and x . This is equivalent { } ⊆j − { } { } × 0 { } to the existence of sets A IRn 1 such that Q = A IR. Because the C are all j (cid:104) (cid:105) j j j cones with vertex at 0, we have x, Dχ = 0 and hence Cj −(cid:104) − (cid:105) aD χ = x x , Dχ . (8.4) n Cj 0 Cj Thus | − | | | ≤ x x | | D χ | | 0 Dχ (8.5) n Cj x Cj 0 and then (cid:90) (cid:90) (cid:88) (cid:88) 2 2 | | − | | α D χ = t1 n α (cid:90) D χ (8.6) j n Cj,t (cid:88)j n Cj j=0 B(x0,ρ) j=0 B(x0,ρt) − ≤ t2 nρ 2 | | | | α Dχ (8.7) x j Cj 0 j=0 B(x0,ρt) − t2 nρF { } = | | ( C , B(x , ρt)) (8.8) S j 0 x 0 ≤ ρnt C | | . (8.9) x 0 Thus D χ = lim D χ = 0, j = 0, 1, 2. (8.10) n Qj i→∞ n Cj,ti (cid:90) (cid:90) However, for almost all s < t, by Theorem 2.6, | − | H − ≤ | | χ χ d n 1 D χ = 0 (8.11) B Qj,s Qj,t B × n Qj R R (s,t) ⊆ − where χ (y) = χ (y, r). This implies the existence of sets A IRn 1 such that for Qj,r Qj j almost all r and s we have 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: so

text: F

text: F

text: F

text: C

text: B(x

text: ρ))

text: n

text: C

text: B(x

text: ρt))

text: ρn

text: C

text: B(x

text: S

text: j,t

text: S

text: j

text: S

text: j

text: The

text: argument

text: in

text: the

text: proof

text: of

text: Theorem

text: implies

text: the

text: existence

text: of

text: a

text: sequence

text: t

text: converging

text: i

text: to

text: such

text: that

text: C

text: converges

text: to

text: cones

text: Q

text: each

text: with

text: a

text: vertex

text: at

text: x

text: and

text: that

text: V-minimize

text: F

text: j,i

text: j

text: over

text: IRn.

text: S

text: It

text: remains

text: to

text: prove

text: that

text: Q

text: are

text: cylinders

text: with

text: axes

text: through

text: and

text: x

text: This

text: is

text: equivalent

text: ⊆j

text: to

text: the

text: existence

text: of

text: sets

text: A

text: IRn

text: such

text: that

text: Q

text: A

text: IR.

text: Because

text: the

text: C

text: are

text: all

text: j

text: j

text: j

text: j

text: cones

text: with

text: vertex

text: at

text: we

text: have

text: x,

text: Dχ

text: and

text: hence

text: Cj

text: aD

text: χ

text: x

text: x

text: Dχ

text: n

text: Cj

text: Cj

text: Thus

text: x

text: x

text: D

text: χ

text: Dχ

text: n

text: Cj

text: x

text: Cj

text: and

text: then

text: α

text: D

text: χ

text: n

text: α

text: D

text: χ

text: j

text: n

text: Cj,t

text: n

text: Cj

text: nρ

text: α

text: Dχ

text: x

text: j

text: Cj

text: nρF

text: C

text: B(x

text: ρt))

text: S

text: j

text: x

text: ρnt

text: C

text: x

text: Thus

text: D

text: χ

text: lim

text: D

text: χ

text: j

text: n

text: Qj

text: i→∞

text: n

text: Cj,ti

text: However,

text: for

text: almost

text: all

text: s

text: t,

text: by

text: Theorem

text: H

text: χ

text: χ

text: d

text: n

text: D

text: χ

text: B

text: Qj,s

text: Qj,t

text: B

text: n

text: Qj

text: R

text: R

text: (s,t)

text: where

text: χ

text: (y)

text: χ

text: (y,

text: r).

text: This

text: implies

text: the

text: existence

text: of

text: sets

text: A

text: IRn

text: such

text: that

text: for

text: Qj,r

text: Qj

text: j

text: almost

text: all

text: r

text: and

text: s

text: we

text: have

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − { } { } in IRn 1, then there exists (cid:15) > 0, R > 0, and sets E coinciding with A outside some ⊆ B j j compact set H ˜ such that R F { } B ≤ F { } B − ( E , ˜ ) ( A , ˜ ) (cid:15). (8.15) S j R S j R (cid:40) Let T > 0 and set × − | | E ( T, T ) in x < T j n M = | | (8.16) j Q outside x < T j n (cid:16){ } { } (cid:17) ×(cid:16)− (cid:17) for j = 0, 1, 2, giving M = Q outside H [ T, T ]. Hence j j F { } B × − ≤ F { } B × − Q , ˜ [ T, T ] M , ˜ [ T, T ] . (8.17) S j R S j R (cid:16) (cid:17) (cid:16) (cid:17) However, we have F { } B × − F { } B Q , ˜ [ T, T ] = 2T A , ˜ (8.18) S j R S j R and (cid:16) (cid:17) (cid:16) (cid:17) (cid:88) 2 F { } B × − ≤ F { } B − S Mj , ˜R [ T, T ] 2T S (cid:16) Ej , ˜R(cid:17) + 2ωn−1Rn 1 αj (cid:88) (8.19) j=0 2 ≤ F { } B − − 2T A , ˜ 2T (cid:15) + 2ω − Rn 1 α (8.20) S j R (cid:80)n 1 j j=0 This contradicts (8.17) for suﬃciently large T , say, T > ωn−1 Rn−1 2 α . (cid:15) j=0 j If we weaken our deﬁnition of minimality by abandoning the volume constraint again, then we are able to prove the converse. We expect it is true with the volume constraint, but Figure 7 illustrates the diﬃculty in generalizing the following proof. { } { }× − × Theorem 8.4 Suppose Q = A IR are permissible cylinders in IRn = IRn 1 IR without j j { } − { } the volume constraint condition. Then A is minimal in IRn 1 implies Q is minimal in j j IRn. { } − { } Proof. Suppose A is minimal in IRn 1 and let M be permissible Caccioppoli sets in IRn { j} j B − coinciding with Q outside some compact set K. Recall that ˜ denotes the ball in IRn 1 j R centered at 0 with radius R, and choose T > 0 such that ⊆ B × − K ˜ ( T, T ). (8.21) T 29 ##LTLine##

text: in

text: IRn

text: then

text: there

text: exists

text: R

text: and

text: sets

text: E

text: coinciding

text: with

text: A

text: outside

text: some

text: B

text: j

text: j

text: compact

text: set

text: H

text: such

text: that

text: R

text: F

text: B

text: F

text: B

text: E

text: A

text: S

text: j

text: R

text: S

text: j

text: R

text: Let

text: T

text: and

text: set

text: E

text: T,

text: T

text: in

text: x

text: T

text: j

text: n

text: M

text: j

text: Q

text: outside

text: x

text: T

text: j

text: n

text: for

text: j

text: giving

text: M

text: Q

text: outside

text: H

text: T,

text: T

text: Hence

text: j

text: j

text: F

text: B

text: F

text: B

text: Q

text: T,

text: T

text: M

text: T,

text: T

text: S

text: j

text: R

text: S

text: j

text: R

text: However,

text: we

text: have

text: F

text: B

text: F

text: B

text: Q

text: T,

text: T

text: A

text: S

text: j

text: R

text: S

text: j

text: R

text: and

text: F

text: B

text: F

text: B

text: S

text: Mj

text: ˜R

text: T,

text: T

text: S

text: Ej

text: αj

text: F

text: B

text: A

text: 2ω

text: Rn

text: α

text: S

text: j

text: R

text: j

text: This

text: contradicts

text: for

text: suﬃciently

text: large

text: T

text: say,

text: T

text: ωn−1

text: α

text: j

text: If

text: we

text: weaken

text: our

text: deﬁnition

text: of

text: minimality

text: by

text: abandoning

text: the

text: volume

text: constraint

text: again,

text: then

text: we

text: are

text: able

text: to

text: prove

text: the

text: converse.

text: We

text: expect

text: it

text: is

text: true

text: with

text: the

text: volume

text: constraint,

text: but

text: Figure

text: illustrates

text: the

text: diﬃculty

text: in

text: generalizing

text: the

text: following

text: proof.

text: Theorem

text: Suppose

text: Q

text: A

text: IR

text: are

text: permissible

text: cylinders

text: in

text: IRn

text: IRn

text: IR

text: without

text: j

text: j

text: the

text: volume

text: constraint

text: condition.

text: Then

text: A

text: is

text: minimal

text: in

text: IRn

text: implies

text: Q

text: is

text: minimal

text: in

text: j

text: j

text: IRn.

text: Proof.

text: Suppose

text: A

text: is

text: minimal

text: in

text: IRn

text: and

text: let

text: M

text: be

text: permissible

text: Caccioppoli

text: sets

text: in

text: IRn

text: j}

text: j

text: B

text: coinciding

text: with

text: Q

text: outside

text: some

text: compact

text: set

text: K.

text: Recall

text: that

text: denotes

text: the

text: ball

text: in

text: IRn

text: j

text: R

text: centered

text: at

text: with

text: radius

text: R,

text: and

text: choose

text: T

text: such

text: that

text: B

text: K

text: T,

text: T

text: T

text: ##LTLine##


text: Figure 12: An improvement when three consecutive sectors have only two ﬂuids. one of the angles is smaller than the corresponding Γ and then (after renaming the indices ij if necessary) we can invoke Proposition 9.7 followed by Proposition 9.6 to achieve the desired result. Thus, we turn immediately to the V-minimal case. The key observation in the V-minimal case is that we can actually improve Almgren’s Vol- ume Adjustment Lemma by removing any lack of uniformity when our conﬁguration consists solely of cones. Indeed, we suppose toward a contradiction that we have a V-minimal conﬁg- uration of cones which does not satisfy the angle condition. In this case, it follows from the D-minimal proof that we can lower the energy by some amount within B if we temporarily 1 ignore the volume constraint. On the other hand, by considering our sectors on a large enough disk, we can restore the volume constraint by adding or subtracting rectangles along the bound- aries of the sectors at a cost which is bounded by twice the width of the rectangle times the largest σ . See Figure 13. Of course, since we can choose our disk to be as large as we like, our ij rectangles can have arbitrarily small width, and therefore we can ﬁx the volume constraint with a loss to our energy which is as small as we like. The arbitrarily small width that we can have for these rectangles also guarantees that even if one of our sectors is very thin, by shrinking the width of the rectangle if necessary, we do not have to worry about having an intersection with more than one of the rays bounding our sector. Thus, the original conﬁguration could not pos- sibly have been the V-minimizer on our large disk and that gives us the desired contradiction. Remark 9.10 (Rectangles are not optimal, but very convenient) The competing vari- ations built by rectangles could be immediately improved by using smoother connections to the old boundary, however, we prefer the explicit construction presented in the proof. We note that Futer, Gnepp, McMath, Munson, Ng, Pahk, and Yoder [FGMMNPY] studied planar cones that are minimizing, which is similar to some of the result above, however proved by a calibration argument as in [LM1994]. Using Morgan’s result that there are only ﬁnitely many triple points in the tangent plane [Mo1998], we may use our results to conclude that there are only ﬁnitely many triple points on the blow up sphere ∂B . Classifying this ﬁnite number 1 of free boundary points now becomes a variant of the Coloring Problem. 38 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: Figure

text: An

label: caption

text: improvement

text: when

text: three

text: consecutive

text: sectors

text: have

text: only

text: two

text: ﬂuids.

text: one

text: of

text: the

text: angles

text: is

text: smaller

text: than

text: the

text: corresponding

text: Γ

text: and

text: then

text: (after

text: renaming

text: the

text: indices

text: ij

text: if

text: necessary)

text: we

text: can

text: invoke

text: Proposition

text: followed

text: by

text: Proposition

text: to

text: achieve

text: the

text: desired

text: result.

text: Thus,

text: we

text: turn

text: immediately

text: to

text: the

text: V-minimal

text: case.

text: The

text: key

text: observation

text: in

text: the

text: V-minimal

text: case

text: is

text: that

text: we

text: can

text: actually

text: improve

text: Almgren’s

text: Vol-

text: ume

text: Adjustment

text: Lemma

text: by

text: removing

text: any

text: lack

text: of

text: uniformity

text: when

text: our

text: conﬁguration

text: consists

text: solely

text: of

text: cones.

text: Indeed,

text: we

text: suppose

text: toward

text: a

text: contradiction

text: that

text: we

text: have

text: a

text: V-minimal

text: conﬁg-

text: uration

text: of

text: cones

text: which

text: does

text: not

text: satisfy

text: the

text: angle

text: condition.

text: In

text: this

text: case,

text: it

text: follows

text: from

text: the

text: D-minimal

text: proof

text: that

text: we

text: can

text: lower

text: the

text: energy

text: by

text: some

text: amount

text: within

text: B

text: if

text: we

text: temporarily

text: ignore

text: the

text: volume

text: constraint.

text: On

text: the

text: other

text: hand,

text: by

text: considering

text: our

text: sectors

text: on

text: a

text: large

text: enough

text: disk,

text: we

text: can

text: restore

text: the

text: volume

text: constraint

text: by

text: adding

text: or

text: subtracting

text: rectangles

text: along

text: the

text: bound-

text: aries

text: of

text: the

text: sectors

text: at

text: a

text: cost

text: which

text: is

text: bounded

text: by

text: twice

text: the

text: width

text: of

text: the

text: rectangle

text: times

text: the

text: largest

text: σ

text: See

text: Figure

text: Of

text: course,

text: since

text: we

text: can

text: choose

text: our

text: disk

text: to

text: be

text: as

text: large

text: as

text: we

text: like,

text: our

text: ij

text: rectangles

text: can

text: have

text: arbitrarily

text: small

text: width,

text: and

text: therefore

text: we

text: can

text: ﬁx

text: the

text: volume

text: constraint

text: with

text: a

text: loss

text: to

text: our

text: energy

text: which

text: is

text: as

text: small

text: as

text: we

text: like.

text: The

text: arbitrarily

text: small

text: width

text: that

text: we

text: can

text: have

text: for

text: these

text: rectangles

text: also

text: guarantees

text: that

text: even

text: if

text: one

text: of

text: our

text: sectors

text: is

text: very

text: thin,

text: by

text: shrinking

text: the

text: width

text: of

text: the

text: rectangle

text: if

text: necessary,

text: we

text: do

text: not

text: have

text: to

text: worry

text: about

text: having

text: an

text: intersection

text: with

text: more

text: than

text: one

text: of

text: the

text: rays

text: bounding

text: our

text: sector.

text: Thus,

text: the

text: original

text: conﬁguration

text: could

text: not

text: pos-

text: sibly

text: have

text: been

text: the

text: V-minimizer

text: on

text: our

text: large

text: disk

text: and

text: that

text: gives

text: us

text: the

text: desired

text: contradiction.

text: Remark

text: (Rectangles

text: are

text: not

text: optimal,

text: but

text: very

text: convenient)

text: The

text: competing

text: vari-

text: ations

text: built

text: by

text: rectangles

text: could

text: be

text: immediately

text: improved

text: by

text: using

text: smoother

text: connections

text: to

text: the

text: old

text: boundary,

text: however,

text: we

text: prefer

text: the

text: explicit

text: construction

text: presented

text: in

text: the

text: proof.

text: We

text: note

text: that

text: Futer,

text: Gnepp,

text: McMath,

text: Munson,

text: Ng,

text: Pahk,

text: and

text: Yoder

text: [FGMMNPY]

text: studied

text: planar

text: cones

text: that

text: are

text: minimizing,

text: which

text: is

text: similar

text: to

text: some

text: of

text: the

text: result

text: above,

text: however

text: proved

text: by

text: a

text: calibration

text: argument

text: as

text: in

text: Using

text: Morgan’s

text: result

text: that

text: there

text: are

text: only

text: ﬁnitely

text: many

text: triple

text: points

text: in

text: the

text: tangent

text: plane

text: we

text: may

text: use

text: our

text: results

text: to

text: conclude

text: that

text: there

text: are

text: only

text: ﬁnitely

text: many

text: triple

text: points

text: on

text: the

text: blow

text: up

text: sphere

text: ∂B

text: Classifying

text: this

text: ﬁnite

text: number

text: of

text: free

text: boundary

text: points

text: now

text: becomes

text: a

text: variant

text: of

text: the

text: Coloring

text: Problem.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: { } Figure 2: Permissible sets E . j See Figure 2. The full energy functional which sums surface tension, wetting energy, and potential energy (cid:32) (cid:33) (cid:90) (cid:90) (cid:90) due to gravity is given by: (cid:88) 2 F { } | | H − ( E ) := α Dχ + β χ d n 1 + ρ g z dV , (3.3) SWP j j Ej j Ej j j=0 Ω ∂Ω Ej As we scale inward we can eliminate the wetting energy entirely and view our solution restricted (cid:32) (cid:33) (cid:90) (cid:90) to an interior ball as a minimizer of (cid:88)an energy given by: 2 F { } | | ( E ) := α Dχ + ρ g z dV . (3.4) SP j j Ej j j=0 Ω Ej (cid:32) (cid:90) (cid:90) (cid:33) (cid:48) ⊂⊂ Of course this energy we will freq(cid:88)uently consider on subdomains, so for Ω Ω we deﬁne: 2 F { } (cid:48) | | ( E , Ω ) := α Dχ + ρ g z dV . (3.5) SP j j (cid:48) Ej j ∩ (cid:48) j=0 Ω Ej Ω Massari showed that this energy functional is lower semicontinuous in [Mas] under certain F ≡ assumptions on the constants. (In fact he showed it for , but where β 0 is allowed.) F SWP j The lower semicontinuity of ensures that this Dirichlet problem is well-posed, although it SP does not guarantee that the Dirichlet data is attained in the usual sense. In fact, a minimizer can actually have any Dirichlet data, but if it does not match up with the given data, then it must pay for an interface at the boundary. Summarizing these statements from [Mas] we can say: Theorem 3.2 (Massari’s Existence Theorem) If ≥ α 0, j ≥ | − | (3.6) α + α β β , i j i j 7 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: Permissible

text: sets

text: E

text: j

text: See

text: Figure

text: The

text: full

text: energy

text: functional

text: which

text: sums

text: surface

text: tension,

text: wetting

text: energy,

text: and

text: potential

text: energy

text: due

text: to

text: gravity

text: is

text: given

text: by:

text: F

text: H

text: E

text: α

text: Dχ

text: β

text: χ

text: d

text: n

text: ρ

text: g

text: z

text: dV

text: SWP

text: j

text: j

text: Ej

text: j

text: Ej

text: j

text: Ω

text: ∂Ω

text: Ej

text: As

text: we

text: scale

text: inward

text: we

text: can

text: eliminate

text: the

text: wetting

text: energy

text: entirely

text: and

text: view

text: our

text: solution

text: restricted

text: to

text: an

text: interior

text: ball

text: as

text: a

text: minimizer

text: of

text: energy

text: given

text: by:

text: F

text: E

text: α

text: Dχ

text: ρ

text: g

text: z

text: dV

text: SP

text: j

text: j

text: Ej

text: j

text: Ω

text: Ej

text: Of

text: course

text: this

text: energy

text: we

text: will

text: consider

text: on

text: subdomains,

text: so

text: for

text: Ω

text: Ω

text: we

text: deﬁne:

text: F

text: E

text: Ω

text: α

text: Dχ

text: ρ

text: g

text: z

text: dV

text: SP

text: j

text: j

text: Ej

text: j

text: Ω

text: Ej

text: Ω

text: Massari

text: showed

text: that

text: this

text: energy

text: functional

text: is

text: lower

text: semicontinuous

text: in

text: [Mas]

text: under

text: certain

text: F

text: assumptions

text: on

text: the

text: constants.

text: (In

text: fact

text: he

text: showed

text: it

text: for

text: but

text: where

text: β

text: is

text: allowed.)

text: F

text: SWP

text: j

text: The

text: lower

text: semicontinuity

text: of

text: ensures

text: that

text: this

text: Dirichlet

text: problem

text: is

text: well-posed,

text: although

text: it

text: SP

text: does

text: not

text: guarantee

text: that

text: the

text: Dirichlet

text: data

text: is

text: attained

text: in

text: the

text: usual

text: sense.

text: In

text: fact,

text: a

text: minimizer

text: can

text: actually

text: have

text: any

text: Dirichlet

text: data,

text: but

text: if

text: it

text: does

text: not

text: match

text: up

text: with

text: the

text: given

text: data,

text: then

text: it

text: must

text: pay

text: for

text: an

text: interface

text: at

text: the

text: boundary.

text: Summarizing

text: these

text: statements

text: from

text: [Mas]

text: we

text: can

text: say:

text: Theorem

text: (Massari’s

text: Existence

text: Theorem)

text: If

text: α

text: j

text: α

text: α

text: β

text: β

text: i

text: j

text: i

text: j

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##