text: 11 � ��� ������ ����� ������ ����� � ��� ������ ����� ������ ����� ������ - - - ����-� - ��� � ������ � � � � � � � �������� ��� ��� ��� ��� ��� without mirror, with phonons without mirror, without phonons � ��� with mirror, with phonons - - - with mirroΩr, without phonΩons Ω Ω ����� ���� ���� ω-ω���(� - ) ���� ���� ���� ��� ���� � ���Ω�( - ) � �� � � �� � �� � FIG. 7: Left: Incoherent component of the RF spectrum for a single TLS (blue) and the eﬀective TLS incorporating surface-induced modiﬁcations (red). Right: Ratio of coherent emission for all four cases (with/without mirror, with/without the phonon environment) as a function of the (normalised) eﬀective Rabi − frequency. Ω denotes the saturation Rabi frequency for γpt = 0.001 ps 1. See text for a discussion. s 0 even when driving far below saturation. The level at surface), as the problem then straightforwardly maps which this fraction plateaus is phonon coupling strength onto the case of several optical dipoles in a shared (free and temperature dependent54. By contrast, in the ab- space) electromagnetic environment – a problem which sence of phonons, almost all light is coherently scattered has been studied extensively, see, e.g., Ref.7. Future work at weak enough driving. The close agreement between might investigate the role of geometry in conﬁgurations the two curves in each pair bears testament to the fact with N > 1 emitters, possibly resulting in the enhance- that the surface-modiﬁed emitter largely behaves like a ment of Dicke superradiance of an ensemble of solid state bare emitter once the eﬀective Rabi frequency has been emitters31,55, or the use of mirrors to bring about other corrected for (with the slight remaining discrepancy due collective eﬀects in the light matter interaction, for ex- to modiﬁcations of the natural lifetime). Indeed, plot- ample inspired by a recent proposal for engineering the ting this ratio directly as a function of the laser driving quantum-enhanced absorption of light56 or by harnessing ﬁeld amplitude reveals sizeable horizontal shifts between sub-radiant collective states57,58. these two curves in each pair (not shown). Another interesting avenue for future work might be VI. Summary and Discussion the study of charged quantum dots featuring excited trion states. In addition to the optical dipole, the im- age approach would then feature a separate permanent We have extended the method of images – traditionally dipole. To a ﬁrst approximation, we would expect this developed for capturing spontaneous emission in atomic second dipole to be static, meaning it would not radi- ensembles near reﬂective interfaces – to the case of a ate and only modify the spectrum via energetic shifts. driven solid-state emitter near a metal surface. We have However, one might speculate whether the Coulomb in- developed two approaches: a half-sided cavity and image teraction of the three charges involved in the trion state dipole, and shown that the latter agrees with the former, could slightly ‘wiggle’ this dipole, making some radiative but only when additional ‘selection rules’ are introduced contribution to the overall spectrum conceivable. to constrain the dynamics to the relevant subspace. Both our approaches agree with a Green’s function treatment Acknowledgements in the absence of a vibrational environment. Through a rigorous derivation, we ﬁnd that the emitter can in- The authors thank Peter Kirton, Fabio Biancalana, deed still be described as an eﬀective (phonon-dressed) and David Gershoni for valuable suggestions. D.S. two-level system with appropriately modiﬁed properties, thanks SUPA for ﬁnancial support, T.S. acknowledges even in the presence of a phonon bath and for a driven studentship funding from EPSRC under grant no EP/G03673X/1, system. Our calculated RF spectrum corroborates this B. D. G. thanks the Royal Society, and E. M. G. acknowl- observation. edges support from the Royal Society of Edinburgh and We note that image dipole approach not only neces- the Scottish Government. sitates a larger Hilbert space but also involved a more Appendix cumbersome ME derivation than the half-sided cavity ap- proach. This begs the questions whether such an image A. Eigenbasis Dissipators approach remains useful. 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text: without

text: mirror,

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text: phonons

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text: Right:

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text: 9 tion represents diagonal Lamb shift terms. Their small D. Eﬀective TLS in the energy eigenbasis energetic shifts can be absorbed into the bare TLS transi- tion frequency. We thus focus on the oﬀ-diagonal element As stated in the introduction, previous literature treat- which is of the form: ing spontaneous emission from initially excited emitters considered the transition from the symmetrically excited to the ground state, as this choice yields matching re- 1G (cid:48) V12 = 2 12(q∆r)γ0pt(ω ) , (47) sults with o√ther {m|e(cid:105)tho| d(cid:105)s4|,7.(cid:105) W| e(cid:105)}follow t|h(cid:105)is app| ro(cid:105)√a|ch a(cid:105)nd adopt the basis e , s , a , g with s = ( 0 X + | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) − | (cid:105) | (cid:105) 1 2 G X1 02 )/ 2 and a = ( 01 X2 X1 02 )/ 2, see where the fun(cid:18)ction 12 is (cid:19) Fig. 5. In this basis, our full polaron ME reads: G 3 −sin(x) − cos(x) cos(x) d − i (cid:48) 12(x) = 2 x2 x3 + x . (48) dtρSP (t) = (cid:126)[HSP , ρSP (t)] (50) + Ds (ρ ) + Da (ρ ) + Ds (ρ ) , pn SP pt SP pt SP Again, this corresponds to the same energy shift term we have previously encountered in Sec. . After diagonal- where the dissipator terms are explicitly given in Ap- (cid:48) ising the Hamiltonian, the frequency of the symmetric pendix . Here, H denotes the system diagonalised SP excited to ground state transition (in the polaron frame) Hamiltonian [including the energy shift term Eq. (47)]. is then given by The ME photonic dissipator separates into a symmetric | (cid:105) ↔ | (cid:105) ↔ | (cid:105) channel ( g s e ) and an antisymmetric one | (cid:105) ↔ | (cid:105) ↔ | (cid:105) (cid:48) (cid:48) ( g a e ). Courtesy of the fully correlated ω˜ = ω + V , (49) 12 phonon bath, phonons also only act in the symmetric channel. √ √ Since Ω = Ω , the symmetric channel Rabi frequency 1 2 exactly matching the transition frequency Eq. (31) of becomes Ω := (Ω + Ω )/ 2 = 2Ω = Ω and hence sg 1 2 cav the half-sided cavity model. we obtain the same phonon rates as in the half-sided cav- ity approach5. Furthermore, the a√ntisymmetric channel − | (cid:105) Rabi frequency Ω := (Ω Ω )/ 2 = 0, meaning that e a 1 2 the laser ﬁeld is completely decoupled from the antisym- metric state. Consistency with the Green’s function and half-sided cavity approach demands that we restrict the dynam- ics of our four-dimensional Hilbert space to the subspace | (cid:105) {| (cid:105) | (cid:105)} s spanned by the states g , s , i.e. the larger Hilbert space only served to let us calculate the correct proper- 2V12 ties of this single transition. Fully decoupling the anti- | (cid:105) a symmetric singly and the doubly excited states from the dynamics is achieved by disabling the laser driving on | (cid:105) ↔ | (cid:105) the s e transition. For ﬁnite temperature photon (cid:54) environments with N(ω) = 0, we also need to remove dis- sipative photon absorption channels, by dropping the an- tisymmetric dissipator term Da (ρ ) from the ME and | (cid:105) pt S|P(cid:105) ↔ | (cid:105) g explicitly removing the dissipative s e operator. The image approach can thus be reduced to an eﬀec- tive TLS model featuring the same Rabi frequency, SE FIG. 5: Energy level diagram for the two emitter | (cid:105) | (cid:105) rate, and transition frequency as the half-sided cavity ap- system. The symmetric ( s ) and antisymmetric ( a ) proach – i.e. displaying full equivalence between the two levels are shifted up and down by V , respectively. The 12 representations. black arrows indicate the laser driving; the In Fig. 6, we summarise the key results from the pre- antisymmetric state is decoupled. Blue and red wavy vious sections: We show the transition frequency and SE lines indicate photon emission from the antisymmetric rate for the all four cases considered in this Article along- and symmetric channel, respectively. As discussed in side their schematic depictions. The driving term is not the text, it is necessary to disable driving on the | (cid:105) ↔ | (cid:105) s e transition (black dashed) to recover the | (cid:105) ↔ | (cid:105) eﬀective two level-system g s . For environments permitting photon absorption, the dashed wavy 5 The last equality holds due to the diﬀerence in density of modes transitions also need to be explicitly disabled. appearing in the derivation of the Rabi frequency in both models. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: tion

text: represents

text: diagonal

text: Lamb

text: shift

text: terms.

text: Their

text: small

text: D.

label: section

text: Eﬀective

text: TLS

text: in

text: the

text: energy

text: eigenbasis

text: energetic

text: shifts

text: can

text: be

text: absorbed

text: into

text: the

text: bare

text: TLS

text: transi-

text: tion

text: frequency.

text: We

text: thus

text: focus

text: on

text: the

text: oﬀ-diagonal

text: element

text: As

text: stated

text: in

text: the

text: introduction,

text: previous

text: literature

text: treat-

text: which

text: is

text: of

text: the

text: form:

text: ing

text: spontaneous

text: emission

text: from

text: initially

text: excited

text: emitters

text: considered

text: the

text: transition

text: from

text: the

text: symmetrically

text: excited

text: to

text: the

text: ground

text: state,

text: as

text: this

text: choice

text: yields

text: matching

text: re-

label: equation

text: sults

text: with

text: o√ther

text: W|

text: app|

text: adopt

text: the

text: basis

text: e

text: s

text: a

text: g

text: with

text: s

text: X

text: G

text: and

text: a

text: see

text: where

text: the

text: is

text: Fig.

text: In

text: this

text: basis,

text: our

text: full

text: polaron

text: ME

text: reads:

text: G

text: −sin(x)

text: cos(x)

text: cos(x)

text: d

text: i

text: x

text: dtρSP

text: (t)

text: ρSP

text: (t)]

text: Ds

text: (ρ

text: Da

text: (ρ

text: Ds

text: (ρ

text: pn

text: SP

text: pt

text: SP

text: pt

text: SP

text: Again,

text: this

text: corresponds

text: to

text: the

text: same

text: energy

text: shift

text: term

text: we

text: have

text: previously

text: encountered

text: in

text: Sec.

text: After

text: diagonal-

text: where

text: the

text: dissipator

text: terms

text: are

text: explicitly

text: given

text: in

text: Ap-

text: ising

text: the

text: Hamiltonian,

text: the

text: frequency

text: of

text: the

text: symmetric

text: pendix

text: Here,

text: H

text: denotes

text: the

text: system

text: diagonalised

text: SP

text: excited

text: to

text: ground

text: state

text: transition

text: (in

text: the

text: polaron

text: frame)

text: Hamiltonian

text: [including

text: the

text: energy

text: shift

text: term

text: Eq.

text: is

text: then

text: given

text: by

text: The

text: ME

text: photonic

text: dissipator

text: separates

text: into

text: a

text: symmetric

text: channel

text: g

text: s

text: e

text: and

text: an

text: antisymmetric

text: one

text: g

text: a

text: e

text: Courtesy

text: of

text: the

text: fully

text: correlated

text: ω˜

text: ω

text: V

text: phonon

text: bath,

text: phonons

text: also

text: only

text: act

text: in

text: the

text: symmetric

text: channel.

text: Since

text: Ω

text: Ω

text: the

text: symmetric

text: channel

text: Rabi

text: frequency

text: exactly

text: matching

text: the

text: transition

text: frequency

text: Eq.

text: of

text: becomes

text: Ω

text: (Ω

text: Ω

text: 2Ω

text: Ω

text: and

text: hence

text: sg

text: cav

text: the

text: half-sided

text: cavity

text: model.

text: we

text: obtain

text: the

text: same

text: phonon

text: rates

text: as

text: in

text: the

text: half-sided

text: cav-

text: ity

text: Furthermore,

text: the

text: a√ntisymmetric

text: channel

text: Rabi

text: frequency

text: Ω

text: (Ω

text: Ω

text: meaning

text: that

text: e

text: a

text: the

text: laser

text: ﬁeld

text: is

text: completely

text: decoupled

text: from

text: the

text: antisym-

text: metric

text: state.

text: Consistency

text: with

text: the

text: Green’s

text: function

text: and

text: half-sided

text: cavity

text: approach

text: demands

text: that

text: we

text: restrict

text: the

text: dynam-

text: ics

text: of

text: our

text: four-dimensional

text: Hilbert

text: space

text: to

text: the

text: subspace

text: s

text: spanned

text: by

text: the

text: states

text: g

text: s

text: i.e.

text: the

text: larger

text: Hilbert

text: space

text: only

text: served

text: to

text: let

text: us

text: calculate

text: the

text: correct

text: proper-

text: ties

text: of

text: this

text: single

text: transition.

text: Fully

text: decoupling

text: the

text: anti-

text: a

text: symmetric

text: singly

text: and

text: the

text: doubly

text: excited

text: states

text: from

text: the

text: dynamics

text: is

text: achieved

text: by

text: disabling

text: the

text: laser

text: driving

text: on

text: the

text: s

text: e

text: transition.

text: For

text: ﬁnite

text: temperature

text: photon

text: environments

text: with

text: N(ω)

text: we

text: also

text: need

text: to

text: remove

text: dis-

text: sipative

text: photon

text: absorption

text: channels,

text: by

text: dropping

text: the

text: an-

text: tisymmetric

text: dissipator

text: term

text: Da

text: (ρ

text: from

text: the

text: ME

text: and

text: pt

text: g

text: explicitly

text: removing

text: the

text: dissipative

text: s

text: e

text: operator.

text: The

text: image

text: approach

text: can

text: thus

text: be

text: reduced

text: to

text: an

text: eﬀec-

text: tive

text: TLS

text: model

text: featuring

text: the

text: same

text: Rabi

text: frequency,

text: SE

text: FIG.

text: Energy

text: level

text: diagram

text: for

text: the

text: two

text: emitter

text: rate,

text: and

text: transition

text: frequency

text: as

text: the

text: half-sided

text: cavity

text: ap-

text: system.

text: The

text: symmetric

text: s

text: and

text: antisymmetric

text: a

text: proach

text: i.e.

text: displaying

text: full

text: equivalence

text: between

text: the

text: two

text: levels

text: are

text: shifted

text: up

text: and

text: down

text: by

text: V

text: respectively.

text: The

text: representations.

text: black

text: arrows

text: indicate

text: the

text: laser

text: driving;

text: the

text: In

text: Fig.

text: we

text: summarise

text: the

text: key

text: results

text: from

text: the

text: pre-

text: antisymmetric

text: state

text: is

text: decoupled.

text: Blue

text: and

text: red

text: wavy

text: vious

text: sections:

text: We

text: show

text: the

text: transition

text: frequency

text: and

text: SE

text: lines

text: indicate

text: photon

text: emission

text: from

text: the

text: antisymmetric

text: rate

text: for

text: the

text: all

text: four

text: cases

text: considered

text: in

text: this

text: Article

text: along-

text: and

text: symmetric

text: channel,

text: respectively.

text: As

text: discussed

text: in

text: side

text: their

text: schematic

text: depictions.

text: The

text: driving

text: term

text: is

text: not

text: the

text: text,

text: it

text: is

text: necessary

text: to

text: disable

text: driving

text: on

text: the

text: s

text: e

text: transition

text: (black

text: dashed)

text: to

text: recover

text: the

text: eﬀective

text: two

text: level-system

text: g

text: s

text: For

text: environments

text: permitting

text: photon

text: absorption,

text: the

text: dashed

text: wavy

text: The

text: last

text: equality

text: holds

text: due

text: to

text: the

text: diﬀerence

text: in

text: density

text: of

text: modes

text: transitions

text: also

text: need

text: to

text: be

text: explicitly

text: disabled.

text: appearing

text: in

text: the

text: derivation

text: of

text: the

text: Rabi

text: frequency

text: in

text: both

text: models.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 15 To test the overall impact of the embedding selection problem on the quan- tum hardware’s performance, we experimentally compare three criteria previ- ously proposed in the literature. We generated 100 instances for each problem size and 50 embeddings for each instance, and selected 1 embedding according to each of the criteria. Speciﬁcally, we consider the following criteria: 1. Selecting the embedding with the smallest number of physical qubits (PQ); 2. Selecting the embedding with the shortest longest chain (LCh); 3. Selecting the embedding with chains of equal length (STD). Whereas the ﬁrst two criteria are straightforward, heuristic ME algorithms do not necessarily return embeddings with chains of equal length as required for the third criterion. To account for the selection of an embedding with chains of equal size, we calculate the standard deviation of the chain size for each embedding and select the embedding with the minimum standard deviation. Further, we consider the empirical selection criteria for which we select the embedding that yield the highest success probability. Figure 6a shows (a) Embedding selection comparison (b) Mean number of qubits used (c) Mean size of longest chain (d) Mean STD of chain size Fig. 6 Comparison of diﬀerent embedding selection criteria for instances with ∈ logical density d [75, 85]. Each point indicates the mean across 100 instances of each problem size. a) shows the comparison in terms of the mean success probability for the DW2X device performing 1000 annealing runs. Additional results show comparison in terms of (b) the mean number of physical qubits, (c) the mean size of the largest chain, and (d) the mean STD of chain size. the embedding selection comparison for the three criteria and the empirical ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: To

text: test

text: the

text: overall

text: impact

text: of

text: the

text: embedding

text: selection

text: problem

text: on

text: the

text: quan-

text: tum

text: hardware’s

text: performance,

text: we

text: experimentally

text: compare

text: three

text: criteria

text: previ-

text: ously

text: proposed

text: in

text: the

text: literature.

text: We

text: generated

text: instances

text: for

text: each

text: problem

text: size

text: and

text: embeddings

text: for

text: each

text: instance,

text: and

text: selected

text: embedding

text: according

text: to

text: each

text: of

text: the

text: criteria.

text: Speciﬁcally,

text: we

text: consider

text: the

text: following

text: criteria:

label: list

text: Selecting

text: the

text: embedding

text: with

text: the

text: smallest

text: number

text: of

text: physical

text: qubits

text: (PQ);

text: Selecting

text: the

text: embedding

text: with

text: the

text: shortest

text: longest

text: chain

text: (LCh);

text: Selecting

text: the

text: embedding

text: with

text: chains

text: of

text: equal

text: length

text: (STD).

text: Whereas

text: the

text: ﬁrst

text: two

text: criteria

text: are

text: straightforward,

text: heuristic

text: ME

text: algorithms

text: do

text: not

text: necessarily

text: return

text: embeddings

text: with

text: chains

text: of

text: equal

text: length

text: as

text: required

text: for

text: the

text: third

text: criterion.

text: To

text: account

text: for

text: the

text: selection

text: of

text: an

text: embedding

text: with

text: chains

text: of

text: equal

text: size,

text: we

text: calculate

text: the

text: standard

text: deviation

text: of

text: the

text: chain

text: size

text: for

text: each

text: embedding

text: and

text: select

text: the

text: embedding

text: with

text: the

text: minimum

text: standard

text: deviation.

text: Further,

text: we

text: consider

text: the

text: empirical

text: selection

text: criteria

text: for

text: which

text: we

text: select

text: the

text: embedding

text: that

text: yield

text: the

text: highest

text: success

text: probability.

text: Figure

text: shows

text: (a)

text: Embedding

text: selection

text: comparison

text: (b)

text: Mean

text: number

text: of

text: qubits

text: used

text: (c)

text: Mean

text: size

text: of

text: longest

text: chain

text: (d)

text: Mean

text: STD

text: of

text: chain

text: size

text: Fig.

text: Comparison

text: of

text: diﬀerent

text: embedding

text: selection

text: criteria

text: for

text: instances

text: with

text: logical

text: density

text: d

text: Each

text: point

text: indicates

text: the

text: mean

text: across

text: instances

text: of

text: each

text: problem

text: size.

text: a)

text: shows

text: the

text: comparison

text: in

text: terms

text: of

text: the

text: mean

text: success

text: probability

text: for

text: the

text: device

text: performing

text: annealing

text: runs.

text: Additional

text: results

text: show

text: comparison

text: in

text: terms

text: of

text: (b)

text: the

text: mean

text: number

text: of

text: physical

text: qubits,

text: (c)

text: the

text: mean

text: size

text: of

text: the

text: largest

text: chain,

text: and

text: (d)

text: the

text: mean

text: STD

text: of

text: chain

text: size.

text: the

text: embedding

text: selection

text: comparison

text: for

text: the

text: three

text: criteria

text: and

text: the

text: empirical

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: Single-View 3D Reconstruction of Correlated Gamma-Neutron Sources Mateusz Monterial, Student Member, IEEE, Peter Marleau, Member, IEEE, and Sara A. Pozzi, Senior Member, IEEE Abstract— We describe a new method of 3D image recon- medium. Hal Anger was the ﬁrst to develop a gamma camera struction of neutron sources that emit correlated gammas (e.g. through the use of multichannel collimators to modulate Cf-252, Am-Be). This category includes a vast majority of neutron incident radiation [7]. The same principle can be applied with sources important in nuclear threat search, safeguards and more complex coded aperture masks, which are analogous to non-proliferation. Rather than requiring multiple views of the source this technique relies on the source’s intrinsic property superimposed pinhole cameras, to image both thermal and fast of coincidence gamma and neutron emission. As a result only neutrons [8], [9]. a single-view measurement of the source is required to perform The technique introduced in this paper is an extension of the 3D reconstruction. In principle, any scatter camera sensitive the second category of radiation cameras which track multiple to gammas and neutrons with adequate timing and interaction scatters to reconstruct source location. In gamma ray imaging, location resolution can perform this reconstruction. Using a neutron double scatter technique, we can calculate a conical these are called Compton cameras [5], and are a mature surface of possible source locations. By including the time to technology with commercially available portable cameras [10]. a correlated gamma we further constrain the source location in The neutron scatter camera functions in an analogous way to three-dimensions by solving for the source-to-detector distance the Compton camera, but with the use of the time-of-ﬂight along the surface of said cone. As a proof of concept we applied (TOF) between the ﬁrst two scatters to determine the incident these reconstruction techniques on measurements taken with the the Mobile Imager of Neutrons for Emergency Responders neutron energy [11]. (MINER). Two Cf-252 sources measured at 50 and 60 cm from The discussion so far has been limited to 2D imaging the center of the detector were resolved in their varying depth systems, but in principle any radiation camera can produce with average radial distance relative resolution of 26%. To 3D reconstruction of a source. The most common approach is demonstrate the technique’s potential with an optimized system to take multiple 2D images from different views, and combine we simulated the measurement in MCNPX-PoliMi assuming timing resolution of 200 ps (from 2 ns in the current system) them to form 3D rendering of the source. This technique and source interaction location resolution of 5 mm (from 3 cm). is used in Single Photon Emission Computed Tomography These simulated improvements in scatter camera performance (SPECT) and Positron Emission Tomography (PET) to image resulted in radial distance relative resolution decreasing to an radioisotopes inside a patient [1]. More recently researchers at average of 11%. Lawrence Berkeley National Laboratory have used a variation of this technique, combined with 3D rendering of physical I. INTRODUCTION space, to reconstruct source locations in real time [12]. All T of these techniques require multiple views of the source and HE practice of radiation imaging is well established some freedom of movement. in ﬁelds as diverse as medicine [1], [2], astronomy Single-sided 3D imaging has been demonstrated in Comp- [o¨nfelder2004ooo¨nfelder20044], and nuclear safeguards and ton cameras by taking advantage of the parallax effect [13]. non-proliferation [5], [6]. Radiation imaging cameras function However, parallax techniques require a large solid angle cov- by either: 1) modulating the incident radiation, or 2) track- erage to function as a modality at all, whereas our method ing the multiple scatters of incident particle in the detector only requests solid angle coverage to increase efﬁciency. Furthermore, our technique would function at any distance This material is based upon work supported by the U.S. Department of Homeland Security Grant Award Number: 2012-DN-130-NF0001. The views with a portable system, even if it required long dwell times. and conclusions contained in this document are those of the authors and should Parallax by comparison would be restricted by system size and not be interpreted as representing the ofﬁcial policies, either expressed or require close enough source-to-detector distances to function implied, of the U.S. Department of Homeland Security. Sandia National Laboratories is multi-program laboratory managed and in the ﬁrst place. This makes our technique potentially valuable operated by Sandia Corporation, a wholly owned subsidiary of Lockheed for nuclear inspection, emergency response and treaty veriﬁ- Martin Corporation, for the U.S. Department of Energy’s National Nuclear cation, where multiple views of the object of interest may be Security Administration under contract DE-AC04-94AL85000. SAND number 2016-10226 J. restricted, even though the location of the source is known. This work was funded in-part by the Consortium for Veriﬁcation Technol- A single system can image both neutrons and gammas if ogy under Department of Energy National Nuclear Security Administration the underlying detectors are sensitive to both particle types. award number DE-NA0002534. M. Monterial and S. A. Pozzi are with the Nuclear Engineering and Such scatter camera systems were constructed as radiation Radiological Sciences Department, University of Michigan, Ann Arbor, MI telescopes for astronomy [4], and more recently for stand- 48109 USA (e-mail: mateuszm@umich.edu). off detection in nuclear security applications [14]. We employ P. Marleau is with Sandia National Laboratories, Livermore, CA 94550 USA. this class of scatter camera to project a cone of possible

text: Single-View

label: title

text: Reconstruction

text: of

text: Correlated

text: Gamma-Neutron

text: Sources

text: Mateusz

label: author

text: Monterial,

text: Student

text: Member,

text: IEEE,

text: Peter

text: Marleau,

text: Member,

text: IEEE,

text: and

text: Sara

text: A.

text: Pozzi,

text: Senior

text: Member,

text: IEEE

text: Abstract—

text: We

label: abstract

text: describe

text: a

text: new

text: method

text: of

text: image

text: recon-

text: medium.

text: Hal

text: Anger

text: was

text: the

text: ﬁrst

text: to

text: develop

text: a

text: gamma

text: camera

text: struction

text: of

text: neutron

text: sources

text: that

text: emit

text: correlated

text: gammas

text: (e.g.

text: through

text: the

text: use

text: of

text: multichannel

text: collimators

text: to

text: modulate

text: Am-Be).

text: This

text: category

text: includes

text: a

text: vast

text: majority

text: of

text: neutron

text: incident

text: radiation

text: The

text: same

text: principle

text: can

text: be

text: applied

text: with

text: sources

text: important

text: in

text: nuclear

text: threat

text: search,

text: safeguards

text: and

text: more

text: complex

text: coded

text: aperture

text: masks,

text: which

text: are

text: analogous

text: to

text: non-proliferation.

text: Rather

text: than

text: requiring

text: multiple

text: views

text: of

text: the

text: source

text: this

text: technique

text: relies

text: on

text: the

text: source’s

text: intrinsic

text: property

text: superimposed

text: pinhole

text: cameras,

text: to

text: image

text: both

text: thermal

text: and

text: fast

text: of

text: coincidence

text: gamma

text: and

text: neutron

text: emission.

text: As

text: a

text: result

text: only

text: neutrons

text: a

text: single-view

text: measurement

text: of

text: the

text: source

text: is

text: required

text: to

text: perform

text: The

text: technique

text: introduced

text: in

text: this

text: paper

text: is

text: an

text: extension

text: of

text: the

text: reconstruction.

text: In

text: principle,

text: any

text: scatter

text: camera

text: sensitive

text: the

text: second

text: category

text: of

text: radiation

text: cameras

text: which

text: track

text: multiple

text: to

text: gammas

text: and

text: neutrons

text: with

text: adequate

text: timing

text: and

text: interaction

text: scatters

text: to

text: reconstruct

text: source

text: location.

text: In

text: gamma

text: ray

text: imaging,

text: location

text: resolution

text: can

text: perform

text: this

text: reconstruction.

text: Using

text: a

text: neutron

text: double

text: scatter

text: technique,

text: we

text: can

text: calculate

text: a

text: conical

text: these

text: are

text: called

text: Compton

text: cameras

text: and

text: are

text: a

text: mature

text: surface

text: of

text: possible

text: source

text: locations.

text: By

text: including

text: the

text: time

text: to

text: technology

text: with

text: commercially

text: available

text: portable

text: cameras

text: a

text: correlated

text: gamma

text: we

text: further

text: constrain

text: the

text: source

text: location

text: in

text: The

text: neutron

text: scatter

text: camera

text: functions

text: in

text: an

text: analogous

text: way

text: to

text: three-dimensions

text: by

text: solving

text: for

text: the

text: source-to-detector

text: distance

text: the

text: Compton

text: camera,

text: but

text: with

text: the

text: use

text: of

text: the

text: time-of-ﬂight

text: along

text: the

text: surface

text: of

text: said

text: cone.

text: As

text: a

text: proof

text: of

text: concept

text: we

text: applied

text: (TOF)

text: between

text: the

text: ﬁrst

text: two

text: scatters

text: to

text: determine

text: the

text: incident

text: these

text: reconstruction

text: techniques

text: on

text: measurements

text: taken

text: with

text: the

text: the

text: Mobile

text: Imager

text: of

text: Neutrons

text: for

text: Emergency

text: Responders

text: neutron

text: energy

text: (MINER).

text: Two

text: sources

text: measured

text: at

text: and

text: cm

text: from

text: The

text: discussion

text: so

text: far

text: has

text: been

text: limited

text: to

text: imaging

text: the

text: center

text: of

text: the

text: detector

text: were

text: resolved

text: in

text: their

text: varying

text: depth

text: systems,

text: but

text: in

text: principle

text: any

text: radiation

text: camera

text: can

text: produce

text: with

text: average

text: radial

text: distance

text: relative

text: resolution

text: of

text: To

text: reconstruction

text: of

text: a

text: source.

text: The

text: most

text: common

text: approach

text: is

text: demonstrate

text: the

text: technique’s

text: potential

text: with

text: an

text: optimized

text: system

text: to

text: take

text: multiple

text: images

text: from

text: different

text: views,

text: and

text: combine

text: we

text: simulated

text: the

text: measurement

text: in

text: MCNPX-PoliMi

text: assuming

text: timing

text: resolution

text: of

text: ps

text: (from

text: ns

text: in

text: the

text: current

text: system)

text: them

text: to

text: form

text: rendering

text: of

text: the

text: source.

text: This

text: technique

text: and

text: source

text: interaction

text: location

text: resolution

text: of

text: mm

text: (from

text: cm).

text: is

text: used

text: in

text: Single

text: Photon

text: Emission

text: Computed

text: Tomography

text: These

text: simulated

text: improvements

text: in

text: scatter

text: camera

text: performance

text: (SPECT)

text: and

text: Positron

text: Emission

text: Tomography

text: (PET)

text: to

text: image

text: resulted

text: in

text: radial

text: distance

text: relative

text: resolution

text: decreasing

text: to

text: an

text: radioisotopes

text: inside

text: a

text: patient

text: More

text: recently

text: researchers

text: at

text: average

text: of

text: Lawrence

text: Berkeley

text: National

text: Laboratory

text: have

text: used

text: a

text: variation

text: of

text: this

text: technique,

text: combined

text: with

text: rendering

text: of

text: physical

text: I.

text: INTRODUCTION

text: space,

text: to

text: reconstruct

text: source

text: locations

text: in

text: real

text: time

text: All

text: T

text: of

text: these

text: techniques

text: require

text: multiple

text: views

text: of

text: the

text: source

text: and

text: HE

text: practice

text: of

text: radiation

text: imaging

text: is

text: well

text: established

text: some

text: freedom

text: of

text: movement.

text: in

text: ﬁelds

text: as

text: diverse

text: as

text: medicine

text: astronomy

text: Single-sided

text: imaging

text: has

text: been

text: demonstrated

text: in

text: Comp-

text: and

text: nuclear

text: safeguards

text: and

text: ton

text: cameras

text: by

text: taking

text: advantage

text: of

text: the

text: parallax

text: effect

text: non-proliferation

text: Radiation

text: imaging

text: cameras

text: function

text: However,

text: parallax

text: techniques

text: require

text: a

text: large

text: solid

text: angle

text: cov-

text: by

text: either:

text: modulating

text: the

text: incident

text: radiation,

text: or

text: track-

text: erage

text: to

text: function

text: as

text: a

text: modality

text: at

text: all,

text: whereas

text: our

text: method

text: ing

text: the

text: multiple

text: scatters

text: of

text: incident

text: particle

text: in

text: the

text: detector

text: only

text: requests

text: solid

text: angle

text: coverage

text: to

text: increase

text: efﬁciency.

text: Furthermore,

text: our

text: technique

text: would

text: function

text: at

text: any

text: distance

text: This

text: material

text: is

text: based

text: upon

text: work

text: supported

text: by

text: the

text: U.S.

text: Department

text: of

text: Homeland

text: Security

text: Grant

text: Award

text: Number:

text: The

text: views

text: with

text: a

text: portable

text: system,

text: even

text: if

text: it

text: required

text: long

text: dwell

text: times.

text: and

text: conclusions

text: contained

text: in

text: this

text: document

text: are

text: those

text: of

text: the

text: authors

text: and

text: should

text: Parallax

text: by

text: comparison

text: would

text: be

text: restricted

text: by

text: system

text: size

text: and

text: not

text: be

text: interpreted

text: as

text: representing

text: the

text: ofﬁcial

text: policies,

text: either

text: expressed

text: or

text: require

text: close

text: enough

text: source-to-detector

text: distances

text: to

text: function

text: implied,

text: of

text: the

text: U.S.

text: Department

text: of

text: Homeland

text: Security.

text: Sandia

text: National

text: Laboratories

text: is

text: multi-program

text: laboratory

text: managed

text: and

text: in

text: the

text: ﬁrst

text: place.

text: This

text: makes

text: our

text: technique

text: potentially

text: valuable

text: operated

text: by

text: Sandia

text: Corporation,

text: a

text: wholly

text: owned

text: subsidiary

text: of

text: Lockheed

text: for

text: nuclear

text: inspection,

text: emergency

text: response

text: and

text: treaty

text: veriﬁ-

text: Martin

text: Corporation,

text: for

text: the

text: U.S.

text: Department

text: of

text: Energy’s

text: National

text: Nuclear

text: cation,

text: where

text: multiple

text: views

text: of

text: the

text: object

text: of

text: interest

text: may

text: be

text: Security

text: Administration

text: under

text: contract

text: SAND

text: number

text: J.

text: restricted,

text: even

text: though

text: the

text: location

text: of

text: the

text: source

text: is

text: known.

text: This

text: work

text: was

text: funded

text: in-part

text: by

text: the

text: Consortium

text: for

text: Veriﬁcation

text: Technol-

text: A

text: single

text: system

text: can

text: image

text: both

text: neutrons

text: and

text: gammas

text: if

text: ogy

text: under

text: Department

text: of

text: Energy

text: National

text: Nuclear

text: Security

text: Administration

text: the

text: underlying

text: detectors

text: are

text: sensitive

text: to

text: both

text: particle

text: types.

text: award

text: number

text: M.

text: Monterial

text: and

text: S.

text: A.

text: Pozzi

text: are

text: with

text: the

text: Nuclear

text: Engineering

text: and

text: Such

text: scatter

text: camera

text: systems

text: were

text: constructed

text: as

text: radiation

text: Radiological

text: Sciences

text: Department,

text: University

text: of

text: Michigan,

text: Ann

text: Arbor,

text: MI

text: telescopes

text: for

text: astronomy

text: and

text: more

text: recently

text: for

text: stand-

text: USA

text: (e-mail:

text: mateuszm@umich.edu).

text: off

text: detection

text: in

text: nuclear

text: security

text: applications

text: We

text: employ

text: P.

text: Marleau

text: is

text: with

text: Sandia

text: National

text: Laboratories,

text: Livermore,

text: CA

text: USA.

text: this

text: class

text: of

text: scatter

text: camera

text: to

text: project

text: a

text: cone

text: of

text: possible


text: ν˜1R,2(I,3) h, h′ ν˜R(I) V 1 h′ ν˜R(I) h, h′ ν˜1R(I) V 1,2,3 ν˜R,I ν 1 ν˜R(I) ν 1 Z′ χ˜ 1 ν˜R(I) ν¯ 1 ν˜1R,I ν¯ Figure 7. Feynman diagrams of the dominant annihilation and co-annihilation channels of the right-handed (cid:48) sneutrino ν˜ into the CP-even Higgs bosons (h, h ), the SM vector bosons (V = W, Z) and νν¯. 1 ◦ The inner region (|b| < 2 ) is avoided because this region of the sky that is most contaminated with strong gamma-ray point sources and with very large uncertainties in the diﬀuse emission. (cid:48) The B − L neutralinos (either B˜ or η˜ ) cannot account for such gamma-ray excess for the 2 following two reasons: (i) The B − L neutralino masses are constrained by relic abundance to be larger than 100 GeV. (ii) As the B − L neutralinos are Majorana fermions, their annihilation cross − − section can be approximated as (cid:104)σannv(cid:105) (cid:39) bv2 (no s-wave), which should be of order 10 26 cm3 s 1 at the freeze out temperature, where the velocity of the DM v is of order 0.1 c, where c is the speed of F − light. In our galactic halo the velocity of the DM particles is much smaller (v ∼ 10 3 c). Therefore, − − the B −L neutralino annihilation cross sections in the galactic halo are of order 10 30 cm3 s 1, which are quite small to account for the GCE. 4 Right-handed Sneutrino Dark Matter We now turn to the lightest right-handed sneutrino LSP, ν˜ . As in the previous section, we will 1 investigate the relic abundance constraints imposed on the parameter space in this scenario, then we analyse the prediction of this candidate for the direct detection and gamma-rays ﬂux from the GC. 4.1 Right-handed Sneutrino Relic Abundance The relevant interactions for dominant channels of right-handed sneutrino annihilation are given – 12 – ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: h,

text: h′

text: ν˜R(I)

text: V

text: h′

text: ν˜R(I)

text: h,

text: h′

text: V

text: ν˜R,I

text: ν

text: ν˜R(I)

text: ν

text: Z′

text: χ˜

text: ν˜R(I)

text: ν¯

text: ν¯

text: Figure

text: Feynman

label: caption

text: diagrams

text: of

text: the

text: dominant

text: annihilation

text: and

text: co-annihilation

text: channels

text: of

text: the

text: right-handed

text: sneutrino

text: ν˜

text: into

text: the

text: CP-even

text: Higgs

text: bosons

text: (h,

text: h

text: the

text: SM

text: vector

text: bosons

text: (V

text: W,

text: Z)

text: and

text: νν¯.

text: The

text: inner

text: region

text: (|b|

text: is

text: avoided

text: because

text: this

text: region

text: of

text: the

text: sky

text: that

text: is

text: most

text: contaminated

text: with

text: strong

text: gamma-ray

text: point

text: sources

text: and

text: with

text: very

text: large

text: uncertainties

text: in

text: the

text: diﬀuse

text: emission.

text: The

text: B

text: L

text: neutralinos

text: (either

text: B˜

text: or

text: η˜

text: cannot

text: account

text: for

text: such

text: gamma-ray

text: excess

text: for

text: the

text: following

text: two

text: reasons:

text: (i)

text: The

text: B

text: L

text: neutralino

text: masses

text: are

text: constrained

text: by

text: relic

text: abundance

text: to

text: be

text: larger

text: than

text: GeV.

text: (ii)

text: As

text: the

text: B

text: L

text: neutralinos

text: are

text: Majorana

text: fermions,

text: their

text: annihilation

text: cross

text: section

text: can

text: be

text: approximated

text: as

text: (no

text: s-wave),

text: which

text: should

text: be

text: of

text: order

text: s

text: at

text: the

text: freeze

text: out

text: temperature,

text: where

text: the

text: velocity

text: of

text: the

text: DM

text: v

text: is

text: of

text: order

text: c,

text: where

text: c

text: is

text: the

text: speed

text: of

text: F

text: light.

text: In

text: our

text: galactic

text: halo

text: the

text: velocity

text: of

text: the

text: DM

text: particles

text: is

text: much

text: smaller

text: (v

text: c).

text: Therefore,

text: the

text: B

text: −L

text: neutralino

text: annihilation

text: cross

text: sections

text: in

text: the

text: galactic

text: halo

text: are

text: of

text: order

text: s

text: which

text: are

text: quite

text: small

text: to

text: account

text: for

text: the

text: GCE.

text: Right-handed

text: Sneutrino

text: Dark

text: Matter

text: We

text: now

text: turn

text: to

text: the

text: lightest

text: right-handed

text: sneutrino

text: LSP,

text: ν˜

text: As

text: in

text: the

text: previous

text: section,

text: we

text: will

text: investigate

text: the

text: relic

text: abundance

text: constraints

text: imposed

text: on

text: the

text: parameter

text: space

text: in

text: this

text: scenario,

text: then

text: we

text: analyse

text: the

text: prediction

text: of

text: this

text: candidate

text: for

text: the

text: direct

text: detection

text: and

text: gamma-rays

text: ﬂux

text: from

text: the

text: GC.

text: Right-handed

text: Sneutrino

text: Relic

text: Abundance

text: The

text: relevant

text: interactions

text: for

text: dominant

text: channels

text: of

text: right-handed

text: sneutrino

text: annihilation

text: are

text: given

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: MŽ

text: @GeVD

text: Ν1

text: Figure

text: The

text: thermal

text: relic

text: abundance

text: of

text: right-handed

text: sneutrino

text: LSP

text: as

text: a

text: function

text: of

text: its

text: mass.

text: The

text: gray

text: triangles

text: denote

text: to

text: the

text: excluded

text: points

text: due

text: to

text: LUX

text: upper

text: bound.

text: Horizontal

text: lines

text: correspond

text: to

text: the

text: Planck

text: limits

text: on

text: DM

text: abundance.

text: where

text: (in

text: the

text: case

text: of

text: the

text: lightest

text: sneutrino

text: being

text: ν˜R)

text: are

text: as

text: follows

text: 2µ

text: g

text: v

text: Γ

text: R

text: R

text: Y

text: Y

text: g˜g

text: νmn

text: νln

text: BL

text: m

text: Γ

text: m

text: Γ

text: u

text: d

text: guu¯h

text: v

text: sin

text: β

text: gdd¯h

text: v

text: cos

text: β

text: m

text: Γ

text: m

text: Γ

text: vu

text: vd

text: From

text: these

text: expressions,

text: one

text: ﬁnds

text: that

text: on

text: the

text: one

text: hand

text: the

text: coupling

text: g

text: v

text: GeV

text: BL

text: and

text: the

text: coupling

text: g

text: is

text: quite

text: suppressed

text: due

text: to

text: very

text: small

text: mixing

text: Γ

text: /Γ

text: and

text: up/down

text: Yukawa

text: qq¯h

text: coupling,

text: therefore,

text: the

text: eﬀective

text: coupling

text: due

text: to

text: h

text: exchange

text: in

text: eq.

text: is

text: of

text: order

text: cos

text: β)

text: GeV

text: On

text: the

text: other

text: hand

text: the

text: coupling

text: g

text: Y

text: v

text: GeV

text: and

text: the

text: cou-

text: ν˜Rν˜Rh

text: ν

text: pling

text: gdd¯h

text: cos

text: β

text: therefore,

text: the

text: eﬀective

text: coupling

text: due

text: to

text: h

text: exchange

text: is

text: of

text: order

text: GeV

text: In

text: this

text: case,

text: the

text: eﬀective

text: coupling

text: of

text: ν˜1

text: to

text: proton

text: and

text: neutrino,

text: is

text: about

text: three

text: order

text: of

text: magnitudes

text: larger

text: than

text: the

text: eﬀective

text: coupling

text: of

text: the

text: neutralinos

text: fχ˜1

text: Thus,

text: one

text: would

text: expect

text: p,n

text: a

text: larger

text: spin-independent

text: cross

text: section

text: for

text: sneutrino

text: DM

text: that

text: may

text: even

text: exceed

text: the

text: LUX

text: limits.

text: In

text: ﬁgure

text: we

text: show

text: the

text: spin-independent

text: cross

text: section

text: of

text: the

text: scattering

text: between

text: the

text: ν˜

text: DM

text: and

text: proton

text: as

text: a

text: function

text: of

text: its

text: mass.

text: Here

text: we

text: used

text: all

text: the

text: allowed

text: points

text: of

text: the

text: scan

text: in

text: ﬁgure

text: that

text: lead

text: to

text: a

text: viable

text: ν˜

text: DM.

text: As

text: expected,

text: some

text: points

text: exceed

text: the

text: experimental

text: limits

text: of

text: LUX,

text: however,

text: a

text: signiﬁcant

text: number

text: of

text: benchmarks

text: predicts

text: cross

text: sections

text: that

text: can

text: be

text: probed

text: in

text: near

text: future.

text: Right-handed

text: Sneutrino

text: and

text: Galactic

text: Center

text: γ-rays

text: Ν

text: ##LTFigure##

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text: (cid:101) (cid:101) (cid:62) (cid:62) ω ω follows by Lemma 3.4 since j i. Inequalities (2) can be proved analogously with the help of rj ri Lemma 4.5 (1). Inequality (3) follows straightforward from (1) and (2). (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) || || (cid:62) || || (cid:62) (cid:101) || || (cid:62)(cid:101) || || || || || || Since s r 1, we have r s by symmetry and therefore s = r . α,ω α,ω α,ω α,ω α,ω α,ω || || || || (cid:101) By Lemma 3.4, this is equivalent to s = r . This and the ﬁrst two statements of Lemma 4.6 (cid:101) (cid:101) (cid:101) (cid:101)0 α,ω (cid:101) i α(cid:101),ω (cid:101) (cid:101) imply that α = α = α , and ω = ω = ω , in particular, j = i. Therefore, s is trivial in s0 ri rj rj r(cid:101)i (cid:101)s0 0 (cid:104)(cid:104) (cid:105)(cid:105)(cid:101) (cid:101) (cid:101) N/ r , and the index i is determined by α = α . One can prove in the same way that r is trivial i s0 ri i (cid:101) (cid:104)(cid:104) (cid:105)(cid:105) (cid:101)(cid:104)(cid:104) (cid:105)(cid:105) (cid:104)(cid:104) (cid:105)(cid:105) (cid:101) (cid:101) in N/ s . Thus, s = r . From Section we know that N is a free group. So by Theorem 4.1, 0 0 N i N (cid:101) ± ± s is conjugate to r 1 in N. Finally, r is conjugate to s 1 in H, and the proof of Proposition 2.1 in the 0 i || || (cid:62) case r 1 is completed. α,ω (cid:101) 5 Proof of Proposition 2.1 for r with non-positive α-ω-length (cid:101) 5.1 Properties of r with non-positive α-ω-length || || Lemma 5.1. Let r be an element in N with r < 1. Then the reduced word representing r in basis α,ω B +(α ) contains only b ’s. r i ∈ ∈ Proof. We have ω < α . By deﬁnition, r G−∞ and r G ∞. r r ,ωr αr, • − B− Let r be the reduced word representing r in the basis (ω ) of G−∞ . Recall that r ,ωr B− { | ∈ } ∪ { | (cid:54) (cid:54) (cid:54) } (ω ) = b i I y 1 m n, (cid:96) ω , r i m,(cid:96) r { − } where I = ω k + 1, . . . , ω . r r • B Let r+ be the reduced word representing r in the basis +(α ) of G ∞. Recall that r αr, B { | ∈ } ∪ { | (cid:54) (cid:54) (cid:62) } +(α ) = b j J y 1 m n, (cid:96) α , r j m,(cid:96) r { − } where J = α , . . . , α + k 1 . r r ∈ ∈ ≡ − For each i I, there is a unique j J such that j i mod k. Then r+ can be obtained from r by − − − replacements of all letters b by b u −1 u −1 . . . u 1 followed by reduction (see Remark 2.3). The second i j j k j 2k i −∞ − −∞ − indices of y-letters appearing in these replacements lie in the interval ( , max J k] = ( , α 1]. r − −∞ ⊆ −∞ − The second indices of y-letters of r lie in the interval ( , ω ] ( , α 1]. Therefore, the second r r −∞ − indices of y-letters of r+ (if exist) lie in ( , α 1]. Hence, r+ does not have y-letters. r (cid:101) 5.2 Conclusion of the proof for r with non-positive α-ω-length (cid:101) (cid:101) (cid:101) ∈ (cid:101) As in Section , we can replace the elements r, s N from Proposition 2.1 by suitable elements r, s. In || || (cid:101) (cid:101) || || (cid:62) (cid:101) this section, we consider the case r < 1. Assuming s 1, we immediately get a contradiction α,ω α,ω || || by applying Lemma 4.6 with reversed roles of r and s. Thus, we have s < 1. α,ω 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ω

text: ω

text: follows

text: by

text: Lemma

text: since

text: j

text: i.

text: Inequalities

text: can

text: be

text: proved

text: analogously

text: with

text: the

text: help

text: of

text: rj

text: ri

text: Lemma

text: Inequality

text: follows

text: straightforward

text: from

text: and

text: Since

text: s

text: r

text: we

text: have

text: r

text: s

text: by

text: symmetry

text: and

text: therefore

text: s

text: r

text: α,ω

text: α,ω

text: α,ω

text: α,ω

text: α,ω

text: α,ω

text: By

text: Lemma

text: this

text: is

text: equivalent

text: to

text: s

text: r

text: This

text: and

text: the

text: ﬁrst

text: two

text: statements

text: of

text: Lemma

text: α,ω

text: i

text: imply

text: that

text: α

text: α

text: α

text: and

text: ω

text: ω

text: ω

text: in

text: particular,

text: j

text: i.

text: Therefore,

text: s

text: is

text: trivial

text: in

text: ri

text: rj

text: rj

text: N/

text: r

text: and

text: the

text: index

text: i

text: is

text: determined

text: by

text: α

text: α

text: One

text: can

text: prove

text: in

text: the

text: same

text: way

text: that

text: r

text: is

text: trivial

text: i

text: ri

text: i

text: in

text: N/

text: s

text: Thus,

text: s

text: r

text: From

text: Section

text: we

text: know

text: that

text: N

text: is

text: a

text: free

text: group.

text: So

text: by

text: Theorem

text: N

text: i

text: N

text: s

text: is

text: conjugate

text: to

text: r

text: in

text: N.

text: Finally,

text: r

text: is

text: conjugate

text: to

text: s

text: in

text: H,

text: and

text: the

text: proof

text: of

text: Proposition

text: in

text: the

text: i

text: case

text: r

text: is

text: completed.

text: α,ω

text: Proof

text: of

text: Proposition

text: for

text: r

text: with

text: non-positive

text: α-ω-length

text: Properties

text: of

text: r

text: with

text: non-positive

text: α-ω-length

text: Lemma

text: Let

text: r

text: be

text: an

text: element

text: in

text: N

text: with

text: r

text: Then

text: the

text: reduced

text: word

text: representing

text: r

text: in

text: basis

text: α,ω

text: B

text: +(α

text: contains

text: only

text: b

text: ’s.

text: r

text: i

text: Proof.

text: We

text: have

text: ω

text: α

text: By

text: deﬁnition,

text: r

text: G−∞

text: and

text: r

text: G

text: r

text: r

text: ,ωr

text: αr,

text: B−

text: Let

text: r

text: be

text: the

text: reduced

text: word

text: representing

text: r

text: in

text: the

text: basis

text: (ω

text: of

text: G−∞

text: Recall

text: that

text: r

text: ,ωr

text: B−

text: (ω

text: b

text: i

text: I

text: y

text: m

text: n,

text: ω

text: r

text: i

text: r

text: where

text: I

text: ω

text: k

text: ω

text: r

text: r

text: B

text: Let

text: r+

text: be

text: the

text: reduced

text: word

text: representing

text: r

text: in

text: the

text: basis

text: +(α

text: of

text: G

text: Recall

text: that

text: r

text: αr,

text: B

text: +(α

text: b

text: j

text: J

text: y

text: m

text: n,

text: α

text: r

text: j

text: r

text: where

text: J

text: α

text: α

text: k

text: r

text: r

text: For

text: each

text: i

text: I,

text: there

text: is

text: a

text: unique

text: j

text: J

text: such

text: that

text: j

text: i

text: mod

text: k.

text: Then

text: r+

text: can

text: be

text: obtained

text: from

text: r

text: by

text: replacements

text: of

text: all

text: letters

text: b

text: by

text: b

text: u

text: −1

text: u

text: −1

text: u

text: followed

text: by

text: reduction

text: (see

text: Remark

text: The

text: second

text: i

text: j

text: j

text: k

text: j

text: i

text: indices

text: of

text: y-letters

text: appearing

text: in

text: these

text: replacements

text: lie

text: in

text: the

text: interval

text: max

text: J

text: k]

text: α

text: r

text: The

text: second

text: indices

text: of

text: y-letters

text: of

text: r

text: lie

text: in

text: the

text: interval

text: ω

text: α

text: Therefore,

text: the

text: second

text: r

text: r

text: indices

text: of

text: y-letters

text: of

text: r+

text: (if

text: exist)

text: lie

text: in

text: α

text: Hence,

text: r+

text: does

text: not

text: have

text: y-letters.

text: r

text: Conclusion

text: of

text: the

text: proof

text: for

text: r

text: with

text: non-positive

text: α-ω-length

text: As

text: in

text: Section

text: we

text: can

text: replace

text: the

text: elements

text: r,

text: s

text: N

text: from

text: Proposition

text: by

text: suitable

text: elements

text: r,

text: s.

text: In

text: this

text: section,

text: we

text: consider

text: the

text: case

text: r

text: Assuming

text: s

text: we

text: immediately

text: get

text: a

text: contradiction

text: α,ω

text: α,ω

text: by

text: applying

text: Lemma

text: with

text: reversed

text: roles

text: of

text: r

text: and

text: s.

text: Thus,

text: we

text: have

text: s

text: α,ω

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text: ##LTLine##


text: ∗ We give an algorithm which ﬁnds a word r representing r such that the smallest index of letters used ∗ in r is maximum possible. We call such index the α-limit of r and denote it by α . In other words, α r r is the largest index such that r is an element of G ∞. The following algorithm rewrites an arbitrary αr, ∈ \{ } ∗ B word r N 1 into the presentation r of r written in basis +(α ) (see Lemma 3.2). For a word r in r { | ∈ Z} ∪ { | (cid:54) (cid:54) ∈ Z} the alphabet b i y 1 m n, i , let min(r) denote the minimal index of letters of r. i m,i ∈ \{ } Algorithm 3.1. Let r N 1 . Suppose that r is given as a ﬁnite word in letters b , b , . . . and i i+1 (cid:54) (cid:54) ∈ y , y , . . . (1 m n). In particular, r G ∞. Let r[0] be the reduced word representing r in m,i m,i+1 i, B basis +(i). Increasing i if necessary, we may assume that i = min(r[0]). (1) Let r[1] be the word obtained from r[0] by replacement of each occurrence of the letter b by i − b u 1 followed by free reduction. Then r[1] presents r in the following basis of G ∞: i+k i i, { } ∪ { | (cid:54) (cid:54) (cid:54) } b , b , . . . , b y 1 m n, i l . ((cid:63)) i+1 i+2 i+k m,l (cid:54) (cid:54) (1a) If r[1] does not contain a letter y (1 m n), we reset r[0] := r[1], i := min(r[1]), and go m,i back to (1). Clearly, the new i is larger than the old one. (cid:54) (cid:54) ∗ (1b) If r[1] contains a letter y (1 m n), the algorithm ends with r = r[0] and α = i. m,i r ∗ B Lemma 3.2. The output r of Algorithm 3.1 coincides with the word representing r in Basis +(α ), r where α is the α-limit of r. r Proof. We shall prove that the algorithm ends and that the integer assigned to α by the algorithm is r really the α-limit of r. The algorithm ends since the length of r[0] decreases with each iteration of (1) which does not terminate the algorithm. This can be veriﬁed in the following way: If we arrive in (1a), then the word (cid:54) (cid:54) r[1] does not contain a letter y (1 m n). Recall that r[1] was obtained from r[0] by replacing all m,i − (cid:54) (cid:54) b with b u 1 and reducing the resulting word. The fact that there is no letter y (1 m n) left i i+k i m,i in r[1] means that each b in r[0] occurred in a subword of the form b u . Therefore r[1] is shorter than i i i r[0]. Finally, we show that r is not an element of G ∞, where i as in (1b). Observe that i+1, ∗ (cid:104) | (cid:54) (cid:54) (cid:105) G ∞ = G ∞ y 1 m n . i, i+1, m,i (cid:54) (cid:54) ∈ By (1b), r, written in basis ((cid:63)), uses a letter y (1 m n). Therefore r / G ∞ and α = i. m,i i+1, r \{ } Corollary 3.3. Let r be an element from N 1 given as a word in some b-left basis. Starting Algorithm 3.1 with r, we either get the same presentation or a presentation of shorter length. 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: give

text: an

text: algorithm

text: which

text: ﬁnds

text: a

text: word

text: r

text: representing

text: r

text: such

text: that

text: the

text: smallest

text: index

text: of

text: letters

text: used

text: in

text: r

text: is

text: maximum

text: possible.

text: We

text: call

text: such

text: index

text: the

text: α-limit

text: of

text: r

text: and

text: denote

text: it

text: by

text: α

text: In

text: other

text: words,

text: α

text: r

text: r

text: is

text: the

text: largest

text: index

text: such

text: that

text: r

text: is

text: an

text: element

text: of

text: G

text: The

text: following

text: algorithm

text: rewrites

text: an

text: arbitrary

text: αr,

text: B

text: word

text: r

text: N

text: into

text: the

text: presentation

text: r

text: of

text: r

text: written

text: in

text: basis

text: +(α

text: (see

text: Lemma

text: For

text: a

text: word

text: r

text: in

text: r

text: Z}

text: Z}

text: the

text: alphabet

text: b

text: i

text: y

text: m

text: n,

text: i

text: let

text: min(r)

text: denote

text: the

text: minimal

text: index

text: of

text: letters

text: of

text: r.

text: i

text: m,i

text: Algorithm

text: Let

text: r

text: N

text: Suppose

text: that

text: r

text: is

text: given

text: as

text: a

text: ﬁnite

text: word

text: in

text: letters

text: b

text: b

text: and

text: i

text: y

text: y

text: m

text: n).

text: In

text: particular,

text: r

text: G

text: Let

text: be

text: the

text: reduced

text: word

text: representing

text: r

text: in

text: m,i

text: i,

text: B

text: basis

text: +(i).

text: Increasing

text: i

text: if

text: necessary,

text: we

text: may

text: assume

text: that

text: i

text: Let

text: be

text: the

text: word

text: obtained

text: from

text: by

text: replacement

text: of

text: each

text: occurrence

text: of

text: the

text: letter

text: b

text: by

text: i

text: b

text: u

text: followed

text: by

text: free

text: reduction.

text: Then

text: presents

text: r

text: in

text: the

text: following

text: basis

text: of

text: G

text: i+k

text: i

text: i,

text: b

text: b

text: b

text: y

text: m

text: n,

text: i

text: l

text: i+k

text: m,l

text: If

text: does

text: not

text: contain

text: a

text: letter

text: y

text: m

text: n),

text: we

text: reset

text: i

text: and

text: go

text: m,i

text: back

text: to

text: Clearly,

text: the

text: new

text: i

text: is

text: larger

text: than

text: the

text: old

text: one.

text: If

text: contains

text: a

text: letter

text: y

text: m

text: n),

text: the

text: algorithm

text: ends

text: with

text: r

text: and

text: α

text: i.

text: m,i

text: r

text: B

text: Lemma

text: The

text: output

text: r

text: of

text: Algorithm

text: coincides

text: with

text: the

text: word

text: representing

text: r

text: in

text: Basis

text: +(α

text: r

text: where

text: α

text: is

text: the

text: α-limit

text: of

text: r.

text: r

text: Proof.

text: We

text: shall

text: prove

text: that

text: the

text: algorithm

text: ends

text: and

text: that

text: the

text: integer

text: assigned

text: to

text: α

text: by

text: the

text: algorithm

text: is

text: r

text: really

text: the

text: α-limit

text: of

text: r.

text: The

text: algorithm

text: ends

text: since

text: the

text: length

text: of

text: decreases

text: with

text: each

text: iteration

text: of

text: which

text: does

text: not

text: terminate

text: the

text: algorithm.

text: This

text: can

text: be

text: veriﬁed

text: in

text: the

text: following

text: way:

text: If

text: we

text: arrive

text: in

text: then

text: the

text: word

text: does

text: not

text: contain

text: a

text: letter

text: y

text: m

text: n).

text: Recall

text: that

text: was

text: obtained

text: from

text: by

text: replacing

text: all

text: m,i

text: b

text: with

text: b

text: u

text: and

text: reducing

text: the

text: resulting

text: word.

text: The

text: fact

text: that

text: there

text: is

text: no

text: letter

text: y

text: m

text: n)

text: left

text: i

text: i+k

text: i

text: m,i

text: in

text: means

text: that

text: each

text: b

text: in

text: occurred

text: in

text: a

text: subword

text: of

text: the

text: form

text: b

text: u

text: Therefore

text: is

text: shorter

text: than

text: i

text: i

text: i

text: Finally,

text: we

text: show

text: that

text: r

text: is

text: not

text: an

text: element

text: of

text: G

text: where

text: i

text: as

text: in

text: Observe

text: that

text: G

text: G

text: y

text: m

text: n

text: i,

text: m,i

text: By

text: r,

text: written

text: in

text: basis

text: uses

text: a

text: letter

text: y

text: m

text: n).

text: Therefore

text: r

text: G

text: and

text: α

text: i.

text: m,i

text: r

text: Corollary

text: Let

text: r

text: be

text: an

text: element

text: from

text: N

text: given

text: as

text: a

text: word

text: in

text: some

text: b-left

text: basis.

text: Starting

text: Algorithm

text: with

text: r,

text: we

text: either

text: get

text: the

text: same

text: presentation

text: or

text: a

text: presentation

text: of

text: shorter

text: length.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Theorem 4.1 (Magnus’ Freiheitssatz (cf. [Mag30])). Let F be a free group on a basis X, and let ∈ g be a cyclically reduced word in F with respect to X, containing a letter x X. Then the subgroup \{ } (cid:104)(cid:104) (cid:105)(cid:105) generated by X x is naturally embedded into the group F/ g . F (cid:104)(cid:104) (cid:105)(cid:105) (cid:104)(cid:104) (cid:105)(cid:105) ∈ By abuse of notat(cid:101)ion, we write A/ a instead of A/ a , where A is a group and a A. (cid:101) A (cid:101) ∈ (cid:101) → (cid:104)(cid:104) (cid:105)(cid:105) Corollary 4.2. Let r N be a suitable element. Then we get the embeddings G ∞ (cid:44) N/ r and αr+1, (cid:101) → (cid:104)(cid:104) (cid:105)(cid:105) G−∞ − (cid:44) N/ r . ,ωr 1 (cid:101) (cid:101) (cid:101) B (cid:101) Proof. By the deﬁnition of α-limits, r written in basis +(α ) contains at least one letter in Y , and r (cid:101) αr (cid:101) B by Deﬁnition 3.7, the suitable element r is cyclically reduced in this basis. We extend +(α ) to a free (cid:101) r (cid:101) { | (cid:54) (cid:54) } basis of N by adding the letters y 1 m n, i < α . The presentation of r will not change by m,i r (cid:101) → (cid:104)(cid:104) (cid:105)(cid:105) that. Now, G ∞ (cid:44) N/ r follows immediately from Theorem 4.1. The other embedding follows αr+1, analogously. ∈ || || (cid:62) B Lemma 4.3. Suppose that r N satisﬁes r 1. Then r, written in the basis +(α ) of G ∞, α,ω r αr, (cid:62) contains at least one letter y with l ω . m,l r (cid:54) Proof. By assumption, α ω . To the contrary, suppose that r r ∈ (cid:104) | (cid:54) (cid:54) (cid:105) r b , b , . . . , b − , y , y , . . . , y − 1 m n . αr αr+1 αr+k 1 m,αr m,αr+1 m,ωr 1 Then, using relations b = b − u − , we obtain i i k i k ∈ (cid:104) | (cid:54) (cid:54) (cid:105) r b − , . . . , b − , y − , y − , . . . , y − 1 m n . αr k αr 1 m,αr k m,αr k+1 m,ωr 1 ∈ Hence r G−∞ − . A contradiction. ,ωr 1 4.2 The structure of some quotients of N (cid:101) (cid:101) (cid:101) This section and the next one are very similar to [BS08, Section 4 and 5], but due to some important (cid:104)(cid:104) (cid:105)(cid:105) changes we cannot skip them. Our aim in this subsection is to present N/ r , r , . . . , r as an i i+1 j (cid:101) (cid:101) amalgamated product. We denote w = b u . i i i ∈ || || (cid:62) (cid:54) Lemma 4.4. Let r N be a suitable element with r 1, and let i, j be two integers with i j. (cid:101) (cid:101) α,ω − We denote(cid:101)s(cid:101)= α an(cid:101)d t = ω 1. Th(cid:101)en(cid:101)we have:(cid:101) (cid:101) rj rj (cid:104)(cid:104) (cid:105)(cid:105) ∼ (cid:104)(cid:104) (cid:105)(cid:105) ∗ (cid:104)(cid:104) (cid:105)(cid:105) (1) N/ r , r , . . . , r = G−∞ / r , r , . . . , r − G ∞/ r . i i+1 j ,t i i+1 j 1 s, j Gs,t w − = b , t k+1 t+1 . . . (cid:101) (cid:101) (cid:101) w = b t t+k (cid:104)(cid:104) (cid:105)(cid:105) (2) G ∞ naturally embeds into N/ r , r , . . . , r . s+1, i i+1 j 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Theorem

text: (Magnus’

text: Freiheitssatz

text: (cf.

text: Let

text: F

text: be

text: a

text: free

text: group

text: on

text: a

text: basis

text: X,

text: and

text: let

text: g

text: be

text: a

text: cyclically

text: reduced

text: word

text: in

text: F

text: with

text: respect

text: to

text: X,

text: containing

text: a

text: letter

text: x

text: X.

text: Then

text: the

text: subgroup

text: generated

text: by

text: X

text: x

text: is

text: naturally

text: embedded

text: into

text: the

text: group

text: F/

text: g

text: F

text: By

text: abuse

text: of

text: we

text: write

text: A/

text: a

text: instead

text: of

text: A/

text: a

text: where

text: A

text: is

text: a

text: group

text: and

text: a

text: A.

text: A

text: Corollary

text: Let

text: r

text: N

text: be

text: a

text: suitable

text: element.

text: Then

text: we

text: get

text: the

text: embeddings

text: G

text: N/

text: r

text: and

text: G−∞

text: N/

text: r

text: ,ωr

text: B

text: Proof.

text: By

text: the

text: deﬁnition

text: of

text: α-limits,

text: r

text: written

text: in

text: basis

text: +(α

text: contains

text: at

text: least

text: one

text: letter

text: in

text: Y

text: and

text: r

text: αr

text: B

text: by

text: Deﬁnition

text: the

text: suitable

text: element

text: r

text: is

text: cyclically

text: reduced

text: in

text: this

text: basis.

text: We

text: extend

text: +(α

text: to

text: a

text: free

text: r

text: basis

text: of

text: N

text: by

text: adding

text: the

text: letters

text: y

text: m

text: n,

text: i

text: α

text: The

text: presentation

text: of

text: r

text: will

text: not

text: change

text: by

text: m,i

text: r

text: that.

text: Now,

text: G

text: N/

text: r

text: follows

text: immediately

text: from

text: Theorem

text: The

text: other

text: embedding

text: follows

text: analogously.

text: B

text: Lemma

text: Suppose

text: that

text: r

text: N

text: satisﬁes

text: r

text: Then

text: r,

text: written

text: in

text: the

text: basis

text: +(α

text: of

text: G

text: α,ω

text: r

text: αr,

text: contains

text: at

text: least

text: one

text: letter

text: y

text: with

text: l

text: ω

text: m,l

text: r

text: Proof.

text: By

text: assumption,

text: α

text: ω

text: To

text: the

text: contrary,

text: suppose

text: that

text: r

text: r

text: r

text: b

text: b

text: b

text: y

text: y

text: y

text: m

text: n

text: αr

text: αr+k

text: m,αr

text: m,ωr

text: Then,

text: using

text: relations

text: b

text: b

text: u

text: we

text: obtain

text: i

text: i

text: k

text: i

text: k

text: r

text: b

text: b

text: y

text: y

text: y

text: m

text: n

text: αr

text: k

text: αr

text: m,αr

text: k

text: m,αr

text: m,ωr

text: Hence

text: r

text: G−∞

text: A

text: contradiction.

text: ,ωr

text: The

text: structure

text: of

text: some

text: quotients

text: of

text: N

text: This

text: section

text: and

text: the

text: next

text: one

text: are

text: very

text: similar

text: to

text: Section

text: and

text: but

text: due

text: to

text: some

text: important

text: changes

text: we

text: cannot

text: skip

text: them.

text: Our

text: aim

text: in

text: this

text: subsection

text: is

text: to

text: present

text: N/

text: r

text: r

text: r

text: as

text: an

text: i

text: j

text: amalgamated

text: product.

text: We

text: denote

text: w

text: b

text: u

text: i

text: i

text: i

text: Lemma

text: Let

text: r

text: N

text: be

text: a

text: suitable

text: element

text: with

text: r

text: and

text: let

text: i,

text: j

text: be

text: two

text: integers

text: with

text: i

text: j.

text: α,ω

text: We

text: α

text: t

text: ω

text: rj

text: rj

text: N/

text: r

text: r

text: r

text: G−∞

text: r

text: r

text: r

text: G

text: r

text: i

text: j

text: ,t

text: i

text: j

text: s,

text: j

text: Gs,t

text: w

text: b

text: t

text: w

text: b

text: t

text: t+k

text: G

text: naturally

text: embeds

text: into

text: N/

text: r

text: r

text: r

text: i

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##