text: (cid:0) (cid:1) A. Deﬁnitions and Notation { } F − − Let [n] denote the set 1, 2, . . . , n . Let B = q beFthe ﬁnite gu,α(x) = Tr u(x α) /(x α). (1) ﬁeld of q elements, for some prime power q. Let F = be an ≥ qt extension ﬁeld of B, where t 1. We refer to the elements of F as symbols and the elements of B as sub-symbols. The ﬁeld By the deﬁnition of a trace function, the next lemma follows F may∼also be viewed as a vector space of dimension t over B, in a straightforward manner. i.e. F = Bt, and hence each symbol in F may be represented Lemma 1 ([9]). The polynomial g (x) deﬁned in (1) satisﬁes C u,α as a vector of length t over B. A linear [n, k] code over F the following properties. − − is a subspace of Fn of dimension k. Each element of a code is (a) deg(g ) = qt 1 1; (b) g (α) = u. C C⊥ u,α u,α referred to as a codeword. The dual of a code , denoted , is − − ≤ − C − By Lemma 1 (a), deg(g ) = qt 1 1 r 1. Therefore, the orthogonal complement of , and has dimension r = n k. u,α C⊥ Deﬁnition 1. Let F[x] denote the ring of polynomials over F. the polynomCial gu,α(x) corres{ponds to a }codeword of and is ⊆ a check for . Now let U (cid:0)= u , . . . , u(cid:1) be a basis of F over A Reed-Solomon code RS(A, k) Fn of dimension k over a 1 t ﬁnite ﬁeld F w(cid:110)i(cid:0)th evaluation points(cid:1) A = {α , α , . . . , α } (cid:111)⊆ F B, and set 1 2 n is deﬁned as: (cid:52) − ∗ − ∗ ∈ g (x) = g ∗(x) = Tr u (x α ) /(x α ), i [t]. i ui,α i RS(A, k) = f(α1), . . . , f(αn) : f ∈ F[x], deg(f) < k . These t polynomials correspond to t codewords of C⊥. Therefore, (cid:88) we obtain t equations of the form A generalized Reed-Solomon code, GRS(A, k, λ), where λ = ∈ (cid:0) (cid:1) (λ , . . . , λ ) Fn, is deﬁned similarly to a Reed-Solomon ∗ ∗ − ∈ 1 n g (α )f(α ) = g (α)f(α), i [t]. (2) code, except that the codeword corresponding to a polynomial i i (cid:54) ∈ ∈ \{ ∗} f is deﬁned as λ f(α ), . . . , λ f(α ) , λ = 0 for all i α A α 1 1 n n i [n]. It is well known that the dual of a Reed-Solomon code ≤ | | A key step in the Guruswami-Wootters repair scheme is to RS(A, k), for any n F , is a generalized Reed-Solomon − apply the trace function to both sides of (2) to obtain t different code GRS(A, n k, λ), for some multiplier vector λ (see [4, Chp. 10]). Whenever clear from the context, we use f(x) to repai(cid:0)r equations (cid:1) (cid:88) (cid:0) (cid:1) − denote a polynomial of degree at most k 1, which corresponds t(cid:80)o a codeword of the Reed-Solomon code C =−RS(A,−k), −and Tr gi(α∗)f(α∗) = − Tr gi(α)f(α) , i ∈ [t]. (3) g(x) to denote a polynomial of degree at most r 1 =Cn⊥ k 1, α∈A\{α∗} which corresponds to a codeword of the dual code . Since α∈A g(α)(λαf(α))C= 0, we refer to the p|oly|nomial g(x) as a According to Lemma 1 (b), gi(α∗) = ui, for i = 1, . . . , t. check poly∈nomial for . Note that when n = F , we have λα = 1 Moreover, by the linearity of the trace function, we can rewrite (3) for all α F. In general, as recovering f(α) is equivalent to as f(cid:0)ollows. F(cid:1)or i = 1, .(cid:88). . , t, (cid:0) (cid:1) (cid:16) (cid:17) recovering λ f(α), to simplify the notation, we omit the factor α λ in our derivations. α ∗ − − ∗ × f(α) B. The Guruswami-Wootters Repair Scheme for One Erasure Tr uif(α ) = ∈ \{ ∗} Tr ui(α α ) Tr α − α∗ . (4) ∈ α A α (cid:16) (cid:17) Suppose that the polynomial f(x) F[x] corresponds to a C codeword of the Reed-Solomon code = RS(A, k) and that The right-hand side sums of the equations (4) may be com- ∗ ∗ ∈ f(α ) is the erased symbol, where α A is an evaluation point puted by downloading the re(cid:0)construct(cid:1)ion trace Tr f−(α)∗ from of the code. ∈ \{ ∗} α α the node storing f(α), for each α A α . As a consequence, Given that F is an extension ﬁeld of B of degree t, i.e. ∗ ∗ F(cid:80)= F and B = F , for some prime power q, one may deﬁne the t independent traces Tr uif(α ) , i = 1, . . . , t, of f(α ) qt q ∈ can be determined by downloading one sub-symbol from each the ﬁeld trace of any symbol α F according to Tr (α) = − ∗ − F/B of the n 1 available nodes, and the erased symbol f(α ) may ti=01 αqi. The trace belongs to B. When clear from the context, subsequently be recovered from its t independent traces. The we omit the subscript F/B. The key points in the repair scheme following theorem summarizes this brief discussion. proposed by Guruswami and Wootters [9] can be summarized ≥ − as follows. First, each symbol in F can be recovered from its t Theorem 1 ([9]). When r qt 1, there exists a repair scheme (cid:80) − independent traces. More precisely, given a basis u , u , . . . , u for Reed-Solomon codes with a repair bandwidth of (n 1) log q ∈ 1 2 t ≥ − | 2 of F over B, any α F can be uniquely determined given the bits. More generally, if r qs, where (t s) t, by setting B = ∈ ⊥ F − − v{al⊥ue}s of Tr(ui α) for i [t], i.e. α = ti=1 Tr{(ui}α)ui , where qt−s one can devise a repair scheme with bandwidth (n 1)(t u t is the dual (trace-orthogonal) basis of u t (see, for s) log q bits, which is optimal when n = qt and r = qs. i i=1 i ≥i=1 − 2 instance [17, Ch. 2, Def. 2.30]). Second, when r qt 1, the { } In general, any set of t polynomials g (x), . . . , g (x) trace function also provide checks that generate repair equations − 1 (cid:80) t each of which has degree at most r 1 and such that with coefﬁcients that are linearly dependent over B, which keeps rank {g (α∗), . . . , g (α∗)} = t(cid:0) can be used to re(cid:1)pair f(α∗). the repair cost low. q 1 t g(xN)ot∈e tFha[xt ]ththeatchseactikssfyodf eCg(ga)re≤prrec−ise1ly. Fthoorsre ≥polqytn−o1m, iwales Tsuhbe-sryempabiorlsb,awndhweriedtbhα i=(cid:52)n rtahniskqca{sge1(eαq)u,a.ls. .b, g=t(α)}α.∈MA\o{rαe∗o}vbeαr, may deﬁne repair checks via the trace function as follows. For in this case bα equals the number of sub-symbols downloaded each u ∈ F and α ∈ F, we introduce the polynomial during the repair process from the node storing f(α). ##LTLine## ##LTLine##

label: equation

text: A.

text: Deﬁnitions

text: and

text: Notation

text: F

text: Let

text: [n]

text: denote

text: the

text: set

text: n

text: Let

text: B

text: q

text: beFthe

text: ﬁnite

text: gu,α(x)

text: Tr

text: u(x

text: α)

text: /(x

text: α).

text: ﬁeld

text: of

text: q

text: elements,

text: for

text: some

text: prime

text: power

text: q.

text: Let

text: F

text: be

text: an

text: qt

text: extension

text: ﬁeld

text: of

text: B,

text: where

text: t

text: We

text: refer

text: to

text: the

text: elements

text: of

text: F

text: as

text: symbols

text: and

text: the

text: elements

text: of

text: B

text: as

text: sub-symbols.

text: The

text: ﬁeld

text: By

text: the

text: deﬁnition

text: of

text: a

text: trace

text: function,

text: the

text: next

text: lemma

text: follows

text: F

text: may∼also

text: be

text: viewed

text: as

text: a

text: vector

text: space

text: of

text: dimension

text: t

text: over

text: B,

text: in

text: a

text: straightforward

text: manner.

text: i.e.

text: F

text: Bt,

text: and

text: hence

text: each

text: symbol

text: in

text: F

text: may

text: be

text: represented

text: Lemma

text: The

text: polynomial

text: g

text: (x)

text: deﬁned

text: in

text: satisﬁes

text: C

text: u,α

text: as

text: a

text: vector

text: of

text: length

text: t

text: over

text: B.

text: A

text: linear

text: [n,

text: k]

text: code

text: over

text: F

text: the

text: following

text: properties.

text: is

text: a

text: subspace

text: of

text: Fn

text: of

text: dimension

text: k.

text: Each

text: element

text: of

text: a

text: code

text: is

text: (a)

text: deg(g

text: qt

text: (b)

text: g

text: (α)

text: u.

text: C

text: C⊥

text: u,α

text: u,α

text: referred

text: to

text: as

text: a

text: codeword.

text: The

text: dual

text: of

text: a

text: code

text: denoted

text: is

text: C

text: By

text: Lemma

text: (a),

text: deg(g

text: qt

text: r

text: Therefore,

text: the

text: orthogonal

text: complement

text: of

text: and

text: has

text: dimension

text: r

text: n

text: k.

text: u,α

text: C⊥

text: Deﬁnition

text: Let

text: F[x]

text: denote

text: the

text: ring

text: of

text: polynomials

text: over

text: F.

text: the

text: polynomCial

text: gu,α(x)

text: corres{ponds

text: to

text: a

text: }codeword

text: of

text: and

text: is

text: a

text: check

text: for

text: Now

text: let

text: U

text: u

text: be

text: a

text: basis

text: of

text: F

text: over

text: A

text: Reed-Solomon

text: code

text: RS(A,

text: k)

text: Fn

text: of

text: dimension

text: k

text: over

text: a

text: t

text: ﬁnite

text: ﬁeld

text: F

text: evaluation

text: A

text: {α

text: α

text: α

text: F

text: B,

text: and

text: set

text: n

text: is

text: deﬁned

text: as:

text: g

text: (x)

text: g

text: ∗(x)

text: Tr

text: u

text: (x

text: α

text: /(x

text: α

text: i

text: [t].

text: i

text: ui,α

text: i

text: RS(A,

text: k)

text: f(αn)

text: f

text: F[x],

text: deg(f)

text: k

text: These

text: t

text: polynomials

text: correspond

text: to

text: t

text: codewords

text: of

text: C⊥.

text: Therefore,

text: we

text: obtain

text: t

text: equations

text: of

text: the

text: form

text: A

text: generalized

text: Reed-Solomon

text: code,

text: GRS(A,

text: k,

text: λ),

text: where

text: λ

text: (λ

text: λ

text: Fn,

text: is

text: deﬁned

text: similarly

text: to

text: a

text: Reed-Solomon

text: n

text: g

text: (α

text: )f(α

text: g

text: (α)f(α),

text: i

text: [t].

text: code,

text: except

text: that

text: the

text: codeword

text: corresponding

text: to

text: a

text: polynomial

text: i

text: i

text: f

text: is

text: deﬁned

text: as

text: λ

text: f(α

text: λ

text: f(α

text: λ

text: for

text: all

text: i

text: α

text: A

text: α

text: n

text: n

text: i

text: [n].

text: It

text: is

text: well

text: known

text: that

text: the

text: dual

text: of

text: a

text: Reed-Solomon

text: code

text: A

text: key

text: step

text: in

text: the

text: Guruswami-Wootters

text: repair

text: scheme

text: is

text: to

text: RS(A,

text: k),

text: for

text: any

text: n

text: F

text: is

text: a

text: generalized

text: Reed-Solomon

text: apply

text: the

text: trace

text: function

text: to

text: both

text: sides

text: of

text: to

text: obtain

text: t

text: different

text: code

text: GRS(A,

text: n

text: k,

text: λ),

text: for

text: some

text: multiplier

text: vector

text: λ

text: (see

text: Chp.

text: Whenever

text: clear

text: from

text: the

text: context,

text: we

text: use

text: f(x)

text: to

text: equations

text: denote

text: a

text: polynomial

text: of

text: degree

text: at

text: most

text: k

text: which

text: corresponds

text: a

text: codeword

text: of

text: the

text: Reed-Solomon

text: code

text: C

text: =−RS(A,−k),

text: −and

text: Tr

text: gi(α∗)f(α∗)

text: Tr

text: gi(α)f(α)

text: i

text: [t].

text: g(x)

text: to

text: denote

text: a

text: polynomial

text: of

text: degree

text: at

text: most

text: r

text: =Cn⊥

text: k

text: α∈A\{α∗}

text: which

text: corresponds

text: to

text: a

text: codeword

text: of

text: the

text: dual

text: code

text: Since

text: α∈A

text: g(α)(λαf(α))C=

text: we

text: refer

text: to

text: the

text: p|oly|nomial

text: g(x)

text: as

text: a

text: According

text: to

text: Lemma

text: (b),

text: gi(α∗)

text: ui,

text: for

text: i

text: t.

text: check

text: poly∈nomial

text: for

text: Note

text: that

text: when

text: n

text: F

text: we

text: have

text: λα

text: Moreover,

text: by

text: the

text: linearity

text: of

text: the

text: trace

text: function,

text: we

text: can

text: rewrite

text: for

text: all

text: α

text: F.

text: In

text: general,

text: as

text: recovering

text: f(α)

text: is

text: equivalent

text: to

text: as

text: i

text: t,

text: recovering

text: λ

text: f(α),

text: to

text: simplify

text: the

text: notation,

text: we

text: omit

text: the

text: factor

text: α

text: λ

text: in

text: our

text: derivations.

text: α

text: f(α)

text: B.

text: The

text: Guruswami-Wootters

text: Repair

text: Scheme

text: for

text: One

text: Erasure

text: Tr

text: uif(α

text: Tr

text: ui(α

text: α

text: Tr

text: α

text: α∗

text: α

text: A

text: α

text: Suppose

text: that

text: the

text: polynomial

text: f(x)

text: F[x]

text: corresponds

text: to

text: a

text: C

text: codeword

text: of

text: the

text: Reed-Solomon

text: code

text: RS(A,

text: k)

text: and

text: that

text: The

text: right-hand

text: side

text: sums

text: of

text: the

text: equations

text: may

text: be

text: com-

text: f(α

text: is

text: the

text: erased

text: symbol,

text: where

text: α

text: A

text: is

text: an

text: evaluation

text: point

text: puted

text: by

text: downloading

text: the

text: trace

text: Tr

text: f−(α)∗

text: from

text: of

text: the

text: code.

text: α

text: α

text: the

text: node

text: storing

text: f(α),

text: for

text: each

text: α

text: A

text: α

text: As

text: a

text: consequence,

text: Given

text: that

text: F

text: is

text: an

text: extension

text: ﬁeld

text: of

text: B

text: of

text: degree

text: t,

text: i.e.

text: F

text: and

text: B

text: F

text: for

text: some

text: prime

text: power

text: q,

text: one

text: may

text: deﬁne

text: the

text: t

text: independent

text: traces

text: Tr

text: uif(α

text: i

text: t,

text: of

text: f(α

text: qt

text: q

text: can

text: be

text: determined

text: by

text: downloading

text: one

text: sub-symbol

text: from

text: each

text: the

text: ﬁeld

text: trace

text: of

text: any

text: symbol

text: α

text: F

text: according

text: to

text: Tr

text: (α)

text: F/B

text: of

text: the

text: n

text: available

text: nodes,

text: and

text: the

text: erased

text: symbol

text: f(α

text: may

text: αqi.

text: The

text: trace

text: belongs

text: to

text: B.

text: When

text: clear

text: from

text: the

text: context,

text: subsequently

text: be

text: recovered

text: from

text: its

text: t

text: independent

text: traces.

text: The

text: we

text: omit

text: the

text: subscript

text: F/B.

text: The

text: key

text: points

text: in

text: the

text: repair

text: scheme

text: following

text: theorem

text: summarizes

text: this

text: brief

text: discussion.

text: proposed

text: by

text: Guruswami

text: and

text: Wootters

text: can

text: be

text: summarized

text: as

text: follows.

text: First,

text: each

text: symbol

text: in

text: F

text: can

text: be

text: recovered

text: from

text: its

text: t

text: Theorem

text: When

text: r

text: qt

text: there

text: exists

text: a

text: repair

text: scheme

text: independent

text: traces.

text: More

text: precisely,

text: given

text: a

text: basis

text: u

text: u

text: u

text: for

text: Reed-Solomon

text: codes

text: with

text: a

text: repair

text: bandwidth

text: of

text: (n

text: log

text: q

text: t

text: of

text: F

text: over

text: B,

text: any

text: α

text: F

text: can

text: be

text: uniquely

text: determined

text: given

text: the

text: bits.

text: More

text: generally,

text: if

text: r

text: qs,

text: where

text: (t

text: s)

text: t,

text: by

text: setting

text: B

text: F

text: v{al⊥ue}s

text: of

text: Tr(ui

text: α)

text: for

text: i

text: [t],

text: i.e.

text: α

text: Tr{(ui}α)ui

text: where

text: qt−s

text: one

text: can

text: devise

text: a

text: repair

text: scheme

text: with

text: bandwidth

text: (n

text: u

text: t

text: is

text: the

text: dual

text: (trace-orthogonal)

text: basis

text: of

text: u

text: t

text: (see,

text: for

text: s)

text: log

text: q

text: bits,

text: which

text: is

text: optimal

text: when

text: n

text: qt

text: and

text: r

text: qs.

text: i

text: i

text: instance

text: Ch.

text: Def.

text: Second,

text: when

text: r

text: qt

text: the

text: In

text: general,

text: any

text: set

text: of

text: t

text: polynomials

text: g

text: (x),

text: g

text: (x)

text: trace

text: function

text: also

text: provide

text: checks

text: that

text: generate

text: repair

text: equations

text: t

text: each

text: of

text: which

text: has

text: degree

text: at

text: most

text: r

text: and

text: such

text: that

text: with

text: coefﬁcients

text: that

text: are

text: linearly

text: dependent

text: over

text: B,

text: which

text: keeps

text: rank

text: {g

text: (α∗),

text: g

text: (α∗)}

text: can

text: be

text: used

text: to

text: f(α∗).

text: the

text: repair

text: cost

text: low.

text: q

text: t

text: g(xN)ot∈e

text: tFha[xt

text: ]ththeatchseactikssfyodf

text: Fthoorsre

text: iwales

text: Tsuhbe-sryempabiorlsb,awndhweriedtbhα

text: .b,

text: g=t(α)}α.∈MA\o{rαe∗o}vbeαr,

text: may

text: deﬁne

text: repair

text: checks

text: via

text: the

text: trace

text: function

text: as

text: follows.

text: For

text: in

text: this

text: case

text: bα

text: equals

text: the

text: number

text: of

text: sub-symbols

text: downloaded

text: each

text: u

text: F

text: and

text: α

text: F,

text: we

text: introduce

text: the

text: polynomial

text: during

text: the

text: repair

text: process

text: from

text: the

text: node

text: storing

text: f(α).

text: ##LTLine##

text: ##LTLine##


text: 11 Based on the master equation with components (56), of astronomical scales, the mechanism of collectively we perform numerical simulations in nondimensional enhanced quantum gravitational radiation considered → time t Γ t initially excited and subsequently relaxed in above may open up a future prospect of a lab-sized the x-direction, with harmonic modes n = n = 0, 2, 4, 6, gravitational wave transmitter. Based on the ongo- 1 n = n = 0 and a total particle number N = 1, 2, 3, 4, 5 ing rapid development of high-Q nanomechanical res- 2 3 shown in Figs. 1–3. The collective behaviour of the “su- onators demonstrated in the quantum regime [47–50], perradiant” spontaneous emission of gravitons, due to one could envisage a high-density cluster of nanores- the quadratic dependence of the particle occupations (of onators in such a correlated state that they behave like bosonic origin) as well as modal numbers (of quadrupole a system of N identical harmonic oscillators with fre- origin) in Eqs. (57) and (58), is particularly evident in quency ω. Supposing these oscillators occupy around Fig. 1, using an initial single-mode Fock state. Milder the n-th harmonic mode, then following discussions of ampliﬁcation of emission power with particle numbers Eq. (56), the maximum spontaneous decay rate due to are also seen in Figs. 2 and 3, where the initial states collective gravitational radiation is approximately given may be described as maximally mixed and maximally by Γ = N2n2Γ. For example, a future such micro- max entangled respectively. The quantum states are enumer- fabricated cluster consisting of up to N = one mole of | (cid:105) ated with p for p = 1, 2, . . . p with ascending eigen nanoresonators at ω/2π = 10 GHz excited with n = 1000 max energies and then the particle number occupations of could in principle have an observable peak decay rate of higher harmonic modes. Thus, with N = 5 there are up to Γ = 1 Hz via the superradiant spontaneous | (cid:105) max p = 56 even-mode occupation states N , N . . . , N emissions of gravitons. Furthermore, such gravitons max 0 1 6 with N = N = N = 0, starting from the ground could also be detected using a similar cluster of nanores- | 1(cid:105) | 3 5 (cid:105) state 1 = 5, 0, 0, 0, 0, 0, 0 , then the ﬁrst excited state onators instead of an ensemble of atoms as a gravitational | (cid:105) | (cid:105) | (cid:105) | (cid:105) 2 = 4, 0, 1, 0, 0, 0, 0 through 28 = 2, 0, 1, 0, 0, 0, 2 to radiation receiver described in Ref. [46]. The quantum | (cid:105) | (cid:105) the highest state 56 = 0, 0, 0, 0, 0, 0, 5 . nature of gravity could then be probed through the quan- tum properties of the nanoresonators imparted by the While classical sources of gravitational waves are absorbed gravitons. (1,56) (56,56) (1,56) (56,56) (1,56) (56,56) 0.048 ( a ) 0.19 (b) 0.48 (c) (d) ––––– N = 5 0.024 t = 0.005 0.097 t = 0.05 0.24 t = 0.5 ––––– N = 4 ––––– N = 3 0 0 0 ––––– N = 2 ––––– N = 1 (1,1) (56,1) (1,1) (56,1) (1,1) (56,1) t | | FIG. 3. Plots (a)–(c) show the simulation of ρ (cid:48) (t). Here, 5 scalar bosons in a harmonic trap are initially equally and p,p fully distributed in the density matrix for harmonic modes n = 0, 2, 4, 6 as a maximally entangled state. While releasing a continuously decreasing gravitational wave, they spontaneously decay towards the ground state in the bottom left corner, where all 5 particles occupy the n = 0 mode. Plot (d) shows the average radiation power per particle for a similar initial state with diﬀerent particle numbers. VI. CONCLUSION of gravitons mediating spacetime at large and matter, both bosons and fermions, of interest. Motivated by the need for a better understanding of For physically common states subject to a potential, the fundamental process for quantum matter to deco- we have explicitly demonstrated that the abstract master here and dissipate through spontaneous emission and ex- equation describing the general non-Markovian gravita- change of gravitons with the ubiquitous ﬂuctuating gravi- tional decoherence of matter formulated in Ref. [16] can tational environment, we have extended a recently estab- indeed be reduced, free from UV-cutoﬀ, to a more con- lished theory of quantum gravitational decoherence [16], crete Lindblad form, structurally identical to the fam- now complete with the dynamical origin and consequence ily of quantum optical master equations widely applied ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Based

text: on

text: the

text: master

text: equation

text: with

text: components

text: of

text: astronomical

text: scales,

text: the

text: mechanism

text: of

text: collectively

text: we

text: perform

text: numerical

text: simulations

text: in

text: nondimensional

text: enhanced

text: quantum

text: gravitational

text: radiation

text: considered

text: time

text: t

text: Γ

text: t

text: initially

text: excited

text: and

text: subsequently

text: relaxed

text: in

text: above

text: may

text: open

text: up

text: a

text: future

text: prospect

text: of

text: a

text: lab-sized

text: the

text: x-direction,

text: with

text: harmonic

text: modes

text: n

text: n

text: gravitational

text: wave

text: transmitter.

text: Based

text: on

text: the

text: ongo-

text: n

text: n

text: and

text: a

text: total

text: particle

text: number

text: N

text: ing

text: rapid

text: development

text: of

text: high-Q

text: nanomechanical

text: res-

text: shown

text: in

text: Figs.

text: The

text: collective

text: behaviour

text: of

text: the

text: “su-

text: onators

text: demonstrated

text: in

text: the

text: quantum

text: regime

text: perradiant”

text: spontaneous

text: emission

text: of

text: gravitons,

text: due

text: to

text: one

text: could

text: envisage

text: a

text: high-density

text: cluster

text: of

text: nanores-

text: the

text: quadratic

text: dependence

text: of

text: the

text: particle

text: occupations

text: (of

text: onators

text: in

text: such

text: a

text: correlated

text: state

text: that

text: they

text: behave

text: like

text: bosonic

text: origin)

text: as

text: well

text: as

text: modal

text: numbers

text: (of

text: quadrupole

text: a

text: system

text: of

text: N

text: identical

text: harmonic

text: oscillators

text: with

text: fre-

text: origin)

text: in

text: Eqs.

text: and

text: is

text: particularly

text: evident

text: in

text: quency

text: ω.

text: Supposing

text: these

text: oscillators

text: occupy

text: around

text: Fig.

text: using

text: an

text: initial

text: single-mode

text: Fock

text: state.

text: Milder

text: the

text: n-th

text: harmonic

text: mode,

text: then

text: following

text: discussions

text: of

text: ampliﬁcation

text: of

text: emission

text: power

text: with

text: particle

text: numbers

text: Eq.

text: the

text: maximum

text: spontaneous

text: decay

text: rate

text: due

text: to

text: are

text: also

text: seen

text: in

text: Figs.

text: and

text: where

text: the

text: initial

text: states

text: collective

text: gravitational

text: radiation

text: is

text: approximately

text: given

text: may

text: be

text: described

text: as

text: maximally

text: mixed

text: and

text: maximally

text: by

text: Γ

text: For

text: example,

text: a

text: future

text: such

text: micro-

text: max

text: entangled

text: respectively.

text: The

text: quantum

text: states

text: are

text: enumer-

text: fabricated

text: cluster

text: consisting

text: of

text: up

text: to

text: N

text: one

text: mole

text: of

text: ated

text: with

text: p

text: for

text: p

text: p

text: with

text: ascending

text: eigen

text: nanoresonators

text: at

text: ω/2π

text: GHz

text: excited

text: with

text: n

text: max

text: energies

text: and

text: then

text: the

text: particle

text: number

text: occupations

text: of

text: could

text: in

text: principle

text: have

text: an

text: observable

text: peak

text: decay

text: rate

text: of

text: higher

text: harmonic

text: modes.

text: Thus,

text: with

text: N

text: there

text: are

text: up

text: to

text: Γ

text: Hz

text: via

text: the

text: superradiant

text: spontaneous

text: max

text: p

text: even-mode

text: occupation

text: states

text: N

text: N

text: N

text: emissions

text: of

text: gravitons.

text: Furthermore,

text: such

text: gravitons

text: max

text: with

text: N

text: N

text: N

text: starting

text: from

text: the

text: ground

text: could

text: also

text: be

text: detected

text: using

text: a

text: similar

text: cluster

text: of

text: nanores-

text: state

text: then

text: the

text: ﬁrst

text: excited

text: state

text: onators

text: instead

text: of

text: an

text: ensemble

text: of

text: atoms

text: as

text: a

text: gravitational

text: through

text: to

text: radiation

text: receiver

text: described

text: in

text: Ref.

text: The

text: quantum

text: the

text: highest

text: state

text: nature

text: of

text: gravity

text: could

text: then

text: be

text: probed

text: through

text: the

text: quan-

text: tum

text: properties

text: of

text: the

text: nanoresonators

text: imparted

text: by

text: the

text: While

text: classical

text: sources

text: of

text: gravitational

text: waves

text: are

text: absorbed

text: gravitons.

text: a

text: (b)

text: (c)

text: (d)

text: N

text: t

text: t

text: t

text: N

text: N

text: N

text: N

text: t

text: FIG.

text: Plots

text: (a)–(c)

text: show

text: the

text: simulation

text: of

text: ρ

text: (t).

text: Here,

text: scalar

text: bosons

text: in

text: a

text: harmonic

text: trap

text: are

text: initially

text: equally

text: and

text: p,p

text: fully

text: distributed

text: in

text: the

text: density

text: matrix

text: for

text: harmonic

text: modes

text: n

text: as

text: a

text: maximally

text: entangled

text: state.

text: While

text: releasing

text: a

text: continuously

text: decreasing

text: gravitational

text: wave,

text: they

text: spontaneously

text: decay

text: towards

text: the

text: ground

text: state

text: in

text: the

text: bottom

text: left

text: corner,

text: where

text: all

text: particles

text: occupy

text: the

text: n

text: mode.

text: Plot

text: (d)

text: shows

text: the

text: average

text: radiation

text: power

text: per

text: particle

text: for

text: a

text: similar

text: initial

text: state

text: with

text: diﬀerent

text: particle

text: numbers.

text: VI.

text: CONCLUSION

text: of

text: gravitons

text: mediating

text: spacetime

text: at

text: large

text: and

text: matter,

text: both

text: bosons

text: and

text: fermions,

text: of

text: interest.

text: Motivated

text: by

text: the

text: need

text: for

text: a

text: better

text: understanding

text: of

text: For

text: physically

text: common

text: states

text: subject

text: to

text: a

text: potential,

text: the

text: fundamental

text: process

text: for

text: quantum

text: matter

text: to

text: deco-

text: we

text: have

text: explicitly

text: demonstrated

text: that

text: the

text: abstract

text: master

text: here

text: and

text: dissipate

text: through

text: spontaneous

text: emission

text: and

text: ex-

text: equation

text: describing

text: the

text: general

text: non-Markovian

text: gravita-

text: change

text: of

text: gravitons

text: with

text: the

text: ubiquitous

text: ﬂuctuating

text: gravi-

text: tional

text: decoherence

text: of

text: matter

text: formulated

text: in

text: Ref.

text: can

text: tational

text: environment,

text: we

text: have

text: extended

text: a

text: recently

text: estab-

text: indeed

text: be

text: reduced,

text: free

text: from

text: UV-cutoﬀ,

text: to

text: a

text: more

text: con-

text: lished

text: theory

text: of

text: quantum

text: gravitational

text: decoherence

text: crete

text: Lindblad

text: form,

text: structurally

text: identical

text: to

text: the

text: fam-

text: now

text: complete

text: with

text: the

text: dynamical

text: origin

text: and

text: consequence

text: ily

text: of

text: quantum

text: optical

text: master

text: equations

text: widely

text: applied

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 in the quantum optics problems. This enables investi- theoretical framework reported here has recently been gations of the theory and phenomenology of quantum applied and further extended to address the possible gravity to beneﬁt from a wealth of novel characteris- detection of stochastic gravitational waves using corre- tics and solution strategies in the ﬁeld of quantum op- lated atoms [46] and potential observation of spacetime tics [19, 20, 51–54]. One such possibility in terms of the ﬂuctuations through gravitational lensing [64]. newly identiﬁed superradiance of gravitational waves by Another future objective would be to go beyond the a system of coherence particles has been theoretically de- perturbative formulation so as to accommodate larger scribed and numerically illustrated in Sec. . spacetime ﬂuctuations and curved background or none. Extension in this direction could allow the quantum-to- Our general framework may serve to clarify various classical transition in the early universe with graviton conceptual issues encountered in the phenomenological productions to be more accurately analyzed. This may approach to quantum gravity [26, 55–60], with ﬁrst- be initiated by generalizing our non-Markovian master principles insights, and to guide further analytical tools, equation (30) to accommodate cosmological perturba- mathematical techniques, and modelling methodologies tions [65], in addition to its existing gravitational ﬂuc- for possible detections of quantum gravity eﬀects in the tuations in vacuum. A qualitative study of quantum-to- laboratory [61] and observatory [62] on the ground or classical transition may follow from the resulting decoher- in space [4]. In the context of the cosmological stochas- ence of the content of the early universe in the presence of tic gravitational waves, since the universe is considered cosmological perturbations. An additional rationale for spatially ﬂat with a low entropy on exit from inﬂa- this ﬁnal remark is that the development of open quan- tion [63], our theory may describe short-time graviton tum gravitational systems towards background indepen- radiation and reception by a distribution of coherent dence [5–8, 66] might even help navigate the search for states having potentially unexpected but important col- an ultimate full quantum theory of gravity with compat- lective properties including quantum nonlinearity, non- ible and accessible low energy eﬀects like gravitational locality, and entanglement [17, 18]. In this regard, the decoherence and radiance. ACKNOWLEDGMENTS This work was supported by the Carnegie Trust for the Universities of Scotland (T. O.) and by the EPSRC GG-Top Project and the Cruickshank Trust (C. W.). Appendix A: Derivation of the gravitational quantum optical master equation (cid:90) To derive Eq. (52), we ﬁrst introduce ∞ (cid:88) (cid:48) − τ˜ (k, t) = ds τ (k, t )e iks ij ij 0 (cid:48) † − − − = π Fij(n, n , k) an(cid:48)ane i(ωn ωn(cid:48))t δ(k ωn + ωn(cid:48)) (A1) (cid:48) n,n ≥ − ≥ using Eqs. (50) and (51). Note that since(cid:88)k 0, we have nonzero δ(k ωn + ωn(cid:48)) = 0 only if ωn ωn(cid:48). The following relations then hold (cid:88) † − † τ (k, t)τ˜ (k, t) ρ = δ(k ω(∆n)) A (n, ∆n, k) A (m, ∆m, k) ρ (A2) ij ij ij ij (cid:88) † − † τ˜ (k, t) ρ τ (k, t) = δ(k ω(∆n)) A (m, ∆m, k) ρ A (n, ∆n, k) (A3) ij ij ij ij (cid:88) † − † ρ τ˜ (k, t)τ (k, t) = δ(k ω(∆n)) ρ A (m, ∆m, k) A (n, ∆n, k) (A4) ij ij ij ij † − † τ (k, t) ρ τ˜ (k, t) = δ(k ω(∆n)) A (n, ∆n, k) ρ A (m, ∆m, k) (A5) ij ij ij ij summing over n, m, ∆n, ∆m subject to ω(∆n) = ω(∆m) in terms of the operators √ † A (n, ∆n, k) = πF (n + ∆n, n, k) a a . (A6) ij ij n n+∆n ##LTLine## ##LTLine## ##LTLine##

text: in

text: the

text: quantum

text: optics

text: problems.

text: This

text: enables

text: investi-

text: theoretical

text: framework

text: reported

text: here

text: has

text: recently

text: been

text: gations

text: of

text: the

text: theory

text: and

text: phenomenology

text: of

text: quantum

text: applied

text: and

text: further

text: extended

text: to

text: address

text: the

text: possible

text: gravity

text: to

text: beneﬁt

text: from

text: a

text: wealth

text: of

text: novel

text: characteris-

text: detection

text: of

text: stochastic

text: gravitational

text: waves

text: using

text: corre-

text: tics

text: and

text: solution

text: strategies

text: in

text: the

text: ﬁeld

text: of

text: quantum

text: op-

text: lated

text: atoms

text: and

text: potential

text: observation

text: of

text: spacetime

text: tics

text: One

text: such

text: possibility

text: in

text: terms

text: of

text: the

text: ﬂuctuations

text: through

text: gravitational

text: lensing

text: newly

text: identiﬁed

text: superradiance

text: of

text: gravitational

text: waves

text: by

text: Another

text: future

text: objective

text: would

text: be

text: to

text: go

text: beyond

text: the

text: a

text: system

text: of

text: coherence

text: particles

text: has

text: been

text: theoretically

text: de-

text: perturbative

text: formulation

text: so

text: as

text: to

text: accommodate

text: larger

text: scribed

text: and

text: numerically

text: illustrated

text: in

text: Sec.

text: spacetime

text: ﬂuctuations

text: and

text: curved

text: background

text: or

text: none.

text: Extension

text: in

text: this

text: direction

text: could

text: allow

text: the

text: quantum-to-

text: Our

text: general

text: framework

text: may

text: serve

text: to

text: clarify

text: various

text: classical

text: transition

text: in

text: the

text: early

text: universe

text: with

text: graviton

text: conceptual

text: issues

text: encountered

text: in

text: the

text: phenomenological

text: productions

text: to

text: be

text: more

text: accurately

text: analyzed.

text: This

text: may

text: approach

text: to

text: quantum

text: gravity

text: with

text: ﬁrst-

text: be

text: initiated

text: by

text: generalizing

text: our

text: non-Markovian

text: master

text: principles

text: insights,

text: and

text: to

text: guide

text: further

text: analytical

text: tools,

text: equation

text: to

text: accommodate

text: cosmological

text: perturba-

text: mathematical

text: techniques,

text: and

text: modelling

text: methodologies

text: tions

text: in

text: addition

text: to

text: its

text: existing

text: gravitational

text: ﬂuc-

text: for

text: possible

text: detections

text: of

text: quantum

text: gravity

text: eﬀects

text: in

text: the

text: tuations

text: in

text: vacuum.

text: A

text: qualitative

text: study

text: of

text: quantum-to-

text: laboratory

text: and

text: observatory

text: on

text: the

text: ground

text: or

text: classical

text: transition

text: may

text: follow

text: from

text: the

text: resulting

text: decoher-

text: in

text: space

text: In

text: the

text: context

text: of

text: the

text: cosmological

text: stochas-

text: ence

text: of

text: the

text: content

text: of

text: the

text: early

text: universe

text: in

text: the

text: presence

text: of

text: tic

text: gravitational

text: waves,

text: since

text: the

text: universe

text: is

text: considered

text: cosmological

text: perturbations.

text: An

text: additional

text: rationale

text: for

text: spatially

text: ﬂat

text: with

text: a

text: low

text: entropy

text: on

text: exit

text: from

text: inﬂa-

text: this

text: ﬁnal

text: remark

text: is

text: that

text: the

text: development

text: of

text: open

text: quan-

text: tion

text: our

text: theory

text: may

text: describe

text: short-time

text: graviton

text: tum

text: gravitational

text: systems

text: towards

text: background

text: indepen-

text: radiation

text: and

text: reception

text: by

text: a

text: distribution

text: of

text: coherent

text: dence

text: might

text: even

text: help

text: navigate

text: the

text: search

text: for

text: states

text: having

text: potentially

text: unexpected

text: but

text: important

text: col-

text: an

text: ultimate

text: full

text: quantum

text: theory

text: of

text: gravity

text: with

text: compat-

text: lective

text: properties

text: including

text: quantum

text: nonlinearity,

text: non-

text: ible

text: and

text: accessible

text: low

text: energy

text: eﬀects

text: like

text: gravitational

text: locality,

text: and

text: entanglement

text: In

text: this

text: regard,

text: the

text: decoherence

text: and

text: radiance.

text: ACKNOWLEDGMENTS

text: This

text: work

text: was

text: supported

text: by

text: the

text: Carnegie

text: Trust

text: for

text: the

text: Universities

text: of

text: Scotland

text: (T.

text: O.)

text: and

text: by

text: the

text: EPSRC

text: GG-Top

text: Project

text: and

text: the

text: Cruickshank

text: Trust

text: (C.

text: W.).

text: Appendix

text: A:

text: Derivation

text: of

text: the

text: gravitational

text: quantum

text: optical

text: master

text: equation

text: To

text: derive

text: Eq.

text: we

text: ﬁrst

text: introduce

text: τ˜

text: (k,

text: t)

text: ds

text: τ

text: (k,

text: t

text: )e

text: iks

text: ij

text: ij

text: π

text: Fij(n,

text: n

text: k)

text: i(ωn

text: δ(k

text: ωn

text: n,n

text: using

text: Eqs.

text: and

text: Note

text: that

text: we

text: have

text: nonzero

text: δ(k

text: ωn

text: only

text: if

text: ωn

text: The

text: following

text: relations

text: then

text: hold

text: τ

text: (k,

text: t)τ˜

text: (k,

text: t)

text: ρ

text: δ(k

text: ω(∆n))

text: A

text: (n,

text: ∆n,

text: k)

text: A

text: (m,

text: ∆m,

text: k)

text: ρ

text: ij

text: ij

text: ij

text: ij

text: τ˜

text: (k,

text: t)

text: ρ

text: τ

text: (k,

text: t)

text: δ(k

text: ω(∆n))

text: A

text: (m,

text: ∆m,

text: k)

text: ρ

text: A

text: (n,

text: ∆n,

text: k)

text: ij

text: ij

text: ij

text: ij

text: ρ

text: τ˜

text: (k,

text: t)τ

text: (k,

text: t)

text: δ(k

text: ω(∆n))

text: ρ

text: A

text: (m,

text: ∆m,

text: k)

text: A

text: (n,

text: ∆n,

text: k)

text: ij

text: ij

text: ij

text: ij

text: τ

text: (k,

text: t)

text: ρ

text: τ˜

text: (k,

text: t)

text: δ(k

text: ω(∆n))

text: A

text: (n,

text: ∆n,

text: k)

text: ρ

text: A

text: (m,

text: ∆m,

text: k)

text: ij

text: ij

text: ij

text: ij

text: summing

text: over

text: n,

text: m,

text: ∆n,

text: ∆m

text: subject

text: to

text: ω(∆n)

text: ω(∆m)

text: in

text: terms

text: of

text: the

text: operators

text: A

text: (n,

text: ∆n,

text: k)

text: πF

text: (n

text: ∆n,

text: n,

text: k)

text: a

text: a

text: ij

text: ij

text: n

text: n+∆n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Tencor RS100 and Jandel 4-point probe systems as well as the ﬁlm thickness measured by x-ray reﬂectivity (XRR). XRR was performed using Cu Kα radiation in a Bede MetrixL diﬀractometer from Jordan Valley or a Panalytical X’Pert diﬀractometer. Film thicknesses by XRR were cross-calibrated by Rutherford backscattering spectrometry (RBS) measurements ◦ using a 1.52 MeV He+ ion beam in a rotating random mode at a backscatter angle of 170 . In all cases, the contribution of the adhesion layers to the sheet resistance was obtained by independent measurements and taken into account in the determination of the PGM thin ﬁlm resistivity. The diﬀractometers mentioned above were also used to assess the ﬁlm crystallinity using x-ray diﬀraction (XRD). Surface roughnesses were measured by atomic force microscopy (AFM) using a Bruker IconPT microscope. Lateral correlation lengths of the surface roughness were obtained from the autocorrelation function. The microstructure of the ﬁlms was determined from plan-view transmission electron microscopy (TEM) images using Tecnai F30 and Titan3 G2 microscopes. Based on these images, the mean linear grain boundary intercept distance (the average linear distance between grain boundaries) was determined [42, 43]. Due to the almost columnar nature of the microstructure and their expected relatively weak contribution to the resistivity, grain boundaries parallel to the surface were neglected. Electronic structures of the PGMs, Ru, Rh, Pd, Os, Ir, and Pt, as well as of Cu were obtained by ﬁrst-principles calculations based on density functional theory as imple- mented in the Quantum Espresso package [44]. Projector augmented wave [45] potentials with the Perdew–Burke–Ernzerhof generalized gradient [46] approximation of the exchange- × × correlation functional have been used together with a 40 40 40 Monkhorst–Pack k-point sampling grid and an energy cutoﬀ of 80 Ry to ensure the convergence of the total energy diﬀerences (10 – 12 eV). The Fermi surface S (k) was determined from the calculated elec- F,n tron energy as a function of the wave vector k for each band with index n. In addition, using the obtained Fermi surfaces and electronic densities of states, scattering times due to electron–phonon interactions have been calculated using standard ﬁrst-order perturbation theory [47]. Details of these calculations can be found in Ref. [34]. III. MATERIAL PROPERTIES AND RESISTIVITY OF PLATINUM-GROUP METAL THIN FILMS All ﬁlms were polycrystalline as deposited. The Θ-2Θ XRD patterns were consistent with 4

text: Tencor

text: and

text: Jandel

text: probe

text: systems

text: as

text: well

text: as

text: the

text: ﬁlm

text: thickness

text: measured

text: by

text: x-ray

text: reﬂectivity

text: (XRR).

text: XRR

text: was

text: performed

text: using

text: Cu

text: Kα

text: radiation

text: in

text: a

text: Bede

text: MetrixL

text: diﬀractometer

text: from

text: Jordan

text: Valley

text: or

text: a

text: Panalytical

text: X’Pert

text: diﬀractometer.

text: Film

text: thicknesses

text: by

text: XRR

text: were

text: cross-calibrated

text: by

text: Rutherford

text: backscattering

text: spectrometry

text: (RBS)

text: measurements

text: using

text: a

text: MeV

text: He+

text: ion

text: beam

text: in

text: a

text: rotating

text: random

text: mode

text: at

text: a

text: backscatter

text: angle

text: of

text: In

text: all

text: cases,

text: the

text: contribution

text: of

text: the

text: adhesion

text: layers

text: to

text: the

text: sheet

text: resistance

text: was

text: obtained

text: by

text: independent

text: measurements

text: and

text: taken

text: into

text: account

text: in

text: the

text: determination

text: of

text: the

text: PGM

text: thin

text: ﬁlm

text: resistivity.

text: The

text: diﬀractometers

text: mentioned

text: above

text: were

text: also

text: used

text: to

text: assess

text: the

text: ﬁlm

text: crystallinity

text: using

text: x-ray

text: diﬀraction

text: (XRD).

text: Surface

text: roughnesses

text: were

text: measured

text: by

text: atomic

text: force

text: microscopy

text: (AFM)

text: using

text: a

text: Bruker

text: IconPT

text: microscope.

text: Lateral

text: correlation

text: lengths

text: of

text: the

text: surface

text: roughness

text: were

text: obtained

text: from

text: the

text: autocorrelation

text: function.

text: The

text: microstructure

text: of

text: the

text: ﬁlms

text: was

text: determined

text: from

text: plan-view

text: transmission

text: electron

text: microscopy

text: (TEM)

text: images

text: using

text: Tecnai

text: and

text: microscopes.

text: Based

text: on

text: these

text: images,

text: the

text: mean

text: linear

text: grain

text: boundary

text: intercept

text: distance

text: (the

text: average

text: linear

text: distance

text: between

text: grain

text: boundaries)

text: was

text: determined

text: Due

text: to

text: the

text: almost

text: columnar

text: nature

text: of

text: the

text: microstructure

text: and

text: their

text: expected

text: relatively

text: weak

text: contribution

text: to

text: the

text: resistivity,

text: grain

text: boundaries

text: parallel

text: to

text: the

text: surface

text: were

text: neglected.

text: Electronic

text: structures

text: of

text: the

text: PGMs,

text: Ru,

text: Rh,

text: Pd,

text: Os,

text: Ir,

text: and

text: Pt,

text: as

text: well

text: as

text: of

text: Cu

text: were

text: obtained

text: by

text: ﬁrst-principles

text: calculations

text: based

text: on

text: density

text: functional

text: theory

text: as

text: imple-

text: mented

text: in

text: the

text: Quantum

text: Espresso

text: package

text: Projector

text: augmented

text: wave

text: potentials

text: with

text: the

text: Perdew–Burke–Ernzerhof

text: generalized

text: gradient

text: approximation

text: of

text: the

text: exchange-

text: correlation

text: functional

text: have

text: been

text: used

text: together

text: with

text: a

text: Monkhorst–Pack

text: k-point

text: sampling

text: grid

text: and

text: an

text: energy

text: cutoﬀ

text: of

text: Ry

text: to

text: ensure

text: the

text: convergence

text: of

text: the

text: total

text: energy

text: diﬀerences

text: eV).

text: The

text: Fermi

text: surface

text: S

text: (k)

text: was

text: determined

text: from

text: the

text: calculated

text: elec-

text: F,n

text: tron

text: energy

text: as

text: a

text: function

text: of

text: the

text: wave

text: vector

text: k

text: for

text: each

text: band

text: with

text: index

text: n.

text: In

text: addition,

text: using

text: the

text: obtained

text: Fermi

text: surfaces

text: and

text: electronic

text: densities

text: of

text: states,

text: scattering

text: times

text: due

text: to

text: electron–phonon

text: interactions

text: have

text: been

text: calculated

text: using

text: standard

text: ﬁrst-order

text: perturbation

text: theory

text: Details

text: of

text: these

text: calculations

text: can

text: be

text: found

text: in

text: Ref.

text: III.

text: MATERIAL

text: PROPERTIES

text: AND

text: RESISTIVITY

text: OF

text: PLATINUM-GROUP

text: METAL

text: THIN

text: FILMS

text: All

text: ﬁlms

text: were

text: polycrystalline

text: as

text: deposited.

text: The

text: Θ-2Θ

text: XRD

text: patterns

text: were

text: consistent

text: with


text: the expected crystal structures of the stable phases (hcp for Ru, fcc for all other metals) with ◦ (partial) texture [(001) for Ru, (111) for other metals]. Post deposition annealing at 420 C in forming gas for 20 min improved both the crystallinity and led to strong texturing [Fig. 1(a)]. The out-of-plane Scherrer crystallite size of annealed ﬁlms [Fig 1(b)] was typically of the order of the ﬁlm thickness for ﬁlms up to about 15 nm and deviated slightly towards smaller values for thicker ﬁlms. This indicates that the microstructure of the ﬁlms was (nearly) columnar. Figure 2 shows the root-mean-square (RMS) roughness of Cu (TaN/Cu/TaN), Ru (both Ru/SiO and TaN/Ru/TaN), Ir, Pd, and Pt ﬁlms as a function of the ﬁlm thickness after 2 ◦ post deposition annealing at 420 C. The roughness of the annealed ﬁlms increased with increasing ﬁlm thickness but remained below 0.4 nm even for 30 nm thick PGM ﬁlms. Cu ﬁlms were slightly rougher with RMS values of 0.5 to 0.6 nm for the thickest ﬁlms. XRR measurements (not shown) indicated that the roughness of the top surface was very similar to that of buried interfaces (typically also 0.3 to 0.5 nm). The lateral correlation length ξ of the surface roughness (obtained with Gaussian correlation statistics) was between 10 and 15 nm for all ﬁlms with insigniﬁcant diﬀerences between materials/stacks and only little coarsening in the studied thickness range up to 30 nm. Figure 3 shows the resistivity of Cu (TaN/Cu/TaN), Ru (Ru/SiO and TaN/Ru/TaN), 2 ◦ Pd, Ir, and Pt as a function of the ﬁlm thickness. All ﬁlms were annealed at 420 C for 20 min in forming gas. Cu showed a strong increase with decreasing ﬁlm thickness, as observed previously and ascribed to the combination of surface and grain boundary scattering. Note that the Cu resistivity values were close to the ones reported in the literature for scaled Cu interconnect lines of the same critical dimension [48, 49]. By contrast, all PGM thin ﬁlms showed a much weaker thickness dependence of the resistivity than Cu. For ﬁlms with thicknesses of 10 nm and above, the resistivities were much higher than for Cu owing to the higher bulk resistivities of PGMs. However, for ﬁlm thicknesses of about 5 nm and below, the resistivities of Ru and Ir became comparable and even lower than the resistivity of Cu. From a technological point of view, this resistivity crossover renders PGMs, in particular Ru and Ir highly interesting for scaled interconnects with critical dimensions of 10 nm and below, where the current combination of Cu, diﬀusion barriers, and adhesion liner layers may be outperformed by barrierless Ru or Ir metallization. Indeed, scaled Ru ﬁlled interconnect structures have already shown ﬁrst promising results 5

text: the

text: expected

text: crystal

text: structures

text: of

text: the

text: stable

text: phases

text: (hcp

text: for

text: Ru,

text: fcc

text: for

text: all

text: other

text: metals)

text: with

text: (partial)

text: texture

text: for

text: Ru,

text: for

text: other

text: metals].

text: Post

text: deposition

text: annealing

text: at

text: C

text: in

text: forming

text: gas

text: for

text: min

text: improved

text: both

text: the

text: crystallinity

text: and

text: led

text: to

text: strong

text: texturing

text: [Fig.

text: The

text: out-of-plane

text: Scherrer

text: crystallite

text: size

text: of

text: annealed

text: ﬁlms

text: [Fig

text: was

text: typically

text: of

text: the

text: order

text: of

text: the

text: ﬁlm

text: thickness

text: for

text: ﬁlms

text: up

text: to

text: about

text: nm

text: and

text: deviated

text: slightly

text: towards

text: smaller

text: values

text: for

text: thicker

text: ﬁlms.

text: This

text: indicates

text: that

text: the

text: microstructure

text: of

text: the

text: ﬁlms

text: was

text: (nearly)

text: columnar.

text: Figure

text: shows

text: the

text: root-mean-square

text: (RMS)

text: roughness

text: of

text: Cu

text: (TaN/Cu/TaN),

text: Ru

text: (both

text: Ru/SiO

text: and

text: TaN/Ru/TaN),

text: Ir,

text: Pd,

text: and

text: Pt

text: ﬁlms

text: as

text: a

text: function

text: of

text: the

text: ﬁlm

text: thickness

text: after

text: post

text: deposition

text: annealing

text: at

text: C.

text: The

text: roughness

text: of

text: the

text: annealed

text: ﬁlms

text: increased

text: with

text: increasing

text: ﬁlm

text: thickness

text: but

text: remained

text: below

text: nm

text: even

text: for

text: nm

text: thick

text: PGM

text: ﬁlms.

text: Cu

text: ﬁlms

text: were

text: slightly

text: rougher

text: with

text: RMS

text: values

text: of

text: to

text: nm

text: for

text: the

text: thickest

text: ﬁlms.

text: XRR

text: measurements

text: (not

text: shown)

text: indicated

text: that

text: the

text: roughness

text: of

text: the

text: top

text: surface

text: was

text: very

text: similar

text: to

text: that

text: of

text: buried

text: interfaces

text: (typically

text: also

text: to

text: nm).

text: The

text: lateral

text: correlation

text: length

text: ξ

text: of

text: the

text: surface

text: roughness

text: (obtained

text: with

text: Gaussian

text: correlation

text: statistics)

text: was

text: between

text: and

text: nm

text: for

text: all

text: ﬁlms

text: with

text: insigniﬁcant

text: diﬀerences

text: between

text: materials/stacks

text: and

text: only

text: little

text: coarsening

text: in

text: the

text: studied

text: thickness

text: range

text: up

text: to

text: nm.

text: Figure

text: shows

text: the

text: resistivity

text: of

text: Cu

text: (TaN/Cu/TaN),

text: Ru

text: (Ru/SiO

text: and

text: TaN/Ru/TaN),

text: Pd,

text: Ir,

text: and

text: Pt

text: as

text: a

text: function

text: of

text: the

text: ﬁlm

text: thickness.

text: All

text: ﬁlms

text: were

text: annealed

text: at

text: C

text: for

text: min

text: in

text: forming

text: gas.

text: Cu

text: showed

text: a

text: strong

text: increase

text: with

text: decreasing

text: ﬁlm

text: thickness,

text: as

text: observed

text: previously

text: and

text: ascribed

text: to

text: the

text: combination

text: of

text: surface

text: and

text: grain

text: boundary

text: scattering.

text: Note

text: that

text: the

text: Cu

text: resistivity

text: values

text: were

text: close

text: to

text: the

text: ones

text: reported

text: in

text: the

text: literature

text: for

text: scaled

text: Cu

text: interconnect

text: lines

text: of

text: the

text: same

text: critical

text: dimension

text: By

text: contrast,

text: all

text: PGM

text: thin

text: ﬁlms

text: showed

text: a

text: much

text: weaker

text: thickness

text: dependence

text: of

text: the

text: resistivity

text: than

text: Cu.

text: For

text: ﬁlms

text: with

text: thicknesses

text: of

text: nm

text: and

text: above,

text: the

text: resistivities

text: were

text: much

text: higher

text: than

text: for

text: Cu

text: owing

text: to

text: the

text: higher

text: bulk

text: resistivities

text: of

text: PGMs.

text: However,

text: for

text: ﬁlm

text: thicknesses

text: of

text: about

text: nm

text: and

text: below,

text: the

text: resistivities

text: of

text: Ru

text: and

text: Ir

text: became

text: comparable

text: and

text: even

text: lower

text: than

text: the

text: resistivity

text: of

text: Cu.

text: From

text: a

text: technological

text: point

text: of

text: view,

text: this

text: resistivity

text: crossover

text: renders

text: PGMs,

text: in

text: particular

text: Ru

text: and

text: Ir

text: highly

text: interesting

text: for

text: scaled

text: interconnects

text: with

text: critical

text: dimensions

text: of

text: nm

text: and

text: below,

text: where

text: the

text: current

text: combination

text: of

text: Cu,

text: diﬀusion

text: barriers,

text: and

text: adhesion

text: liner

text: layers

text: may

text: be

text: outperformed

text: by

text: barrierless

text: Ru

text: or

text: Ir

text: metallization.

text: Indeed,

text: scaled

text: Ru

text: ﬁlled

text: interconnect

text: structures

text: have

text: already

text: shown

text: ﬁrst

text: promising

text: results


text: surface or interface; R is the reﬂection coeﬃcient (0 < R < 1) of a charge carrier at a grain boundary. In general, p can take diﬀerent values at the top and bottom interface, e.g. when the surface and interface roughnesses are strongly diﬀerent, as described by the model of Soﬀer [58]. However, given the observation that surface and buried interface roughnesses in the stacks considered here are low and very similar, we will assume that a single parameter p can describe both interfaces of the metal ﬁlms. A. Film thickness dependence of the linear grain boundary distance While surface scattering depends directly on the ﬁlm thickness, grain boundary scattering depends on the average linear distance between grain boundaries, l, along the transport direction. Therefore, a quantitative model of the thin ﬁlm resistivity as a function of ﬁlm thickness requires the knowledge of the thickness dependence of l in polycrystalline ﬁlms. Historically, it has often been assumed that l is identical or proportional to the ﬁlm thickness and this assumption has often been used to model the thickness dependence of the resistivity. However, it has been pointed out that such simple relations are generally not valid [59]. We have therefore experimentally determined the average linear grain boundary distance using the intercept method [42, 43] from plan-view transmission electron micrographs for 5, ◦ 10, and 30 nm thick Ru/SiO (as deposited and after annealing at 420 C), TaN/Ru/TaN 2 (annealed), Ir/SiO (annealed), and TaN/Cu/TaN (annealed) thin ﬁlms. Figure 5 shows 2 both sample TEM images as well as the deduced ﬁlm thickness dependence of the mean linear grain boundary intercept length. While for TaN/Cu/TaN and annealed Ru/SiO the 2 mean linear intercept length was close to the ﬁlm thickness, other stacks clearly showed a ∼ saturating eﬀect for 30 nm thick ﬁlms. Linear intercept lengths for in-between thicknesses were obtained by piecewise linear interpolation. Other more nonlinear interpolation schemes did not have any signiﬁcant eﬀects on the modeling results discussed below. B. Semiclassical modeling results and discussion Figure 6 shows the experimental thickness dependence of the resistivities of Ru/SiO , 2 TaN/Ru/TaN, Ir/SiO , and TaN/Cu/TaN, together with the best ﬁts using the Mayadas– 2 ◦ Shatzkes model in Eq. (3). All ﬁlms were annealed at 420 C for 20 min except for an addi- tional data set of as deposited Ru/SiO . To obtain best ﬁts, the experimentally determined 2 thickness dependences of the linear distance between grain boundaries for the diﬀerent ma- 9

text: surface

text: or

text: interface;

text: R

text: is

text: the

text: reﬂection

text: coeﬃcient

text: R

text: of

text: a

text: charge

text: carrier

text: at

text: a

text: grain

text: boundary.

text: In

text: general,

text: p

text: can

text: take

text: diﬀerent

text: values

text: at

text: the

text: top

text: and

text: bottom

text: interface,

text: e.g.

text: when

text: the

text: surface

text: and

text: interface

text: roughnesses

text: are

text: strongly

text: diﬀerent,

text: as

text: described

text: by

text: the

text: model

text: of

text: Soﬀer

text: However,

text: given

text: the

text: observation

text: that

text: surface

text: and

text: buried

text: interface

text: roughnesses

text: in

text: the

text: stacks

text: considered

text: here

text: are

text: low

text: and

text: very

text: similar,

text: we

text: will

text: assume

text: that

text: a

text: single

text: parameter

text: p

text: can

text: describe

text: both

text: interfaces

text: of

text: the

text: metal

text: ﬁlms.

text: A.

label: section

text: Film

text: thickness

text: dependence

text: of

text: the

text: linear

text: grain

text: boundary

text: distance

text: While

text: surface

text: scattering

text: depends

text: directly

text: on

text: the

text: ﬁlm

text: thickness,

text: grain

text: boundary

text: scattering

text: depends

text: on

text: the

text: average

text: linear

text: distance

text: between

text: grain

text: boundaries,

text: l,

text: along

text: the

text: transport

text: direction.

text: Therefore,

text: a

text: quantitative

text: model

text: of

text: the

text: thin

text: ﬁlm

text: resistivity

text: as

text: a

text: function

text: of

text: ﬁlm

text: thickness

text: requires

text: the

text: knowledge

text: of

text: the

text: thickness

text: dependence

text: of

text: l

text: in

text: polycrystalline

text: ﬁlms.

text: Historically,

text: it

text: has

text: often

text: been

text: assumed

text: that

text: l

text: is

text: identical

text: or

text: proportional

text: to

text: the

text: ﬁlm

text: thickness

text: and

text: this

text: assumption

text: has

text: often

text: been

text: used

text: to

text: model

text: the

text: thickness

text: dependence

text: of

text: the

text: resistivity.

text: However,

text: it

text: has

text: been

text: pointed

text: out

text: that

text: such

text: simple

text: relations

text: are

text: generally

text: not

text: valid

text: We

text: have

text: therefore

text: experimentally

text: determined

text: the

text: average

text: linear

text: grain

text: boundary

text: distance

text: using

text: the

text: intercept

text: method

text: from

text: plan-view

text: transmission

text: electron

text: micrographs

text: for

text: and

text: nm

text: thick

text: Ru/SiO

text: (as

text: deposited

text: and

text: after

text: annealing

text: at

text: C),

text: TaN/Ru/TaN

text: (annealed),

text: Ir/SiO

text: (annealed),

text: and

text: TaN/Cu/TaN

text: (annealed)

text: thin

text: ﬁlms.

text: Figure

text: shows

text: both

text: sample

text: TEM

text: images

text: as

text: well

text: as

text: the

text: deduced

text: ﬁlm

text: thickness

text: dependence

text: of

text: the

text: mean

text: linear

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text: EPJ Web of Conferences 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 -0.5 -0.5 -0.5 -0.5 -1-.51 ttssteesppe==p=11802 q2=1 -1-.51 ttssteesppe==p=11802 q2=1 -1-.51 tttssseeeppp===111024 q2=1 -1-.51 tttssseeeppp===111024 q2=1 -4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2 1 1 1 1 0.5 0.5 0.5 0.5 0 0 0 0 -0.5 -0.5 -0.5 -0.5 -1-.51 ttssteesppe==p=11802 q2=4 -1-.51 ttssteesppe==p=11802 q2=4 -1-.51 tttssseeeppp===111024 q2=4 -1-.51 tttssseeeppp===111024 q2=4 -4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2 Figure 5. Illustration of the signal in F with the inclusion of the CEDM term. The data in the top row are for × 3 (cid:126)p = (1,0,0) 2π/La in the following order: insertion on u and d quarks for the a12m310 ensemble followed by insertion on u and d quarks for the a09m310 ensemble. The plots in the second row are in the same order except × with (cid:126)p = (2,0,0) 2π/La. 1 1 1 1 0.5 ttssteesppe==p=11802 0.5 ttssteesppe==p=11802 0.5 tttssseeeppp===111024 q2=1 0.5 tttssseeeppp===111024 q2=1 0 0 0 0 -0.5 -0.5 -0.5 -0.5 -1 q2=1 -1 q2=1 -1 -1 -1.5 -1.5 -1.5 -1.5 -4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2 1 1 1 1 0.5 ttssteesppe==p=11802 0.5 ttssteesppe==p=11802 0.5 tttssseeeppp===111024 0.5 tttssseeeppp===111024 0 0 0 0 -0.5 -0.5 -0.5 -0.5 -1 q2=4 -1 q2=4 -1 q2=4 -1 q2=4 -1.5 -1.5 -1.5 -1.5 -4 -2 0 2 4 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 τ - tsep/2 τ - tsep/2 τ - tsep/2 τ - tsep/2 Figure 6. Il×lustration of the signal in F3 with the inclusion of the γ5 term. The data in the top row are for (cid:126)p = (1,0,0) 2π/La in the following order: insertion on u and d quarks for the a12m310 ensemble followed by insertion on u and d quarks for the a09m310 ensemble. The plots in the second row are in the same order except with (cid:126)p = (2,0,0). an acceptable signal-to-noise ratio can be obtained to estimate F with O(1) errors, our ﬁrst goal for 3 the CEDM calculations. There are theoretical reasons to expect that the connected contribution of the γ5 operator is, with small corrections, proportional to that of the CEDM operator [13]. In Fig. 7, we show the ratio of the two contributions and ﬁnd that this expectation is actually realized. To conclude, the data so far show that an acceptable signal-to-noise ratio can be obtained in the connected contributions to F3 for both the CEDM and γ5 operator insertion. We also ﬁnd that The ratio of the connected contribution of the γ5 to CEDM term is a constant. This implies that the mixing of the γ5 term with the CEDM term can be cast as part of the multiplicative renormalization of the CEDM operator. Controlling its O(1/a2) UV divergent coeﬃcient is a hurdle for lattice theories that respect chiral symmetry (domain wall or overlap fermions) and those that don’t such as Wilson- u-d ,U3F u-d ,U3F u-d , 5γ ,U3F u-d , 5γ ,U3F u-d ,D3F u-d ,D3F u-d , 5γ ,D3F u-d , 5γ ,D3F u-d ,U3F u-d ,U3F u-d , 5γ ,U3F u-d , 5γ ,U3F u-d ,D3F u-d ,D3F u-d , 5γ ,D3F u-d , 5γ ,D3F ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: XXXX 6 [23] M. Majzoobi, F. Koushanfar, and Potkonjak, “Lightweight secure PUFs,” in Proc. IEEE/ACM Int. Conf. Computer-Aided Design, 2008, pp. 670– 673.

text: XXXX

label: reference

text: M.

text: Majzoobi,

text: F.

text: Koushanfar,

text: and

text: Potkonjak,

text: “Lightweight

text: secure

text: PUFs,”

text: in

text: Proc.

text: IEEE/ACM

text: Int.

text: Conf.

text: Computer-Aided

text: Design,

text: pp.


text: Vulnerability of Deep Reinforcement Learning to Policy Induction Attacks 11 memory of the game at the sampled instances of the previous experiment. It is noteworthy that due to random initializations, the exploration mechanism, and the stochastic nature of SGD, even similar Q-networks trained on the same set of observations will obtain diﬀerent sets of weights. The second Q-network was tested to measure its vulnerability to the adversarial examples obtained from the last experiment. Figure 5 shows that more than 70% of the perturbations obtained from both FGSM and JSMA methods also aﬀect the second network, hence verifying the transferability of adversarial examples between DQNs. 100 FGSM JSMA 80 60 40 20 0 1.26 1.28 1.3 1.32 1.34 1.36 No. iterations x 1e6 Fig. 5: Transferability of adversarial examples in DQN 5.3 Performance of Proposed Policy Induction Attack Our ﬁnal experiment tests the performance of our proposed exploitation mechanism. In this experiment, we consider an adversary whose reward value is the exact opposite of the game score, meaning that it aims to devise a policy that maximizes the number of lost games. To obtain this policy, we trained an adversarial DQN on the game, whose reward value was the negative of the value obtained from target DQN’s reward function. With the adversarial policy at hand, a target DQN was setup to train on the game environment to maximize the original reward function. The game environment was modiﬁed to allow per- turbation of pixel values in game frames by the adversary. A second DQN was also setup to train on the target’s observations to provide an estimation of the target DQN to enable blackbox crafting of adversarial example. At every obser- vation, the adversarial policy obtained in the initialization phase was consulted to calculate the action that would satisfy the adversary’s goal. Then, the JSMA algorithm was utilized to generate the adversarial example that would cause the output of the replica DQN network to be the action selected by the adversarial policy. This example was then passed to the target DQN as its observation. Fig- ure 6 compares the performance of unperturbed and attacked DQNs in terms of their reward values, measured as the diﬀerence of current game score with the average score. It can be seen that the reward value for the targeted agent rapidly falls below the unperturbed case and maintains the trend of losing the % etaR sseccuS refsnarT ##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##

text: Vulnerability

text: of

text: Deep

text: Reinforcement

text: Learning

text: to

text: Policy

text: Induction

text: Attacks

text: memory

text: of

text: the

text: game

text: at

text: the

text: sampled

text: instances

text: of

text: the

text: previous

text: experiment.

text: It

text: is

text: noteworthy

text: that

text: due

text: to

text: random

text: initializations,

text: the

text: exploration

text: mechanism,

text: and

text: the

text: stochastic

text: nature

text: of

text: SGD,

text: even

text: similar

text: Q-networks

text: trained

text: on

text: the

text: same

text: set

text: of

text: observations

text: will

text: obtain

text: diﬀerent

text: sets

text: of

text: weights.

text: The

text: second

text: Q-network

text: was

text: tested

text: to

text: measure

text: its

text: vulnerability

text: to

text: the

text: adversarial

text: examples

text: obtained

text: from

text: the

text: last

text: experiment.

text: Figure

text: shows

text: that

text: more

text: than

text: of

text: the

text: perturbations

text: obtained

text: from

text: both

text: FGSM

text: and

text: JSMA

text: methods

text: also

text: aﬀect

text: the

text: second

text: network,

text: hence

text: verifying

text: the

text: transferability

text: of

text: adversarial

text: examples

text: between

text: DQNs.

text: FGSM

text: JSMA

text: No.

text: iterations

text: x

text: Fig.

text: Transferability

text: of

text: adversarial

text: examples

text: in

text: DQN

text: Performance

text: of

text: Proposed

text: Policy

text: Induction

text: Attack

text: Our

text: ﬁnal

text: experiment

text: tests

text: the

text: performance

text: of

text: our

text: proposed

text: exploitation

text: mechanism.

text: In

text: this

text: experiment,

text: we

text: consider

text: an

text: adversary

text: whose

text: reward

text: value

text: is

text: the

text: exact

text: opposite

text: of

text: the

text: game

text: score,

text: meaning

text: that

text: it

text: aims

text: to

text: devise

text: a

text: policy

text: that

text: maximizes

text: the

text: number

text: of

text: lost

text: games.

text: To

text: obtain

text: this

text: policy,

text: we

text: trained

text: an

text: adversarial

text: DQN

text: on

text: the

text: game,

text: whose

text: reward

text: value

text: was

text: the

text: negative

text: of

text: the

text: value

text: obtained

text: from

text: target

text: DQN’s

text: reward

text: function.

text: With

text: the

text: adversarial

text: policy

text: at

text: hand,

text: a

text: target

text: DQN

text: was

text: setup

text: to

text: train

text: on

text: the

text: game

text: environment

text: to

text: maximize

text: the

text: original

text: reward

text: function.

text: The

text: game

text: environment

text: was

text: modiﬁed

text: to

text: allow

text: per-

text: turbation

text: of

text: pixel

text: values

text: in

text: game

text: frames

text: by

text: the

text: adversary.

text: A

text: second

text: DQN

text: was

text: also

text: setup

text: to

text: train

text: on

text: the

text: target’s

text: observations

text: to

text: provide

text: an

text: estimation

text: of

text: the

text: target

text: DQN

text: to

text: enable

text: blackbox

text: crafting

text: of

text: adversarial

text: example.

text: At

text: every

text: obser-

text: vation,

text: the

text: adversarial

text: policy

text: obtained

text: in

text: the

text: initialization

text: phase

text: was

text: consulted

text: to

text: calculate

text: the

text: action

text: that

text: would

text: satisfy

text: the

text: adversary’s

text: goal.

text: Then,

text: the

text: JSMA

text: algorithm

text: was

text: utilized

text: to

text: generate

text: the

text: adversarial

text: example

text: that

text: would

text: cause

text: the

text: output

text: of

text: the

text: replica

text: DQN

text: network

text: to

text: be

text: the

text: action

text: selected

text: by

text: the

text: adversarial

text: policy.

text: This

text: example

text: was

text: then

text: passed

text: to

text: the

text: target

text: DQN

text: as

text: its

text: observation.

text: Fig-

text: ure

text: compares

text: the

text: performance

text: of

text: unperturbed

text: and

text: attacked

text: DQNs

text: in

text: terms

text: of

text: their

text: reward

text: values,

text: measured

text: as

text: the

text: diﬀerence

text: of

text: current

text: game

text: score

text: with

text: the

text: average

text: score.

text: It

text: can

text: be

text: seen

text: that

text: the

text: reward

text: value

text: for

text: the

text: targeted

text: agent

text: rapidly

text: falls

text: below

text: the

text: unperturbed

text: case

text: and

text: maintains

text: the

text: trend

text: of

text: losing

text: the

text: etaR

text: sseccuS

text: refsnarT

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