text: 3 through d(χT)/dT. At T > 50 K, the χ(T) data can be ﬁtted with the Curie-Weiss law (χ(T) = C/(T-θ)) satis- factorily using the Curie constant C = 1.31 and the Curie- Weiss temperature (θ = 9.5 K), as shown by the red solid line in Fig. 3. The eﬀective moment of µ = 3.23 µ eff B per Ni2+ extracted from Curie constant is higher than the expected spin-only value of µ = 2.83 µ for S = 1, calc B which suggests the existence of a partially unquenched orbital contribution. The ﬁtted value of θ = 9.5 K sug- gests the existence of an average ferromagnetic (FM) cou- pling among spins at high temperature but AFM order- ∼ ing occurs at T 23.5 K, which suggests that the mag- N netic interactions must consider couplings beyond nearest neighbor spins and have diﬀerent signs containing both ferromagnetic (FM) and AFM couplings, as veriﬁed later by our ab initio studies in the following. The χ(T) data above 200 K were also ﬁtted by high temperature series (HTS) expansion up to 8th order.19 The ﬁtting parame- ters are found to be g = 2.02 and the exchange interaction (J/k ) = -6.6 K. B FIG. 5. (Color online) (a) Isothermal magnetization M(H) curves of Ni NbBO for the magnetic ﬁeld parallel to diﬀer- 2 6 ent crystallographic axis (b) dM/dH curves of Ni NbBO for 2 6 the magnetic ﬁeld parallel to a-axis at some selected temper- atures. to the a-axis, the magnetization reveals jump at a critical ﬁeld H near H = 36.7 kOe below T at 2 K, which C C N is attributed to a spin-ﬂop transition when the spin sus- ceptibility changes abruptly to a higher level, i.e., the or- FIG. 4. (Color online) The magnetic susceptibilities dered spins originally aligned along the a-direction ﬂop to χ(T) for Ni NbBO single crystal measured in an applied 2 6 the direction perpendicular to the external ﬁeld. As ex- magnetic ﬁeld of 10 kOe parallel to all three crystallo- pected for the ordered spins aligned along the a-direction, graphic axis. The average of single crystal susceptibility we do not observe spin-ﬂop transition manifested in M(H) (χ(T) = (χ +χ +χ )/3) agrees perfectly with those measured a b c using powder sample directly. for H is applied parallel to b- or c- axis. We may summa- rize the H-T phase diagram for Ni NbBO based on the 2 6 magnetic ﬁeld and temperature dependence of M(T,H) Anisotropic magnetic susceptibilities χ(T) for Ni NbBO 2 6 with ﬁeld applied parallel to a-axis, as shown in Fig. 6. single crystals were measured in an applied magnetic A small ﬁeld dependence of T is also shown, where the ﬁeld of 10 kOe parallel to all three crystallographic N boundary of spin-ﬂop transition is indicated according to axes, as shown in Fig. 4. There is no deviation be- the onsets of dM/dH peaks shown in Fig. 5(b). tween χ (T) and χ (T) throughout the measured ZFC FC temperature range. Below T , the anisotropy becomes N signiﬁcantly enhanced as shown in the inset of Fig. 4, which indicates that the spins are aligned along the a-axis C. Theoretical calculations for in the 3D AFM long range ordering. To understand the AFM behavior of Ni NbBO bet- Within ﬁrst-principle density functional theory, we 2 6 ter, we measured magnetization as a function of magnetic ﬁrst calculated the total energy (E ) for the ferromag- FM ﬁeld H along the two crystal orientations above and be- netic state. The total energy per formula unit (f.u.) is low T , as shown in Fig. 5(a), where no ﬁeld or tempera- -72.7578 eV. In order to ﬁnd out the magnetic ground N ture hysteresis were observed. For magnetic ﬁeld parallel state of the system, we have considered various magnetic ##LTFigure## ##LTFigure##

text: through

text: d(χT)/dT.

text: At

text: T

text: K,

text: the

text: χ(T)

text: data

text: can

text: be

text: ﬁtted

text: with

text: the

text: Curie-Weiss

text: law

text: (χ(T)

text: C/(T-θ))

text: satis-

text: factorily

text: using

text: the

text: Curie

text: constant

text: C

text: and

text: the

text: Curie-

text: Weiss

text: temperature

text: (θ

text: K),

text: as

text: shown

text: by

text: the

text: red

text: solid

text: line

text: in

text: Fig.

text: The

text: eﬀective

text: moment

text: of

text: µ

text: µ

text: eff

text: B

text: per

text: extracted

text: from

text: Curie

text: constant

text: is

text: higher

text: than

text: the

text: expected

text: spin-only

text: value

text: of

text: µ

text: µ

text: for

text: S

text: calc

text: B

text: which

text: suggests

text: the

text: existence

text: of

text: a

text: partially

text: unquenched

text: orbital

text: contribution.

text: The

text: ﬁtted

text: value

text: of

text: θ

text: K

text: sug-

text: gests

text: the

text: existence

text: of

text: an

text: average

text: ferromagnetic

text: (FM)

text: cou-

text: pling

text: among

text: spins

text: at

text: high

text: temperature

text: but

text: AFM

text: order-

text: ing

text: occurs

text: at

text: T

text: K,

text: which

text: suggests

text: that

text: the

text: mag-

text: N

text: netic

text: interactions

text: must

text: consider

text: couplings

text: beyond

text: nearest

text: neighbor

text: spins

text: and

text: have

text: diﬀerent

text: signs

text: containing

text: both

text: ferromagnetic

text: (FM)

text: and

text: AFM

text: couplings,

text: as

text: veriﬁed

text: later

text: by

text: our

text: ab

text: initio

text: studies

text: in

text: the

text: following.

text: The

text: χ(T)

text: data

text: above

text: K

text: were

text: also

text: ﬁtted

text: by

text: high

text: temperature

text: series

text: (HTS)

text: expansion

text: up

text: to

text: The

text: ﬁtting

text: parame-

text: ters

text: are

text: found

text: to

text: be

text: g

text: and

text: the

text: exchange

text: interaction

text: (J/k

text: K.

text: B

text: FIG.

text: (Color

text: online)

text: (a)

text: Isothermal

text: magnetization

text: M(H)

text: curves

text: of

text: Ni

text: NbBO

text: for

text: the

text: magnetic

text: ﬁeld

text: parallel

text: to

text: diﬀer-

text: ent

text: crystallographic

text: axis

text: (b)

text: dM/dH

text: curves

text: of

text: Ni

text: NbBO

text: for

text: the

text: magnetic

text: ﬁeld

text: parallel

text: to

text: a-axis

text: at

text: some

text: selected

text: temper-

text: atures.

text: to

text: the

text: a-axis,

text: the

text: magnetization

text: reveals

text: jump

text: at

text: a

text: critical

text: ﬁeld

text: H

text: near

text: H

text: kOe

text: below

text: T

text: at

text: K,

text: which

text: C

text: C

text: N

text: is

text: attributed

text: to

text: a

text: spin-ﬂop

text: transition

text: when

text: the

text: spin

text: sus-

text: ceptibility

text: changes

text: abruptly

text: to

text: a

text: higher

text: level,

text: i.e.,

text: the

text: or-

text: FIG.

text: (Color

text: online)

text: The

text: magnetic

text: susceptibilities

text: dered

text: spins

text: originally

text: aligned

text: along

text: the

text: a-direction

text: ﬂop

text: to

text: χ(T)

text: for

text: Ni

text: NbBO

text: single

text: crystal

text: measured

text: in

text: an

text: applied

text: the

text: direction

text: perpendicular

text: to

text: the

text: external

text: ﬁeld.

text: As

text: ex-

text: magnetic

text: ﬁeld

text: of

text: kOe

text: parallel

text: to

text: all

text: three

text: crystallo-

text: pected

text: for

text: the

text: ordered

text: spins

text: aligned

text: along

text: the

text: a-direction,

text: graphic

text: axis.

text: The

text: average

text: of

text: single

text: crystal

text: susceptibility

text: we

text: do

text: not

text: observe

text: spin-ﬂop

text: transition

text: manifested

text: in

text: M(H)

text: (χ(T)

text: (χ

text: +χ

text: +χ

text: agrees

text: perfectly

text: with

text: those

text: measured

text: a

text: b

text: c

text: using

text: powder

text: sample

text: directly.

text: for

text: H

text: is

text: applied

text: parallel

text: to

text: b-

text: or

text: c-

text: axis.

text: We

text: may

text: summa-

text: rize

text: the

text: H-T

text: phase

text: diagram

text: for

text: Ni

text: NbBO

text: based

text: on

text: the

text: magnetic

text: ﬁeld

text: and

text: temperature

text: dependence

text: of

text: M(T,H)

text: Anisotropic

text: magnetic

text: susceptibilities

text: χ(T)

text: for

text: Ni

text: NbBO

text: with

text: ﬁeld

text: applied

text: parallel

text: to

text: a-axis,

text: as

text: shown

text: in

text: Fig.

text: single

text: crystals

text: were

text: measured

text: in

text: an

text: applied

text: magnetic

text: A

text: small

text: ﬁeld

text: dependence

text: of

text: T

text: is

text: also

text: shown,

text: where

text: the

text: ﬁeld

text: of

text: kOe

text: parallel

text: to

text: all

text: three

text: crystallographic

text: N

text: boundary

text: of

text: spin-ﬂop

text: transition

text: is

text: indicated

text: according

text: to

text: axes,

text: as

text: shown

text: in

text: Fig.

text: There

text: is

text: no

text: deviation

text: be-

text: the

text: onsets

text: of

text: dM/dH

text: peaks

text: shown

text: in

text: Fig.

text: tween

text: χ

text: (T)

text: and

text: χ

text: (T)

text: throughout

text: the

text: measured

text: ZFC

text: FC

text: temperature

text: range.

text: Below

text: T

text: the

text: anisotropy

text: becomes

text: N

text: signiﬁcantly

text: enhanced

text: as

text: shown

text: in

text: the

text: inset

text: of

text: Fig.

text: which

text: indicates

text: that

text: the

text: spins

text: are

text: aligned

text: along

text: the

text: a-axis

text: C.

text: Theoretical

text: calculations

text: for

text: in

text: the

text: AFM

text: long

text: range

text: ordering.

text: To

text: understand

text: the

text: AFM

text: behavior

text: of

text: Ni

text: NbBO

text: bet-

text: Within

text: ﬁrst-principle

text: density

text: functional

text: theory,

text: we

text: ter,

text: we

text: measured

text: magnetization

text: as

text: a

text: function

text: of

text: magnetic

text: ﬁrst

text: calculated

text: the

text: total

text: energy

text: (E

text: for

text: the

text: ferromag-

text: FM

text: ﬁeld

text: H

text: along

text: the

text: two

text: crystal

text: orientations

text: above

text: and

text: be-

text: netic

text: state.

text: The

text: total

text: energy

text: per

text: formula

text: unit

text: (f.u.)

text: is

text: low

text: T

text: as

text: shown

text: in

text: Fig.

text: where

text: no

text: ﬁeld

text: or

text: tempera-

text: eV.

text: In

text: order

text: to

text: ﬁnd

text: out

text: the

text: magnetic

text: ground

text: N

text: ture

text: hysteresis

text: were

text: observed.

text: For

text: magnetic

text: ﬁeld

text: parallel

text: state

text: of

text: the

text: system,

text: we

text: have

text: considered

text: various

text: magnetic

text: ##LTFigure##

text: ##LTFigure##


text: etats eueuQ tseuqeR knowledge about the model, which may not be possible to A. MDP Framework have from an unpredictable environment. Numerical results S A An MDP is deﬁned by a tuple of < , , P, R > where show the convergence as well as the efﬁciency of the proposed S A is a state space, is an action space, P is a transition learning algorithm. probability function, and R is a reward function. The state S space of our admission control for the access point on a (cid:110) (cid:111) II. SYSTEM MODEL balloon is deﬁned as follows: S (cid:44) ∈ E ∈ X Admission Control s = (e, x) : e ; x , (1) Policy Optimal Policy E { } Pannel where = 0, 1, . . . , E is the energy state space λ X λso Battery s w{ hose elemen}ts represent energy levels in the battery. = λe ... λb thxeb,evxecn, xtssw, xheen ias raeqsuetesotfisevferonmts ainnowthheicrhacxcbe,sxs cp,oainntd, axsusaerre, λ RF Rectenna λc and a satellite, respectively. xe is an event for energy arrival. When the system is at state e, if an event x happens, an Fig. 2: The system model. accept/reject decision must be made. Thus, we can deﬁne the (cid:110) (cid:111) action space as follows: We consider energy management problem for a balloon with an access point as shown in Fig. 2. Speciﬁcally, the A (cid:44) ∈ { } a : a 0, 1 , (2) access point is to receive and transmit data from users. The access point has a battery to store energy harvested from (cid:26) solar and RF. When a request is sent to the access point, where it will check the amount of energy remaining in the battery 1, if the arriving request is accepted and apply an admission control policy by deciding whether a = 0, if the arriving request is rejected the request will be accepted or rejected. If the request is (cid:48) accepted, a certain amount of energy from the battery will To derive the transition probability function P(s, a, s ), be used to receive data and transmit data to the next hop. we consider a discrete time system by using uniformization Different requests have different QoS requirements. Therefore, technique [6] with a uniformization parameter u obtained as we divide requests into three classes, i.e., requests from other follows: balloons (i.e., class-1), requests from users on the ground (class-2), and requests from satellites or aircrafts (class-3). u = λ + λ + λ + λ . (3) b c s e The arrival processes of requests from class-1, class-2, and class-3 are assumed to follow the Poisson distribution with mean rates λ , λ , and λ , respectively. When a request is Based on the uniformization parameter u, we can determine accepted, the bballcoon willsreceive an immediate reward (i.e., the probabilities of the events as follows. In the event e, the probabilities of a request arriving from other balloons, a user, revenue). The immediate revenues for accepting requests from class-1, class-2, and class-3 are r , r , and r , respectively. and a satellite/an aircraft are λb/u, λc/u, λs/u, respectively. In the Loon network, balloons bare cassumeds to be equipped The probability of energy arrival is λe/u. Then, we can derive the transition probability matrix for the system. However, with solar panels [4] for harvesting energy from sunlight. to do so, we need to know the environment parameters, Additionally, we assume that the balloons can be equipped e.g., successful energy harvesting probability, requests’ arrival with a rectenna [5] to harvest energy from RF waves. We rates. These parameters are generally not known in advance assume that the energy arrival from both solar panel and RF and building the model with complete information may not be rectenna follows the Poisson distribution with mean rates λ , e possible. Therefore, we apply a learning algorithm based on and the successful energy harvesting probability is psu. The e simulation-based method [7]. The main idea of the simulation- maximum capacity of the battery is E. At each time epoch, based method is based on a “simulator” that can simulate when the access point receives a request, it will consult with the environment by generating environment parameters (e.g., the admission control policy. The admission control policy a successful energy harvesting probability and arrival rates). will determine a decision to accept or reject the request based Then we use the parameters from simulations to derive the on the request’ class and the current energy level in the battery. admission control policy for the access point. Based on the In this paper, we are interested in maximizing the proﬁt for simulation-based method, the transition probability function the service provider in the terms of average reward for the can be deﬁned as follows: balloon. (cid:48) III. PROBLEM FORMULATION P(s, a, s ) = penvp(s)p(a), (4) In this Section, we will formulate the optimization problem as Markov decision process (MDP) and study a learning where p is environment parameter (e.g., the successful env algorithm to obtain the optimal policy for balloons. energy harvesting probability), p(s) is the probability that the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: etats

label: paragraph

font: default

text: eueuQ

text: tseuqeR

text: knowledge

text: about

text: the

text: model,

text: which

text: may

text: not

text: be

text: possible

text: to

text: A.

label: section

text: MDP

text: Framework

text: have

text: from

text: an

text: unpredictable

text: environment.

text: Numerical

text: results

text: S

text: A

text: An

text: MDP

text: is

text: deﬁned

text: by

text: a

text: tuple

text: of

text: P,

text: R

text: where

text: show

text: the

text: convergence

text: as

text: well

text: as

text: the

text: efﬁciency

text: of

text: the

text: proposed

text: S

text: A

text: is

text: a

text: state

text: space,

text: is

text: an

text: action

text: space,

text: P

text: is

text: a

text: transition

text: learning

text: algorithm.

text: probability

text: function,

text: and

text: R

text: is

text: a

text: reward

text: function.

text: The

text: state

text: S

text: space

text: of

text: our

text: admission

text: control

text: for

text: the

text: access

text: point

text: on

text: a

label: equation

text: II.

text: SYSTEM

text: MODEL

text: balloon

text: is

text: deﬁned

text: as

text: follows:

text: S

text: E

text: X

text: Admission

text: Control

text: s

text: (e,

text: x)

text: e

text: x

text: Policy

text: Optimal

text: Policy

text: E

text: Pannel

text: where

text: E

text: is

text: the

text: energy

text: state

text: space

text: λ

text: X

text: λso

text: Battery

text: s

text: w{

text: hose

text: elemen}ts

text: represent

text: energy

text: levels

text: in

text: the

text: battery.

text: λe

text: λb

text: thxeb,evxecn,

text: xtssw,

text: xheen

text: ias

text: raeqsuetesotfisevferonmts

text: ainnowthheicrhacxcbe,sxs

text: cp,oainntd,

text: axsusaerre,

text: λ

text: RF

text: Rectenna

text: λc

text: and

text: a

text: satellite,

text: respectively.

text: xe

text: is

text: an

text: event

text: for

text: energy

text: arrival.

text: When

text: the

text: system

text: is

text: at

text: state

text: e,

text: if

text: an

text: event

text: x

text: happens,

text: an

text: Fig.

text: The

text: system

text: model.

text: accept/reject

text: decision

text: must

text: be

text: made.

text: Thus,

text: we

text: can

text: deﬁne

text: the

text: action

text: space

text: as

text: follows:

text: We

text: consider

text: energy

text: management

text: problem

text: for

text: a

text: balloon

text: with

text: an

text: access

text: point

text: as

text: shown

text: in

text: Fig.

text: Speciﬁcally,

text: the

text: A

text: a

text: a

text: access

text: point

text: is

text: to

text: receive

text: and

text: transmit

text: data

text: from

text: users.

text: The

text: access

text: point

text: has

text: a

text: battery

text: to

text: store

text: energy

text: harvested

text: from

text: solar

text: and

text: RF.

text: When

text: a

text: request

text: is

text: sent

text: to

text: the

text: access

text: point,

text: where

text: it

text: will

text: check

text: the

text: amount

text: of

text: energy

text: remaining

text: in

text: the

text: battery

text: if

text: the

text: arriving

text: request

text: is

text: accepted

text: and

text: apply

text: an

text: admission

text: control

text: policy

text: by

text: deciding

text: whether

text: a

text: if

text: the

text: arriving

text: request

text: is

text: rejected

text: the

text: request

text: will

text: be

text: accepted

text: or

text: rejected.

text: If

text: the

text: request

text: is

text: accepted,

text: a

text: certain

text: amount

text: of

text: energy

text: from

text: the

text: battery

text: will

text: To

text: derive

text: the

text: transition

text: probability

text: function

text: P(s,

text: a,

text: s

text: be

text: used

text: to

text: receive

text: data

text: and

text: transmit

text: data

text: to

text: the

text: next

text: hop.

text: we

text: consider

text: a

text: discrete

text: time

text: system

text: by

text: using

text: uniformization

text: Different

text: requests

text: have

text: different

text: QoS

text: requirements.

text: Therefore,

text: technique

text: with

text: a

text: uniformization

text: parameter

text: u

text: obtained

text: as

text: we

text: divide

text: requests

text: into

text: three

text: classes,

text: i.e.,

text: requests

text: from

text: other

text: follows:

text: balloons

text: (i.e.,

text: requests

text: from

text: users

text: on

text: the

text: ground

text: and

text: requests

text: from

text: satellites

text: or

text: aircrafts

text: u

text: λ

text: λ

text: λ

text: λ

text: b

text: c

text: s

text: e

text: The

text: arrival

text: processes

text: of

text: requests

text: from

text: and

text: are

text: assumed

text: to

text: follow

text: the

text: Poisson

text: distribution

text: with

text: mean

text: rates

text: λ

text: λ

text: and

text: λ

text: respectively.

text: When

text: a

text: request

text: is

text: Based

text: on

text: the

text: uniformization

text: parameter

text: u,

text: we

text: can

text: determine

text: accepted,

text: the

text: bballcoon

text: willsreceive

text: an

text: immediate

text: reward

text: (i.e.,

text: the

text: probabilities

text: of

text: the

text: events

text: as

text: follows.

text: In

text: the

text: event

text: e,

text: the

text: probabilities

text: of

text: a

text: request

text: arriving

text: from

text: other

text: balloons,

text: a

text: user,

text: revenue).

text: The

text: immediate

text: revenues

text: for

text: accepting

text: requests

text: from

text: and

text: are

text: r

text: r

text: and

text: r

text: respectively.

text: and

text: a

text: satellite/an

text: aircraft

text: are

text: λb/u,

text: λc/u,

text: λs/u,

text: respectively.

text: In

text: the

text: Loon

text: network,

text: balloons

text: bare

text: cassumeds

text: to

text: be

text: equipped

text: The

text: probability

text: of

text: energy

text: arrival

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text: )a( L )a( L )a( L 120 80 40 2D: 20% 51 × 51 75 × 75 101 × 101 0 0 2 4 6 8 10 time (s) 25 20 15 10 2D: 70% 5 51 × 51 75 × 75 101 × 101 0 0 2 4 6 8 10 time (s) 60 50 40 30 20 3D: 90% 10 31×31×31 45×45×45 55×55×55 0 0 2 4 6 8 10 time (s) FIG. 7: Time dependence of the width L (in units of the lattice constant a) of the averaged rotational excitation distributions formed by one rotational excitation placed at t = 0 in the centre of 2D and 3D lattices with diﬀerent size. The upper and the lower panels show that the excitations diﬀuse to the edges of the lattice, while the middle plane illustrates that the excitation is localized within the lattice and is not aﬀected by the lattice boundaries. 26 ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )a(

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text: )2a( 2σ r )2a( 2σ r )2a( 2σ r )sm( emit 60 125 9.0 100 8.5 40 7.0 75 5.5 21 25 31 35 size (n) 50 20 3D 3D, ideal 25 0.0 0.1 0.2 0.3 0.00 0.01 0.02 0.03 0.04 0.05 time (s) time (s) 102 t t 1 2 101 3D, disordered 100 10−2 10−1 100 time (s) FIG. 9: Time dependence of σ2 = (cid:104)r2(cid:105) − (cid:104)r(cid:105)2 for a rotational excitation initially placed in the r middle of a 3D lattice. Upper left panel: 3D ideal lattice (circles), disordered lattice with 50 % of vacancies (squares) and disordered lattice with 85 % of vacancies with size 31 × 31 × 31 sites. Upper right panel: 3D ideal lattice with size 21 × 21 × 21 (squares); 25 × 25 × 25 (triangles); 31 × 31× 31 (circles); 35 × 35 × 35 (diamonds). Inset: Lattice size dependence of time t marking the 1 deviation of the wave packet dynamics from that in an ideal inﬁnitely lattice due to the boundary eﬀects. Lower panel: 3D disordered lattice of size 55×55×55 with 90 % of vacancies. The symbols represent the numerical calculations; the full curves are the analytical ﬁts σ2 = Dt2; the dashed r curves are the linear ﬁts σ2 = bt + c. r 28 ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: 2σ

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text: (triangles);

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text: Inset:

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text: The

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text: the

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text: ##LTFigure##

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text: Phys. Rev. B 71, 174203 (2005). [21] N. Y. Yao, C. R. Laumann, S. Gopalakrishnan, M. Knap, M. Mu¨eller, E. A. Demler, and M. D. Lukin, Phys. Rev. Lett. 113, 243002 (2014). [22] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Ko- tochigova, P. S. Julienne, D. S. Jin, and J. Ye, Science 322, 231 (2008). [23] M. H. G. de Miranda, A. Chotia, B. Neyenhuis, D. Wang, G. Qu´em´ener, S. Ospelkaus, J. L. Bohn, J. Ye, and D. S. Jin, Nat. Phys. 7 502 (2011). [24] A. Chotia, B. Neyenhuis, S. A. Moses, B. Yan, J. P. Covey, M. Foss-Feig, A. M. Rey, D. S. Jin, and J. Ye, Phys. Rev. Lett. 108 080405 (2012). [25] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin and J. Ye, Nature 501, 521 (2013). [26] K. R. A. Hazzard, B. Gadway, M. Foss-Feig, B. Yan, S. A. Moses, J. P. Covey, N. Y. Yao, M. D. Lukin, J. Ye, D. S. Jin, and A. M. Rey, Phys. Rev. Lett. 113, 195302 (2014). [27] F. Herrera, M. Litinskaya, and R. V. Krems, Phys. Rev. A 82, 033428 (2010). [28] P. Xiang, M. Litinskaya, E. A. Shapiro, and R. V. Krems, New J. Phys. 15, 063015 (2013). [29] D. DeMille, Phys. Rev. Lett. 88, 067901 (2002). [30] Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A 48, 1687 (1993). [31] S. R. Jackson, T. J. Khoo, and F. W. Strauch, Phys. Rev. A 86, 022335 (2012). [32] P. Xiang, M. Litinskaya and R. V. Krems, Phys. Rev. A 85, 061401(R) (2012). [33] F. Herrera and R. V. Krems, Phys. Rev. A 84, 051401(R) (2011). [34] F. Herrera, K. W. Madison, R. V. Krems, and M. Berciu, Phys. Rev. Lett. 110, 223002 (2013). [35] G. Grynberg and C. Robilliard, Phys. Rep. 355, 335 (2001). [36] Quantum particles in 2D disordered potential are expected to undergo Anderson localization, provided the weak-localization correction is ﬁnite and negative. Depending on the miscrocopic details of the disorder potential, this may or may not be the case. Quoting from Ref. [37]: “Scaling theory does not pretend to be an exact theory, there is thus real interest in knowing whether there is localization in 2 dimesions for speciﬁc systems”. [37] C. A. Mu¨ller and D. Delande, “Disorder and interference: localization phenomena”, Chapter 9 in “Les Houches 2009 - Session XCI: Ultracold Gases and Quantum Information”, edited by C. Miniatura et al. (Oxford University Press, 2011). [38] L. J. Rooﬂ and J. L. Skinnerb, J. Chem. Phys. 89, 3279 (1989). 33

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text: Phys.

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text: V.

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text: F.

text: Herrera

text: and

text: R.

text: V.

text: Krems,

text: Phys.

text: Rev.

text: A

text: F.

text: Herrera,

text: K.

text: W.

text: Madison,

text: R.

text: V.

text: Krems,

text: and

text: M.

text: Berciu,

text: Phys.

text: Rev.

text: Lett.

text: G.

text: Grynberg

text: and

text: C.

text: Robilliard,

text: Phys.

text: Rep.

text: Quantum

text: particles

text: in

text: disordered

text: potential

text: are

text: expected

text: to

text: undergo

text: Anderson

text: localization,

text: provided

text: the

text: weak-localization

text: correction

text: is

text: ﬁnite

text: and

text: negative.

text: Depending

text: on

text: the

text: miscrocopic

text: details

text: of

text: the

text: disorder

text: potential,

text: this

text: may

text: or

text: may

text: not

text: be

text: the

text: case.

text: Quoting

text: from

text: Ref.

text: “Scaling

text: theory

text: does

text: not

text: pretend

text: to

text: be

text: an

text: exact

text: theory,

text: there

text: is

text: thus

text: real

text: interest

text: in

text: knowing

text: whether

text: there

text: is

text: localization

text: in

text: dimesions

text: for

text: speciﬁc

text: systems”.

text: C.

text: A.

text: Mu¨ller

text: and

text: D.

text: Delande,

text: “Disorder

text: and

text: interference:

text: localization

text: phenomena”,

text: Chapter

text: in

text: “Les

text: Houches

text: Session

text: XCI:

text: Ultracold

text: Gases

text: and

text: Quantum

text: Information”,

text: edited

text: by

text: C.

text: Miniatura

text: et

text: al.

text: (Oxford

text: University

text: Press,

text: L.

text: J.

text: Rooﬂ

text: and

text: J.

text: L.

text: Skinnerb,

text: J.

text: Chem.

text: Phys.


text: (3 cos2 θ (cid:48) − 1) nn t (cid:48) = α × (cid:48) . (4) n,n |n − n |3 with α = 52.12 Hz. The matrix elements (2) also lead to the delocalized character of the rotational excitations. In general, the |g(cid:105) → |e(cid:105) excitation generates the following many-body excited state: (cid:88) (cid:89) |ψ (cid:105) = C |e (cid:105) |g (cid:105). (5) exc n n i (cid:54) n i=n · √ In an ideal lattice, the coeﬃcients C = eiap n/ N and |ψ (cid:105) ⇒ |ψ (p)(cid:105) represents a n exc exc rotational Frenkel exciton with wave vector p [40] , i.e. a rotational excitation completely delocalized over the entire lattice. If the lattice is disordered, the excited state is a localized coherent superposition of the Frenkel excitons with diﬀerent p [28]. When the lattice is partially ﬁlled with molecules, the excitation energy of the molecular ensemble depends on the location of molecules in the excited state. The excited states of the many-body system of molecules in a disordered lattice are the eigenstates of the following Hamiltonian: (cid:88) (cid:88) (cid:88) † † Hˆ = w c c + t (cid:48)c c (cid:48), (6) n n n n,n n n (cid:48) n n n where the operator c removes an excitation from site n, w is the excitation energy of n n a molecule in site n, if the site is populated, and zero otherwise. The transition amplitudes (cid:48) t (cid:48) are given by Eq. (2) if both of the sites n and n are populated by molecules and zero n,n otherwise. It is this disorder in the transition amplitudes t (cid:48) that localizes the delocalized n,n excitations. In order to compute the time-evolution of the rotational excitations, we diagonalize the † matrix of the Hamiltonian (6) in the site-representation basis |n(cid:105) ≡ c |0(cid:105), where the vacuum n 7 ##LTLine## ##LTLine##

text: θ

text: nn

text: t

text: α

text: n,n

text: |n

text: n

text: with

text: α

text: Hz.

text: The

text: matrix

text: elements

text: also

text: lead

text: to

text: the

text: delocalized

text: character

text: of

text: the

text: rotational

text: excitations.

text: In

text: general,

text: the

text: excitation

text: generates

text: the

text: following

text: many-body

text: excited

text: state:

text: |ψ

text: C

text: |e

text: |g

text: exc

text: n

text: n

text: i

text: n

text: i=n

text: In

text: an

text: ideal

text: lattice,

text: the

text: coeﬃcients

text: C

text: eiap

text: n/

text: N

text: and

text: |ψ

text: |ψ

text: represents

text: a

text: n

text: exc

text: exc

text: rotational

text: Frenkel

text: exciton

text: with

text: wave

text: vector

text: p

text: i.e.

text: a

text: rotational

text: excitation

text: completely

text: delocalized

text: over

text: the

text: entire

text: lattice.

text: If

text: the

text: lattice

text: is

text: disordered,

text: the

text: excited

text: state

text: is

text: a

text: localized

text: coherent

text: superposition

text: of

text: the

text: Frenkel

text: excitons

text: with

text: diﬀerent

text: p

text: When

text: the

text: lattice

text: is

text: partially

text: ﬁlled

text: with

text: molecules,

text: the

text: excitation

text: energy

text: of

text: the

text: molecular

text: ensemble

text: depends

text: on

text: the

text: location

text: of

text: molecules

text: in

text: the

text: excited

text: state.

text: The

text: excited

text: states

text: of

text: the

text: many-body

text: system

text: of

text: molecules

text: in

text: a

text: disordered

text: lattice

text: are

text: the

text: eigenstates

text: of

text: the

text: following

text: Hamiltonian:

text: Hˆ

text: w

text: c

text: c

text: t

text: c

text: n

text: n

text: n

text: n,n

text: n

text: n

text: n

text: n

text: n

text: where

text: the

text: operator

text: c

text: removes

text: an

text: excitation

text: from

text: site

text: n,

text: w

text: is

text: the

text: excitation

text: energy

text: of

text: n

text: n

text: a

text: molecule

text: in

text: site

text: n,

text: if

text: the

text: site

text: is

text: populated,

text: and

text: zero

text: otherwise.

text: The

text: transition

text: amplitudes

text: t

text: are

text: given

text: by

text: Eq.

text: if

text: both

text: of

text: the

text: sites

text: n

text: and

text: n

text: are

text: populated

text: by

text: molecules

text: and

text: zero

text: n,n

text: otherwise.

text: It

text: is

text: this

text: disorder

text: in

text: the

text: transition

text: amplitudes

text: t

text: that

text: localizes

text: the

text: delocalized

text: n,n

text: excitations.

text: In

text: order

text: to

text: compute

text: the

text: time-evolution

text: of

text: the

text: rotational

text: excitations,

text: we

text: diagonalize

text: the

text: matrix

text: of

text: the

text: Hamiltonian

text: in

text: the

text: site-representation

text: basis

text: c

text: where

text: the

text: vacuum

text: n

text: ##LTLine##

text: ##LTLine##


text: width L is deﬁned as the length of the lattice (for a 1D lattice), the radius of a circle (for (cid:112) a 2D lattice) or the radius of a sphere (for a 3D lattice) containing 90 % of the excitation probability. The standard deviation is deﬁned as σ = (cid:104)r2(cid:105) − (cid:104)r(cid:105)2, where r2 = x2 for r a 1D lattice, r2 = x2 + y2 for a 2D lattice and r2 = x2 + y2 + z2 for a 3D lattice. The distribution width characterizes the physical spread of the rotational excitation over the molecular ensemble. However, this quantity may be signiﬁcantly aﬀected by ﬂuctuations due to speciﬁc disorder realizations. The standard deviation depends on the shape of the distribution but is less aﬀected by the ﬂuctuations. We will use the standard deviation in order to characterize non-ambiguously the dynamical properties such as the time dependence of the distributions. III. QUANTUM WALK OF ROTATIONAL EXCITATIONS As originally proposed by DeMille [29], ultracold molecules trapped in an optical lattice can be selectively excited by applying a gradient of a DC electric ﬁeld that separates the rotational energy levels of molecules in diﬀerent lattice sites by a diﬀerent magnitude. This method can, in principle, be used to excite rotationally a single molecule in a speciﬁc lattice site. If the gradient of the ﬁeld is removed, the rotational excitation thus generated may be transferred to molecules in other lattice sites by resonant energy transfer. The dynamics of this energy transfer can be probed by applying another gradient of the DC electric ﬁeld at time t and detecting either the excited molecules or molecules remaining in the ground state using resonantly-enhanced multiphoton ionization (REMPI). The REMPI detection is a destructive measurement. If the optical lattice is partially populated with randomly distributed molecules, repeating the experiment multiple times to determine the dependence of rotational energy transfer on t is equivalent to averaging over diﬀerent realizations of disorder (i.e. diﬀerent distributions of empty lattice sites). In order to simulate the outcome of such measurements, we consider an optical lattice partially populated with randomly distributed KRb molecules. We compute the time evolu- tion of a rotational excitation placed in a single lattice site at t = 0 and average the results at each t over multiple random distributions of empty lattice sites with a ﬁxed concentra- tion of vacancies. Figures 1, 2 and 3 illustrate the characteristic distributions of rotational excitations as a result of quantum walk of a single rotational excitation initially placed in the centre of a 1D, 2D (square) or 3D (cubic) lattice, respectively. In the following sec- 9 ##LTLine##

text: width

text: L

text: is

text: deﬁned

text: as

text: the

text: length

text: of

text: the

text: lattice

text: (for

text: a

text: lattice),

text: the

text: radius

text: of

text: a

text: circle

text: (for

text: a

text: lattice)

text: or

text: the

text: radius

text: of

text: a

text: sphere

text: (for

text: a

text: lattice)

text: containing

text: of

text: the

text: excitation

text: probability.

text: The

text: standard

text: deviation

text: is

text: deﬁned

text: as

text: σ

text: where

text: for

text: r

text: a

text: lattice,

text: for

text: a

text: lattice

text: and

text: for

text: a

text: lattice.

text: The

text: distribution

text: width

text: characterizes

text: the

text: physical

text: spread

text: of

text: the

text: rotational

text: excitation

text: over

text: the

text: molecular

text: ensemble.

text: However,

text: this

text: quantity

text: may

text: be

text: signiﬁcantly

text: aﬀected

text: by

text: ﬂuctuations

text: due

text: to

text: speciﬁc

text: disorder

text: realizations.

text: The

text: standard

text: deviation

text: depends

text: on

text: the

text: shape

text: of

text: the

text: distribution

text: but

text: is

text: less

text: aﬀected

text: by

text: the

text: ﬂuctuations.

text: We

text: will

text: use

text: the

text: standard

text: deviation

text: in

text: order

text: to

text: characterize

text: non-ambiguously

text: the

text: dynamical

text: properties

text: such

text: as

text: the

text: time

text: dependence

text: of

text: the

text: distributions.

text: III.

text: QUANTUM

text: WALK

text: OF

text: ROTATIONAL

text: EXCITATIONS

text: As

text: originally

text: proposed

text: by

text: DeMille

text: ultracold

text: molecules

text: trapped

text: in

text: an

text: optical

text: lattice

text: can

text: be

text: selectively

text: excited

text: by

text: applying

text: a

text: gradient

text: of

text: a

text: DC

text: electric

text: ﬁeld

text: that

text: separates

text: the

text: rotational

text: energy

text: levels

text: of

text: molecules

text: in

text: diﬀerent

text: lattice

text: sites

text: by

text: a

text: diﬀerent

text: magnitude.

text: This

text: method

text: can,

text: in

text: principle,

text: be

text: used

text: to

text: excite

text: rotationally

text: a

text: single

text: molecule

text: in

text: a

text: speciﬁc

text: lattice

text: site.

text: If

text: the

text: gradient

text: of

text: the

text: ﬁeld

text: is

text: removed,

text: the

text: rotational

text: excitation

text: thus

text: generated

text: may

text: be

text: transferred

text: to

text: molecules

text: in

text: other

text: lattice

text: sites

text: by

text: resonant

text: energy

text: transfer.

text: The

text: dynamics

text: of

text: this

text: energy

text: transfer

text: can

text: be

text: probed

text: by

text: applying

text: another

text: gradient

text: of

text: the

text: DC

text: electric

text: ﬁeld

text: at

text: time

text: t

text: and

text: detecting

text: either

text: the

text: excited

text: molecules

text: or

text: molecules

text: remaining

text: in

text: the

text: ground

text: state

text: using

text: resonantly-enhanced

text: multiphoton

text: ionization

text: (REMPI).

text: The

text: REMPI

text: detection

text: is

text: a

text: destructive

text: measurement.

text: If

text: the

text: optical

text: lattice

text: is

text: partially

text: populated

text: with

text: randomly

text: distributed

text: molecules,

text: repeating

text: the

text: experiment

text: multiple

text: times

text: to

text: determine

text: the

text: dependence

text: of

text: rotational

text: energy

text: transfer

text: on

text: t

text: is

text: equivalent

text: to

text: averaging

text: over

text: diﬀerent

text: realizations

text: of

text: disorder

text: (i.e.

text: diﬀerent

text: distributions

text: of

text: empty

text: lattice

text: sites).

text: In

text: order

text: to

text: simulate

text: the

text: outcome

text: of

text: such

text: measurements,

text: we

text: consider

text: an

text: optical

text: lattice

text: partially

text: populated

text: with

text: randomly

text: distributed

text: KRb

text: molecules.

text: We

text: compute

text: the

text: time

text: evolu-

text: tion

text: of

text: a

text: rotational

text: excitation

text: placed

text: in

text: a

text: single

text: lattice

text: site

text: at

text: t

text: and

text: average

text: the

text: results

text: at

text: each

text: t

text: over

text: multiple

text: random

text: distributions

text: of

text: empty

text: lattice

text: sites

text: with

text: a

text: ﬁxed

text: concentra-

text: tion

text: of

text: vacancies.

text: Figures

text: and

text: illustrate

text: the

text: characteristic

text: distributions

text: of

text: rotational

text: excitations

text: as

text: a

text: result

text: of

text: quantum

text: walk

text: of

text: a

text: single

text: rotational

text: excitation

text: initially

text: placed

text: in

text: the

text: centre

text: of

text: a

text: (square)

text: or

text: (cubic)

text: lattice,

text: respectively.

text: In

text: the

text: following

text: sec-

text: ##LTLine##


text: (cid:18) (cid:19) Rˆ∗ = ρˆ∗ S2∗ ∗ (cid:18)δˆ fo(cid:19)r δˆ > 0 M p + S 2 2 − − ∗ S∗ −δˆ = 1 (1 ρˆ ) 1 ∗ for δˆ < 0, p + S 1 1 ∗ ∗ − where ρˆ = p1p2+p∗1S2 ∗ and δˆ = X Y . p1p2+p1S2+p2S1 (1) (1) Deﬁne ≤ Z = 1 if X Y 1 1 = 0 otherwise. ≤ Then R = E(Z) = P (X Y ). Application of the Rao-Blackwell theorem gives an unbiased 1 1 estimator of R as (cid:88) (cid:88) (cid:88) 1 Y(1) min(W1,W2) W2 R˜ = + f(x/X , S ) + f(x/X , S )f(y/Y , S ) if X < Y U n (cid:88) (cid:88) (1) 1 (1) 1 (1) 2 (1) (1) 1 x=X +1 x=Y y=x (1) (1) 1 min(W1,W2) W2 = + f(x/X , S )f(y/Y , S ) if X = Y n (cid:88) (1)(cid:88)1 (cid:88) (1) 2 (1) (1) 1 x=X y=x (1) 1 W2 min(W1,W2) W2 = f(y/Y , S ) + f(x/X , S )f(y/Y , S ) if X > Y n (1) 2 (1) 1 (1) 2 (1) (1) 1 y=X x=X +1 y=x (1) (1) (cid:80) (cid:80) where, W = X +S , W = Y +S . The variance of this unbiased estimator will be smaller 1 (1) 1 2 (1) 2 than the unbiased estimator 1 n1 n2 I(X < Y ). n1n2 i=1 j=1 i j 4 Simulation Study 4.1 Discussion on simulation results relating to R(t) and R (t). s 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Rˆ∗

text: ρˆ∗

text: δˆ

text: M

text: p

text: S

text: S∗

text: −δˆ

text: ρˆ

text: for

text: δˆ

text: p

text: S

text: where

text: ρˆ

text: and

text: δˆ

text: X

text: Y

text: Deﬁne

text: Z

text: if

text: X

text: Y

text: otherwise.

text: Then

text: R

text: E(Z)

text: P

text: (X

text: Y

text: Application

text: of

text: the

text: Rao-Blackwell

text: theorem

text: gives

text: an

text: unbiased

text: estimator

text: of

text: R

text: as

text: R˜

text: f(x/X

text: S

text: f(x/X

text: S

text: )f(y/Y

text: S

text: if

text: X

text: Y

text: U

text: n

text: x=X

text: x=Y

text: y=x

text: f(x/X

text: S

text: )f(y/Y

text: S

text: if

text: X

text: Y

text: n

text: x=X

text: y=x

text: f(y/Y

text: S

text: f(x/X

text: S

text: )f(y/Y

text: S

text: if

text: X

text: Y

text: n

text: y=X

text: x=X

text: y=x

text: where,

text: W

text: X

text: +S

text: W

text: Y

text: +S

text: The

text: variance

text: of

text: this

text: unbiased

text: estimator

text: will

text: be

text: smaller

text: than

text: the

text: unbiased

text: estimator

text: I(X

text: Y

text: i

text: j

text: Simulation

text: Study

text: Discussion

text: on

text: simulation

text: results

text: relating

text: to

text: R(t)

text: and

text: R

text: (t).

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: true value of R, 2nd and 3rd rows indicate average of estimate of MLE and UE of R, and 4th and 5th rows indicate MSEs of MLE and UE of R respectively. We observe that in almost all cases, MLE is better that UE for R in mean square error sense. Therefore, as soon as we entered to unbiased class, we are losing some eﬃciency. It is to be noticed that MLE in this case is not an unbiased estimator. Moreover, MLE has a computational ease. 5 Concluding Remark This paper takes into account the inferential aspects of reliability with the two-parameter geometric lifetime. The continuous distributions are widely referenced probability laws used in reliability and life testing for continuous data. When the lives of some equipments and components are being measured by the number of completed cycles of operations or strokes, or in case of periodic monitoring of continuous data, the discrete distribution is a natural choice. At the same time, if a minimum warranty life cycle of the items are provided, the two-parameter geometric distribution is the simplest but an important choice. Under this set up estimators of reliability functions- under mission time as well as under stress-strength set up, have been viewed. It is interesting to note that, unlike the case of the two-parameter exponential, the estimators of the parameters of the two-parameter geometric distribution are not complete. As a result, we only get unbiased estimators of the reliability functions for the two-parameter geometric set up. In most of the situations, MLE gives better result than the UE in mean square sense. As soon as we entered to unbiased class, we are loosing some eﬃciency. If one is ready to sacriﬁce the unbiased criteria of the estimator, the MLE in this case is preferable. It is to be noticed here that MLE is not an unbiased estimator. Moreover, MLE has a computational ease. 17

text: true

text: value

text: of

text: R,

text: and

text: rows

text: indicate

text: average

text: of

text: estimate

text: of

text: MLE

text: and

text: UE

text: of

text: R,

text: and

text: and

text: rows

text: indicate

text: MSEs

text: of

text: MLE

text: and

text: UE

text: of

text: R

text: respectively.

text: We

text: observe

text: that

text: in

text: almost

text: all

text: cases,

text: MLE

text: is

text: better

text: that

text: UE

text: for

text: R

text: in

text: mean

text: square

text: error

text: sense.

text: Therefore,

text: as

text: soon

text: as

text: we

text: entered

text: to

text: unbiased

text: class,

text: we

text: are

text: losing

text: some

text: eﬃciency.

text: It

text: is

text: to

text: be

text: noticed

text: that

text: MLE

text: in

text: this

text: case

text: is

text: not

text: an

text: unbiased

text: estimator.

text: Moreover,

text: MLE

text: has

text: a

text: computational

text: ease.

text: Concluding

text: Remark

text: This

text: paper

text: takes

text: into

text: account

text: the

text: inferential

text: aspects

text: of

text: reliability

text: with

text: the

text: two-parameter

text: geometric

text: lifetime.

text: The

text: continuous

text: distributions

text: are

text: widely

text: referenced

text: probability

text: laws

text: used

text: in

text: reliability

text: and

text: life

text: testing

text: for

text: continuous

text: data.

text: When

text: the

text: lives

text: of

text: some

text: equipments

text: and

text: components

text: are

text: being

text: measured

text: by

text: the

text: number

text: of

text: completed

text: cycles

text: of

text: operations

text: or

text: strokes,

text: or

text: in

text: case

text: of

text: periodic

text: monitoring

text: of

text: continuous

text: data,

text: the

text: discrete

text: distribution

text: is

text: a

text: natural

text: choice.

text: At

text: the

text: same

text: time,

text: if

text: a

text: minimum

text: warranty

text: life

text: cycle

text: of

text: the

text: items

text: are

text: provided,

text: the

text: two-parameter

text: geometric

text: distribution

text: is

text: the

text: simplest

text: but

text: an

text: important

text: choice.

text: Under

text: this

text: set

text: up

text: estimators

text: of

text: reliability

text: functions-

text: under

text: mission

text: time

text: as

text: well

text: as

text: under

text: stress-strength

text: set

text: up,

text: have

text: been

text: viewed.

text: It

text: is

text: interesting

text: to

text: note

text: that,

text: unlike

text: the

text: case

text: of

text: the

text: two-parameter

text: exponential,

text: the

text: estimators

text: of

text: the

text: parameters

text: of

text: the

text: two-parameter

text: geometric

text: distribution

text: are

text: not

text: complete.

text: As

text: a

text: result,

text: we

text: only

text: get

text: unbiased

text: estimators

text: of

text: the

text: reliability

text: functions

text: for

text: the

text: two-parameter

text: geometric

text: set

text: up.

text: In

text: most

text: of

text: the

text: situations,

text: MLE

text: gives

text: better

text: result

text: than

text: the

text: UE

text: in

text: mean

text: square

text: sense.

text: As

text: soon

text: as

text: we

text: entered

text: to

text: unbiased

text: class,

text: we

text: are

text: loosing

text: some

text: eﬃciency.

text: If

text: one

text: is

text: ready

text: to

text: sacriﬁce

text: the

text: unbiased

text: criteria

text: of

text: the

text: estimator,

text: the

text: MLE

text: in

text: this

text: case

text: is

text: preferable.

text: It

text: is

text: to

text: be

text: noticed

text: here

text: that

text: MLE

text: is

text: not

text: an

text: unbiased

text: estimator.

text: Moreover,

text: MLE

text: has

text: a

text: computational

text: ease.


text: exponential distribution, [X], the integer part of X, has a two-parameter geometric distribution [see Kalbﬂeish and Prentice [5, Ch. 3]]. The reliability of a component when X follows two- parameter geometric distribution is given by − R(t) = θt r; t = r, r + 1, r + 2, .... (1.2) Laurent [6] and Tate [7] obtained the uniformly minimum variance unbiased estimator (UMVUE) of the reliability function for the two-parameter exponential model. Diﬀerent estimators of this reliability function have been discussed in Sinha [8]. If a system consists of m identical components each follows two-parameter geometric distribu- tion, then the reliability of k-out-of-m system is given by (cid:18) (cid:19) (cid:88) m ≥ m − − R (t) = P (X − t) = R(t)i[1 R(t)]m i; t = r, r + 1, r + 2, ... (1.3) s (m k+1) i i=k Special cases of R (t) give series system (for k = m) and parallel system (for k = 1). s ≤ In the stress-strength setup, R = P (X Y ) originated in the context of the reliability of a component of strength Y subjected to a stress X. The component fails if at any time the ≤ applied stress is greater than its strength and there is no failure when X Y . Thus R is a measure of the reliability of the component. Many authors considered the problem of estimation of R in continuous setup in the past. Particularly, for the two-parameter exponential set up, Beg [9] derived the MLE and the UMVUE of R. In the discrete setup, the reference list is very limited. Maiti [10] has considered stress (or demand) X and strength (or supply) Y as 3

text: exponential

text: distribution,

text: [X],

text: the

text: integer

text: part

text: of

text: X,

text: has

text: a

text: two-parameter

text: geometric

text: distribution

text: [see

text: Kalbﬂeish

text: and

text: Prentice

text: Ch.

text: The

text: reliability

text: of

text: a

text: component

text: when

text: X

text: follows

text: two-

text: parameter

text: geometric

text: distribution

text: is

text: given

text: by

text: R(t)

text: θt

text: r;

text: t

text: r,

text: r

text: r

text: Laurent

text: and

text: Tate

text: obtained

text: the

text: uniformly

text: minimum

text: variance

text: unbiased

text: estimator

text: (UMVUE)

text: of

text: the

text: reliability

text: function

text: for

text: the

text: two-parameter

text: exponential

text: model.

text: Diﬀerent

text: estimators

text: of

text: this

text: reliability

text: function

text: have

text: been

text: discussed

text: in

text: Sinha

text: If

text: a

text: system

text: consists

text: of

text: m

text: identical

text: components

text: each

text: follows

text: two-parameter

text: geometric

text: distribu-

text: tion,

text: then

text: the

text: reliability

text: of

text: k-out-of-m

text: system

text: is

text: given

text: by

text: m

text: m

text: R

text: (t)

text: P

text: (X

text: t)

text: R(t)]m

text: i;

text: t

text: r,

text: r

text: r

text: s

text: (m

text: i

text: i=k

text: Special

text: cases

text: of

text: R

text: (t)

text: give

text: series

text: system

text: (for

text: k

text: m)

text: and

text: parallel

text: system

text: (for

text: k

text: s

text: In

text: the

text: stress-strength

text: setup,

text: R

text: P

text: (X

text: Y

text: originated

text: in

text: the

text: context

text: of

text: the

text: reliability

text: of

text: a

text: component

text: of

text: strength

text: Y

text: subjected

text: to

text: a

text: stress

text: X.

text: The

text: component

text: fails

text: if

text: at

text: any

text: time

text: the

text: applied

text: stress

text: is

text: greater

text: than

text: its

text: strength

text: and

text: there

text: is

text: no

text: failure

text: when

text: X

text: Y

text: Thus

text: R

text: is

text: a

text: measure

text: of

text: the

text: reliability

text: of

text: the

text: component.

text: Many

text: authors

text: considered

text: the

text: problem

text: of

text: estimation

text: of

text: R

text: in

text: continuous

text: setup

text: in

text: the

text: past.

text: Particularly,

text: for

text: the

text: two-parameter

text: exponential

text: set

text: up,

text: Beg

text: derived

text: the

text: MLE

text: and

text: the

text: UMVUE

text: of

text: R.

text: In

text: the

text: discrete

text: setup,

text: the

text: reference

text: list

text: is

text: very

text: limited.

text: Maiti

text: has

text: considered

text: stress

text: (or

text: demand)

text: X

text: and

text: strength

text: (or

text: supply)

text: Y

text: as


text: )t( R dna )t(R ot gnitaler snoitaluclaC :4 elbaT s 8 = m ,2 = k ,52 = t ,8.0 = θ ,52 = c ,51 = r 21.loC 11.loC 01.loC 9.loC 8.loC 7.loC 6.loC 5.loC 4.loC 3.loC 2.loC 1.loC n 192.36 52672.0 37.221 48332.0 75912.0 90902.0 408.26 01621.0 70.721 06111.0 99501.0 73701.0 01 495.17 83952.0 18.811 05132.0 51122.0 90902.0 806.27 78021.0 81.021 04111.0 96701.0 73701.0 51 019.57 34452.0 93.811 81432.0 19422.0 90902.0 924.77 48911.0 07.811 89211.0 90901.0 73701.0 02 589.97 23242.0 70.811 57622.0 61912.0 90902.0 369.18 99511.0 46.711 57011.0 91801.0 73701.0 52 574.88 75722.0 38.411 22022.0 89512.0 90902.0 651.09 51211.0 90.411 56901.0 03801.0 73701.0 05 839.39 27712.0 29.111 30412.0 99112.0 90902.0 460.59 94901.0 63.111 32801.0 16701.0 73701.0 001 529.69 79212.0 67.011 42112.0 11012.0 90902.0 965.79 92801.0 64.011 96701.0 53701.0 73701.0 002 )t( R dna )t(R ot gnitaler snoitaluclaC :5 elbaT s 8 = m ,2 = k ,52 = t ,8.0 = θ ,52 = c ,02 = n 21.loC 11.loC 01.loC 9.loC 8.loC 7.loC 6.loC 5.loC 4.loC 3.loC 2.loC 1.loC r 526.88 26100.0 568.49 58100.0 28100.0 93000.0 18.101 17400.0 78.101 65500.0 25500.0 77300.0 0 649.97 40900.0 50.311 41900.0 58800.0 55300.0 929.68 66310.0 300.601 63410.0 02410.0 25110.0 5 434.27 41250.0 93.021 61940.0 57640.0 90030.0 783.87 37040.0 58.211 67930.0 58830.0 81530.0 01 956.67 51842.0 27.811 16722.0 73912.0 90902.0 993.87 65711.0 14.911 66011.0 97701.0 73701.0 51 179.121 17097.0 91.711 18457.0 68057.0 84597.0 691.98 11443.0 77.831 04723.0 51913.0 86723.0 02 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )t(

text: R

text: dna

text: )t(R

text: ot

text: gnitaler

text: snoitaluclaC

text: elbaT

text: s

text: m

text: k

text: t

text: θ

text: c

text: r

text: n

text: )t(

text: R

text: dna

text: )t(R

text: ot

text: gnitaler

text: snoitaluclaC

text: elbaT

text: s

text: m

text: k

text: t

text: θ

text: c

text: n

text: r

text: ##LTLine##

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text: Table 18: MSEs of Estimator R | r r 5 10 15 20 2 1 0.952381 0.321076 0.102261 0.033509 0.954791 0.300429 0.107198 0.038337 5 0.953994 0.313513 0.107466 0.033587 0.001314 0.012567 0.005071 0.001269 0.001318 0.013776 0.006272 0.001443 0.999985 0.952381 0.312076 0.102261 0.999870 0.952055 0.296181 0.103630 10 0.999989 0.951173 0.309150 0.103415 × − 1.49867 10 7 0.001352 0.012463 0.004663 × − 6.16543 10 9 0.001373 0.013512 0.005783 1 .999985 .952381 .312076 0.999998 0.999865 0.952267 0.302442 15 1 0.999983 0.951499 0.315536 × − × − 1.01544 10 10 2.91485 10 7 0.001478 0.012921 × − × − 2.37772 10 17 4.87421 10 8 0.001480 0.014082 1 1 0.999985 0.952381 1 .999998 0.999858 0.952217 20 1 1 0.999986 0.951531 × − × − × − 4.40539 10 13 1.37763 10 10 1.86115 10 7 0.001436 × − × − × − 2.43492 10 24 2.37772 10 17 8.33376 10 9 0.001436 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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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