text: 2 1000 %) 5.0 d) ( 4.8 a 47 pW F mr 100 1.1 nW or 4.6 (Δφ 3.6 nW fact 4.4 ation 10 mFano 44..02 devi 1 mu 3.8 n ni 3.6 Alla Mi 0 50 100 150 200 0.1 1 10 100 1000 Experimental realization Probe time τ (µs) Figure S2. Left: minimum Fano factor F obtained in 200 Figure S1. Log-log scale of the measured Allan deviation of successive experiments. R(cid:80)ight: histogram. the phase as a function of the probe time window τ for diﬀer- ent probe powers at the location of the lattice trap. (Points) ∞ | where α normalizes p(N ϕ ) = 1 and Experimental data (Solid lines) Expected theoretical quan- (cid:88) N=0 1...l tum noise for the the overall quantum eﬃciency of detection q = 0.40 and probe power. ∞ | | | p(N ϕ − ) = p(N N − )p(N − ϕ − ). l 1...l 1 l l 1 l 1 1...l 1 is modeled as a Markov process, so that each atom has a Nl−1=0 (S18) ﬁxed probability P to “disappear”, i.e. the probability to have N atoms in the trap in the i-th step given that the l atom number was Nl−1 in the previous step is governed For Fig. 3, we assume a uniform probability p(N0) for by a binomial distr(cid:18)ibution(cid:19): Ni < 4400 and calibrate P from ﬁtting eq. (S14) to the mean atom number decay recorded in 200 separate ex- periments. In Fig. S2 we present the statistics of the | N − − − p(Nl Nl−1) = Nl l1 (1 P)NlPNi−l Nl, (S13) aocfhtiheeveadtoFmannoumfabcetorrisn otbhtoasiene2d00beyxpBearyimeseiannts.estimation Atom number calibration method so that on average the atom number evolves as To accurately determine the number of trapped atoms | (cid:105) → | (cid:48)(cid:105) a single sideband resonant with the 4 4 transition (cid:104) (cid:105) − − Nl = N0e lP = N0e ∆ttP . (S14) is used. In brief, the idea behind the mea|su(cid:48)r(cid:105)ement is as follows. An excited atom in any of the 4 states can | (cid:105) | (cid:105) decay either to the 3 and 4 manifold. The branch- Also in each step, we obtain a new measurement of the ing ratio for this decay is independent of the Zeeman optical phase shift ϕ proportional to the current atom m (cid:48) sublevel. Because of the large ground state hyper- F | (cid:105) number N with an added random Gaussian shot noise l ﬁne splitting Cs atoms in the 3 state are far-detuned contribution ε with var ε = (δϕ)2: from the probe laser frequency and therefore do inter- act with the probe light only very weakly, i.e., atoms in this state do not absorb any photons from the probe ϕ = kN + ε, (S15) l l to an excellent approximation. The number of trapped | (cid:105) atoms in 4 will decrease with time during probing and so that ϕ is normally distributed: l the probe transmission will increase accordingly. To ﬁnd the number of atoms we need to determine the number of p(ϕl|Nl) = √ 1 e−12 (ϕl−vakrNεl)2 . (S16) scattering events it takes for the probe to reach full trans- 2π var ε mission. To model this, we consider a simple three level scheme and derive a diﬀerential equation for the optical Starting with an initial estimation of the atom prob- depth, d, as this will directly give the transmission using | (cid:105) ability p(N ), in each step we use the result of the new the Lambert-Beer law. The number of atoms in the 4 0 phase measurement ϕ to update our estimation: state N changes due to scattering events. We have with l 4 Φin(Φout) as the input (o(cid:16)utput) photon ﬂu(cid:17)x respectively, | | | p(N ϕ ) = α p(ϕ N ) p(N ϕ − ), (S17) i 1...l l l l 1...l 1 dN −1 − 4 = Φ (t) Φ (t) . 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text: 3 1.0 from which we ﬁnd for the sample transmission T(t) = − e d(t) the expression for the ﬁt model in Fig. S3. 0.8 n 4000 1 missio 0.6 3000 otons T(Nat, αat, t) = 1 + [exp (αatNat) − 1] exp (−αatΦint/k); ns ph (S24) Tra 0.4 2000 ed er 1000 t at allowing us to deduce both Nat and αat. c 0.2 0 S Since the method is used to calibrate the atom num- 0.0 0.1 0.2 0.3 0.4 0.5 ber, possible systematic eﬀects need to be addressed. Time (ms) 0.0 Probe laser detuning, polarization, population redistri- bution among Zeeman sub-levels and inhomogeneous 0.0 0.1 0.2 0.3 0.4 0.5 broadening of the probe transition by trap light do not Time (ms) inﬂuence the number of spontaneous emission cycles Figure S3. Calibration measurement of the absolute atom needed to bleach the sample and hence do not change number in the lattice trap. An atom number of N = the estimated atom number. The possible dark state in ± ± at | (cid:105) 1606 4stat 160sys atoms is determined through optical the Zeeman manifold of the 4 level, where population pumping into a dark state via the integration of the total num- could be trapped, is avoided by a suitably oriented mag- ber of probe photons required to bleach the ensemble. The | (cid:105) → | (cid:105) netic ﬁeld. Modiﬁcations to the excited state decay rate probe light is resonant with the atomic 4 4(cid:48) transition caused by the close proximity to the dielectric surface and the ﬁber-guided probe power at the atoms is 5.0 pW. The of the ﬁber are small (below the percent level) and not transmission is deduced from an heterodyne measurement of the two quadratures of the probe ﬁeld. Signals are recorded expected to change the branching ratio to ﬁrst order. 10 ms after the sub-Doppler cooling phase. The shown data Other systematic eﬀects like radiation trapping, which is an average over 178 runs. The model ﬁt estimates an asso- has compromised the accuracy of the method in previous ciated on-resonanc|e optical d(cid:105)e→pth| per atom (cid:105)of 1.64 % on the implementations with cold atoms, as well as collective closed transition ( 4, m = 4 5(cid:48), m = 5 ). F F back-scattering are largely suppressed by the extreme 1D geometry and the incommensurate lattice and probe wavelengths. In practice, the largest systematic error by far stems from the calibration uncertainty of the overall On average, it takes k = 2.4 scattering events to pump | (cid:105) | (cid:105) quantum eﬃciency q and hence the input photon ﬂux to an atom from 4 to 3 as can be easily derived from the | (cid:48)(cid:105) the atomic sample, which amounts to 10% in our exper- partial decay rates (7/12Γ, 5/12Γ) of the 4 states. The iment. We note that this calibration error is not subject output photon ﬂux follows Lambert-Beer’s law to ﬂuctuations between consecutive experimental runs. Several of the above mentioned parameters and eﬀects − Φ (t) = Φ (t) exp( d(t)), (S20) systematically lower the speed of the optical pumping out in transient and hence bias the value for the ﬁtted single where atom coupling strength αat. Similarly, atom number variations from run to run conspire to soften the sharp ≡ d(t) N (t)α (S21) transition from almost complete absorption to full trans- at at mission when raw signals are averaged over many realiza- tions. We observe a signiﬁcant diﬀerence between cou- is the optical depth, and α denotes the per-atom op- pling strength extracted from a ﬁt to an averaged trace, at tical depth on the used transition. Combining eqs. (S19) as shown in Fig. S3, and the average coupling strength to (S21), we obtain the desired diﬀerential equation for from ﬁts to individual traces. The atom number esti- d(t): mate, instead, is robust against interchange of ﬁtting and averaging operations. Comparing the average of inferred coupling strength values from individual ﬁts to the cou- d −α − − d(t) = at Φ (t) [1 exp( d(t))] , (S22) pling strength from dispersive phase shift measurements dt k in we ﬁnd a remaining relative deviation of 25%. Simpliﬁed model for atom number estimator Assuming(cid:104) a co(cid:16)nstant inpu(cid:17)t ﬂux, the solution(cid:105)is variance − − To arrive at the formula for the variance of the atom d(t) = ln 1 + ed(t=0) 1 exp ( α Φ t/k) , (S23) at in number estimator in eq. 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label: paragraph

font: default

text: from

text: which

text: we

text: ﬁnd

text: for

text: the

text: sample

text: transmission

text: T(t)

text: e

text: d(t)

text: the

text: expression

text: for

text: the

text: ﬁt

text: model

text: in

text: Fig.

text: n

label: equation

text: missio

text: otons

text: T(Nat,

text: αat,

text: t)

text: [exp

text: (αatNat)

text: exp

text: (−αatΦint/k);

text: ns

text: ph

text: Tra

text: ed

text: er

text: t

text: at

text: allowing

text: us

text: to

text: deduce

text: both

text: Nat

text: and

text: αat.

text: c

text: S

text: Since

text: the

text: method

text: is

text: used

text: to

text: calibrate

text: the

text: atom

text: num-

text: ber,

text: possible

text: systematic

text: eﬀects

text: need

text: to

text: be

text: addressed.

text: Time

text: (ms)

text: Probe

text: laser

text: detuning,

text: polarization,

text: population

text: redistri-

text: bution

text: among

text: Zeeman

text: sub-levels

text: and

text: inhomogeneous

text: broadening

text: of

text: the

text: probe

text: transition

text: by

text: trap

text: light

text: do

text: not

text: Time

text: (ms)

text: inﬂuence

text: the

text: number

text: of

text: spontaneous

text: emission

text: cycles

text: Figure

text: Calibration

text: measurement

text: of

text: the

text: absolute

text: atom

text: needed

text: to

text: bleach

text: the

text: sample

text: and

text: hence

text: do

text: not

text: change

text: number

text: in

text: the

text: lattice

text: trap.

text: An

text: atom

text: number

text: of

text: N

text: the

text: estimated

text: atom

text: number.

text: The

text: possible

text: dark

text: state

text: in

text: at

text: atoms

text: is

text: determined

text: through

text: optical

text: the

text: Zeeman

text: manifold

text: of

text: the

text: level,

text: where

text: population

text: pumping

text: into

text: a

text: dark

text: state

text: via

text: the

text: integration

text: of

text: the

text: total

text: num-

text: could

text: be

text: trapped,

text: is

text: avoided

text: by

text: a

text: suitably

text: oriented

text: mag-

text: ber

text: of

text: probe

text: photons

text: required

text: to

text: bleach

text: the

text: ensemble.

text: The

text: netic

text: ﬁeld.

text: Modiﬁcations

text: to

text: the

text: excited

text: state

text: decay

text: rate

text: probe

text: light

text: is

text: resonant

text: with

text: the

text: atomic

text: transition

text: caused

text: by

text: the

text: close

text: proximity

text: to

text: the

text: dielectric

text: surface

text: and

text: the

text: ﬁber-guided

text: probe

text: power

text: at

text: the

text: atoms

text: is

text: pW.

text: The

text: of

text: the

text: ﬁber

text: are

text: small

text: (below

text: the

text: percent

text: level)

text: and

text: not

text: transmission

text: is

text: deduced

text: from

text: an

text: heterodyne

text: measurement

text: of

text: the

text: two

text: quadratures

text: of

text: the

text: probe

text: ﬁeld.

text: Signals

text: are

text: recorded

text: expected

text: to

text: change

text: the

text: branching

text: ratio

text: to

text: ﬁrst

text: order.

text: ms

text: after

text: the

text: sub-Doppler

text: cooling

text: phase.

text: The

text: shown

text: data

text: Other

text: systematic

text: eﬀects

text: like

text: radiation

text: trapping,

text: which

text: is

text: an

text: average

text: over

text: runs.

text: The

text: model

text: ﬁt

text: estimates

text: an

text: asso-

text: has

text: compromised

text: the

text: accuracy

text: of

text: the

text: method

text: in

text: previous

text: ciated

text: on-resonanc|e

text: optical

text: per

text: atom

text: on

text: the

text: implementations

text: with

text: cold

text: atoms,

text: as

text: well

text: as

text: collective

text: closed

text: transition

text: m

text: m

text: F

text: F

text: back-scattering

text: are

text: largely

text: suppressed

text: by

text: the

text: extreme

text: geometry

text: and

text: the

text: incommensurate

text: lattice

text: and

text: probe

text: wavelengths.

text: In

text: practice,

text: the

text: largest

text: systematic

text: error

text: by

text: far

text: stems

text: from

text: the

text: calibration

text: uncertainty

text: of

text: the

text: overall

text: On

text: average,

text: it

text: takes

text: k

text: scattering

text: events

text: to

text: pump

text: quantum

text: eﬃciency

text: q

text: and

text: hence

text: the

text: input

text: photon

text: ﬂux

text: to

text: an

text: atom

text: from

text: to

text: as

text: can

text: be

text: easily

text: derived

text: from

text: the

text: the

text: atomic

text: sample,

text: which

text: amounts

text: to

text: in

text: our

text: exper-

text: partial

text: decay

text: rates

text: of

text: the

text: states.

text: The

text: iment.

text: We

text: note

text: that

text: this

text: calibration

text: error

text: is

text: not

text: subject

text: output

text: photon

text: ﬂux

text: follows

text: Lambert-Beer’s

text: law

text: to

text: ﬂuctuations

text: between

text: consecutive

text: experimental

text: runs.

text: Several

text: of

text: the

text: above

text: mentioned

text: parameters

text: and

text: eﬀects

text: Φ

text: (t)

text: Φ

text: (t)

text: exp(

text: d(t)),

text: systematically

text: lower

text: the

text: speed

text: of

text: the

text: optical

text: pumping

text: out

text: in

text: transient

text: and

text: hence

text: bias

text: the

text: value

text: for

text: the

text: ﬁtted

text: single

text: where

text: atom

text: coupling

text: strength

text: αat.

text: Similarly,

text: atom

text: number

text: variations

text: from

text: run

text: to

text: run

text: conspire

text: to

text: soften

text: the

text: sharp

text: d(t)

text: N

text: (t)α

text: transition

text: from

text: almost

text: complete

text: absorption

text: to

text: full

text: trans-

text: at

text: at

text: mission

text: when

text: raw

text: signals

text: are

text: averaged

text: over

text: many

text: realiza-

text: tions.

text: We

text: observe

text: a

text: signiﬁcant

text: diﬀerence

text: between

text: cou-

text: is

text: the

text: optical

text: depth,

text: and

text: α

text: denotes

text: the

text: per-atom

text: op-

text: pling

text: strength

text: extracted

text: from

text: a

text: ﬁt

text: to

text: an

text: averaged

text: trace,

text: at

text: tical

text: depth

text: on

text: the

text: used

text: transition.

text: Combining

text: eqs.

text: as

text: shown

text: in

text: Fig.

text: and

text: the

text: average

text: coupling

text: strength

text: to

text: we

text: obtain

text: the

text: desired

text: diﬀerential

text: equation

text: for

text: from

text: ﬁts

text: to

text: individual

text: traces.

text: The

text: atom

text: number

text: esti-

text: d(t):

text: mate,

text: instead,

text: is

text: robust

text: against

text: interchange

text: of

text: ﬁtting

text: and

text: averaging

text: operations.

text: Comparing

text: the

text: average

text: of

text: inferred

text: coupling

text: strength

text: values

text: from

text: individual

text: ﬁts

text: to

text: the

text: cou-

text: d

text: −α

text: d(t)

text: at

text: Φ

text: (t)

text: exp(

text: d(t))]

text: pling

text: strength

text: from

text: dispersive

text: phase

text: shift

text: measurements

text: dt

text: k

text: in

text: we

text: ﬁnd

text: a

text: remaining

text: relative

text: deviation

text: of

text: Simpliﬁed

text: model

text: for

text: atom

text: number

text: estimator

text: a

text: ﬂux,

text: the

text: variance

text: To

text: arrive

text: at

text: the

text: formula

text: for

text: the

text: variance

text: of

text: the

text: atom

text: d(t)

text: ln

text: exp

text: α

text: Φ

text: t/k)

text: at

text: in

text: number

text: estimator

text: in

text: eq.

text: we

text: start

text: by

text: relating

text: the

text: phase

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text: 2 of time according to an agreement and report signiﬁcant peak assumed to be connected to all the homes in the residential load reduction [18]. When the air conditioner is turned OFF, community. In an open setup, the grid controller is assumed the customers can experience unbounded variations away from to be communicating directly with the air conditioners. In this their desired thermostat set point. setup it is also assumed that the grid controller knows the In this paper, we consider power throttling type of control power consumption proﬁle as well as usage pattern of each for the air conditioners. Instead of just considering a basic 2- air conditioner in each home. In a more private setup, a home state model, we develop a generic K-state model as mandated controller can be installed in each home which can then act by [16]. For example, in a 3-state model, air conditioner can as an additional interface between the grid controller and the be turned OFF, ON or operated at 50% of the rated power. We air conditioner. In this setup, home controller communicates study the problem of peak load reduction by controlling the with the air conditioner according to the commands received air conditioners in a residential smart grid community, while from the grid controller. The power consumption proﬁle and considering DRM plans which are easily comprehensible. Our usage pattern of the air conditioner is known only to the DRM plans thus clearly specify the “maximum inconvenience home controller. It should be noted that the presence of a duration” (maximum time duration in which air conditioner is home controller in the system model only ensures privacy of denied demanded operation at full rated power in a particular individual household data. Moreover, since we are considering day) as well as “maximum inconvenience severity” (maximum only one ﬂexible load in this paper, therefore, the functionality temperature variation from the thermostat set point). We of a home controller and the physical interface on the air develop an algorithm that tells us the effectiveness of DRM conditioner might also be thought of as a single interface. plans and throttlable states in reducing the peak load on the B. Load Model grid. We simulate various DRM plans and the amount of peak reduction each plan can offer with different number of We consider the power consumption and usage patterns throttlable states. Simulation results show that with a basic 2- of a smart grid community over a period of 24 hours at a state ON/OFF model, peak load reduction of up to 16.5% can granularity of 5 minutes (hence yielding a total of T = 288 be achieved with a DRM plan with maximum inconvenience time slots in a day). For the purposes of this paper we consider duration of 1 hour and maximum inconvenience severity of 3 all non-air conditioning loads to be essential loads, which have oF from the thermostat set point of 65 oF. The amount of peak ﬁxed scheduling needs and ﬁxed power consumption. The grid load reduction increases to 21% when a third state capable of cannot control these loads in any possible way and is required operation at 50% rated power is added. However, increasing to provide them with the necessary power at the exact time of the number of states to 5 yields marginal returns over a 3-state their operation. Examples of such loads include refrigerators, model resulting in 22% peak load reduction. We also study the televisions, computers, lights etc. impact of maximum inconvenience duration and maximum C. Power Consumption Model of Air conditioner inconvenience severity parameters on peak load reduction. The results in this paper can provide useful guidelines for This model is required in order to compute the power developing appropriate DRM plans, incentives and ﬁnancial consumption of air conditioner. Let, PAC denote the rated j rewards for the smart grid users. power of air conditioner of user j. At any time instant t the air The rest of the paper is organized as follows. The system conditioner is assumed to be operating in one of the possible ∈ model, load model, power consumption and thermal model states k [1, K]. In subsequent discussion k = 1 will always of air conditioner and description of DRM plans is given in correspond to the state in which the air conditioner is OFF, Section II. The optimization problem and DRM algorithms whereas k = K will always correspond to the state in which are given in Section III. Simulation results are presented in the air conditioner is being operated at the rated power. The Section IV while the paper is concluded in Section V. power pAC(t) consumed by the air conditioner of user j in j,k II. SYSTEM MODEL, LOAD MODEL, POWER state k is then given by, CONSUMPTION AND THERMAL LOAD MODEL OF AIR − k 1 × CONDITIONER AND DRM PLANS pAC(t) = − PAC, k = 1, ..., K (1) j,k K 1 j A. System Model We consider a smart grid community comprising of J homes For example, an air conditioner that can operate in 5-states (also called users, consumers, customers) where each home is has the capability to throttle power at 0%, 25%, 50%, 75% equipped with an air conditioner. The air conditioner is as- and 100% of the rated power. sumed to be capable of power throttling and can operate in K D. Thermal Load Model of Air conditioner different states. We assume a uni-directional power ﬂow from the grid to the consumers. We assume a physical/electrical The room temperature variations resulting from throttling interface (also called demand response enabling device) [16] air conditioner power can be estimated using a thermal load connected to the air conditioner capable of bi-directional model of air conditioner. Let α and β denote respectively j j information exchange with the controller. This interface is also the start time and end time of the interval in which user j assumed to be capable of operating the air conditioner in the has demanded air conditioner. Let WAC(t) denote a binary j desired state according to the DRM signal received from the variable indicating the demand status of user j’s air conditioner grid. In our system model, there is a grid controller that is in time slot t, ##LTLine##

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text: ##LTLine##