a quantum network of clocks .pdf

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A quantum network of clocks

P. Komar,1,E. M. Kessler,1,2,M. Bishof,3 L. Jiang,4 A. S. Srensen,5 J. Ye,3 and M. D. Lukin11Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA

2ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA3JILA, NIST, Department of Physics, University of Colorado, Boulder, CO 80309, USA

4Department of Applied Physics, Yale University New Haven, CT 06520, USA5QUANTOP, Danish National Research Foundation Centre of Quantum Optics,

Niels Bohr Institute, DK-2100 Copenhagen, Denmark(Dated: October 24, 2013)

The development of precise atomic clocks hasled to many scientific and technological advancesthat play an increasingly important role in mod-ern society. Shared timing information consti-tutes a key resource for positioning and naviga-tion with a direct correspondence between timingaccuracy and precision in applications such as theGlobal Positioning System (GPS). By combiningprecision metrology and quantum networks, wepropose here a quantum, cooperative protocol forthe operation of a network consisting of geograph-ically remote optical atomic clocks. Using non-local entangled states, we demonstrate an opti-mal utilization of the global network resources,and show that such a network can be operatednear the fundamental limit set by quantum the-ory yielding an ultra-precise clock signal. Fur-thermore, the internal structure of the network,combined with basic techniques from quantumcommunication, guarantees security both from in-ternal and external threats. Realization of such aglobal quantum network of clocks may allow con-struction of a real-time single international time

scale (world clock) with unprecedented stabilityand accuracy.

With the advances of highly phase coherent lasers,optical atomic clocks containing multiple atoms havedemonstrated stability that reaches the standard quan-tum limit (SQL) set by the available atom number withina clock [1, 2]. Reaching beyond SQL, we stand togain a significant improvement of clock performance bypreparing atoms in quantum correlated states (e.g., spinsqueezed states[3]). Here we describe a new approach tomaximize the performance of a network composed of mul-tiple clocks allowing to gain advantage of all rescourcesavailable at each node. Several recent advances in preci-

sion metrology and quantum science make this approachrealistic. On the one hand, capabilities to maintain phasecoherent optical links spanning the entire visible spec-trum and over macroscopic distances have been demon-strated, with the capability of delivering the most stableoptical oscillator from one color or location to another[4,5]. On the other hand, quantum communications and

These authors contributed equally to this work

FIG. 1. The concept of world-wide quantum clock net-work. a) Illustration of a cooperative clock operation pro-tocol in which individual parties (e.g., satellite based atomicclocks from different countries) jointly allocate their respec-tive resources in a global network involving entangled quan-tum states. This guarantees an optimal use of the globalresources, achieving an ultra-precise clock signal limited onlyby the fundamental bounds of quantum metrology and, in ad-dition, guaranteeing secure distribution of the clock signal. b)In addition to locally operating the individual clocks, the dif-ferent nodes (i.e., satellites) employ network-wide entangledstates to interrogate their respective local oscillators (LOs).The acquired information is sent to a particular node servingas a center where it is used to stabilize a center of mass modeof the different LOs. This yields an ultra-precise clock signalaccessible to all network members.

entanglement techniques are enabling distant quantumobjects to be connected in a quantum network [68], thatcan enable novel, extraordinary capabilities. Combiningthese two technological frontiers, we show here that adistributed network composed of quantum-limited clocks

arXiv:1310.60

45v1

[quant-ph]2

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separated by large distances as appropriate, e.g., forthe satellite-based clocks possibly operated by differentnations can be operated as an ultimate world clock,where all members combine their individual resources ina quantum coherent way to achieve greater clock stabilityand distribute this international time scale in real timefor all.

The distributed achitecture allows each participant of

the network to profit from a stability of the local clocksignal that is enhanced by a factor proportional to thetotal number of parties (as compared to an indepen-dent operation of the individual clocks) without losingsovereignty or compromising security. This cooperativegain strongly incentivizes joining the collaborative net-work while retaining robustness against disruptions ofcommunication channels by allowing the parties to fallback to individual clock operation. Our scheme is supe-rior to an alternative approach of disseminating the timesignal from a single location containing all qubits, sinceerrors arising from imperfect phase links can be largelyreduced by relying on the stabilized and locally avail-

able local oscillators. We demonstrate that by preparingquantum-correlated states of remote clocks, the networkcan yield the best possible clock signal allowed by quan-tum theory for the combined resources. Furthermore, en-abled through the use of quantum communication tech-niques, such a network can be made secure, such thatonly parties contributing to its operation may enjoy thebenefit of an ultra-precise clock signal. Besides servingas a real-time clock for the international time scale, theproposed quantum network also represents a large-scalequantum sensor that can be used to probe the fundamen-tal laws of physics, including relativity and connectionsbetween space-time and quantum physics.

THE CONCEPT OF QUANTUM CLOCK

NETWORK

Fig. 1 illustrates the basic concept for the proposedquantum clock network. We consider a set ofKatomicclocks (constituting the nodes of the network), each basedon a large number of atoms (clock qubits) serving asthe frequency reference 0 at different geographical lo-cations. In our approach, each clock has its own inde-pendently operated local oscillator (LO),Ej(t) eijt,with detuningj =j 0, (j = 1, 2 . . . K ). It keeps thetime by interrogating its qubits periodically, and uses themeasurement data to stabilize the LO frequency at thereference frequency of the atomic transition. However,as opposed to the conventional approach, in which eachLO interrogates its own independent qubits, we considerthe situation in which each network node allocates someof its qubits to form entangled states stretching acrossall nodes. When interrogated within a properly designedmeasurement scheme, such entangled network states pro-vide ultra-precise information about the deviation of thecenter-of-mass (COM) frequencyCOM=

jj/Kof all

local oscillators from the atomic resonance.Each clock cycle consists from three stages: prepara-

tion of the clock atom state (initialization), interrogationby the LOs (measurement) and correction of the laserfrequency according to the measurement outcome (feed-back). In the further analysis, we assume, for conve-nience, that in each interrogation cycle one of the nodesplays the role of an alternating center, which initiates

each Ramsey cycle and collects the measurement datafrom the other nodes via classical channels [Fig. 1 b)],as well as LO signals via optical links, to feedback theCOM signal. (In practice, it is straightforward to devisea similar network with same functionality and a flat hier-archical structure where no center is needed, see Supple-mentary Information). This information, in turn, can beutilized in a feedback cycle to yield a Heisenberg-limitedstability of the COM clock signal generated by the net-work, which is subsequently distributed to the individualnodes in a secure fashion. As a result, after a few cycles,the LOs corresponding to each individual node achievean accuracy and stability effectively resulting from inter-

rogating atoms in the entire network.

PREPARATION OF NETWORK-WIDE

ENTANGLED STATES

In the initialization stage of each clock cycle, entangledstates spanning across the nodes at different geographicalpositions of the network are prepared. In the following,we describe exemplarily how a single network-wide GHZstate can be prepared. The entangled states employedin the proposed quantum network protocol which aredescribed in the following section consist of products

of GHZ states of different size. They can be prepared byrepetition of the protocol that we now describe.For simplicity, we assume that each node j (j =

1, . . . K ) contains an identical number n of clock qubitswhich we label as 1j , 2j , . . . nj (in the SupplementaryInformation we discuss the case where the nodes con-tain different amounts of clock qubits). Further, weassume, for convenience, that the center node (j =1) has access to additional 2(K 1) ancilla qubitsa2, . . . , aK, b2, . . . , bK besides the n clock atoms (aslightly more complicated procedure allows to refrainfrom the use of ancilla qubits, see Supplementary Infor-mation). The entangling procedure starts at the centerwith the creation of a fully entangled state of one half of

the ancilla qubits{bj}, and its first clock qubit 11. Thiscan be realized, e.g. with a single qubit /2-rotation (onqubit 11) and a series of controlled not (CNOT) gates[9] (between 11 and eachbj). The result is a GHZ state,

[|00 . . . 011,b2,b3,...bK +i|11 . . . 111,b2,b3,...bK ]/

2. In par-allel, the center uses the other half of the ancillas {aj}to create single EPR pairs with each node j= 1, eitherby directly sending flying qubits and converting them tostationary qubits, or by using quantum repeater tech-niques to prepare high-fidelity entanglement [10]. As a


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FIG. 2. Entangled state preparation between distant nodes.a) The center node (j = 1) initiates the initialization se-quence by preparing a local GHZ state accross the qubits

{bj}K

j=2 and 11

, as well as (K 1) EPR pairs on the qubitpairs{(aj , 1j)}Kj=2. Quantum teleportation expands this GHZstate to the first qubit within each of the individual nodes.b) Originating from the teleported qubits, the nodes grow theGHZ state to involve all the desired local qubits by employ-ing local entangling operations. The procedure results in acommon GHZ states over all atoms of the nodes.

result of this procedure, one part of the pair is storedat the center node (qubit aj), while the other one isstored at the jth node (qubit 1j), forming the states

[|00aj,1j + |11aj ,1j ]/

2 for every j (see Fig.2).

Next, the center performsK

1 separate Bell measure-

ments on its ancilla qubit pairs {(bj , aj)}. This teleportsthe state of qubit bj to qubit 1j (j = 2, . . . K ), up to alocal single-qubit rotation, which is performed after themeasurement outcomes are sent to the node via classi-cal channels. The result of the teleportations is a collec-tive GHZ state 1

2|00 . . . 011,12,...1K+i|11 . . . 111,12,...1K ,

stretching across the first qubits of all K nodes.

In the final step of entangling, all nodes (includingthe center) extend the entanglement to all of their re-maining clock qubits. To do this, each nodej performsa series of CNOT gates controlled on 1j and targetingqubits 2j , 3j , . . . nj . At the end of the protocol the dif-

ferent nodes share a common GHZ state [|0

+ i|1]/

2,

where|0 and|1 are product states of all qubits{ij :i = 1, 2, . . . n, j = 1, 2, . . . K } being in|0 or|1, re-spectively. As discussed below, in practice the entangle-ment distribution can be done either via polarization- orfrequency-entangled photons with frequency difference inthe microwave domain, in which case the ancillary qubitsinvolved in the entanglement distribution will be differentfrom the clock qubits. Typically, as part of the prepara-tion process, time delays arise between the initializationof different clock qubits. Its detrimental effects can be

entirely avoided by proper local timing or prior prepara-tion of entanglement, as discussed in the SupplementaryInformation.

INTERROGATION

The use of entangled resources (in form of network-

wide GHZ-like states) during the interrogation phase en-ables an optimal use of the available resources via thefollowing procedure. Assume we have a total ofNqubitsat our disposal which are equally distributed between theKnodes (indexed j = 1, . . . K ) and prepared in a non-

local GHZ state [|0+i|1]/2, where |0(1) |0(1)N.During the interrogation time T, a clock qubit at node

j picks up a relative phase j = jT. Due to the non-local character of the state, these phases accumulate inthe total state of the atoms [|0 + iei|1]/2, where thecollective phase after the interrogation time Tis given as

=K

j=1

N

Kj = N COMT, (1)

where COM = COM 0. To extract the phase infor-mation picked up by the different GHZ states, after eachinterrogation phase, the individual nodes j measure theirrespective qubits in the x-basis, and evaluate the parityof all measurement outcomespj . Subsequently, the nodessend this information to the center node via a classicalchannel, where the total parity p =

jpj is evaluated,

and the phase information is extracted [14, 15]. Note,that only the full set{pj |j = 1 . . . K } contains informa-tion. This can be interpreted as only the center nodeholding the key, namely its own measurement outcome

p1, to decode the phase information sent from the nodes.The proportionality with N in Eq. (1) represents thequantum enhancement in the estimation ofCOM. How-ever, for realistic laser noise spectra, this suggested en-hancement is corrupted by the increase of uncontrolledphase slips for a single GHZ state[11]: Whenever afterthe Ramsey time the phase which due to the laserfrequency fluctuations constitutes a random variable it-self falls out of the interval [, ] the estimation fails.This limitation restricts the maximal Ramsey time tovalues T < ( N LO)1, preventing any quantum gain inthe estimation.

To circumvent this problem, we use entangled statesconsisting of products of successively larger GHZ ensem-bles, see SI and[12]. In this approach, interrogated net-work atoms are split into several independent, sharedgroups. We write the number of the first group ofatoms as N = 2M1K, for some natural number M.Furthermore, the network shares additional groups ofatoms, each containing 2iK (i = 0, . . . M 2) equallydistributed between the nodes and prepared in GHZstates. Finally, each node has a small number of un-correlated atoms interrogated by LOs. Using a protocolreminiscent of the phase estimation algorithm[9, 12, 13]


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these states allow to directly assess the bits Zi {0, 1}of the binary fraction representation of the laser phaseLO = COMT = 2[(Z1 1)21 +Z222 +Z323 . . .].This yields an estimate of LO with Heisenberg-limitedaccuracy, up to a logarithmic correction, see SI:

LO= 8

log(N)/N, (2)

even for Ramsey times beyond the limits of the laserfrequency fluctuations [T >( N 1LO)], whereNrepresentthe total number of clock atoms employed in the scheme.The logarithmic correction arises due to the number ofparticles required to realize this (incoherent) version ofthe phase estimation algorithm.

FEEDBACK

The measured value of the phase LO, gives an es-timate on the COM detuning COM after each Ramsey

cycle, which is subsequently used by the center node tostabilize the COM laser signal. To this end the centergenerates the COM of the frequencies. Every node sendsits local oscillator fieldEi to the center via phase-stableoptical links, and the center synthesizes the COM fre-quencyCOM by averaging the j frequencies with equalweights [16]. This can be implemented via heterodynebeat of the local oscillator in the center against each in-coming laser signal, resulting inKbeat frequencies. Syn-thesizing these beat frequencies allows the local oscilla-tor of the central node to phase track COM. The centerdistributes the stabilized clock signal to different mem-bers of the network by sending individual error signalsj = COM + (j

COM) to all nodesj , respectively, and

corrects its own LO as well, accordingly. Alternatively,the center can be operated to provide restricted feedbackinformation to the nodes, see SI.

STABILITY ANALYSIS

In this section, we demonstrate that the proposedquantum clock network achieves the best clock signal, al-lowed by quantum theory for the available resources, i.e.the total atom number. Rather than individually oper-ating their respective LOs the joint use of resources al-lows the network to directly interrogate and stabilize theCOM mode of the lasers. To quantify this cooperativegain, we compare networks of different types (classical orquantum mechanical interrogation of the respective LOs)and degrees of cooperation (no cooperation, classical, orquantum cooperation).

First, we analyze the stability of the proposed quan-tum clock network, corresponding to the case of quantuminterrogation and cooperation (curve a in Fig.3). In thiscase, the analysis resulting in Eq. (2) suggests that nearHeisenberg-limited scaling with a total atom number can

0.01 1 100 104 106 1081 10-4

5 10-4

0.001

0.005

0.010

0.050

0.100

Averging time LO

Allande

viation

LO

0y

(a)

(b)

(c)

(d) (e)

(a) quantum/quantum

(b) quantum/classical

(c) quantum/non-coop

(d) classical/classical

(e) classical/non-coop

LO

(Ni)

LO

(Ni)

LO

(i)

K

K

N

Averaging time

interrogation / cooperation

(a) quantum / quantum

(b) quantum / classical

(c) quantum / non

(d) classical / classical

(e) classical / non

|| ||r

| , , .. .| , , .. .

| , | ,

| | | |

| |

|| |

| | |

FIG. 3. Performance of different operation schemes.Comparison of the achievable (rescaled) Allan deviationLO0y using clock networks of different types and de-

grees of cooperation. (a) the proposed protocol realizingquantum interrogation and cooperation, (b) quantum inter-rogation and classical cooperation, (c) quantum interrogationand no cooperation, (d) classical interrogation and classical

cooperation, (e) classical interrogation and no cooperation (cf.text). The dotted base line represents the fundamental boundarising from the finite width of the clock atoms transition[compare Eq. (4)]. This optimal stability can be attainedonly via cooperation between the nodes. The quantum clocknetwork (a) represents the optimal form of cooperation, andreaches this boundary faster than any other operational mode.Parameters are N= 1000, K= 10, i = 10

4LO.

be achieved for the entangled clock network. In particu-lar, for a given total particle numberNand for averagingtimes shorter than the timescale set by individual qubitnoise < 1/(iN) (where i is the atomic linewidth),the network operation achieves a Heisenberg-limited Al-lan deviation (ADEV) of the COM laser mode

y() = 1

0

n02MK

1

log(N)

0N

1

, (3)

up to small numerical corrections. Here, the number ofGHZ copies per group n0 log(N) (N n02M+1K) isfound after optimization [cf. SI], and gives rise to a loga-rithmic correction in the total particle number. The 1/scaling results from the effective cancellation of the lowfrequency part of the laser noise spectrum, achieved bythe cascaded protocol described above, possibly in com-

bination with additional stages of uncorrelated interroga-tions using varying Ramsey times [17, 18], see[12]. Thisallows the cycle time T (which is assumed to be equalto the interrogation time) to be extended to the totalavailable measurement time .

Eventually, for large averaging times >1/(iN) theRamsey time becomes fundamentally limited by individ-ual noise processes (T 1/(iN)). As a result, the 1/Nscaling breaks down, and the ADEV returns to the squareroot scaling with both the employed particle number and


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averaging time,

y() 10

N

i

, (4)

up to constant numerical factors. Eq. (4) results fromfundamental quantum metrological bounds[19], and rep-resents the best conceivable clock stability in the pres-ence of individual particle decoherence which, in a net-

work, can only be achieved via cooperation. Indepen-dently operating a clock, in contrast, can only achievea stability scaling with the local number of atoms, i.e.y()

K/N.

Fig.3illustrates the comparison of entangled clock net-work with other approaches. A network in which the Knodes cooperate classically (curve b in Fig.3), by locallymeasuring the individual phase deviation j , and com-bining the outcomes via classical channels, outperformsindividually operated clocks (curve c) by a factor of

K

(for both cases, assuming optimal quantum interrogationfor individual nodes[12, 20]). The quantum network pro-tocol (curve a) increases this cooperative advantage by

an additional factor of K for short averaging times,reaching Heisenberg-limit. The ADEV converges to thefundamental bound [Eq. (4)]Ktimes faster compared tothe case of classical cooperation (curve b). Although anoptimal, classical, local protocol (e.g. [17, 18]), combinedwith classical cooperation (curve d), eventually reachesthe same bound [Eq. (4)], this approach is atom-shotnoise limited, and hence its stability is reduced by a fac-tor of

N for short averaging times [compare Eq. (3)]

compared to the quantum network protocol. Hence, theoptimal stability [Eq. (4)] is reached at averaging timesthat are Ntimes longer than for the proposed quantumnetwork. Naturally, all of the above approaches are su-

perior to a classical scheme without cooperation (curvee).As a specific example, we first consider ion clocks that

can currently achieve a stability of 2.8 1015 after 1 sof averaging time [22]. The entangled states of up to14 ions has already been demonstrated [24] as was theentanglement of remote ions [40]. We consider a net-work of ten clocks, each containing ten ions. Using Al+

(0 = 2 1121 THz, i = 2 8 mHz), we find thatthe quantum cooperative protocol can reach 4 1017fractional frequency uncertainty after 1 s. Even morepronounced improvement could potentially be achievedusing e.g. Yb+ ions, due to the long coherence time(2.2

104 s) of its octupole clock transition.

The quantum gain could be even more pronouncedfor neutral atomic clocks. For a network consisting often clocks similar to the one operated in JILA [1], eachcontaining 1000 neutral atoms with central frequency0 = 2 429 THz and linewidth i = 2 1 mHz,the quantum cooperative scheme can achieve a stabil-ity of 2 1018 after 1s averaging, and is an orderof magnitude better than the best classical cooperativescheme. Future advances, allowing to employ clock tran-sitions with linewidths of a few tens of Hz (such as

erbium), could possibly allow for further improvement,achieving fractional frequency uncertainty beyond 1020after 100 s. This level of stability is in the same orderof magnitude then the required sensitivity to successfullyuse the network as a gravitational interferometer [44].

SECURITY

A network with such precise time-keeping capabilitiescan be subject to both internal and external attacks. Ef-fectively countering them is crucial to establish a reliableground for cooperation. We consider the network secureif the implemented countermeasures can prevent exter-nal parties from benefiting from the network (eavesdrop-ping), as well as effectively detect any malicious activitiesof any of the members (sabotage).

Sabotage describes the situation where one of thenodes intended or unintended operates in a dam-aging manner. For example, one node could try sendingfalse LO frequencies or wrong measurement bits in thehope of corrupting the collective measurement outcomes.In order to detect such malicious participants, the cen-tral node can occasionally perform assessment tests ofthe different nodes by teleporting an uncorrelated qubitstate [|0 + ei|1]/2, where is a randomly chosenphase known only to the center. By checking for statis-tical discrepancies between the measurement results andthe detuning of the LO signal sent by the node underscrutiny, the center can rapidly and reliably determinewhether the particular node is operating properly (SeeFig.4a and Supplementary Information).

Eavesdropping, i.e., the unauthorized attempt to ac-cess the stabilizedCOM frequency, can be prevented byencoding the classical channels, over which the centerand the nodes exchange feedback signals, using quantumkey distribution protocols [21]. Our protocol can keepthe stabilized signal hidden from outsiders by mixing thefeedback signal with the LO signal at each node onlyafter the non-stabilized LO has been sent to the center(see Fig.4b and SI). As a result, even if all LO signals areintercepted, the eavesdropper is able to access only thenon-stabilized COM signal. Furthermore, the center ex-clusively can decode the measurement results sent by theindividual nodes using its own measurement outcomes as

mentioned above. As a result, the stabilized COM signalremains accessible exclusively to parties involved in thecollaboration.

Finally, we note that a distributed operation offers sig-nificant security advantages over an alternative approachof having all resources combined in one place from wherethe signal is distributed. In case of a physical attack ofthe network, disabling the center or the communicationlinks, the nodes can fall back to an independent clockoperation using their local resources.


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FIG. 4. Schematics of security countermeasures. a) Thecenter node can choose to test any node j by teleporting adisentangled qubit with a certain phase rotation. A properlyoperating node creates a local GHZ state [|0+ ei|1]/2from the sent qubit, measures the parity of the GHZ state,and sends it to the center. The measured parity holds infor-mation on the phase = +, where is the accumulatedphase of the LO at the node. The center verifies by com-paring it with the classically determined phase of the sent LOsignal with respect to the COM signal.b) Eavesdropping can be prevented by prescribing that onlythe non-stabilized LO signals are sent through classical chan-

nels and encoding the radio frequency feedback signal withphase modulation according to a shared secret key.

OUTLOOK

One of the advantages of the proposed quantum clocknetwork involves its ability to maintain and synchronizethe time standards across multiple parties in the real-time. Unlike the current world time standard, where theindividual signals from different clocks are averaged andcommunicated with a time delay (a so called paper clock),in our quantum clock network all participants have accessto the ultra-stable signal at any time. Furthermore, byhaving full access to their local clocks the different partieskeep their full sovereignty and ensure security, as opposedto a joint operation of a single clock.

Realization of the full-scale network of the type de-scribed here will require a number of technological ad-vances in both metrology and experimental quantum in-formation science. The remote entanglement can be im-plemented by using recently demonstrated techniques forindividual atom-photon entanglement [2529]. Since theteleportation protocol requires quantum links capable of

sharing EPR pairs with sufficiently high repetition rateand fidelity, entanglement purification [30]and quantumrepeater techniques [10]will likely be required. In prac-tice, qubits used for entanglement distribution may notbe ideal for clocks. However, as noted previously remoteentanglement does not need to involve coherent qubits atoptical frequencies (e.g., polarization entanglement canbe used). In such a case, the use of hybrid approaches,

combining different systems for entanglement and localclock operations, may be warranted. It might also be in-teresting to explore if high-fidelity entangled EPR pairscan be used to create remote entangled states of spin-squeezed type [3, 31, 32], possibly by following the ap-proach for cat state preparation in remote optical cavi-ties[33], or using local, collective interactions and repeti-tive teleportation[3438]. In addition, while space-basedcommunication networks will be capable of maintainingoptical phase coherence for the links between clocks, wenote that establishing ground-space coherent optical linksremains a technical challenge and requires an intense re-search effort which has recently started [39]. Finally, ifthe entire network is spanned by satellites in space, theon-board local oscillators can further benefit from themuch lower noise level compared to ground-based clocks.

If realized, such a quantum network of clocks canhave important scientific, technological, and social con-sequences. Besides creating a world platform for timeand frequency metrology, such a network may find im-portant applications to a range of technological advancesfor earth science[41] and to the test and search for thefundamental laws of nature, including relativity and theconnection between quantum and gravitational physics[4245].

SUPPLEMENTARY INFORMATION

Appendix A: GHZ cascade in a network ofK clocks

Here, we discuss the details of using quantum corre-lated states constructed out ofN = K n qubits, equallydistributed among K clocks, namely the GHZ state ofthe form

[|00 . . . 0 + ei|11 . . . 1]/

2, (A1)

where|q q . . . q =|qN , q {0, 1}. Entanglement hastwo effects here: First, it makes the phase of such a GHZstate,, sensitive to the accumulated phase of the center-of-massof all the K independent local oscillators, (each

located at one of the clocks) LO=K

j=1(j)/K, where

(j) =T0

dt ((j)(t) 0) is the accumulated phase ofthe LO at clockj , during the interrogation time T, here(j)(t) is the instantaneous frequency of the LO, while 0is the transition frequency of the clock qubit. Second, it


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increases the sensitivity, due to quantum enhancement: K

j

N/Ki

Ui,j

|0 + ei|1 /2 =

= [|0 + ei(+NCOM)|1]/

2, (A2)

where Ui,j =

|0

0

|+ei

(j)

|1

1

| is the time evolution

operator during the interrogation time, acting on the ithqubit at clockj , and |0 and |1 are product states of allqubits being in|0 or|1, respectively.

1. Parity measurement

By setting the initial phase of the GHZ state, , to0 and /2 in two parallel instances, we effectively mea-

sure the real and imaginary part ofeiNCOM , and thus

get an estimate on the value ofNCOM up to 2 phaseshifts. The most cost-effective way to do this is to mea-sure all qubits in the localx-basis. In this basis, the state

Eq. (A2) is written as

12

|+|2

N+ ei

|+ + |2

N, (A3)

where = +NCOM, and| = |0|12 . The abovestate can be expanded in a sum:

1

2(N+1)/2

q{+,}N

Nj=1

qj

+ ei

|q1, q2, . . . q N,

(A4)where we labeled all qubits with k

{1, 2, . . . N

}, irre-

spective of which clock they belong to. The probabilityof a certain outcome q= (q1, q2, . . . q N), (qj {+, }),is

P(q) = 12N+1

|1 +p(q)ei|2, (A5)

wherep(q) =N

j=1 qj is the parity of the sum of all mea-surement bits. Now, the clocks send their measurementbits to the center node, which evaluates p. This parity isthe global observable that is sensitive to the accumulatedphase, since its distribution is

P(p= ) =1

cos()

2 . (A6)

The above procedure is identical to the parity measure-ment scheme described in[14].

2. Cascaded GHZ scheme

Provided with N qubits distributed equally among Kclocks, we imagine that each clock separates its qubits

FIG. 5. GHZ cascade protocol for K clocks. Each allo-cates qubits for different levels of the protocol: In level 0,n1/Kqubits are put into an uncorrelated ensemble. In leveli, (i= 1, 2 . . .M ), each clock allocatesn02

i1 qubits for creat-ingn0 parallel instances of GHZ states with 2

i1K entangledqubits. Due to the exponential scaling of the degree of entan-glement, most of the total available qubits are used in higherlevels of the cascade. This is a necessary condition to achieveHeisenberg scaling, up to logarithmic factors.

intoM+1 different groups. The 0th group containsn1/Kuncorrelated qubits, and the ith group (i = 1, 2 . . . M )contains n0independent instances of 2i

1 qubits that areentangled with the other groups of 2i1 qubits in eachclock. In other words, there are n0 independent copiesof GHZ states with a total of 2i1Kqubits entangled ontheith level of the cascade (i 1) (See Fig.5). This waythe total number of qubits can be written as

N=n1+ n0

Mi=1

2i1K n02MK (A7)

where we assumed n1 N.The purpose of this cascaded scheme is to directly as-

sess the digits Y1 and{Zj : j = 2, 3, . . . } in the binaryfraction representation of the phase

LO mod [, ] =2K

Y1+

i=1

Zi+1/2i

, (A8)

where mod [, ] = (x + ) mod 2 , Y1 {0, 1, 2 . . . K 1} and Zi {0, 1}. The 0th level of the


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cascade estimates 0 =K

j=1

(j) mod [, ] /K,

and every ith level after that estimates i = K2i1LO

mod [, ]. From these estimates one can determinethe digits,

Y1 = [K(0+ ) (1+ )] /(2), (A9)Zi = [2(i1+ ) (i+ )] /(2), (A10)

fori = 2, 3, . . . M .The last group (i = M) contains GHZ states with

the most entangled qubits. These are the ones with thefastest evolving phase, and therefore they provide thebest resolution on LO. Since there aren0 independentinstances, their phase M = 2

i=1 ZM+i/2

i is knownup to the uncertainty,2Mpr= 1n0 ,

Assuming that all lower digits{Y1, Zj |j = 2 . . . M }have been determined correctly, this results in the totalmeasurement uncertainty for LO:

2LOpr= 2Mpr(2M1K)2

=4n0N2

, (A11)

where, for the moment, we neglected individual qubitnoise and assumed LO [, ]. However, in gen-eral, the estimation of the lower digits will not be per-fect. In the following Section we investigate the effect oftheserounding errorson the final measurement accuracy.From this analysis we find the optimal number of copiesn0 andn1.

3. Rounding errors

Whenever|est0

0|

> /K, or|esti

i|

> /2 (fori 1), we make a mistake by under- or overestimatingthe number of phase slips Y1 or Zi+1, respectively. Tominimize the effect of this error, we need to optimize howthe total ofNqubits are distributed among the variouslevels of the cascade. In other words we need to findn0,opt andn1,opt.

The probability that a rounding error occurs duringthe estimation ofZi+1 is

Pi,re= 2

/D

d i( i) 2

/D

d 1

s3iexp

2

2s2i

(A12)where = esti i, and i is the conditional den-sity function of esti for a given real i, and s

2i =

Var(esti i) = 1/n0 for i 1, and s20 =20pr =1K2

Kj=1((j))2pr = 1/n1, since((j))2pr = Kn1

for all j. The upper bound for i is obtained by usingthe following upper bound for any binomial distribution:mk

pk(1 p)mk exp

n knp2. (For details, see

Supplementary Materials of [12].) The resulting prob-abilities, after dropping the higher order terms in the

asymptotic expansions, are

P0,re 2K

n1/21 exp

n1

2

2K2

(A13)

Pi,re 4

n1/20 exp

n0

2

8

(i 1) (A14)

The phase shift imposed on the estimate of LO by

a manifested rounding error ofY1 is 2/Kand ofZi is2/(K2i1), for i = 2, 3 . . . M . This results in the totalvariance contribution,

2LOre =

=

2

K

2 P0,re+

Mi=2

Pi1,re(2i+1)2

(A15)

2

K

2 P0,re+ 1

3Pi1,re

. (A16)

We simplify this expression by choosing n1 so thatP0,re 23Pi,re:

n1 = K2n0, (A17)

where max

1, 22n0 log

3K2

8

n1/20

n0, K.

With this choice, we can write the rounding error contri-bution as

2LOre16

K2n1/20 exp

n0

2

8

. (A18)

We note that the amount of extra resources needed forthe 0th level, is marginally small, since the total qubitnumber can be expressed as

N=n1+n0KMi=1

2i1 =n0K(K2+2M2) n0K2M,(A19)

under the assumption K 2M/2.By adding the two error contributions from Eq. (A11)

and Eq. (A28), we obtain the corresponding Allan-variance,

2y() = 1

20 T2LO =:

1

20 [1+ 2] = (A20)

= 1

20 4n0

N2T +

16

K2Tn1/2

0

expn0

2

8(A21)

Now, let us find the optimal value ofn0. We write 1+2,using the new variablex = 82

1n0

, as

1 + 2 = 4

T

8

21

xN2+

32

K21

x1/2exp

1

x

. (A22)

Taking the derivative with respect to x and equating itwith 0, while using the assumption x 1 results in


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9

2 xopt1 1, which can be written as the followingtranscendental equation for the optimal value, xopt,

x1/2opt

2N28K2

exp

1

xopt

. (A23)

The general solution of any equation of the form x =A exp[1/x], in the limit of A 1 and x 1, isx = [log(A)]1 . (For details, see the SupplementaryMaterials of [12].) Using this result we can write

xopt

log

2

8

N2

K2

1 [2 log(N/K)]1(A24)

n0,opt 82

1

xopt

4

2log(N/K) . (A25)

For the realistic case ofN/K 1, indeedxopt 1, andthe corresponding minimal value of 1+ 2 is

[1+ 2]min 1(xopt) =

8

2log(N/K)

N2T . (A26)

This result indicates that, in terms of qubit number, onlya logarithmic extra cost is required to achieve the Heisen-berg limit.

4. Phase slip errors

Although the cascade is designed to detect phase slipsof all levels i = 1, 2 . . . M , a possible phase wrap of leveli = 0 remains undetected. Since the qubits at differentclocks are interrogated independently on the 0th level,each of them estimates the phase of the corresponding

LO, (j)0 (j = 1, 2, . . . K ), and not LO. The probability

of (j)0 falling outside the interval [, ] at least once

during the total measurement time is

Pj,slip = 2 T

d 12LOT

exp

2

2LOT

T

2

3/2

LOTexp

2

2LOT

, (A27)

whereLO is the linewidth of the local oscillator at clockj, corresponding to a white noise spectrum, resulting ina constant phase diffusion over the interrogation timeT,(which assumed to be approximately equal to the cy-cle time). The approximate form above is obtained byneglecting the higher order terms in the asymptotic se-ries expansion under the assumption LOT 1. Oncesuch a phase slip happens, it introduces a 2 phase shift

in (j)0 , and therefore contributes to its overall uncer-

tainty with((j)0 )2 = (2)2Pj,slip. Physically 0 isthe phase of the COM signal, that the center can obtainafter averaging the frequencies of all Klocal oscillators

with equal weights, 0 = COM =K

j=1(j)0 /K, there-

fore 20 = 1K2K

j=1((j)0 )2 = 1K((j)0 )

2,where

we assumed that the LOs are independent but they havethe same linewidth, LO. Since 0 = LO, the abovemeans the following variance contribution

2LOslip=

32

1/2LO

T1/2Kexp

2

2LOT

. (A28)

After adding this error to the previously minimizedprojection and rounding error terms (from Eq. (A26)),we obtain the corresponding Allan-variance, 2y() =1

20([1+ 2]min+ 3), where

[1+ 2]min+ 3 = (A29)

=

8

2log(N/K)

N22LO

21

y+

16

5/2 2LO

K

1

y3/2exp

1

y

,

using the variable y = 22 LOT.Now, let us find the optimal Ramsey time Topt, un-

der the assumption that is sufficiently long. Aftertaking the derivative with respect to y and equatingit with zero, the assumption yopt 1 results in the3

yopt[1+2]min

[1+2]minwhich can be written

as the following transcendental equation,

y3/2opt

3/2

8

LOK

N2

log(N/K)exp

1

y

. (A30)

The asymptotic solution in case ofyopt 1 is (see Sup-plementary of [12])

yopt

log

3/2

8

LOK

N2

log(N/K)

1, (A31)

Topt 2

2

yoptLO

2

2LO

log( LON

2/K)1

(A32)

in the realistic limit ofLO N

2

/K 1. The correspond-ing minimal Allan-variance is

2y() = 1

20

[1+ 2]min+ 3

min

120

LLON2

, (A33)

whereL = 1284 log(N/K)log( LON2/K).

For shortaveraging times, the optimal Ramsey timeisTopt= , instead of Eq. (A32). This makes 3 negligi-ble compared to [1 + 2]min, resulting in a 1/

2 scaling:

2y() = 1

20[1+ 2]

T=min =

1

20

L

N22. (A34)

where L = 8 2 log(N/K). This scaling is more fa-vorable, but it continues to higher values only up to 1LO, where it switches to the 1/ behavior accord-ing to Eq. (A33).

5. Pre-narrowing the linewidth

We can minimize the limiting effect ofLO by narrow-ing the effective linewidth of the local oscillators before-hand. We imagine usingN qubits to locally pre-narrow


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10

the linewidth of all LOs down to an effective linewidtheff indN, before using the rest N N qubits in theGHZ cascade. This eff LO allows the optimal Ram-sey time going above the previous limit, set by 1LO inEq. (A32). This step-by-step linewidth narrowing pro-cedure, using uncorrelated ensembles in every step, isoutlined in [17, 18], and given detailed analysis in [12].Working under the smallN assumption, one can obtain

effas

eff LO

2

2log(LO n)

n

N/n, (A35)

where we imagine usingn qubits in each narrowing step.We find the optimal value ofn to be

nopt 2e2

log(LO), (A36)

by minimizingeff, which yields

[eff]min

LOexp N2

2e log(LO) . (A37)

For a given , we can always imagine carrying outthis pre-narrowing, so that eff <

1, and thereforeEq. (A34) remains valid with the substitution N N N for > 1LO as well. The required number ofqubits, N, is

N 2e2

log(LO)log

LOindN

N. (A38)

due to the exponential dependence in Eq. (A37).

6. Individual qubit dephasing noise

Our scheme, as well as any scheme, is eventually lim-ited by individual qubit noise. Such a noise dephasesGHZ states at an increased rate, compared to uncorre-lated qubits, due to the entanglement, giving the corre-sponding variance contribution for the phase of the GHZ

states in the Mth group,2Mdephasing= 2M1KindT

n0,

after averaging over the n0 independent copies of theGHZ states, each containing 2M1K entangled qubits.The resulting variance contribution for LO is

2LO

dephasing=

indT

n02M1

K

=2indT

N

. (A39)

This term represents a noise floor, which we add toEq. (A34) and obtain our final result for the minimalachievable Allan-variance,

2y() = 1

20

L

N22+

2indN

. (A40)

For long times, the ultimate limit, set by the stan-dard quantum limit,2y() =

120

indN, can be reached by

changing the base of the cascade. Instead of entangling2-times as many qubits in each level of the cascade thanin the previous level, we imagine changing it to a basenumber D . Carrying out the same calculation results inour final result for the achievable Allan-variance:

2y() = 1

20

D

2 2

L

N22+

D

D

1

indN

, (A41)

where L =8

2log(N/K). (See Supplementary of [12]

for details.) The optimal value ofD depends on . Forsmall , Dopt = 2, however for large one can gain afactor of 2 by choosing Dopt = Dmax. Due to natural

constraints,Dmax

N, in which regime, the protocolconsists of only two cascade levels, an uncorrelated 0thlevel, with Nqubits and an entangled 1st level with N qubits.

Appendix B: Security countermeasures

1. Sabotage

In order to detect sabotage, the center can occasion-ally perform assessment tests of the different nodes byteleporting an uncorrelated qubit state [|0 + ei|1]/2,where is a randomly chosen phase known only to thecenter. A properly operating node creates a local GHZstate [|0 + ei|1]/2 from the sent qubit, measuresthe parity of the GHZ state, and sends it to the cen-ter. The measured parity holds information on the phase = +, whereis the accumulated phase of the LO at

the node. Due to the random shift , this appears to berandom to the node, and therefore indistinguishable fromthe result of a regular (non-testing) cycle. On the otherhand, the center can subtract , and recover from thesame measurement results. In the last step, the centerverifies by comparing it with the classically determinedphasecl of the sent LO signal with respect to the COMsignal. The expected statistical deviation of fromcl is

(cl)

KN, while the accuracy of the COM phase

(COM T 0)

K(KKt)N is much smaller, where

Kt is the number of simultaneously tested nodes. In thelikely case ofKt K, this method is precise enough forthe center to discriminate between healthy and unhealthynodes by setting a acceptance range,| cl|

KN.

E.g. the choice of = 4 results in a 4 confidencelevel, meaning only 0.0063% chance for false positives(healthy node detected as unhealthy), and similarly smallchance for false negatives (unhealthy node being unde-

tected) ( 2 1/

N) due to the high precisionwith which is measured. The fact, that the teleportedqubit can be measured only once, also prevents the nodesfrom discovering that it is being tested.


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11

2. Eavesdropping

Eavesdroppers would try to intercept the sent LO sig-nals, and synthesize the stabilized COM for themselves.Our protocol minimizes the attainable information of thisstrategy by prescribing that only the non-stabilized LOsignals are sent through classical channels. This requiresthe feedback to be applied to the LO signal after some of

it has been split off by a beam splitter, and the center tointegrate the generated feedback in time. Alternatively,eavesdroppers could try intercepting the LO signals andthe feedback signals, and gain access to the same informa-tion, the center has. This can be prevented by encodingthe radio frequency feedback signal with phase modula-tion according to a shared secret key. Since such a keycan be shared securely with quantum key distribution,this protocol keeps the feedback signal hidden from out-siders. As a result, even the hardest-working eavesdrop-per, who intercepts all LO signals, is able to access onlythe non-stabilized COM signal, and the stabilized COMsignal remains accessible exclusively to parties involved

in the collaboration.

3. Rotating center role

Since the center works as a hub for all information,ensuring its security has the highest priority. In a sce-nario, where none of the nodes can be trusted enough toplay the permanent role of the center, a rotating stagescheme can be used. By passing the role of the centeraround, the potential vulnerability of the network dueto one untrustworthy site is substantially lowered. Thisrequires a fully connected network and a global schemefor assigning the role of the center.

Appendix C: Network operation

1. Different degree of feedback

Apart from the full feedback, described in the maintext, alternatively, the center can be operated to providerestricted feedback information to the nodes. If the cen-ter sends the averaged error signal COMonly, the LOs atthe nodes will not benefit from the enhanced stability andonly the center can access the stabilized signal. Of coursethe LO at each node will have its own local feedback tokeep it within a reasonable frequency range around theclock transition. Such a safe operational mode makesthe center node the only participant having access to theworld time signal.

As an intermediate possibility, the center can chooseto send regionally averaged feedback signals COM +

jR(j COM)/|R|, uniformly for all j R nodes,where R is a set of nodes, ie. a region. Such a feedbackscheme creates the incentive of cooperation for the nodesin region R. By properly sharing their LO signals with

each other, the nodes can synthesize the regional COMfrequency,

jR(j)/|R|, and steer it with the feedback,

received from the center.

2. Timing

Proper timing of local qubit operations is necessary to

ensure that every qubit in the network is subject to thesameTfree evolution time. The finite propagation timeof light signals introduces delays in the quantum linksand classical channels. Similarly, during the entanglingstep, the finite time required to do CNOT operationsmake the free evolution start at slightly different timesfor different qubits. Since both the initialization and themeasurement are local operations, we can resolve the is-sue of delay by prescribing that the measurement of qubitij(ith qubit at nodej) takes place exactly Ttime after itsinitialization. Occasional waiting times of known lengthcan be echoed out with a -pulse at half time.

In extreme cases, this might cause some qubits to be

measured before others are initialized. However, this isnot a problem, since the portion of the GHZ state thatis alive during the time in question is constantly accu-mulating the j phases from the qubits it consists of.This results in the phenomenon that the total time ofphase accumulation can be much longer than the lengthof individual phase accumulations, provided that the saidinterrogations overlap.

3. More general architectures

So far, we focused on the simplest network structure

with one center initiating every Ramsey cycle and nodeswith equal number of clock qubits.In a more general setup, node j has Nj clock qubits.

IfNj is different for different j , then the nodes will con-tribute the the global GHZ states unequally, resulting inentangled states which consists of differentNj number ofqubits from each site j . Such a state picks up the phase

=j

Njj , (C1)

where j is the phase of the LO at site j relative tothe atomic frequency. As a result, the clock networkmeasures the following collective LO frequency

LO =

jN

jj

jNj

. (C2)

This represents only a different definition of the worldtime (a weighted average of the times at the locations ofthe nodes, instead of a uniform average), but it does notaffect the overall stability.

The initial laser linewidths of the nodes jLO can alsobe different. The stability achievable in this case is


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12

bounded by the stability obtained for a uniform linewidthLO = maxj

jLO. If linewidths are known, the center can

divise the best estimation method which uses linewidthdependent weights in the LO frequency averaging step.

Although it is simple to demonstrate the importantnetwork operational concepts with the architecture withone center, this structure is not a necessary. The quan-tum channels, connecting different nodes, can form a

sparse (but still connected) graph, and the entanglementglobal entanglement can still be achieved by intermedi-

ate nodes acting as repeater stations. This way entangle-ment can be passed along by these intermediate nodes.Moreover, the center can be eliminated from the entan-gling procedure by making the nodes generate local GHZstates, and connect them with their neighbors by bothmeasuring their shared EPR qubit with one of the qubitsform the local GHZ state in the Bell-basis. After commu-nicating the measurement result via classical channels,

and performing the required single qubit operations, aglobal GHZ state is formed.

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ACKNOWLEDGEMENT

We are grateful to Till Rosenband and Vladan Vuleticfor enlightening discussions. This work was supported byNSF, CUA, ITAMP, HQOC, JILA PFC, NIST, DARPAQUSAR, the Alfred P. Sloan Foundation, the Quinessprograms, ARO MURI, and the ERC grant QIOS (grantno. 306576); MB acknowledges support from NDSEGand NSF GRFP. It is dedicated to Rainer Blatt and PeterZoller on the occasion of their 60th birthday, when initialideas for this work were formed.

COMPETING FINANCIAL INTERESTS

The authors declare no competing financial interests.

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