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A quantum network of clocks

Peter Kómár, Eric M. Kessler, Michael Bishof, Liang Jiang, Anders S. Sørensen, Jun Ye, Mikhail D. Lukin

arXiv:1310.6045v1quant-ph

TL;DR

Realistic laser noise can limit the quantum gain of a single-GHZ clock interrogation through phase slips. The paper considers a cooperative quantum clock network, whose protocol improves short-time stability over classical cooperation; long averaging times remain noise-limited.

  • Problem

    Realistic laser noise can make a single-GHZ phase estimate fail through phase slips, limiting Ramsey time and preventing quantum gain.

    Source: INTERROGATION · p. 3

  • Method

    The quantum network protocol cooperatively operates clocks across nodes, with measurements and oscillator signals combined through the network.

    Source: STABILITY ANALYSIS · p. 5, SECURITY · p. 5

  • Results

    At short averaging times, the quantum protocol adds a factor of √K cooperative advantage over classical cooperation and reaches the Heisenberg limit.

    Source: STABILITY ANALYSIS · p. 5

  • Takeaways & Limitations

    If the center or communication links are disabled, nodes can fall back to independent clock operation using local resources.

    Source: SECURITY · p. 5

    For averaging times τ > 1/(γiN), individual noise limits Ramsey time and the 1/N scaling breaks down.

    Source: STABILITY ANALYSIS · p. 4

Abstract

from arXiv · show

The development of precise atomic clocks has led to many scientific and technological advances that play an increasingly important role in modern society. Shared timing information constitutes a key resource for positioning and navigation with a direct correspondence between timing accuracy and precision in applications such as the Global Positioning System (GPS). By combining precision metrology and quantum networks, we propose here a quantum, cooperative protocol for the operation of a network consisting of geographically remote optical atomic clocks. Using non-local entangled states, we demonstrate an optimal utilization of the global network resources, and show that such a network can be operated near the fundamental limit set by quantum theory yielding an ultra-precise clock signal. Furthermore, the internal structure of the network, combined with basic techniques from quantum communication, guarantees security both from internal and external threats. Realization of such a global quantum network of clocks may allow construction of a real-time single international time scale (world clock) with unprecedented stability and accuracy.

THE CONCEPT OF QUANTUM CLOCK NETWORK

The network links geographically separate atomic clocks through entangled states spanning its nodes. Their measurements estimate the center-of-mass frequency, which the network stabilizes and distributes as a shared clock signal.

THE CONCEPT OF QUANTUM CLOCK NETWORK

  • Each node allocates some clock qubits to entangled states spanning the network, enabling information about the center-of-mass frequency deviation.Each clock has its own local oscillator and interrogates its qubits periodically to stabilize the oscillator to the atomic transition.Source: THE CONCEPT OF QUANTUM CLOCK NETWORK · p. 2
  • A node serving as alternating center collects measurement data and local-oscillator signals, then feeds back the center-of-mass signal.A flat hierarchical structure without a center is also described as straightforward to devise.Source: THE CONCEPT OF QUANTUM CLOCK NETWORK · p. 2
  • After a few cycles, the feedback yields a Heisenberg-limited center-of-mass clock signal and individual-node oscillators with network-wide accuracy and stability.The stabilized signal is subsequently distributed securely to the individual nodes.Source: THE CONCEPT OF QUANTUM CLOCK NETWORK · p. 2

PREPARATION OF NETWORK-WIDE ENTANGLED STATES

The preparation protocol creates a GHZ state shared across nodes, then extends it across their clock qubits. It uses central ancillas, entanglement distribution, teleportation, and local gates.

PREPARATION OF NETWORK-WIDE ENTANGLED STATES

INTERROGATION

Network-wide GHZ states accumulate the nodes' local-oscillator phases into a collective phase that encodes center-of-mass detuning. A multiscale set of GHZ groups estimates that phase while addressing phase slips that limit a single large GHZ state.

INTERROGATION

  • During interrogation, each node's qubits accumulate its local phase, and the non-local GHZ state combines these into a collective phase.Source: INTERROGATION · p. 3
  • Nodes measure their qubits in the x-basis and send local parities to the center, where only the full set is used to extract phase information.The center's own parity acts as a key needed to decode the information sent by the nodes.Source: INTERROGATION · p. 3
  • A single GHZ state gives phase sensitivity proportional to the number of network qubits, but realistic laser noise causes phase slips that cap Ramsey time and prevent quantum gain.When the accumulated phase falls outside [−π, π], estimation fails; the stated bound is T < (N~γLO)^−1.Source: INTERROGATION · p. 3
  • Products of successively larger shared GHZ ensembles plus uncorrelated atoms directly assess binary phase bits, yielding Heisenberg-limited phase accuracy up to a logarithmic correction.The scheme is described as operating even for Ramsey times beyond the single-GHZ laser-fluctuation limit.Source: INTERROGATION · pp. 3–4

FEEDBACK

The center uses the estimated phase to stabilize the network's center-of-mass frequency, synthesizing the signal from the local oscillators and distributing correction signals to the nodes.

FEEDBACK

  • The center averages the local-oscillator frequencies with equal weights to synthesize the center-of-mass frequency.It can phase-track this frequency by heterodyning its local oscillator against each incoming laser signal.Source: FEEDBACK · p. 4
  • After each Ramsey cycle, the estimated center-of-mass detuning is used to stabilize the center-of-mass laser signal.Source: FEEDBACK · p. 4
  • The center sends each node an error signal combining the center-of-mass detuning with that node's frequency offset, and corrects its own oscillator.Restricted feedback information to the nodes is also described as an alternative.Source: FEEDBACK · p. 4

STABILITY ANALYSIS

The network jointly interrogates and stabilizes the center-of-mass laser mode, using the total atom number to approach the best clock stability allowed by quantum theory. Quantum cooperation reaches the fundamental bound sooner than classical cooperation, with illustrative ion- and neutral-clock estimates showing the potential stability.

STABILITY ANALYSIS

  • For τ < 1/(γiN), the network achieves near-Heisenberg-limited ADEV scaling with total atom number, with 1/τ scaling enabled by cascaded cancellation of low-frequency laser noise.The optimized protocol uses n0 ∼ log(N) GHZ copies per group, introducing a logarithmic correction in total particle number.Source: STABILITY ANALYSIS · p. 4
  • For averaging times τ > 1/(γiN), individual noise fundamentally limits Ramsey time to T ≤ 1/(γiN); consequently, 1/N scaling breaks down and ADEV returns to square-root scaling with particle number.Source: STABILITY ANALYSIS · p. 4
  • K times faster convergence to the fundamental ADEV bound is achieved by quantum cooperation than by classical cooperation; optimal classical cooperation reaches it N times later.For short averaging times, the quantum network also gains a factor of √K over classical cooperation; classical cooperation gains √K over independent quantum clocks.Source: Figure 3 · p. 4, STABILITY ANALYSIS · p. 5
  • For the parameters ω0 = 2π × 1121 THz and γi = 2π × 8 mHz, the quantum cooperative protocol can reach a projected stability of 4 × 10−17.Source: STABILITY ANALYSIS · p. 5
  • ∼2 × 10−18 stability after 1 s averaging is projected for ten 1000-atom neutral clocks, an order of magnitude better than the best classical cooperative scheme.Future clock transitions with linewidths of a few tens of µHz could possibly reach fractional frequency uncertainty beyond 10−20 after τ ∼100 s.Source: STABILITY ANALYSIS · p. 5

SECURITY

The protocol addresses sabotage by testing nodes and eavesdropping by protecting feedback and restricting what intercepted signals reveal. If the center or communication links are disabled, nodes can resume independent clock operation.

SECURITY

  • Random-phase qubit tests let the center detect a node that sends false oscillator frequencies or incorrect measurement bits by checking measurement statistics against its transmitted oscillator detuning.The center can test nodes by teleporting an uncorrelated qubit with a phase known only to the center.Source: SECURITY · p. 5
  • Quantum key distribution can encode feedback channels, while sending only non-stabilized oscillator signals ensures interception reveals only the non-stabilized COM signal.The center alone can decode node measurement results using its own outcomes, leaving the stabilized COM signal accessible only to collaborating parties.Source: SECURITY · p. 5
  • If an attack disables the center or communication links, nodes can fall back to independent clock operation using local resources.Source: SECURITY · p. 5

OUTLOOK

The proposed network gives participants real-time access to an ultra-stable shared signal while preserving local clock access and sovereignty. Realization requires advances in remote entanglement and coherent links, and could support metrology and tests of fundamental physics.

OUTLOOK

  • Unlike a delayed “paper clock,” the network gives all participants access to the ultra-stable signal in real time while preserving local clock access and sovereignty.The passage contrasts this with averaging and communicating individual clock signals with a time delay.Source: OUTLOOK · p. 6
  • Full-scale realization requires sufficiently high-rate, high-fidelity EPR-pair links; entanglement purification and quantum repeaters will likely be needed.Establishing coherent ground-space optical links remains a technical challenge; hybrid entanglement and local-clock systems may be warranted.Source: OUTLOOK · p. 6
  • If realized, the network may support earth-science applications and tests of relativity and the connection between quantum and gravitational physics.The passage also describes it as a platform for time and frequency metrology.Source: OUTLOOK · p. 6

SUPPLEMENTARY INFORMATION

The network uses GHZ states to make measurements sensitive to the accumulated phase of all clocks’ local oscillators, then combines local measurement outcomes through a center node. A cascade of entangled states estimates phase at multiple scales, with most qubits allocated to higher levels.

Appendix A: GHZ cascade in a network of K clocks

  • A network-wide GHZ state accumulates the phases of the clocks’ local oscillators, and its phase sensitivity is enhanced by quantum correlations.The state evolves under the local-oscillator phases during interrogation.Source: Appendix A · pp. 6–7

1. Parity measurement

  • The clocks measure qubits in the local x-basis and send their outcomes to the center, which evaluates the global parity to estimate phase.Parallel GHZ instances initialized with phases 0 and π/2 measure the real and imaginary parts of the accumulated phase factor.Source: §1 · p. 7

2. Cascaded GHZ scheme

  • Each cascade level uses parallel GHZ copies entangling 2^(i−1)K qubits, while level 0 uses uncorrelated qubits to estimate the phase’s lower digits.Because entanglement grows exponentially across levels, most qubits go to higher levels, a necessary condition for Heisenberg scaling up to logarithmic factors.Source: §2 · pp. 7–8, Figure 5 · p. 7

3. Rounding errors

The cascade’s accuracy depends on avoiding rounding errors when estimating phase digits. Optimizing the qubit allocation limits their effect, with only a logarithmic extra qubit cost for Heisenberg scaling.

3. Rounding errors

  • A digit-estimation error occurs when the phase estimate crosses the stated threshold, causing the corresponding phase-slip digit to be over- or underestimated.The qubit allocation across cascade levels is optimized to minimize this error’s effect.Source: §3 · p. 8
  • Rounding errors shift the estimated local-oscillator phase by 2π/K for Y1 and 2π/(K2^(i−1)) for Zi.These shifts contribute to the total variance.Source: §3 · p. 8
  • The extra resources assigned to the uncorrelated level-0 group are marginally small.The total qubit count is expressed as the cascade allocation plus this level’s overhead.Source: §3 · p. 8
  • Only a logarithmic extra qubit cost is required to achieve the Heisenberg limit.This result holds for the realistic case N/K ≫ 1.Source: §3 · p. 9

4. Phase slip errors

The analysis accounts for undetected level-0 phase wraps and their effect on clock stability. Pre-narrowing the local-oscillator linewidth can extend the useful Ramsey time beyond the phase-slip limit.

4. Phase slip errors

  • The cascade detects phase slips at levels 1 through M, but a phase wrap in level 0 remains undetected because clocks estimate their local-oscillator phases independently.The analysis assumes independent local oscillators with the same linewidth and white-noise phase diffusion.Source: §4 · p. 9
  • For short averaging times, Topt = τ, Γ3 is negligible, and the Allan variance scales as 1/τ^2.The more favorable scaling continues only up to τ ∼ γLO^-1, where it switches to 1/τ behavior.Source: §4 · p. 9

5. Pre-narrowing the linewidth

  • Pre-narrowing the local oscillators’ effective linewidth allows the optimal Ramsey time to exceed the prior limit of approximately γLO^-1.The protocol uses N* qubits for narrowing before the remaining qubits enter the GHZ cascade.Source: §5 · pp. 9–10

6. Individual qubit dephasing noise

The analysis bounds long-time clock stability under individual-qubit dephasing and describes security, feedback, and timing choices for operating the network.

6. Individual qubit dephasing noise

  • Individual-qubit dephasing adds a noise floor to the minimal Allan variance, and long averaging ultimately reaches the standard quantum limit.The GHZ states dephase faster than uncorrelated qubits because of entanglement.Source: §6 · p. 10
  • For small τ, Dopt=2; for large τ, choosing Dopt=Dmax can yield a factor-of-2 gain.The optimal cascade base depends on averaging time.Source: §6 · p. 10

1. Sabotage

  • Random-phase teleportation tests let the center distinguish healthy from unhealthy nodes; with Λ=4, the text reports 4σ confidence and a 0.0063% false-positive chance.False negatives are also described as similarly unlikely, and the single-use teleported qubit prevents nodes from discovering they are being tested.Source: §1 · p. 10

2. Eavesdropping

  • Encoding feedback with a shared secret key keeps the stabilized COM signal inaccessible to outsiders, even if they intercept all LO signals.The text describes quantum key distribution as a way to share the key securely.Source: §2 · p. 11

3. Rotating center role

  • When no node is trusted as permanent center, rotating that role can lower vulnerability to an untrustworthy site, but requires a fully connected network and global role assignment.Source: §3 · p. 11

1. Different degree of feedback

  • Restricted feedback can leave only the center with the stabilized signal; regional feedback instead gives a selected set of nodes a shared regional correction.Nodes in a region can synthesize their regional COM frequency and steer it using the center's feedback.Source: §1 · p. 11

2. Timing

  • Measurement exactly T after initialization equalizes free-evolution time; known waiting intervals can be echoed with a π-pulse halfway through.Overlapping interrogations can make total phase accumulation longer than any individual interrogation.Source: §2 · p. 11

3. More general architectures

The network can use more general connected architectures and unequal node resources; these alter the definition of world time or impose linewidth-dependent stability bounds.

3. More general architectures

  • Unequal clock-qubit counts produce a weighted average of local times rather than a uniform average, without affecting overall stability.Source: §3 · p. 11
  • With differing LO linewidths, achievable stability is bounded by the uniform-linewidth case at the largest linewidth; known linewidths allow weighted frequency averaging.Source: §3 · p. 12
  • A sparse but connected quantum-channel graph can still achieve global entanglement, so a single center is not necessary.Source: §3 · p. 12
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