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arXiv:1911.05279 · Advanced Quantum Technologies

Synchronization and Estimation of Gravity‐Induced Time Difference for Quantum Clocks

Jieci Wang, Tonghua Liu, Jiliang Jing, Songbai Chen

Published May 21, 2019• 8 citations
arXivHTMLPDF

Abstract

Abstract It has recently been reported that, under an operational definition of time, quantum clocks would get entangled through gravitational effects. Here, an alternative scenario is studied: The clocks have different masses and energy gaps, which would produce time differences via gravitational interaction. The proposal of quantum clock synchronization (QCS) for the gravity‐induced time difference is discussed. It is illustrated how the stability of measurement probability in the QCS proposal is influenced by the gravitational interaction induced by the clocks themselves. It is found that the precision of clock synchronization depends on the energy gaps of the clocks and the improvement of precision in quantum metrology is in fact an indicator of entanglement generation. The quantum enhanced estimation of time difference is also presented and it is found that the quantum Fisher information is very sensitive to the distance between the clocks.

DOI
10.1002/qute.201900003

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Abstract It has recently been reported that, under an operational definition of time, quantum clocks would get entangled through gravitational effects. Here, an alternative scenario is studied: The clocks have different masses and energy gaps, which would produce time differences via gravitational interaction. The proposal of quantum clock synchronization (QCS) for the gravity‐induced time difference is discussed. It is illustrated how the stability of measurement probability in the QCS proposal is influenced by the gravitational interaction induced by the clocks themselves. It is found that the precision of clock synchronization depends on the energy gaps of the clocks and the improvement of precision in quantum metrology is in fact an indicator of entanglement generation. The quantum enhanced estimation of time difference is also presented and it is found that the quantum Fisher information is very sensitive to the distance between the clocks.
Atomic clocks achieve extraordinary precision through quantum control over large ensembles of atoms, with precision increasing and atom number advancing steadily. Emerging nuclear clock technology encodes clock states in nuclear rather than electronic transitions, offering a new frontier for precision measurement. Here we study whether a large ensemble of such clock nuclei could be used to observe their mutual gravitational interaction. We investigate the gravitational redshift induced by neighboring atoms, including nuclear energy superpositions as sources of gravity, giving rise to a genuinely quantum gravitational interaction between pairs of nuclei. We find that the resulting clock frequency shift due to mutual gravity may become non-negligible for a macroscopic number of nuclei, for instance in a solid-state thorium clock, and we propose approaches to mitigate electromagnetic field effects by exploiting spatial symmetry and high-precision spectroscopy. We further discuss how energy-induced mass superpositions could be directly probed using external gravitational field sensors. By contrast, observing gravity-induced entanglement involving clocks would require substantially greater experimental advances due to decoherence from spontaneous emission. Taken together, our results suggest that nuclear clocks, either alone or in combination with complementary sensors such as optomechanical systems, may offer a future route toward probing gravity at short distances and testing gravitating quantum source masses beyond the Newtonian limit.
We show that motion and gravity affect the precision of quantum clocks. We consider a localised quantum field as a fundamental model of a quantum clock moving in spacetime and show that its state is modified due to changes in acceleration. By computing the quantum Fisher information we determine how relativistic motion modifies the ultimate bound in the precision of the measurement of time. While in the absence of motion the squeezed vacuum is the ideal state for time estimation, we find that it is highly sensitive to the motion-induced degradation of the quantum Fisher information. We show that coherent states are generally more resilient to this degradation and that in the case of very low initial number of photons, the optimal precision can be even increased by motion. These results can be tested with current technology by using superconducting resonators with tunable boundary conditions.
In general relativity, the picture of space-time assigns an ideal clock to each world line. Being ideal, gravitational effects due to these clocks are ignored and the flow of time according to one clock is not affected by the presence of clocks along nearby world lines. However, if time is defined operationally, as a pointer position of a physical clock that obeys the principles of general relativity and quantum mechanics, such a picture is, at most, a convenient fiction. Specifically, we show that the general relativistic mass-energy equivalence implies gravitational interaction between the clocks, whereas the quantum mechanical superposition of energy eigenstates leads to a nonfixed metric background. Based only on the assumption that both principles hold in this situation, we show that the clocks necessarily get entangled through time dilation effect, which eventually leads to a loss of coherence of a single clock. Hence, the time as measured by a single clock is not well defined. However, the general relativistic notion of time is recovered in the classical limit of clocks.
In this paper, a non-repeating quantum algorithm is introduced that depends on detectable Byzantine’s quantum solution that attains Clk synchronization in the arbitrary faultier processes’ presence through utilizing a double quantum system only. Precisely, it has been shown that general relativistic mass–energy equivalences intimate gravitational relation among Clks, however, energy’s quantum mechanical superposition resulted in a non-fixed metric background. Depending only on assumptions that gravitational principles held in such conditions, it has been shown that the Clks essentially gets involve over time dilation effect that ultimately results in a double Clk coherence loss. Therefore, the measured time through a double Clk is not defined precisely. Although, the time’s general relativistic notion is recuperated in the Clks conventional limits.
We propose a practical scheme of satellite-based quantum clock synchronization with atmospheric dispersion cancellation and by taking into account spacetime background of the Earth. Two frequency entangled pulses are employed to synchronize two clocks, one at a ground station and the other at a satellite. The time discrepancy of the two clocks is introduced into the pulses by moving mirrors and is extracted by measuring the coincidence rate of the pulses in the interferometer. We find that the pulses are distorted due to effects of gravity when they propagate between the Earth and the satellite, resulting in coincidence rate are remarkably affected. We also find that the precision of the clock synchronization are sensitive to the source parameters and the altitude of the satellite. Our proposal can be realized with current technology.
We present an enhanced entangled-quantum-clock protocol that incorporates a quantum phase estimation algorithm to directly estimate proper-time differences as an unknown phase. By employing highly entangled multi-clock states, the achievable uncertainty scales inversely with the total number of quantum clocks, achieving the Heisenberg limit O(1/N)O(1/N)O(1/N) while remaining completely immune to dynamical phase noise. Furthermore, we discuss the feasibility of utilizing this ultra-high precision protocol for gravitational wave detection, highlighting its potential for quantum-enhanced relativistic sensing and fundamental physics tests.