Formulation of Determining the Gravity Potential Difference Using Ultra-High Precise Clocks via Optical Fiber Frequency Transfer Technique

Abstract

Based on gravity frequency shift effect predicted by general relativity theory, this study discusses an approach for determining the gravity potential (geopotential) difference between arbitrary two points P and Q by remote comparison of two precise optical clocks via optical fiber frequency transfer. After synchronization, by measuring the signal’s frequency shift based upon the comparison of bidirectional frequency signals from P and Q oscillators connected with two optical atomic clocks via remote optical fiber frequency transfer technique, the geopotential difference between the two points could be determined, and its accuracy depends on the stabilities of the optical clocks and the frequency transfer comparison technique. Due to the fact that the present stability of optical clocks achieves 1.6×10⁻¹⁸ and the present frequency transfer comparison via optical fiber provides stabilities as high as 10⁻¹⁹ level, this approach is prospective to determine geopotential difference with an equivalent accuracy of 1.5 cm. In addition, since points P and Q are quite arbitrary, this approach may provide an alternative way to determine the geopotential over a continent, and prospective potential to unify a regional height datum system.

Keywords

gravity frequency shift / optical fiber frequency transfer / optical clock / gravity potential

References

Akatsuka T, Takamoto M, Katori H. Optical Lattice Clocks with Non-Interacting Bosons and Fermions. Nature Physics, 2008, 4(12): 954-959.

Bjerhammar A. On a Relativistic Geodesy. Bulletin Géodésique, 1985, 59(3): 207-220.

Bloom B J, Nicholson T L, Williams J R, . An Optical Lattice Clock with Accuracy and Stability at the 10⁻¹⁸ Level. Nature, 2014, 506(7486): 71-75.

Chou C W, Hume D B, Koelemeij J, . Frequency Comparison of Two High-Accuracy Al+ Optical Clocks. Physical Review Letters, 2010, 104 7 070802

Chou C W, Hume D B, Rosenband T, . Optical Clocks and Relativity. Science, 2010, 329(5999): 1630-1633.

Diddams S A, Bergquist J C, Jefferts S R, . Standards of Time and Frequency at the Outset of the 21st Century. Science, 2004, 306(5700): 1318-1324.

Diddams S A, Udem T, Bergquist J C, . An Optical Clock Based on a Single Trapped ¹⁹⁹Hg+ Ion. Science, 2001, 293(5531): 825-828.

Droste S, Ozimek F, Udem T, . Optical-Frequency Transfer over a Single-Span 1 840 km Fiber Link. Physical Review Letters, 2013, 111 11 110801

Dziewonski A M, Anderson D L. Preliminary Reference Earth Model. Physics of the Earth and Planetary Interiors, 1981, 25(4): 297-356.

Flury J. Relativistic Geodesy. Journal of Physics Conference Series, 2016, 723 1 012051

Grosche G, Terra O, Predehl K, . Optical Frequency Transfer via 146 km Fiber Link with 10⁻¹⁹ Relative Accuracy. Optics Letters, 2009, 34(15): 2270-2272.

Grotti J, Koller S, Vogt S, . Geodesy and Metrology with a Transportable Optical Clock. Nature Physics, 2018, 14(5): 437-441.

Guena J, Abgrall M, Rovera D, . Progress in Atomic Fountains at LNE-SYRTE. Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on, 2012, 59(3): 391-409.

Heiskanen W A, Moritz H. Physical Geodesy, 1967, San Francisco: Freeman and Company

Hinkley N, Sherman J A, Phillips N B, . An Atomic Clock with 10⁻¹⁸ Instability. Science, 2013, 341(6151): 1215-1218.

Hofmann-Wellenhof, B., Moritz, H., 2006. Physical Geodesy. Springer

Huntemann N, Okhapkin M, Lipphardt B, . High-Accuracy Optical Clock Based on the Octupole Transition in ¹⁷¹Yb+. Physical Review Letters, 2012, 108 9 090801

Jiang H, Kéfélian F, Crane S, . Long-Distance Frequency Transfer over an Urban Fiber Link Using Optical Phase Stabilization. Journal of the Optical Society of America B, 2008, 25(12): 2029-2035.

Katila T, Riski K J. Measurement of the Interaction between Electromagnetic Radiation and Gravitational Field Using ⁶⁷Zn Mössbauer Spectroscopy. Physics Letters A, 1981, 83(2): 51-54.

Katori H. Optical Lattice Clocks and Quantum Metrology. Nature Photonics, 2011, 5(4): 203-210.

Kéfélian F, Lopez O, Jiang H F, . High-Resolution Optical Frequency Dissemination on a Telecommunications Network with Data Traffic. Optics Letters, 2009, 34(10): 1573-1575.

Li W Y, Liu Y X, Li B, . Hydrocarbon Exploration in the South Yellow Sea Based on Airborne Gravity, China. Journal of Earth Science, 2016, 27(4): 686-698.

Lion G I, Panet I, Wolf P, . Determination of a High Spatial Resolution Geopotential Model Using Atomic Clock Comparisons. Journal of Geodesy, 2017, 91(6): 597-611.

Lisdat C, Grosche G, Quintin N, . A Clock Network for Geodesy and Fundamental Science. Nature Communications, 2016, 7 12443

Lopez O, Haboucha A, Chanteau B, . Ultra-Stable Long Distance Optical Frequency Distribution Using the Internet Fiber Network. Optics Express, 2012, 20 21 23518

Lopez O, Kanj A, Pottie P E, . Simultaneous Remote Transfer of Accurate Timing and Optical Frequency over a Public Fiber Network. Applied Physics B, 2013, 110(1): 3-6.

Ludlow A D, Zelevinsky T, Campbell G K, . Sr Lattice Clock at 1×10⁻¹⁶ Fractional Uncertainty by Remote Optical Evaluation with a Ca Clock. Science, 2008, 319(5871): 1805-1808.

Ma L S, Bartels A, Robertsson L, . Optical Frequency Synthesis and Comparison with Uncertainty at the 10⁻¹⁹ Level. Science, 2004, 303(5665): 1843-1845.

Ma L S, Jungner P, Ye J, . Delivering the Same Optical Frequency at Two Places: Accurate Cancellation of Phase Noise Introduced by an Optical Fiber or other Time-Varying Path. Optics Letters, 1994, 19(21): 1777-1779.

Madej A A, Dubé P, Zhou Z C, . ⁸⁸Sr+ 445-THz Single-Ion Reference at the 10⁻¹⁷ Level via Control and Cancellation of Systematic Uncertainties and Its Measurement against the SI Second. Physical Review Letters, 2012, 109 20 203002

Mai E. Time, Atomic Clocks, and Relativistic Geodesy, 2013, München: Verlag der Bayerischen Akademie der Wissenschaften

Marra G, Slavík R, Margolis H S, . High-Resolution Microwave Frequency Transfer over an 86-km-Long Optical Fiber Network Using a Mode-Locked Laser. Optics Letters, 2011, 36 4 511

Müller H, Peters A, Chu S. A Precision Measurement of the Gravitational Redshift by the Interference of Matter Waves. Nature, 2010, 463(7283): 926-929.

Newbury, N. R., Swann, W. C., Coddington, I., et al., 2007a. Fiber Laser-Based Frequency Combs with High Relative Frequency Stability. Frequency Control Symposium, 2007 Joint with the 21st European Frequency and Time Forum. IEEE International. 980–983. https://doi.org/10.1109/FREQ.2007.4319226

Newbury N R, Williams P A, Swann W C. Coherent Transfer of an Optical Carrier over 251 km. Optics Letters, 2007, 32(21): 3056-3058.

Pound R V, Rebka G A Jr.. Gravitational Red-Shift in Nuclear Resonance. Physical Review Letters, 1959, 3(9): 439-441.

Pound R V, Rebka G A Jr.. Attempts to Detect Resonance Scattering InZn67; The Effect of Zero-Point Vibrations. Physical Review Letters, 1960, 4(8): 397-399.

Pound R V, Rebka G A Jr.. Variation with Temperature of the Energy of Recoil-Free Gamma Rays from Solids. Physical Review Letters, 1960, 4(6): 274-275.

Pound R V, Snider J L. Effect of Gravity on Gamma Radiation. Physical Review, 1965, 140(3B): B788-B803.

Predehl K, Grosche G, Raupach S M F, . A 920-Kilometer Optical Fiber Link for Frequency Metrology at the 19th Decimal Place. Science, 2012, 336(6080): 441-444.

Primas, L. E., Lutes, G. F., Sydnor, R. L., 1988. Fiber Optic Frequency Transfer Link. Proceedings of 42nd Annual Symposium on Frequency Control, June 1–3, 1988, Baltimore, MD. 478–484

Raupach, S. M. F., Grosche, G., 2013. Chirped Frequency Transfer with an Accuracy of 10⁻¹⁸ and Its Application to the Remote Synchronization of Timescales. arXiv: 1308.6725v2 [physics.optics] (2013-9-30)

Raupach S M F, Koczwara A, Grosche G. Optical Frequency Transfer via a 660 km Underground Fiber Link Using a Remote Brillouin Amplifier. Optics Express, 2014, 22(22): 26537-26547.

Rosenband T, Hume D B, Schmidt P O, . Frequency Ratio of Al+ and Hg+ Single-Ion Optical Clocks, Metrology at the 17th Decimal Place. Science, 2008, 319(5871): 1808-1812.

Shen W-B. Relativistic Physical Geodesy: [Dissertation], 1998, Graz: Graz Technical University

Shen, W.-B., 2013a. Orthometric Height Determination Based upon Optical Clocks and Fiber Frequency Transfer Technique. 2013 Saudi International Electronics, Communications and Photonics Conference (SIECPC), April 27–30, 2013, Riyadh, Saudi Arabia. https://doi.org/10.1109/SIECPC.2013.6550987

Shen W-B. Orthometric Height Determination Using Optical Clocks. EGU General Assembly Conference Abstracts, 2013, 15 5214.

Shen W-B, Chao D, Jin B. On Relativistic Geoid. Bollettino di Geodesia e Scienze Affini, 1993, 52(3): 207-216.

Shen W-B, Ning J S, Chao D B, . A Proposal on the Test of General Relativity by Clock Transportation Experiments. Advances in Space Research, 2009, 43(1): 164-166.

Shen W-B, Ning J S, Liu J N, . Determination of the Geopotential and Orthometric Height Based on Frequency Shift Equation. Natural Science, 2011, 3(5): 388-396.

Shen W.-B., Peng, Z., 2012. Gravity Potential Determination Using Remote Optical Fiber. International Symposium on Gravity, Geoid and Height Systems GGHS 2012. Dec. 3, 2012, Venice, Italy

Shen Z Y, Shen W-B, Zhang S X. Formulation of Geopotential Difference Determination Using Optical-Atomic Clocks Onboard Satellites and on Ground Based on Doppler Cancellation System. Geophysical Journal International, 2016, 206(2): 1162-1168.

Shen Z Y, Shen W-B, Zhang S X. Determination of Gravitational Potential at Ground Using Optical-Atomic Clocks on Board Satellites and on Ground Stations and Relevant Simulation Experiments. Surveys in Geophysics, 2017, 38(4): 757-780.

Snider J L. New Measurement of the Solar Gravitational Red Shift. Physical Review Letters, 1972, 28(13): 853-856.

Soffel M, Herold H, Ruder H, . Relativistic Geodesy: The Concept of Asymptotically Fixed Reference Frames. Manuscr. Geod, 1988, 13(3): 139-142.

Soffel M, Herold H, Ruder H, . Relativistic Theory of Gravimetric Measurements and Definition of the Geoid. Manuscr. Geod, 1988, 13: 143-146.

Takano T, Takamoto M, Ushijima I, . Geopotential Measurements with Synchronously Linked Optical Lattice Clocks. Nature Photonics, 2016, 10(10): 662-666.

Tenzer R, Bagherbandi M. Theoretical Deficiencies of Isostatic Schemes in Modeling the Crustal Thickness along the Convergent Continental Tectonic Plate Boundaries. Journal of Earth Science, 2016, 27(6): 1045-1053.

Turneaure J P, Will C M, Farrell B F, . Test of the Principle of Equivalence by a Null Gravitational Red-Shift Experiment. Physical Review D, 1983, 27(8): 1705-1714.

Ushijima I, Takamoto M, Das M, . Cryogenic Optical Lattice Clocks. Nature Photonics, 2015, 9(3): 185-189.

Vessot R F C, Levine M W. A Test of the Equivalence Principle Using a Space-Borne Clock. General Relativity and Gravitation, 1979, 10(3): 181-204.

Vessot R F C, Levine M W, Mattison E M, . Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser. Physical Review Letters, 1980, 45(26): 2081-2084.

Wada M, Watabe K-I, Okubo S, . A Precise Frequency Comparison System Using an Optical Carrier. Electronics and Communications in Japan, 2015, 98: 19-27.

Weinberg S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, 1972, New York: Wiley

Ye J, Peng J-L, Jones R J, . Delivery of High-Stability Optical and Microwave Frequency Standards over an Optical Fiber Network. Journal of the Optical Society of America B, 2003, 20 7 1459