## Abstract

Observers in relative motion or at different gravitational potentials measure disparate clock rates. These predictions of relativity have previously been observed with atomic clocks at high velocities and with large changes in elevation. We observed time dilation from relative speeds of less than 10 meters per second by comparing two optical atomic clocks connected by a 75-meter length of optical fiber. We can now also detect time dilation due to a change in height near Earth’s surface of less than 1 meter. This technique may be extended to the field of geodesy, with applications in geophysics and hydrology as well as in space-based tests of fundamental physics.

Albert Einstein’s theory of relativity forced us to alter our concepts of reality. One of the more startling outcomes of the theory is that we have to give up our notions of simultaneity. This is manifest in the so-called twin paradox (*1*), in which a twin sibling who travels on a fast-moving rocket ship returns home younger than the other twin. This “time dilation” can be quantified by comparing the tick rates of identical clocks that accompany the traveler and the stationary observer. Another consequence of Einstein’s theory is that clocks run more slowly near massive objects. In the range of speeds and length scales encountered in our daily life, relativistic effects are extremely small. For example, if two identical clocks are separated vertically by 1 km near the surface of Earth, the higher clock emits about three more second-ticks than the lower one in a million years. These effects of relativistic time dilation have been verified in several important experiments (*2*–*6*) and are accounted for routinely in satellite-based navigation systems (*7*). The most accurate determinations historically involve velocities near the speed of light (*8*) and changes in elevation of 10^{4} to 10^{7} m (*3*, *4*). Previously, small relativistic shifts (<10^{−16}) could be observed only over short distances in γ-ray Mössbauer spectroscopy (*5*, *9*) and atom interferometry (*6*). However, they might be detected over longer distances by clocks with sufficiently high sensitivity, such as accurate atomic clocks operating in the optical regime, or “optical clocks.” Here we report the detection of relativistic time dilation due to velocities of several meters per second and, separately, due to a change in height of 0.33 m by comparing two optical clocks based on ^{27}Al^{+} ions.

With operating frequencies in the petahertz (1 PHz = 10^{15} Hz) range and natural linewidths at the millihertz level, optical clocks have demonstrated substantial improvements in stability and accuracy over the current microwave time and frequency standards (*10*). We compared two optical atomic clocks based on individual trapped Al^{+} ions, with reported systematic frequency uncertainties of 8.6 × 10^{−18} (*11*) and 2.3 × 10^{−17} (*12*). For comparison, the lowest reported frequency uncertainty of Cs fountain clocks is 3.4 × 10^{−16} (*13*). With the accuracy and the associated sensitivity of these Al^{+} optical clocks, frequency variations below 10^{−16} in the clocks due to relativistic effects can be observed.

Due to the lack of an accessible optical transition in ^{27}Al^{+} for efficient laser cooling and state detection, precision spectroscopy of these ions uses techniques developed in quantum information science. Here, an Al^{+} ion is sympathetically cooled through its Coulomb interaction with an auxiliary “logic” ion that is simultaneously held in the same linear radio-frequency (RF) Paul trap (*14*). The logic ion also helps prepare and detect the internal state of the Al^{+} ion via quantum logic protocols. In this work, the two Al^{+} clocks used a beryllium (^{9}Be^{+}) ion (*14*) and a magnesium (^{25}Mg^{+}) ion (*11*), respectively, as the logic ion. The Al^{+ 1}S_{0}↔^{3}P_{0} clock transition with frequency *f*_{0} near 1.121 PHz has a narrow (∆*f* = 8 mHz) natural linewidth and a corresponding intrinsic quality (Q) factor of *f*_{0}/∆*f* = 1.4 × 10^{17} that permits high sensitivity for detecting small frequency-shifting effects. However, the observed linewidth for the clock transition is limited by the linewidth of the probe laser. We probed the clock transition with a subhertz linewidth laser referenced to a high-finesse optical cavity (*15*). In the Al-Mg clock, with 300 ms probe duration we obtained a narrow, Fourier transform–limited linewidth, realizing a Q factor of 4.2 × 10^{14} with nearly 80% contrast (Fig. 1). This high-Q line provides the basis for high-stability clock operation and sensitivity to small frequency shifts.

The two Al^{+} optical clocks were located in separate laboratories and were compared by transmitting the stable clock signal through a 75-m length of phase-stabilized optical fiber. To observe time dilation due to motion, we set the Al^{+} ion in the Al-Mg clock in motion by applying a small static electric field that shifts the position of the ion slightly away from the null of the confining RF field (*16*). The ion is thereby subject to an RF electric field at *f*_{RF} = 59 MHz (Fig. 2) and undergoes harmonic motion. The ion velocity is adjusted by varying the applied static electric field. In the language of the twin paradox, the moving Al^{+} ion is the traveling twin, and its harmonic motion amounts to many round trips. Time dilation from this motion leads to a fractional frequency shift for the moving clock of (*17*)
*v*_{||} is the velocity of the Al^{+} ion along the wave vector of the probe laser beam *c* is the speed of light, *v* is the ion’s velocity with respect to the laboratory reference frame, and *f*_{0} is the ion’s proper resonant frequency. Angle brackets denote time averages. Because the induced Al^{+} ion motion is harmonic, its contribution to 〈*v*_{||}〉 averages to zero; therefore, any observed change in the ion's transition frequency is due to a change in γ and corresponds to relativistic time dilation (*18*). For *v*/*c* << 1, Eq. 1 can be approximated by δ*f*/*f*_{0} ≈ −〈*v*^{2}〉/2*c*^{2} (*17*). We measured the frequency difference between the two clocks (δ*f*/*f*_{0}) while varying the velocity of the ion motion. The experimental results, which confirm the prediction of Eq. 1, are plotted in Fig. 2.

Differences in gravitational potential can be detected by comparing the tick rate of two clocks. For small height changes on the surface of Earth, a clock that is higher by a distance ∆*h* runs faster by*g* ≈ 9.80 m/s^{2} is the local acceleration due to gravity (*4*). The gravitational shift corresponds to a clock shift of about 1.1 × 10^{−16} per meter of change in height. To observe this shift, we first compared the frequencies of the two Al^{+} clocks at the original height difference of ∆*h* = *h*(Mg-Al) − *h*(Be-Al) = −17 cm, which was measured with a laser level. Then we elevated the optical table on which the Mg-Al clock was mounted, supporting it on platforms that increased the height by 33 cm, and compared the frequencies again. The two measurements consist of approximately 100,000 s of low-height data and 40,000 s of high-height data, and the clocks exhibit (Fig. 3) a fractional frequency change of (4.1 ± 1.6) × 10^{−17}. When this shift is interpreted as a measurement of the change in height of the Al-Mg clock, the result of 37 ± 15 cm agrees well with the known value of 33 cm.

Although ideally 〈*v*_{||}〉 = 0, small linear velocities of the Al^{+} ions can occur because of effects such as slow electrical charging of insulating material in the trap. From Eq. 1, the clock’s frequency (that is, the frequency of the probe laser locked to the moving ion’s clock transition) exhibits a fractional frequency shift^{+} ion is moving at an average velocity 〈*v*_{||}〉 in the propagation direction of a probe laser beam. In the comparison measurements between the Al^{+} clocks, the Doppler effect was carefully constrained by alternate use of probe laser beams counterpropagating with respect to each other (*11*). Any motion of the ion is detected as a difference in the transition frequencies measured by the two laser beams. In the Al-Mg clock, we observed a fractional frequency difference of (1.2 ± 0.7) × 10^{−17} between the two probe directions, which corresponds to the ion moving at a speed of (1.8 ± 1.1) nm/s in the lab frame. However, the clock rate is not significantly affected by a velocity of this magnitude, because it is derived from an average of the two opposite laser-probe directions.

Small relativistic effects reported here have been observed with optical atomic clocks of unprecedented precision and accuracy. With improved accuracy, the sensitivity of optical clocks to small variations in gravitational potential might find applications in geodesy (*19*, *20*), hydrology (*21*), and tests of fundamental physics in space (*22*). The basic components for clock-based geodetic measurements were demonstrated here by comparing two accurate Al^{+} optical clocks through 75 m of noise-canceled fiber and measuring height-dependent clock shifts. In clock-based geodesy (*23*, *24*), accurate optical clocks would be linked to form a network of “inland tide gauges” (*25*) that measure the distance from Earth’s surface to the geoid: the equipotential surface of Earth’s gravity field that matches the global mean sea level. Such a network could operate with high temporal (daily) and geospatial resolution at the clock locations. It would therefore complement geodetic leveling networks, whose update period is typically 10 years or longer, as well as biweekly satellite-generated global geoid maps.

For a network to be useful, clock accuracy must be improved to 10^{−18} or better (*26*–*28*) to allow for height measurements with 1-cm uncertainty. In Al^{+} clocks, improved control of the ion motion is needed to reduce the uncertainty of motional time dilation, and issues of reliability must be addressed, so that the clocks can operate unattended for long periods. High-quality links are also needed to connect the optical clocks. Realistic link demonstrations with telecommunications fiber akin to the links used in this work have shown that optical frequencies can be transmitted across fiber lengths of up to 250 km with inaccuracy below 10^{−18} (*29*–*31*), and continent-scale demonstrations are in progress (*30*). However, intercontinental links may require the faithful transmission of optical carrier frequencies to satellites through the atmosphere, and this is an unsolved problem under active investigation (*32*, *33*).