A Clock Directly Linking Time to a Particle's Mass

A particle with mass-energy E = mc² is represented by a wave oscillating at the Compton frequency ω₀ = mc²/ħ in the particle's rest frame, where c is the speed of light and ħ is Planck's constant h divided by 2π (1). This is the basis of de Broglie's theory of matter waves (2) and underpins modern quantum mechanics and field theory: The time evolution of states is given by wave equations whose plane-wave solutions are proportional toe^–iϕ = exp(–ip_μx^μ/ħ) = exp(–iω₀τ) (1)where p_μ = (–mγ, mγv) and x^μ are the momentum and position four-vector, τ = t/γ is the proper time, γ is the Lorentz factor, and v and t are the laboratory-frame velocity and time. Much has been theorized about the physical reality of quantum states as "oscillators" (3–7), but surprisingly few experiments have been proposed to address this topic (8). Here, we directly address a consequence of Eq. 1 that has deep physical and perhaps even cosmological implications: Because the oscillations of a wave packet accumulate phase ω₀τ just like a clock following the same trajectory, it has been suggested that the passage of time is observable as soon as the universe contains massive particles (1). It ought to be possible to build a Compton clock: a clock referenced to the mass of a single particle.

Advances in timekeeping (9–12) have followed the use of references with higher frequency, increased quality factor Q, and lowered instability from systematic influences. The Compton frequency of a stable particle is high (ω₀/2π = 3 × 10²⁵ Hz for a cesium atom), has virtually infinite quality factor Q = ω₀/Γ, where 1/Γ is the particle's 1/e lifetime, and is one of the most stable quantities in nature. In principle, a Compton-frequency clock could be built by annihilating a particle-antiparticle pair and counting the frequencies of the generated photons. However, this frequency is far beyond modern counting techniques; to access it, we require a method of dividing it into a technically accessible range.

Here, we present a Compton clock by combining an atom interferometer with an optical frequency comb. In a Ramsey-Bordé atom interferometer (Fig. 1) (7, 13), a quantum wave packet initially at rest interacts with a pulse of two counterpropagating laser beams having laboratory-frame frequencies of ω_± and wavenumbers of k_±, respectively. The wave packet absorbs n photons from the first beam while being stimulated to emit n photons into the second beam, without modifying its internal state through multiphoton Bragg diffraction (14, 15). By appropriate choice of the laser intensity and pulse duration, the process occurs with a probability of 50%, splitting the wave packet. While the first partial wave packet remains at rest, the second moves away with a laboratory-frame momentum p = nħ(k₊ + k_­–). It is convenient to introduce a rapidity parameter ρ so that p = mcsinh(ρ) and the Lorentz factor γ = [1 – v²/c²]^–1/2 = cosh(ρ). Energy-momentum conservation (the Bragg condition) provides the relation ω_± = ω_Le^±ρ/2 (2)where ω_L = ω₀sinh(ρ/2)/n. [In the nonrelativistic limit, ω_L ≈ (ω₊ + ω_–)/2 ≡ kc and ω_± ≈ ω_L(1 ± nħk/mc).] Interaction with four laser pulses (Fig. 1) brings the wave packets together for interference. The probability of detecting the particle at the outputs (Fig. 1) is given by cos²(Δϕ/2), where Δϕ is the relative phase of the wave packets. In a relativistic, semiclassical treatment, Eq. 1 shows that the wave packets acquire a relative phase of Δϕ_free = ω₀[τ⁽¹⁾ – τ⁽²⁾] = 2ω₀T(γ⁻¹ – 1) (nonrelativistically, Δϕ_free ≈ –4n²ħk²T/m) during their free evolution between the laser pulses (3), where τ^(1,2) are the proper times on paths 1 and 2. Their difference τ⁽¹⁾ – τ⁽²⁾ is related to the laboratory-frame time by a factor γ⁻¹ – 1, which is determined by the laser frequencies by Eq. 2 and can be used to divide the Compton frequency into a technically accessible range. Synchronizing an oscillator to the divided Compton frequency is accomplished as follows: In the experiment, this oscillator sets the frequency difference ω_m ≡ ω₊ – ω_– between the counterpropagating laser beams. Whenever the particle absorbs a photon, the phase of the photon is added to the wave packet's phase and subtracted for stimulated emission. It can be shown (16) that the sum Δϕ_laser of these phases (eq. S2) cancels the free evolution phase Δϕ_free exactly if, and only if, ω_± satisfy Eq. 2, so that ωm=2nωL2ω0(3)The phase cancellation Δϕ = Δϕ_laser + Δϕ_free = 0 is verified by observing the interference fringes of the particle at the interferometer output and maintained by feedback to the oscillator.

• Download high-res image
• Open in new tab
• Download Powerpoint

Ramsey-Bordé interferometers are routinely used to measure ħ/m (17), but such measurements of ħ/m are not themselves clocks because they are reliant upon an external frequency reference to measure ω_L. We use an optical frequency comb to set ω_L = Nω_m, eliminating the laser frequency as a free parameter. Combined with Eq. 3, we obtain ωm=ω0(2nN2)(4)which is an exact result in this relativistic and semiclassical picture, showing that ω_m is determined solely by the Compton frequency. Corrections from nonclassical paths are negligible here. We are free to choose any value for n and N and lock the oscillator directly to the Compton frequency or a subharmonic without changing the underlying physics. For example, n = 2 and N = 1/2 yields ω₀ = ω_m = 2ω_L and ω_± = ω_L(1 + 2)^±1. In practice, N is chosen so that ω_L and ω_m are accessible to existing technologies. The experiment is a Compton clock by virtue of the fact that it self-references the laser frequency through the optical frequency comb, which ensures that the output frequency ω_m is fully determined by the Compton frequency and known ratios. It is independent of any external standards. The self-referenced atom interferometer serves as a clockwork that links the frequency of our master oscillator to the Compton frequency of the matter wave packets.

Our experiment uses cesium-133 atoms because they are amenable to the techniques of atom optics and are electrically neutral, which reduces systematic effects. Because their internal structure is not relevant to ω_m, the atoms approximate noninteracting point masses. Our atom interferometer has been described in detail elsewhere (15). The clock's operation is illustrated in Fig. 2. A voltage-controlled 10-MHz crystal oscillator O1 [FTS 1050A (Symmetricom, Beverly, MA)] with frequency ω_cryst is the master oscillator for all frequency sources, pulse generators, and lasers involved in the experiment. The frequency comb stabilizes the laser frequency to N_cω_cryst, where N_c = 35,173,594.165 (16). A direct digital synthesizer (DDS) generates ω_m/2 = N_DDSω_cryst at ~82 kHz, and acousto-optical modulators (AOMs) generate ω_± by applying the relation ω_± = ω_L ± ω_m/2 (valid in the nonrelativistic limit), where N_DDS = 2,326,621,801,616/2⁴⁸. The factor N is thus N = N_c/N_DDS = 4,255,305,521.31286. Another AOM shapes the time domain profile of the laser pulses. Our signal to noise is optimal for 5th-order Bragg diffraction (n = 5) with a pulse separation time (Fig. 1) T = 160 ms. The atomic population at the interferometer output is determined by means of fluorescence detection and then processed to yield the interferometer phase Δϕ = Δϕ_laser + Δϕ_free.

• Download high-res image
• Open in new tab
• Download Powerpoint

A feedback loop adjusts ω_cryst so as to maintain Δϕ = 0. Its operation is demonstrated in Fig. 3: The first 10 data points are taken during normal operation. At the 11th data point, we briefly disable the feedback, so that ω_cryst is free-running, and the experiment performs a Ramsey-Bordé photon recoil measurement. The measured signal is then proportional to ω_L² = (N_cω_cryst)² (Eq. 3). Increasing N_c by 100 parts per billion (ppb) increases ω_L and produces a 200-ppb increase in Δϕ_free (Fig. 3, shaded area). At the 13th data point, the feedback is restored. Because the feedback stabilizes ω_cryst to a value inversely proportional to N_c² (Eq. 4), the signal frequency decreases by 400 ppb, equilibrating 200 ppb lower than the first 10 data points (Fig. 3), which is opposite to the observed shift for a recoil measurement.

• Download high-res image
• Open in new tab
• Download Powerpoint

Quantitative agreement is demonstrated by comparing the rest-mass stabilized frequency ω_cryst with a rubidium frequency standard [SRS FS725 (Stanford Research Systems, Sunnyvale, CA)] over 6 hours (Fig. 4A). It averages to 9,999,998.127 Hz with a SD of the mean of 0.015 Hz. A χ² test (Fig. 4B) yields a normalized χ² = 1.4. Our statistical uncertainty is 0.015 Hz × (χ²)^1/2 = 0.018 Hz, or 1.8 ppb. Using Eq. 4 and correcting for systematic effects (16), we obtain the Compton frequency ω₀/2π = (2,993,486,252 ± 12) × 10¹⁶ Hz. The deviation from the expected values (16, 18) is –5.2 ± 4.0 ppb and is consistent with zero within 2σ. The Allan variance (Fig. 4C) is below 10⁻⁸/[τ/(1000 s)]^1/2, where τ is the integration time.

• Download high-res image
• Open in new tab
• Download Powerpoint

Although this accuracy is modest, similar to the first cesium atomic clocks (9), the resolution and accuracy of atom interferometry is advancing rapidly (19). Up to 100-fold improvement in resolution could be obtained by approaching the shot noise limit of our interferometer. Use of higher laser frequencies (which is possible without changing the reference particle), lighter particles, and/or longer interrogation times can possibly lead to a Compton clock that can serve as a primary time standard. The intrinsic stability and Q factor of a stable particle's Compton frequency is unmatched by any other frequency reference. Mass, frequency, time, and length could all be derived from one fundamental unit, defined by a specific particle.

The clock can also be used for the opposite purpose, measuring mass by measuring the Compton frequency. In 2011, the General Conference on Weights and Measures (CGPM-2011) considered a revision to the SI units that would assign an exact value to the Planck constant h (20). The kilogram would then be referenced to the second through the defined values of the Planck constant and the speed of light. Microscopic masses could be related to the fine structure constant (21, 22) or h/M (17); macroscopic masses could be measured by using the Watt balance (23, 24). These methods, however, require auxiliary measurements and/or intricate theory (16). The Compton clock would provide an absolute measurement of the cesium atom's mass to an accuracy of 4.0 ppb, which is competitive with other methods and greater than 10 times more accurate than in the present SI (18). Other microscopic masses can be related to the cesium mass by means of mass spectroscopy.

The link to macroscopic masses can be made by Avogadro spheres: silicon crystals of accurately measured volume V and lattice constant a (25). In theory, the number of atoms contained in such a crystal is N_at = 8V/a³, where 8 is the number of atoms in the unit cell. Because binding energies are negligible, the Compton frequency of the sphere is ω_M = [m(Si)/m(¹³³Cs)]N_at ω_Cs, given by the measured Compton frequency of cesium atoms ω_Cs. This yields its mass as M = ω_Mħ/c². The ratio m(Si)/m(¹³³Cs) is between the effective molar mass of the sphere's material and cesium-133. According to (25), present data yields the spheres' mass with an overall accuracy of 30 ppb so that they would constitute the most accurately calibrated macroscopic masses under the proposed CGPM-2011 redefinition—a testament to the precision achieved in constructing Avogadro spheres.

Although any method for measuring microscopic mass can be used, the Compton clock offers a transparent connection between the second and a microscopic mass on the basis of simple physical principles without requiring auxiliary measurements. It directly realizes a long-standing proposal to measure mass in terms of the Compton frequency (6). The method outlined here offers a different set of systematic effects as compared with Watt balances (23, 24), thus serving as an important test of the overall consistency of the laws of physics and experimental methods. It is based on a body's inertial rather than gravitational mass. In the context of the present SI, the combination of the Avogadro project and the Compton clock serves as a measurement of the Planck constant.

Looking forward, the rapidly developing field of optomechanics (26) might enable measurements of the single-photon recoil energy of a nanomechanical mirror. This could result in a clock referenced to the mass of a mesoscopic object, or a mesoscopic mass standard. Because Bragg diffraction of electrons has already been demonstrated (27), a clock using elementary particles or even antiparticles is possible. Such clocks would be useful for testing CPT symmetry or the Einstein Equivalence Principle for antimatter.

We have demonstrated a clock stabilized to the rest mass of a particle. It highlights the intimate connection between frequency and mass. It proves that massive particles can serve as a frequency reference without requiring their mass to be converted to energy as an explicit illustration of a key principle of quantum mechanics. Furthermore, we have shown that a single massive particle is sufficient to measure time.