Accelerating the averaging rate of atomic ensemble clock stability using atomic phase lock

Much effort has been invested in improving the stability of atomic clocks, which were first demonstrated more than 50 years ago [1]. The stability of a clock, denoted by σ_y (where y is a fractional frequency), is expressed as [2]

$$\begin{eqnarray}&&{{\sigma }_{y}}=\frac{1}{K\cdot Q\cdot {\rm SNR}\cdot \sqrt{n}}=\frac{1}{K\cdot Q\cdot {\rm SNR}}\sqrt{\frac{{{T}_{c}}}{\tau }}\end{eqnarray} \tag{ 1 }$$

where K is a constant of the order unity that depends on the spectrum shape, Q is the quality factor, n is the total number of measurements, T_c is the cycle time and τ denote the total measurement time. SNR is the signal-to-noise ratio of a single measurement of an atomic population ratio. The measurement noise in the SNR may consist of the technical error of detection, quantum projection, fluctuating local oscillator (LO) frequency, fluctuating atomic frequency shifts and loss of atomic coherence. The technical noise is a combination of many noises that are related to signal detection, including laser power and frequency fluctuation, photon shot noise, atom number fluctuations, and electronics noise. In this paper, we mainly discuss technical noise, quantum projection noise (QPN) and LO noise, assuming that all other noises are sufficiently smaller. The description of clock stability expressed in equation (1) improves the clock stability with ${{\tau }^{-1/2}}$ scaling. When the measurement noise is dominated by technical noise or QPN, the stability line expressed as equation (1) is hereafter referred to as the 'technical noise limit,' or 'QPN limit,' respectively. Figure 1 illustrates the stability of the technical limit and QPN limit, assuming the technical limit is larger than the QPN limit.

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From equation (1), we observe that stability can be improved by (1) increasing the Q of the resonance by increasing the probing time or carrier frequency, (2) increasing the SNR and (3) averaging over many measurements (i.e. increasing the n). Past improvements have focused on strategies (1) and (2). For example, the Q of the optical ion clock developed by Bergquist et al [3] was several orders of magnitude higher than that of microwave clocks. Katori et al [4] demonstrated an optical lattice clock that simultaneously increases the Q and SNR. Squeezed [5] or entangled states [6] were proposed to improve the technical noise, reducing it to below the QPN limit [7]. We need to be careful that equation (1) does not account for the effect of the dead time, known as the Dick effect [8]. Dead time is the time expended in any process other than for probing an atom with microwaves. The stability is limited by the Dick effect when the frequency noise of the LO is large compared to the technical noise [7] and the dead time is significant. In this case, reducing the dead time reduces the σ_y by ${{\sigma }_{y}}\propto {{\tau }^{-1}}$ until the SNR limit line is reached (the blue broken line in figure 1(a)). This idea [9] has been recently demonstrated [10].

To accelerate the averaging rate of strategy (3), we have proposed an atomic phase lock (APL) [11]. Principally, the atomic phase lock lowers stability at the fastest rate τ⁻¹ by genuinely monitoring the phase of the atom. Provided that the atomic phase remains coherent and is monitored as such, the stability of the atomic clock should improve by τ⁻¹. However, the stability of atomic clocks normally improves by ${{\tau }^{-1/2}}$ even when employing the Ramsey sequence [12], which measures the atomic phase. This trend occurs because the atomic phase is destroyed (and initialized) after each projection measurement cycle. If the atoms could maintain their phase coherence over many measurement cycles, the stability would reduce much more rapidly (as τ⁻¹), and the stability could be expressed as

$$\begin{eqnarray}&&\sigma _{y}^{APL}=\frac{1}{K\cdot Q\cdot {\rm SNR}\cdot n}\sim \frac{1}{K\cdot {{f}_{0}}\cdot {\rm SNR}\cdot \tau }.\end{eqnarray} \tag{ 2 }$$

where σ_y ^APL is the frequency stability under maintenance of the atomic phase coherence.

Figure 1(a) shows the typical stability of an atomic clock limited by LO noise. In the presence of significant dead time, the stability is limited by the Dick limit [8]. As mentioned in the previous paragraph, if the dead time is sufficiently small, then the stability reduces as τ⁻¹ until it reaches the SNR limit. This means that when limited by LO noise (figure 1(a)), it is not an ideal situation to demonstrate the APL, because observing the τ⁻¹ could just mean the dead time is small enough to detect.

In order to avoid this ambiguity, we have decided to stay in the technical noise limit, where technical noise is larger than LO noise (figure 1(b)), for demonstration of APL. In this case, the observation of the τ⁻¹ dependency only comes from the effect of APL and not from elimination of the Dick effect.

We developed a method that projects only a part of the atoms in order to maintain the coherence of an atomic phase for multiple cycles. We call this method a partial projection measurement (PPM). In principle, the stability ends up at the same value whether you perform a projection measurement part by part or all at once. In other words, projecting $1/{{n}_{cp}}$ part of atoms at a time maintains the coherence of phase for up to n_cp cycles and reduces the stability at its fastest rate by τ⁻¹ for the same n_cp cycles, but SNR is reduced by a factor $1/\sqrt{{{n}_{cp}}}$ in return. Therefore, APL using a PPM results in the same stability as the conventional method where a projection measurement is performed once for all the atoms. In this sense, this paper demonstrates a proof-of-principle experiment of APL, but not the actual improvement of the overall clock performance. We note, however, that APL opens a way to overcome the technical noise limit when an atomic clockʼs SNR is limited by technical noise. For example, if we trap one million atoms but we can project only 10000 atoms at one time, the stability is limited by an SNR of 100, which is a factor ten worse than the QPN limit for one million atoms. If we introduce APL using a PPM, we maintain the phase lock of LO to atoms for up to 100 cycles of partial projections, and the stability approaches the QPN limit for one million atoms.

This paper experimentally demonstrates the APL. Section 2 describes our experimental setup. We used a single ensemble of ions and performed projection measurements on only a portion of the ions at a time (section 3). Section 4 shows the results of three phase measurements obtained by the proposed method without resetting the atomic states. The stability is reduced by τ⁻¹ instead of ${{\tau }^{-1/2}}$ and the long term stability line is lowered by a factor of $\sqrt{3}$, as expected. Section 5 discusses the application of the method and suggests ideas for further improvement.

2.1. Ion trap

This section describes our experimental setup. Figure 2 shows our linear radio frequency (rf) quadrupole trap, in which four cylindrical rods (diameter = 2 mm) were placed at the corner of a square separated by 4 mm (center to center), and the DC bias plates are spaced by 30 mm. Sinusoidal voltage of 10.15 MHz with an amplitude of 330 V_pp and a constant 50 V were applied on rf rods and DC bias plates, respectively. Ytterbium ions of 171 isotope (¹⁷¹Yb⁺) are selectively trapped by a 399 nm photo-ionization laser (not shown in figure 2) and a 370 nm cooling laser.

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Unintentionally, ions were trapped in two locations, delineated by red ovals in figure 2. Each cloud (3 mm length) contains about 2000 ions. Finite-method-element simulations revealed that our rf rods were much more closely spaced compared to the distance between the two bias plates. At such small separation, the DC potential near the trap center is shielded by the rods. Consequently, a small potential barrier (whose origin is unclear) remains in the center, even with a 50 V DC potential on the bias plates. One of the clouds was used for clock measurements.

2.2. Ion cooling

Once trapped, the ions are cooled to about 50 mK by Doppler laser cooling. This 50 mK temperature was obtained by separately measuring the 370 nm cooling transition broadening [13]. The ions were predominantly cooled by the 370 nm laser (13 μW), and the cooling cycle was closed by combining the 935 nm repump laser (6 mW) with 14 GHz modulation of the 370 nm cooling laser [14] (figure 3). The trap area of the vacuum chamber was covered by a single-layer magnetic shield of shielding factor 15, yielding an interior field of 0.04 Gauss. During microwave proving, this residual field was canceled by three pairs of Helmholtz coils. The dark states were destabilized [15] by a 0.4 Gauss field tilted $45{}^\circ$ from the laser axis using Helmholtz coils. The ions were initialized to the $^{2}{{{\rm S}}_{1/2}}(F=0)$ state by a cooling laser that is phase-modulated at 2.1 GHz by an electro optical modulator.

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2.3. Microwave transition

3.1. Partial projection measurement

We now introduce partial projection measurement (PPM). In PPM, a 16 μW diagonal laser is operated at 370 nm (see figure 2) such that only a portion of the ions interacts with the laser. The waists of the ions and diagonal beams are about equal (approximately 200 μm). The basic measurement sequence is as follows.

• (i)  
  Initialize the atomic state (cooling and pumping to $^{2}{{{\rm S}}_{1/2}}(F=0)$).
• (ii)  
  Manipulate the state with microwaves (details are shown in figures 4 and 5).
• (iii)  
  Perform PPM.
• (iv)  
  repeat Steps (ii) and (iii).

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After the first partial measurement, the projected ions became mixed with unprojected ions during the manipulation period before the next measurement. We noted that our ion temperature was high enough so that the ion cloud never crystalized. The measured population is valid provided that the ratio of unprojected to projected ions exceeded the SNR of the measurement.

For example, when there are 10,000 total ions and 100 ions are measured at an SNR of 10 via PPM, 1% of the total ions are projected at each PPM and will give wrong information about the phase in the next measurement. Since these projected ions scatter in the whole ion cloud during step (ii), the ratio of projected ions increases exponentially as $1-0.{{99}^{{{n}_{cp}}}}$, when one repeats the PPM for n_cp times. Since these projected ions result in phase estimation error, one should limit the n_cp to less than 10 measurement so that projection error is less than the technical noise. In other words, we maintain a constant SNR by terminating the APL before the projection noise exceeds the technical noise.

3.2. Check of PPM with Rabi oscillation signal

3.3. Check of PPM with continuous-phase measurement (modified Ramsey method)

We now test the decoherence due to PPM in phase measurement sequences. To monitor the total phase difference accumulated over multiple measurement cycles, we must modify the Ramsey sequence, as well as the projection measurement. Our modified Ramsey sequence is shown in figure 5(a) and proceeds via the following steps:

• (i)  
  Initialize once at the beginning of the measurement.
• (ii)  
  Construct a superposition state by applying a $\pi /2$ rotation (0.75 ms) with $0{}^\circ$ phase.
• (iii)  
  Wait for a free precession time.
• (iv)  
  Transfer the phase to the population by applying a $\pi /2$ rotation with a $90{}^\circ$ phase.
• (v)  
  Perform PPM.
• (vi)  
  Revert the population to the original phase by applying a $3\pi /2$ rotation with $90{}^\circ$ phase.
• (vii)  
  repeat (iii)–(vi).

This sequence is very similar to that proposed in our previous paper [11], but is slightly modified to accommodate technical limitations. Our 12 GHz signal generator can switch phase with the external logic control in only two profiles (in our case, at $0{}^\circ$ and $90{}^\circ$), so the 1/2π rotation at the $270{}^\circ$ phase was replaced with a 3/2π rotation at the $90{}^\circ$ phase. The 12 GHz microwave signal generator was referenced to a hydrogen maser, and the frequency noise was much smaller than the SNR of the measurements. The phase error in a hydrogen maser at 0.1 s averaging time is below 0.02 radians. This implies that the measured zero phase shift between the LO and the atoms should always lie within the measurement noise. Figure 5(b) shows the measured phase over 20 PPM measurements. Each datum is the average of 32 measurements, while the measurement noise (∼0.2 radians) is mainly due to the scattering of the 370 nm laser. Clearly, PPM measures the correct phase over three consecutive measurements, and thereafter deviates from the true phase by more than one sigma. After the third measurement, the atomic phase abruptly loses coherence, and the population ratio rapidly asymptotes toward 70% excitation. Together with the Rabi oscillations measured by PPM, we conclude that our continuous-phase measurement is valid for up to three measurements.

We consider a simple model in which a constant number of ions are projected and lose coherence during each measurement. Prior to the n-th measurement, the proportion of ions whose states are projected (denoted as P_proj ) and is given by,

$$\begin{eqnarray}&&{{P}_{proj}}=1-{{(1-p)}^{n-1}}\end{eqnarray} \tag{ 3 }$$

where p is the ratio of the number of ions that get projected in a single measurement, normalized by the total number of ions. Fitting the data in figure 5 to equation (3), and an additional fitting parameter (the amplitude) yields the solid curve in figure 5. From this fitting, p is estimated as 18%.

3.4. Simulation evaluation of PPM

To elucidate the decoherence rate, we simulated ion motion using a molecular dynamics method [17]. We calculated the motions of 2000 ions trapped in a potential that corresponded to our trap parameters, assuming constant temperature. We confirmed that ions undergo Brownian motion with a mean-squared displacement along the optical axis $\Delta {{z}^{2}}\equiv \langle {{(z-{{z}_{0}})}^{2}}\rangle$ (where z₀ is the initial position at t = 0), given by

$$\begin{eqnarray}&&\Delta {{z}^{2}}=2Dt\end{eqnarray} \tag{ 4 }$$

where D is the diffusion constant of ions and t denotes time.

Simulating this ensemble with T = 50 mK, we obtained $D=3.5\times {{10}^{-6}}[{{m}^{2}}\;{{s}^{-1}}]$. Varying the temperature from 10 mK to 2 K, we found that D and T were related through the Boltzmann constant and the mobility μ, namely,

$$\begin{eqnarray}&&D=\mu {{k}_{B}}T.\end{eqnarray} \tag{ 5 }$$

The fitting gives $\mu =8.62\times {{10}^{18}}[{{m}^{2}}\;{{s}^{-1}}{{J}^{-1}}]$.

Next, we counted the number of ions passing through the region in which the diagonal laser and ions overlap. At T = 50 mK, 17% of the total ions were struck by the measurement laser within 1 ms. This implies that 17% of the ions were projected and decohered during 1 ms measurement time, consistent with the estimates of figure 5. The cause of the abrupt decoherence after the third measurement remains unclear.

This section experimentally compares the stability of the standard method (in which phase is initialized during each cycle) with that of an APL. The APL initializes the phase after each sequence of three PPMs, as shown in figure 6(a). Figure 6(b) shows the standard deviation in the APL results (filled red triangles) and the overall Allan deviation (filled red aquares). APL ($\tau =0.1\sim 0.3$) and APL repetition ($\tau >0.3$) are calculated in different ways. For $\tau =0.1\sim 0.3$, a frequency error $\Delta f$ of LO compared to atomic frequency is estimated from the measured total phase ϕ_n as,

$$\begin{eqnarray}&&\Delta f=\frac{{{\phi }_{n}}}{n{{T}_{FP}}}\end{eqnarray} \tag{ 6 }$$

where n indicates the n-th APL measurement cycle (n = 1, 2, or 3) and ${{T}_{FP}}=0.1s$ is a free precession time of the Ramsey sequence. In figure 6(b), the standard deviation in the n-th measurement scales as τ⁻¹ (red triangles). For $\tau >0.3$, APL cycles are established, and we calculate and plot the Allan deviation from the final third measurement in each cycle (red squares). The triangles and circles in figure 6(b) are calculated from the same data and the slight mismatch at $\tau =0.3s$ is due to the difference between the standard and Allan deviations.

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For comparison, the Allan deviation of the regular Ramsey measurement, in which the phase is initialized during each cycle, is also plotted in figure 6(b). This deviation scales as ${{\tau }^{-1/2}}$. Based on the measured Allan deviations, the stability of APL is improved by a factor of $\sqrt{3}$ relative to the standard Ramsey cycles. Overall, we have demonstrated that APL improves the stability by a factor of $\sqrt{{{n}_{cp}}}$, where n_cp is the number of continuous APL measurement.

The number of valid PPMs can be increased by (1) reducing the temperature of the ions (lowering the diffusion coefficient), (2) trapping more ions (reducing the proportion of the decohered ions in a single measurement), or (3) adopting weak measurements. Since weak measurements ideally allow us to estimate phase but are unable to preserve the phase over multiple measurement cycles, we expect that this last strategy could greatly increase the number of valid PPM. A promising weak measurement scheme is the Faraday rotation, which is discussed in detail in our previous paper [11].

Although this experiment is not a demonstration of the actual improvement of clock performance, as already mentioned, APL via PPM will be useful when the performance of an atomic clock is limited by technical noise. In the past, trapping atoms of more than (SNR)² was futile. APL via PPM opens a path to utilize more number of atoms than (SNR)² for better stability, thereby lowering the stability line in figure 1 to

$$\begin{eqnarray}&&{{\sigma }_{y}}=\frac{1}{\sqrt{{{n}_{cp}}}K\cdot Q\cdot {\rm SNR}}\sqrt{\frac{{{T}_{c}}}{\tau }}.\end{eqnarray} \tag{ 7 }$$

PPM limits the n_cp to ${{N}_{atom}}/{{({\rm SNR})}^{2}}$ (where N_atom is the total number of atoms in the trap) because the number of measured atoms is adjusted to (SNR)² and all of these atoms become decohered during a single measurement. When ${{n}_{cp}}={{N}_{atom}}/{{({\rm SNR})}^{2}}$, equation (7) computes the QPN limit line. Therefore, when APL is based on PPM, the system cannot be improved beyond the QPN limit (figure 1), except perhaps by weak measurement. However, discussion of the weak measurement limit is beyond the scope of this paper.

Since we performed the proof-of-principle experiment in a regime limited by technical noise, we didn't have to feed the atomic signal back to the LO. Our next step would be to perform the APL for the case where LO noise limits the stability, in order to demonstrate the actual improvement of a clock performance. In that case, we would need to evaluate the servo error in the feedback system to the LO frequency carefully.

Recently, the use of multiple atomic traps has been proposed [18, 19]. This scheme shows the same τ⁻¹ scaling and an overall stability that is reduced by a factor of m2^−m , where m is the number of atomic traps. Further stability improvements are conceivable if this scheme could be combined with APL.

This work was supported by the JST PRESTO program and NICT. We thank H Hachisu and T Ido for comments on this manuscript.