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We have developed a microwave frequency standard based on the 15.2 GHz ground-stated hyperfine transition of 113Cd+ ions. Using a laser-cooled ion cloud trapped in a linear quadrupole Paul trap, the clock transition frequency is measured to be 15 199 862 855.0125(87) Hz, with an accuracy at the 10−13 level. The main errors are from the microwave frequency reference used in the experiment. The precision is improved by nearly two orders of magnitude than that reported before.
©2013 Optical Society of America
1. Introduction
Trapped ions have been successfully employed in quantum information processing in recent years [1, 2]. Quantum entanglement between a single trapped Cd+ ion and a single photon has been demonstrated [3]. Due to the long interaction time between the ions and the applied radiation field, trapped ions are also suitable for ultrahigh precision spectroscopy experiment. In the past decades, several atomic clocks have been built based on different trapped ions, for example, the optical atomic clocks based on Al+ [4], Hg+ [5], Sr+ [6] ions and the microwave atomic clocks based on Hg+ [7], Yb+ [8] and Cd+ [9–11] ions. The Al+ optical clock developed by Chou et al. is by far the most precise frequency standard [4]. Alan Madej et al. at the National Research Council in Canada (NRC) have just reported an optical frequency standard based on a trapped single 88Sr+ ion with a fractional uncertainty of 2 × 10−17, measured over a period of two-month [6]. The precision of these ion clocks supersedes that of the current realization of the second, and potentially could lead to its new definition.
Microwave clocks based on trapped ions also play an important role in atomic clock applications. Recently, we have engaged in developing a 113Cd+ trapped ions atomic clock. It is aimed to be a transportable, high-precision microwave frequency standard, potentially applicable to time-keeping clocks comparison at ground stations of the COMPASS satellite navigation system of China. It may also play an important role in several general relativistic experiments. The clock’s frequency stability is expected to be at 10−14τ-1/2 level [11]. Meanwhile, it is designed to be a transportable clock since only one single laser is needed to realize laser cooling, optical pump, and probing, taking advantage of the simple energy structure of 113Cd+ ions [11].
To operate a microwave atomic clock with high precision, one has to measure its frequency at zero external magnetic field. For 113Cd+ ions, several groups have reported their measurements during the past decades. Using the spin polarization transfer from metastable He atoms to Cd+ ions, Hamel and Vienne obtained 15 240(200) MHz [12]; Tanaka et al. [9] and Jelenkovic et al. [10] subsequently obtained 15 199 862 858(2) Hz and 15 199 862 855.0(2) Hz, respectively, both using a microwave-optical double-resonance method in the presence of He buffer gas. Using the laser-microwave double-resonance technique, we measured the spectroscopy of the ground-state hyperfine splitting of laser-cooled 113Cd+ ions trapped in a linear quadruple trap and obtained a value of 15 199 862 854.96(12) Hz [11]. In this paper, a new measurement of the ground-state hyperfine splitting of the 113Cd+ ions is reported, where the precision is pushed to the 10−13 level. In order to achieve this improvement of precision, we have upgraded the experiment setup, (named JMI-2), including the ion trap, the optical part, and magnetic field stabilization compared with the previous experimental setup in Ref [11], (JMI-1). The result we obtained using this new apparatus agrees very well with previous measurements, listed above, and the precision is improved by nearly two orders of magnitude.
2. Experiment setup
The fundamental principle of the 113Cd+ clock and the details of the frequency standard apparatus of JMI-1 can be found in Ref [11], and only the changes of the experiment setups of JMI-2 are reported in detail here. The schematic diagram of the JMI-2 experimental setup is shown in Fig. 1 .
Fig. 1 The schematic diagram of the JMI-2 experimental setup. HWP: half-wave plate; QWP: quarter-wave plate; L: lens; AOM: acousto-optic modulator; M: mirror; OS: optical shutter; PBS: polarizing beamsplitter; EMCCD: electron-multiplying CCD; PMT: photo-multiplier tube detector.
The ions are trapped in a radio frequency (rf), linear, quadrupole Paul trap enclosed in an ultrahigh vacuum chamber with a background pressure of approximately 5 × 10−10 mBar. The main differences between the ion traps of JMI-1 and JMI-2 are listed as follows: (1) The ratio of R/r0 is optimized to be 1.1468 [13], where 2R = 14.22 mm is the outer diameter of the electrode and r0 = 6.2 mm is the radial distance from the axis of the trap to the closest surface of the electrodes. The trap with this optimum dimensions can reduce the rf heating effect and increase the number of trapped ions. (2) Every electrode is separated into three segments. The outer segments work as the endcaps, replacing the endcap rings in the JMI-1 trap. The structure of the endcap rods is shown in Fig. 1. (3) In the JMI-2 trap, an rf field with amplitude of 400 V at a frequency of 1.25 MHz is applied on the diagonal electrodes to confine the ions in the radial direction. A dc voltage of about 100 V is applied on the endcaps to confine the ions in the axial direction. The parameters of the rf field are changed because of the trap dimension difference; (4) One of the four inner electrode segments is designed as a slotted circular waveguide resonator to feed the microwave field. Compared to the more traditional, horn antenna feeding, where the microwave field is fed into the vacuum chamber through a silica window, the new design makes the system much more compact. (5) In the JMI-2 oven, a piece of isotopic enriched 113Cd metal (93%), rather than natural abundance cadmium metal is loaded. This improves the signal-to-noise ratio (SNR) of the clock transition signal.
The laser system is a 214.5 nm frequency-quadrupled tunable diode laser. During measurement, the UV laser frequency has been red-detuned and locked to the cycling transition of 2S1/2 |F = 1> ↔ 2P3/2 |F = 2> by employing a homemade logic circuit and a Fabry-Perot optical spectrum analyzer as a transfer cavity [14]. The energy level of the 113Cd+ ion is shown in Fig. 2 . In the upgraded optical part as shown in Fig. 1, the laser is used to cool the ions when the AOMIis shut off. The laser light is frequency shifted twice by AOMIand II (400 MHz each) and used as the pump laser, pumping the ions into the state 2S1/2 |F = 0, mF = 0>. When ν, the frequency of the applied microwave field is close to the resonant frequency of the ions’ ground-state hyperfine splitting, the population of the ions in the 2S1/2 |F = 1> state will change accordingly. This population change can be probed with the laser after two successive 400 MHz opposite frequency shifts through AOMIandII. Two optical shutters are installed in the setup to block off all laser beams during microwave interrogation. With this optical setup, we can optimize the detuning and the power of the probe laser, and the lasers can be switched on and off more quickly at the nano-second level, thereby improving the SNR of the clock signal. The fluorescence signal is detected with an EMCCD and an ultra-violet photomultiplier operating in the Geiger mode.
The stabilization of the magnetic field is one of the most important factors for high-precision spectroscopic measurement. In the JMI-1 setup, one Helmholtz coil pair is used to create a static magnetic field parallel to the electrodes, and another two pairs are used to null the ambient magnetic fields in other directions. In order to improve the stability of the magnetic field in the JMI-2 setup, outside the three pairs of Helmholtz coils, a slice of silicon-steel sheet is applied to wrap the entire physics package as a facilitating magnetic shield to suppress the ambient magnetic fields fluctuation. Besides that, a fluxgate magnetometer is installed next to the vacuum chamber to measure the temporal variation of the magnetic field in all three dimensions with a nano-Tesla resolution. After the residual magnetic fields are carefully nulled in the directions perpendicular to the electrodes [11], the magnetic field is stabilized by the fluxgate magnetometer via the feedback to the current of the pair of the Helmholtz coils in the electrode direction. With this method, the standard deviation of the magnetic field fluctuation in short term is controlled to less than 20 nT. The 15.2 GHz microwave resonance radiation propagating perpendicularly to the trap axis enables observation of the 2S1/2 |F = 0, mF = 0> ↔ 2S1/2 |F = 1, mF = 0> clock transition. A fraction of the microwave radiation propagates along the trap axis, hence the Zeeman transitions of ΔF = 1, ΔmF = ± 1 can also be measured. The absolute amplitude of the magnetic field where the ions located is measured via the magnetic-field-sensitive transitions of ΔF = 1, ΔmF = ± 1.
3. High-resolution hyperfine spectroscopy of 113Cd+
According to the Rabi-Breit formula, the position of the energy levels is
whereIn this equation, B0 is the applied magnetic field, and EHFS = hνHFS corresponds to the quantum level, where νHFS is the ground state hyperfine frequency at B0 = 0 and h is Planck’s constant. F is the hyperfine quantum number of the ground states, and mF corresponds to different Zeeman substructures. I is the nuclear angular momentum number, and here I = 1/2 for 113Cd+ ions. gI and gJ are the electronic and nuclear g factors, respectively, and μB is the Bohr magneton. The minus and plus signs apply in front of the square root when F = I-S and F = I + S, respectively.When the magnetic field is very low, namely x«1, approximations can be made. For 113Cd+ ions, the 0-0 ground state hyperfine frequency at a given magnetic field can be expressed as
where ν0,0, ν0,1 and ν0,-1 are the 2S1/2 |F = 0, mF = 0> ↔ 2S1/2 |F = 1, mF = 0>, 2S1/2 |F = 0, mF = 0> ↔ 2S1/2 |F = 1, mF = 1>, and 2S1/2 |F = 0, mF = 0> ↔ 2S1/2 |F = 1, mF = −1> transition frequencies at a given magnetic field, respectively. Since gJ and gI are constants, so we can obtain νHFS by fitting Eq. (3) on the experimental data of ν0,0, and ν0,1-ν0,-1 measured at different static magnetic fields. ν0,0 is insensitive to the magnetic field, hence it can be measured by Ramsey interrogation with a high accuracy. ν0,1 and ν0,-1 are measured with Rabi interrogation since the ΔF = 1, ΔmF = ± 1 Zeeman lines are magnetic field sensitive, which makes it difficult to apply the Ramsey’s method due to magnetic field noise. The details of Rabi and Ramsey measurement sequences have been reported in Ref [11]. A brief overview of the measurement is described here. Figure 3 shows two examples of measured Rabi spectroscopy of ΔF = 1, ΔmF = 1 Zeeman transition and the 0-0 Ramsey fringe of clock transition at a given magnetic field.Fig. 3 (a) Spectroscopy of ΔF = 1, ΔmF = 1 Zeeman line of the 113Cd+ ions. The rf pulse width is 0.1 ms. (b) Ramsey fringe of the clock transiton of the 113Cd+ ions with a free precession period of 2 s. The rf pulses’ widths are 400 ms. The gate time of photon counting for both measurements are set at 15 ms.
During the spectroscopic measurement of ν0,1 and ν0,-1, we scan the microwave frequency over 60 kHz about the center frequency ν0,1 and ν0,-1, respectively, with a microwave pulse duration of τ = 0.1 ms. By fitting the measured data shown in Fig. 3(a) to the equation of magnetic dipole transition probability
withwhere b is the Rabi angular frequency, ν is the microwave radiation frequency, and ν0 is the center frequency of the spectrum. The frequency of ν0,1 or ν0,-1 is obtained for a given magnetic field with a statistical error of less than 45 Hz.For the clock transition frequency measurement using Ramsey interrogation, the pulses durations τ are 400 ms, and the free precession period between two pulses is T = 2 s. The data as shown in Fig. 3(b) is fitted to the equation
where Ω0 = 2π(ν-ν0). The frequency of ν0,0 is obtained with a statistical error less than 0.8 mHz.A set of clock transition frequency ν0,0 and ΔF = 1, ΔmF = ± 1 Zeeman transitions ν0,1 and ν0,-1 are measured at different values of magnetic field as described above. The ground state hyperfine splitting of 113Cd+, νHFS is obtained by fitting the experimental data to Eq. (3). During the fitting, the statistical error of ν0,0 is set as y error, and the statistical error of ν0,1-ν0,-1 is set as x error. We obtain the ground state hyperfine splitting of 113Cd+ νHFS = 15199862855.0125(36) Hz, as shown in Fig. 4 . The error of the result is mainly due to statistical error when extrapolating the ground state hyperfine transition frequency to zero magnetic field.
Fig. 4 Clock transition ν0,0 versus the difference of the frequency of ΔmF = ± 1 Zeeman transitions, ν0,1-ν0,-1. The measured data is fitted to Eq. (3), with residuals also shown.
4. Error analysis
In order to determine the precision of the measurement, several sources of errors have to be carefully considered. In this section, we analyze the effect of the second-order Doppler effect, the collision (pressure) shift, the light shift, the residual magnetic field fluctuation, and the uncertainty of the frequency reference. Finally, we give a comparison with previously reported results.
4.1 Second-order Doppler Effect
Since the trapped ion cloud is much smaller than the wavelength of the clock transition and is well-situated in the Lamb-Dicke regime, the first order Doppler effect is entirely suppressed. However, the second-order Doppler frequency shift (SODFS) still exists. In the experiment, the high temperature ion cloud is cooled down via laser cooling. We measured the temperature of 113Cd+ ions to be about 1 ± 0.5 K by the measurement of the Doppler broadening of the D2 line. The measurement method used here is the same as that in [15]. The total fractional SODFS due to both of the secular motion and the micro-motion of the trapped ions is known as [16]
where k is Boltzmann’s constant, T is the kinetic temperature characterizing the secular motion, m is the mass of the ion, c is the speed of light, and Ndk is a coefficient of the SODFS due to the micro-motion averaged across the entire ion cloud. Following the method in [16], we calculated the coefficient Ndk with the parameters of our system and obtained Ndk≈3. Hence, the total SODFS is -(3 ± 1.5) × 10−15 in this measurement.4.2 Influence of the static magnetic field
In the experiment, by measuring the frequencies of the magnetic-field-sensitive transitions ν0,1 and ν0,-1, the absolute amplitude of the magnetic field where the ions located can be measured. Consequently, the frequency uncertainty due to the residual magnetic field in the previous measurement [11] is avoided. In this measurement, the statistical error of ν0,1-ν0,-1 representing the uncertainty of the magnetic field in every measurement is set as x error during the extrapolation, hence the error of the extrapolating the ground state hyperfine transition frequency to zero magnetic field includes the influence of the magnetic field. The errors of ν0,1 and ν0,-1 are less than 45 Hz which leads to an upper bound of 1.5 × 10−13 on the uncertainty in the second-order Zeeman shift.
4.3 Pressure shift
The frequency shift for a given background gas is related to the ground-state valence electron orbital radius, as Vetter derived for hyperfine transition of alkali-like ions [17]. We haven’t measured the pressure shift coefficient for 113Cd+ and there is no prior publication on this subject, to our knowledge. Considering that the vacuum is maintained by an ion pump to be about 5 × 10−10 mbar in our case, we estimate that the pressure shift frequency uncertainty is below 10−14 assuming the pressure shift coefficient is in the same order of 133Cs [18],171Yb+ [8] and 199Hg+ [19]. Hence the pressure shift can be ignored in the present measurement. Detailed measurement will be conducted in the future.
4.4 Blackbody radiation shift
The ambient temperature is about 300 K. The blackbody ac Zeeman shift is estimated to be -(2.4 ± 0.24) × 10−15 at 300 ± 10 K in our experiments [20]. To our knowledge, no prior publication has reported the fractional blackbody ac hyperfine Stark shift for 113Cd+ ions and this shift is still under investigation. However, we find that the magnitude of the blackbody shifts of 133Cs [21], 137Ba+ [22], 171Yb+ [23] and 199Hg+ [22] at 300 K are all below 2 × 10−14. Hence, it is safe to ignore the blackbody shift in this measurement and the frequency uncertainty of this shift is estimated to be less than 2 × 10−14 at this stage. Detailed measurement will be reported in the future.
4.5 Light shift
For the Ramsey and Rabi interrogation of 113Cd+, after the state preparation of the 113Cd+ ions by the pump laser, all lasers are blocked off during microwave interrogation. Under this circumstance, the light shift is estimated to be smaller than the resolution of the spectroscopy obtained and hence can be neglected.
4.6 Frequency uncertainty of the reference
The microwave oscillator of the 113Cd+ spectroscopy measurement is referenced to a commercial cesium clock with a frequency accuracy of 5 × 10−13, calibrated by the National Institute of Metrology of China (NIM). Therefore, the frequency uncertainty due to this microwave oscillator is estimated to be 0.0076 Hz.
Table 1 lists the systematic frequency shifts and uncertainties. In conclusion, the ground-state hyperfine splitting of 113Cd+ is
where the main error is the combination of the statistical error during the extrapolation of the transition frequency and the frequency uncertainty listed in Table 1.
Table 1. Estimated Fractional Systematic Frequency Shifts and Uncertainties.
Figure 5 shows the measurement improvement in the precision of frequency of the ground-state hyperfine splitting of 113Cd+ ions. The four points with error bars in Fig. 5 represent the results from Tanaka et al., Jelenkovic et al., the JMI-1 setup and this measurement, from left to right, respectively.
Fig. 5 Measurements of the ground-state hyperfine splitting of 113Cd+ ions. 1, 2, 3, and 4 represent the results from Tanaka et al., Jelenkovic et al., the JMI-1 setup and the JMI-2 trap, respectively. The inset shows the comparison of the last three measurements.
5. Conclusion
As one of the most important steps of developing a microwave frequency standard based on laser-cooled, 113Cd+ trapped ions, we measured its 2S1/2 |F = 0, mF = 0> ↔ 2S1/2 |F = 1, mF = 0> hyperfine transition at zero magnetic field. The frequency of the clock transition is found to be 15 199 862 855.0125(87) Hz with an uncertainty at the 10−13 level. The main error is from the microwave frequency reference used in the experiment. The value agrees with the previous results very well and is improved in its accuracy by nearly two orders of magnitude, compared to previously reported measurements. The measurement is important for the development of the 113Cd+ trapped ions atomic clock, and the result is also important for cadmium ion atomic structure study [24], such as figuring out the hyperfine anomaly in the S1/2 state of cadmium ions according to the hyperfine splitting of 113Cd+ and 111Cd+ ions, which can improve the understanding of the nuclear structure as reported for Hg+ in [7]. We anticipate that with an improved microwave or optical frequency reference, the accuracy of the measurement can be further improved. The direct comparison of the hyperfine transition frequency and electronic transition frequency, and measuring the stability of this ratio, could be important toward setting a limit on the stability of fundamental constants. Finally, the experiment reported here may even be useful for quantum information processing research based on cadmium ions [3].
Acknowledgments
We acknowledge funding supports from the Major State Basic Research Development Program of China (973 Program) (No. 2010CB922901) and the Tsinghua University Scientific Research Initiative Program (No. 20131080063 and No. 20121080077).
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