Investigation of experimental issues concerning successful operation of quantum-logic-based $^{27}\hbox {Al}^+$ ion optical clock

Abstract

Successful operations of quantum-logic-based $^{27}\hbox {Al}^+$ ion optical clocks have been reported by several groups. But there are still several lingering issues. The first is the proper choice of logic ions, and the second is the proper quantum state identification methods. In this paper, the advantage of using $^{25}\hbox {Mg}^+$ as the logic ion to ensure smaller time-dilation shift is discussed. We also compare several statistical methods to identify the $^{27}\hbox {Al}^+$ ion clock state. Finally, we report the observation of the $^{27}\hbox {Al}^+$ ion ${^1S_0} \rightarrow ^{3}{P_0}$ clock transition based on $^{25}\hbox {Mg}^+$–$^{27}\hbox {Al}^+$ ion pair. As a precondition for quantum logic spectroscopy (QLS), both the stretch (STR) mode and the center of mass (COM) mode of the $^{27}\hbox {Al}^+$ and $^{25}\hbox {Mg}^+$ ion pair are cooled to the vibrational ground state by Raman sideband cooling. The mean phonon number is measured to be 0.10(1) for the STR mode and 0.01(1) for the COM mode, respectively. The heating rate is evaluated to be 0.23(5) phonons/s for the STR mode and 3.0(7) phonons/s for the COM mode, respectively. The clock transition is observed with a full-width-half-maximum (FWHM) of 38(4) Hz at 22 ms interrogation time.

1 Introduction

Atomic clocks play important roles both in fundamental science and everyday life [1, 2]. Given the increasing demands on better frequency standards for diverse applications in measuring time variations of fundamental constants [3, 4], exploring geophysical phenomena [5], detecting gravitational waves [6], testing of general relativity [7], and searching for dark matter [8], more stable and more accurate atomic clocks are needed. In comparison with microwave atomic clocks, optical clocks potentially can have better stability and systematic uncertainty due to higher Q values. Here, Q is defined as the ratio of a clock transition frequency to its linewidth. Optical clocks have seen rapid progress in this century. The state-of-the-art optical clocks have achieved systematic uncertainties just below $10^{-18}$ level [9], while the best Cesium fountain clocks have fractional frequency uncertainties of $1.1 \times 10^{-16}$ [10].

There are several possible routes to develop an optical clock. One is based on trapped ions in a Paul trap, and one is based on neutral atoms trapped in an optical lattice. Another possibility is to use active optical clocks [11]. Potential choices of single ion species include $\hbox {Hg}^+$ [12, 13], $\hbox {Sr}^+$ [14,15,16], $\hbox {Yb}^+$ [3, 17,18,19], $\hbox {Ca}^+$ [20,21,22], $\hbox {In}^+$ [23, 24], and $\hbox {Ba}^+$ [25], while potential choices of optical lattice clocks and active optical clocks include Sr [26,27,28,29,30,31,32], Yb [33,34,35,36], and Hg [37,38,39].

Fig. 1

a The relevant energy levels of $^{27}\hbox {Al}^+$ ion. The symbols for some of the energy levels are $\left| {^1S_0},{F=5/2}, {m_{\mathrm{F}}= 5/2}\right\rangle$$\equiv$$\left| \downarrow \right\rangle _{\mathrm{{Al}}}$, $\left| {^3P_1},{F=7/2}, {m_{\mathrm{F}}= 7/2}\right\rangle$$\equiv$$\left| \uparrow \right\rangle _{\mathrm{{Al}}}$ and $\left| {^3P_0},{F=5/2}, {m_{\mathrm{F}}= 5/2}\right\rangle$$\equiv$$\left| c\right\rangle _{\mathrm{{Al}}}$. The blue sideband (BSB) transition is performed on the logic transition $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$$\leftrightarrow$$\left| \uparrow \right\rangle _{\mathrm{{Al}}}$ at 267.0 nm. The clock transition corresponds to $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$$\leftrightarrow$$\left| c \right\rangle _{\mathrm{{Al}}}$ at 267.4 nm. b The relevant energy levels of $^{25}\hbox {Mg}^+$ ion. The groundstate hyperfine energy levels are defined as $\mid {^2S_{1/2}},F=3, m_{\mathrm{F}}= 3>$$\equiv$$\left| \downarrow \right\rangle _{\mathrm{L}}$ and $\left| {^2S_{1/2}},{F=2}, {m_{\mathrm{F}}= 2}\right\rangle$$\equiv$$\left| \uparrow \right\rangle _{\mathrm{L}}$. The Raman laser is red detuned 16 GHz from the corresponding resonant transition to the level $\left| {^2P_{3/2}},{F=3}, {m_{\mathrm{F}}= 3}\right\rangle$, R-$\upsigma ^+$ denotes Raman-$\upsigma ^+$; R-$\uppi$ denotes Raman-$\uppi$, BR-co denotes co-propagating blue Raman. Doppler represents the Doppler cooling beam which is carried on the transition of $\left| \downarrow \right\rangle _{\mathrm{L}}$$\leftrightarrow$$\left| {^2P_{3/2}},{F=4}, {m_{\mathrm{F}}= 4}\right\rangle$, and repump is the repump beam which is carried on $\left| \uparrow \right\rangle _{\mathrm{L}}$$\leftrightarrow$$\left| {^2P_{3/2}},{F=3}, {m_{\mathrm{F}}= 3}\right\rangle$ transition

Due to the narrow natural linewidth of 8 mHz, the ${^1S_0} \rightarrow {^3P_0}$ transition of $^{27}\hbox {Al}^+$ at 267.4 nm has been recognized as an excellent optical clock transition candidate. Its electric quadrupole shift is negligible (J = 0), therefore, the transition frequency is insensitive to the electric field gradient in the ion trap [40]. Furthermore, black-body radiation contribution to the fractional frequency uncertainty of the $^{27}\hbox {Al}^+$ ion clock is as low as $4 \times 10^{-19}$ at room temperature, which is the smallest among all atomic species currently under investigation for optical clocks [41]. However, $^{27}\hbox {Al}^+$ ions cannot be cooled directly due to the lack of direct cooling lasers at 167 nm. Recently, a two-photon transition cooling scheme based on the transition of $3s^{2}$$^1S_0 \rightarrow 3s3p {^1D_2}$ at 234 nm has been proposed [42], which is still under development. Another obstacle for realizing the $^{27}\hbox {Al}^+$ ion clock is the detection of the clock transition. The clock transition is usually realized by the electronic shelving technique. As for the $^{27}\hbox {Al}^+$, the best candidate transition for the electronic shelving detection of the clock transition is the ${^1S_0} \rightarrow {^3P_1}$ transition, but the lifetime of ${^3P_1}$ state is 305 $\upmu$s, so the fluorescence rate is very small and the detection is very slow.

Both issues can be solved by introducing an auxiliary ion called logic ion to sympathetically cool $^{27}\hbox {Al}^+$ and to map the internal states of $^{27}\hbox {Al}^+$ ion to logic ion’s internal states via the common vibrational phonon mode through quantum logic spectroscopy (QLS) [43]. Using the QLS method, the first $^{27}\hbox {Al}^+$ ion optical clock was successfully demonstrated at the National Institute of Standards and Technology (NIST) [44]. Since then, several groups have been working on QLS-based $^{27}\hbox {Al}^+$ ion optical clocks [45,46,47,48].

There are still lingering issues concerning the successful operation of the QLS-based $^{27}\hbox {Al}^+$ optical clocks. The first is the proper choice of the logic ion. So far $\hbox {Mg}^+$, $\hbox {Be}^+$, and $\hbox {Ca}^+$ have been used as the logic ion. Different logic ions have different masses and different natural linewidths, corresponding to different Doppler cooling limits and different sympathetic cooling rates. The question of which logic ion is better has been raised by many research groups before, and has been investigated before without definite answer [49]. For an $^{27}\hbox {Al}^+$ ion optical clock, the time-dilation shift is one of the largest systematic uncertainties [50]. The time-dilation shift is proportional to the kinetic energy of the $^{27}\hbox {Al}^+$ ion, which is decided by the sympathetic cooling limit and by the heating rate of the ion trap. In this work we compare different logic ions in $^{27}\hbox {Al}^+$ ion optical clock from the point of view of the time-dilation shift due to the heating rate. Our investigation shows that $^{25}\hbox {Mg}^+$ is better as a logic ion. Also, since the heating rate has a large effect on the time-dilation shift, we make an in-depth report on the procedures we used to lower the heating rate of the ion trap.

Another issue is what the best statistical methods to identify the states of $^{27}\hbox {Al}^+$ ion clock transition with larger contrast and better accuracy is. To detect the state of the $^{27}\hbox {Al}^+$ ion, generally the states of the $^{27}\hbox {Al}^+$ ion are repeatedly probed and the fluorescence of the $^{25}\hbox {Mg}^+$ ion is detected and analyzed. To find the best detection method with high fidelity, we used a Monte-Carlo method to simulate several different detection methods. Based on the simulation result we propose the best method for clock transition identification and the best method for clock transition locking.

In this paper, we first describe the principle of the QLS for $^{27}\hbox {Al}^+$ ion optical clock, and discuss the choice of the logic ion. After that we describe the experimental setup of the $^{27}\hbox {Al}^+$ ion optical clock. Details of Raman sideband cooling for $^{25}\hbox {Mg}^+$ and $^{27}\hbox {Al}^+$ ion pair, and blue sideband spectroscopy for the QLS are presented. Different state identification methods are simulated and at last the clock transition spectroscopy is presented.

2 Principle of the quantum logic spectroscopy

2.1 QLS process

For simplicity of the description, we write $\left| {^1S_0},{F=5/2}, {m_{\mathrm{F}}= 5/2}\right\rangle$ state of $^{27}\hbox {Al}^+$ ion as $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$, and ${\left| {^3P_1},{F=7/2}, {m_{\mathrm{F}}= 7/2} \right\rangle }$ state as $\left| \uparrow \right\rangle _{\mathrm{{Al}}}$. The $\left| {^3P_0},{F=5/2}, {m_{\mathrm{F}}= 5/2} \right\rangle$ state is denoted as ${\left| c \right\rangle _{\mathrm{{Al}}}}$. The blue sideband transition (BSB) is performed on the logic transition $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$$\leftrightarrow$$\left| \uparrow \right\rangle _{\mathrm{{Al}}}$ at 267.0 nm. The clock transition is performed on $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$$\leftrightarrow$$\left| c \right\rangle _{\mathrm{{Al}}}$ at 267.4 nm. As for the logic ion of $^{25}\hbox {Mg}^+$ ion, the ground state hyperfine energy level $\left| {^2S_{1/2}},{F=3}, {m_{\mathrm{F}}= 3} \right\rangle$ is denoted as $\left| \downarrow \right\rangle _{\mathrm{L}}$ state, $\left| {^2S_{1/2}},{F=2}, {m_{\mathrm{F}}= 2}\right\rangle$ is denoted as $\left| \uparrow \right\rangle _{\mathrm{L}}$ state (Fig. 1). $\left| \downarrow \right\rangle _{\mathrm{L}}$ and $\left| \uparrow \right\rangle _{\mathrm{L}}$ states are called the bright state and dark state, respectively. Motional states are written as $\left| n \right\rangle$ where n means the phonon number of the logic and spectroscopy ions’ vibrational state (stretch mode or center of mass mode).

Before clock laser interrogation, both the logic ion and spectroscopy ion internal states are prepared in the vibrational ground state. After the initialization, the wave function of the system can be written as ${\Psi _0}\mathrm{{ = }}{\left| \downarrow \right\rangle _{\mathrm{{Al}}}}{\left| \downarrow \right\rangle _{\mathrm{L}}}\left| 0 \right\rangle$. After the clock laser interrogation, the internal state of the $^{27}\hbox {Al}^+$ ion will be excited to the superposition state of ${\left| \downarrow \right\rangle _{\mathrm{{Al}}}}$ and ${\left| c \right\rangle _{\mathrm{{Al}}}}$, and the wave function of the system has the form of

$$\begin{aligned} \begin{aligned} {\Psi _1}&=\left( {\alpha {\left| c \right\rangle _{\mathrm{{Al}}}}+\beta \left| \downarrow \right\rangle _{\mathrm{{Al}}}}\right) {\left| \downarrow \right\rangle _{\mathrm{L}}}\left| 0 \right\rangle \\&=\alpha {\left| c \right\rangle _{\mathrm{{Al}}}}{\left| \downarrow \right\rangle _{\mathrm{L}}}\left| 0 \right\rangle +{\beta \left| \downarrow \right\rangle _{\mathrm{{Al}}}}{\left| \downarrow \right\rangle _{\mathrm{L}}}\left| 0 \right\rangle . \end{aligned} \end{aligned}$$

(1)

Here ${\left| \alpha \right| ^2}+{\left| \beta \right| ^2}=1$, ${\left| \alpha \right| ^2}$ is the transition probability of the clock laser interrogation.

Then, the wave function for the ${\left| c \right\rangle _{\mathrm{{Al}}}}$ and ${\left| \downarrow \right\rangle _{\mathrm{{Al}}}}$ after driving a $\pi$ pulse of the $^{27}\hbox {Al}^+$ ion’s BSB can be written as

$$\begin{aligned} {\Psi _2}=\alpha {\left| c \right\rangle _{\mathrm{{Al}}}}{\left| \downarrow \right\rangle _{\mathrm{L}}}\left| 0 \right\rangle +\beta { \left| \uparrow \right\rangle _{\mathrm{{Al}}}} {\left| \downarrow \right\rangle _{\mathrm{L}}}\left| 1 \right\rangle . \end{aligned}$$

(2)

After the application of a Raman RSB $\pi$ pulse on the logic ion, we obtain the wave function as

$$\begin{aligned} {\Psi _3}=\alpha {\left| c \right\rangle _{\mathrm{{Al}}}}{\left| \downarrow \right\rangle _{\mathrm{L}}}\left| 0 \right\rangle +\beta { \left| \uparrow \right\rangle _{\mathrm{{Al}}}} {\left| \uparrow \right\rangle _{\mathrm{L}}}\left| 0 \right\rangle . \end{aligned}$$

(3)

Finally, the internal states of the spectroscopy ion can be detected by detecting the the internal states of the logic ion, which are distinguished by the fluorescence of the logic ion.

2.2 Comparison of different logic ions

Three different kinds of ions have been used as logic ions for the $^{27}\hbox {Al}^+$ ion optical clock since its first proposal [51]. The $^{27}\hbox {Al}^+$ ion optical clock was first demonstrated with $^{9}\hbox {Be}^+$ as the logic ion [44]. Because of the rather large mass ratio between the $^{27}\hbox {Al}^+$ and $^{9}\hbox {Be}^+$, the $^{27}\hbox {Al}^+$ cannot be cooled efficiently. Then, the $^{25}\hbox {Mg}^+$ was used as the logic ion, because its mass is very close to that of the $^{27}\hbox {Al}^+$. The $^{27}\hbox {Al}^+$ ion optical clock with the $^{25}\hbox {Mg}^+$ as the logic ion has achieved systematic uncertainty of $9.4\times 10^{-19}$ [9] . In addition, $^{40}\hbox {Ca}^+$ has also been suggested as the logic ion for the $^{27}\hbox {Al}^+$ ion optical clock [45, 48, 52].

Because of the relatively low mass of $^{27}\hbox {Al}^+$ compared to other clock ion species, the second-order Doppler shift of the $^{27}\hbox {Al}^+$ is one of the main sources of the systematic uncertainties. One important source for the second-order Doppler shift comes from the heating of the ions due to the electric-field noise of the ion trap [53, 54]. We show here that different logic ions lead to different heating rates of the $^{27}\hbox {Al}^+$ ions, therefore the choice of the logic ion will affect the time-dilation shift.

When the ion has a mean squared velocity of $\left\langle v^2 \right\rangle$, the time-dilation shift is

$$\begin{aligned} {\left( {\frac{{\Delta f}}{f}} \right) _{\mathrm{{TD}}}} = - \frac{\left\langle v^2 \right\rangle }{2{c^2}} = - \frac{E}{{m{c^2}}}. \end{aligned}$$

(4)

Here, E is the total motional mode energy of the ion pair.

Unlike single ion in an ion trap, there are two motional modes in each direction for the ion pair, which are named as “stretch” (STR) mode and “center of mass” (COM) mode, respectively [50, 52]. The secular frequencies of the STR mode and COM mode are determined by the mass ratio of the spectroscopy ion and the logic ion. Take the z direction as an example, the oscillations of the ion pair, and the frequencies of the two modes can be written as [49]

$$\begin{aligned} {q_{\mathrm{{logic}}}}= & {} {z_{\mathrm{c}}}{b_{\mathrm{1}}}\sin \left( {{\omega _{\mathrm{c}}}t + {\phi _{\mathrm{c}}}} \right) + {z_{\mathrm{s}}}{b_{\mathrm{2}}}\cos \left( {{\omega _{\mathrm{s}}}t + {\phi _{\mathrm{s}}}} \right) , \end{aligned}$$

(5)

$$\begin{aligned} {q_{\mathrm{{Al}}}}= & {} \frac{{z_{\mathrm{c}}}{b_{\mathrm{2}}}}{\sqrt{\mu }} \sin \left( {{\omega _{\mathrm{c}}}t + {\phi _{\mathrm{c}}}} \right) - \frac{{z_{\mathrm{s}}}{b_{\mathrm{1}}}}{\sqrt{\mu }}\cos \left( {{\omega _{\mathrm{s}}}t + {\phi _{\mathrm{s}}}} \right) , \end{aligned}$$

(6)

$$\begin{aligned} {\omega _{\mathrm{c}}}= & {} \sqrt{\frac{{1 + \mu - \sqrt{1 - \mu + {\mu ^2}} }}{\mu }} {\omega _{\mathrm{z}}}, \end{aligned}$$

(7)

$$\begin{aligned} {\omega _{\mathrm{s}}}= & {} \sqrt{\frac{{1 + \mu + \sqrt{1 - \mu + {\mu ^2}} }}{\mu }} {\omega _{\mathrm{z}}}. \end{aligned}$$

(8)

Here, $\omega _{\mathrm{{s,c}}}$ and $\phi _{\mathrm{{s,c}}}$ are the angular eigenfrequencies and phases of the ion’s motion, with the subscript c and s representing the COM mode and the STR mode, respectively. The $z_{\mathrm{{s,c}}}$ are the modal amplitudes. $b_{\mathrm{{1,2}}}$ are the components of the normalized eigenvectors of the COM mode. The ${\omega _{\mathrm{z}}}$ is the angular eigenfrequency of a single logic ion along z direction, $\mu$ is the mass ratio between the $^{27}\hbox {Al}^+$ ion and the logic ion which is defined as $\mu =m_{\mathrm{{Al}}}/m_{\mathrm{logic}}$. The normalized eigenvector $b_{\mathrm{{1,2}}}$ satisfies $b_{\mathrm{1}}^2 + b_{\mathrm{2}}^2 = 1$, and can be determined as [49]

$$\begin{aligned} b_{\mathrm{{1}}}^2 = \frac{{1 - \mu + \sqrt{1 - \mu + {\mu ^2}} }}{{2\sqrt{1 - \mu + {\mu ^2}} }}. \end{aligned}$$

(9)

During the clock transition interrogation, the ion pair will be heated by the electric field fluctuations, which in turn increases the energy of the ion pair and both the time-dilation shift itself and its uncertainty. A typical heating rate from the fluctuations of the stochastic electric field, assuming to be homogeneous across the ion crystal, can be written as [49]

$$\begin{aligned} \frac{{\mathrm{d}{E_{\mathrm{c}}}}}{{\mathrm{d}t}}= & {} \frac{{{q^2}{S_{\mathrm{E}}}{\mu }}}{{4{m_{\mathrm{{Al}}}}}}{\left( {{b_{\mathrm{1}}} + \frac{1}{{\sqrt{\mu }}}{b_{\mathrm{2}}}} \right) ^2}, \end{aligned}$$

(10)

$$\begin{aligned} \frac{{\mathrm{d}{E_{\mathrm{s}}}}}{{\mathrm{d}t}}= & {} \frac{{{q^2}{S_{\mathrm{E}}}{\mu }}}{{4{m_{\mathrm{{Al}}}}}}{\left( {{b_{\mathrm{2}}} - \frac{1}{{\sqrt{\mu }}}{b_{\mathrm{1}}}} \right) ^2}. \end{aligned}$$

(11)

Here, $S_{\mathrm{E}}$ is the spectral density of the stochastic electric field, and the drift rate of the time-dilation shift is [49]

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}}{{\mathrm{d}t}}\left( {\frac{{\Delta f}}{f}} \right) _{\mathrm{TD}}&=-\frac{\mathrm{d}}{{\mathrm{d}t}} \frac{{b_{\mathrm{2}}^2{E_{\mathrm{c}}} + b_{\mathrm{1}}^2{E_{\mathrm{s}}}}}{{{m_{\mathrm{{Al}}}}{c^2}}}\\&=\sigma \mu \left[ {{{\left( {{b_{\mathrm{1}}}{b_{\mathrm{2}}} + \frac{{b_{\mathrm{2}}^2}}{{\sqrt{\mu }}}} \right) }^2} + {{\left( {{b_{\mathrm{1}}}{b_{\mathrm{2}}} - \frac{{b_{\mathrm{1}}^2}}{{\sqrt{\mu }}}} \right) }^2}} \right] . \end{aligned} \end{aligned}$$

(12)

Here, $\sigma$ is defined as $-{q^2}{S_{\mathrm{E}}}/{4m_{\mathrm{{Al}}}^2{c^2}}$. So both the heating rate and the drift rate of the time-dilation shift are proportional to $S_{\mathrm{E}}$. When considering the heating rate of the radial mode, two more factors $\varepsilon$ and $\alpha$ have to be introduced, where $\varepsilon =\omega _{\mathrm{p}}/\omega _{\mathrm{z}}$, and $\omega _{\mathrm{p}}$ is the contribution of the rf potential to the radial trap frequencies, $\alpha$ is a parameter indicating the degree of radial asymmetry of the static field. Here, $\omega _{\mathrm{p}}$ and $\omega _{\mathrm{z}}$ are for the single logic ion. The frequencies for the single logic ion along the x and y directions are given by ${\omega _{\mathrm{x}}} =\omega _{\mathrm{z}} \sqrt{\varepsilon ^2 - \alpha }$ and ${\omega _{\mathrm{y}}} = \omega _{\mathrm{z}}\sqrt{\varepsilon ^2 - (1 - \alpha )}$ [49].

Fig. 2

The relationship of the drift rate of the time-dilation (TD) shift and the mass ratio of the $^{27}\hbox {Al}^+$ to different logic ions. $\varepsilon =\omega _{\mathrm{p}}/\omega _{\mathrm{z}}$, $\omega _{\mathrm{p}}$ is the contribution of the rf potential to the radial trap frequencies, $\omega _{\mathrm{z}}$ is the axial trap frequency. The solid line corresponds to the situation of $\varepsilon ^2=4$, the dashed line corresponds to $\varepsilon ^2=3$, and the dotted line corresponds to $\varepsilon ^2=2$. The mass ratios of $^{27}\hbox {Al}^+$ to $^{40}\hbox {Ca}^+$, $^{25}\hbox {Mg}^+$, and $^{9}\hbox {Be}^+$ are 0.675, 1.08, and 3, respectively, which are marked with vertical dashed lines in the figure. The calculation is based on the condition of radially symmetric traps

We have calculated the total drift rate of the time-dilation shift with different mass ratio in a radial symmetric trap, where $\alpha =1/2$. As shown in Fig. 2, the drift rate of the time-dilation shift could achieve a minimum value at the range of the mass ratio between 1.0 and 1.5, which depends on the value of $\varepsilon$. We can see that, much smaller drift rate of the time-dilation shift can be achieved with $^{25}\hbox {Mg}^+$ ion as the logic ion, in comparison with $^{40}\hbox {Ca}^+$ ion and $^{9}\hbox {Be}^+$ ion.

3 Experimental setup

As described in our previous publications [55,56,57], a linear Paul trap is used to trap the $^{25}\hbox {Mg}^+$ and $^{27}\hbox {Al}^+$ ion pair. It consists of four blade electrodes and two end-cap tip electrodes. The blade electrodes supply the radial confinement of trapped ions and the end-cap electrodes supply the axial confinement. The distance from the center of the trap to the blade electrodes is $r_0 = 0.8$ mm, and the radius of curvature of the blades is $r_{\mathrm{e}} = 0.3$ mm and the fluorescence collection solid angle is 0.8 sr [55]. The distance between the center of the trap to the end-cap is $z = 3.0$ mm. The ion trap is driven by a helical resonator with a Q value over 400, and the end-cap is driven by a high voltage power supply directly. Three extra copper rods running along the trap axial direction are used for the compensation of stray electric fields in radial directions. The axial compensation is implemented on the end-cap electrodes.

Two re-entrant fused silica viewports are installed on the vacuum chamber to increase the collection efficiency of the fluorescence, and two sets of imaging lenses are installed outside the viewports for a photon counting module and an electron multiplying CCD. The vacuum chamber with a pressure below $5 \times 10^{-9}$ Pa is maintained by a 40 L/s ion pump, a nonevaporable getter pump, and a titanium sublimation pump. Three pairs of Helmholtz coils powered by a precise current source are mounted around the vacuum chamber to compensate the background magnetic field and define the ion quantization axis with a magnetic field of 6.5 G.

The $^{25}$Mg atoms and the $^{27}$Al atoms are evaporated from a Mg wire and an Al wire through laser ablation. The ablation laser is a commercially available passive Q-switched laser at a wavelength of 532 nm with a maximum pulse energy of 150 $\upmu$J, and a pulse duration of 2 ns. The repetition rate can be adjusted from 0 to 10 kHz. Then the $^{25}\hbox {Mg}^+$ ion is loaded by resonant two-photon ionization of $^{25}$Mg atoms. The ionization laser is a frequency-quadrupled 285 nm tunable diode laser system. The $^{27}\hbox {Al}^+$ ion is also loaded by the resonant two-photon ionization with a 396 nm diode laser. The ionization laser is an extended cavity diode laser (ECDL) with its frequency set to the resonance frequency of the $^{27}$Al ${\left| ^2P_{3/2} ,{F=4} \right\rangle } \leftrightarrow {\left| ^2S_{1/2} ,{F=3} \right\rangle }$ transition which was measured to be 756.547133(3) THz [58].

Two fourth-harmonic-generation (FHG) lasers with a wavelength of 280 nm are used for Raman sideband cooling and Doppler cooling of the trapped $^{25}\hbox {Mg}^+$ ions. The laser frequencies of 280 nm are locked to a high precision wavemeter [59]. The one 280 nm FHG laser that is used for Doppler cooling, repumping and state detection is frequency shifted by two double-passed AOM with a driving frequency of 450 MHz. The other 280 nm FHG laser is red-detuned 16 GHz from the resonance frequency, and is used as the Raman-$\uppi$ laser, the Raman-$\upsigma ^{+}$ laser, and the Raman repump laser.

Two FHG laser are used for spectroscopy on $^{27}\hbox {Al}^+$ ion, one FHG laser with a wavelength of 267.0 nm is used for the $^{27}\hbox {Al}^+$ ion’s logic transition and one FHG laser with a wavelength of 267.4 nm is used for $^{27}\hbox {Al}^+$ ion’s clock transition. The master lasers of the FHG at 267.0 nm and 267.4 nm are locked to two high-finesse 10-cm long ultra-stable Fabry–Pérot reference cavities and both have fractional frequency instabilities of $4.9 \times 10^{-16}$ which is limited by the thermal noise [60].

4 Result and analysis

4.1 Raman sideband cooling

The QLS requires motional ground state cooling of the transfer mode. The Raman sideband cooling is carried on $^{25}\hbox {Mg}^+$ to realized the motional ground state cooling of the ion pair. As discussed in Sect. 2.2, there are STR mode and COM mode along each direction. Although it is different from single ion in ion trap, the cooling strategy of the ion pair can be optimized based on the same strategy used for single ion. Based on our previous investigation on the cooling strategy of single ion [56], we use a cooling sequence consisting of fixed-width RSB pulses with alternating second-order RSB pulses and first-order RSB pulses. In the experiments we optimize the duration as well as the number of the RSB pulses to improve the cooling efficiency.

Fig. 3

a The sequence of one Raman sideband cooling cycle. RSB, red sideband cooling. b The sequence of the Raman sideband cooling for the $^{25}\hbox {Mg}^+$ and $^{27}\hbox {Al}^+$ ion pair. c The sequence for the blue sideband spectroscopy of the $^{27}\hbox {Al}^+$ ion’s logic transition. The operation of BSB on $^{27}\hbox {Al}^+$ ion’s logic transition and RSB on $^{25}\hbox {Mg}^+$ ion’s Raman transition is called QLP. d The branching ratio and the repumping-related transitions of the $^{25}\hbox {Mg}^+$ ion

Both the Doppler and repump lasers are circularly polarized. For Doppler cooling, the laser beam is red detuned by half of the linewidth with respect to the transition from $\left| \downarrow \right\rangle _{\mathrm{L}}$$\leftrightarrow$$\left| {^2P_{3/2}},{F=4}, {m_{\mathrm{F}}= 4}\right\rangle _{\mathrm{L}}$ , which is the cycling transition for Doppler cooling and state detection. R-$\upsigma ^+$ and R-$\uppi$ are the two Raman laser beams used for the Raman sideband cooling and corresponding to the $\left| \uparrow \right\rangle _{\mathrm{L}}$, $\left| \downarrow \right\rangle _{\mathrm{L}}$$\leftrightarrow$$\left| {^2P_{3/2}},{F=3}, {m_{\mathrm{F}}=3}\right\rangle _{\mathrm{L}}$ transitions, respectively. BR-co is a co-propagating blue Raman laser corresponding to the transition of $\left| {^2S_{1/2}},{F=3}, {m_{\mathrm{F}}=2}\right\rangle$$\leftrightarrow$$\left| {^2P_{3/2}},{F=3}, {m_{\mathrm{F}}=3}\right\rangle$. BR-co laser and R-$\upsigma ^+$ are used for the Raman transfer from $\left| {^2S_{1/2}},{F=3}, {m_{\mathrm{F}}=2}\right\rangle$ to $\left| {^2S_{1/2}},{F=2}, {m_{\mathrm{F}}=2}\right\rangle$. All Raman lasers are red detuned by 16 GHz from the resonant frequency. Here R-$\upsigma ^+$ and BR-co are circularly polarized ($\upsigma ^+$) beam and R-$\uppi$ is linearly polarized ($\uppi$) beam.

Before Raman sideband cooling, the $^{25}\hbox {Mg}^+$ and $^{27}\hbox {Al}^+$ ion pair is cooled to the Lamb–Dicke regime by applying Doppler cooling on $^{25}\hbox {Mg}^+$ ion. Then, the Raman sideband cooling is carried on the ion pair. As shown in Fig.  3a, a single Raman sideband cooling process consists of one $\pi$ pulse red sideband (RSB) on $^{25}\hbox {Mg}^+$, two repump processes and one Raman transfer process. The $^{25}\hbox {Mg}^+$ will be derived to ${\left| \uparrow \right\rangle _{\mathrm{L}}}$ from ${\left| \downarrow \right\rangle _{\mathrm{L}}}$ after one RSB process. To ensure continuous RSB cooling, the ${\left| \uparrow \right\rangle _{\mathrm{L}}}$ state should be optically pumped back to the ${\left| \downarrow \right\rangle _{\mathrm{L}}}$ efficiently. When applying the repump laser, the $^{25}\hbox {Mg}^+$ still has a small probability to fall to the $\left| {^2S_{1/2}},{F=3}, {m_{\mathrm{F}}= 2}\right\rangle$ state, as shown in Fig.3d. So the Raman transfer is used to drive the $\left| {^2S_{1/2}},{F=3}, {m_{\mathrm{F}}= 2}\right\rangle$ to a ${\left| \uparrow \right\rangle _{\mathrm{L}}}$ transition and one repump laser to optically pump back to ${\left| \downarrow \right\rangle _{\mathrm{L}}}$ state. Typically, one cycle of the Raman transfer could drive the $^{25}\hbox {Mg}^+$ ion to the ${\left| \downarrow \right\rangle _{\mathrm{L}}}$ state efficiently.

As for the ion pair, both the STR mode and the COM mode are suffering heating all the time, so the Raman sideband cooling should be carried out alternatively to ensure that the two modes both reach the vibrational ground states, as shown in Fig.  3b. The second order RSB of the COM mode and the STR mode are cooled 70 times and then the first order RSB of the COM mode and the STR mode are carried out 40 times.

Fig. 4

The Raman sideband spectroscopy of $^{25}\hbox {Mg}^+$ for the ion pair. The secular motion frequency of the STR mode is 2.75 MHz, and the secular motion frequency of the COM mode is 1.58 MHz. The mean phonon numbers of the STR mode and the COM mode are 0.10(1) and 0.01(1), respectively

Fig. 5

The measurement of the heating rate for the $^{25}\hbox {Mg}^+$–$^{27}\hbox {Al}^+$ ion pair. The square dot is the mean phonon number which is determined through the amplitude ratio of the BSB and RSB of the Raman sideband spectroscopy. The red line is a linear fit of the mean phonon number with the slope represents the heating rate. a The evolution of the phonon number of the COM mode. The heating rate is fitted to be 3.0(7) phonons/s. b The evolution of the phonon number of the STR mode. The heating rate is fitted to be 0.23(5) phonons/s

We optimized the cooling efficiency for the ion pair first. The motional modes of the $^{25}\hbox {Mg}^+$ and $^{27}\hbox {Al}^+$ ion pair are cooled to the vibrational ground states, as shown in Fig.  4. The mean phonon numbers and the heating rates are two important parameters in the QLS, which determine the signal to noise ratio and the coherent time of the QLP. The mean phonon number can be determined from the ratio of the RSB amplitude and the BSB amplitude. The sequence of the STR mode and COM mode will affect the cooling result of the COM mode and the STR mode. The result in Fig. 4 is carried out with the sequence that the RSB of the STR mode is cooled first. So the COM mode has a better cooling result and the mean phonon number of the COM mode is estimated to be ${\bar{n}}_{\mathrm{COM}} = 0.01(1)$. The mean phonon number of the STR mode is determined to be ${\bar{n}}_{\mathrm{STR}} = 0.10(1)$. So the COM mode is chosen as the transfer mode for the QLS.

The heating rate of the motional mode is one of the important parameters that affect the time-dilation shift. To realize a low heating rate, the ion trap was carefully designed, our trap electrodes were meticulously aligned with the help of a high magnification imaging system during the trap installation. The surfaces of the trap electrodes were gold-plated. Electronic filtering for the endcap electrodes was also used to reduce the electronic noise of the ion trap power supply.

The heating rates of the two motional modes are measured by inserting a sequence of delay without any laser interactions after Raman sideband cooling and before the Raman sideband detection. The heating rate of the STR mode is measured to be 0.23(5) phonons/s, and the heating rate of the COM mode is measured to be 3.0(7) phonons/s, as shown in Fig. 5. For comparison, the heating rates in Ref. [50], which has a much smaller trap size, are 0.34(2) phonons/s and 9.12(32) phonons/s for the STR mode and the COM mode, respectively. The heating rates in Ref. [52], which has a similar size with our trap, are 18.6(2.0) phonons/s and 395(59) phonons/s for the STR mode and the COM mode, respectively. Apparently, our setup has relatively low heating rates.

4.2 Quanatum logic spectroscopy

After the ions are cooled to the vibrational ground states, the QLS can be carried out to determine the blue sideband frequency for $^{27}\hbox {Al}^+$ ion’s logic transition. The basic sequence is shown in Fig.  3c. After Doppler cooling and Raman sideband cooling, the ion pair is cooled to the ground state for the transfer mode. Then, one $\pi$ pulse BSB for the $^{27}\hbox {Al}^+$ ion’s logic transition and one $\pi$ pulse RSB for the $^{25}\hbox {Mg}^+$ ion’s Raman transition are carried in sequence. Finally, the internal states of $^{25}\hbox {Mg}^+$ ion are detected by the Doppler cooling laser. The transition probability of the BSB interrogation on the $^{27}\hbox {Al}^+$ ion’s logic transition is determined by repeating measurements multiple times.

Fig. 6

The BSB spectroscopy of $^{27}\hbox {Al}^+$ ion’s logic transition $\left| {^1S_0},{F=5/2}, {m_{\mathrm{F}}=5/2}\right\rangle$$\leftrightarrow$$\left| {^3P_1},{F=7/2}, {m_{\mathrm{F}}=7/2}\right\rangle$. The red line is fitted with $\hbox {sinc}^2$ function with a FWHM of 11.5 kHz

To determine the center frequency of the BSB of the $^{27}\hbox {Al}^+$ ion’s logic transition, we scan the BSB frequency of the $^{27}\hbox {Al}^+$ ion’s logic transition $\left| {^1S_0},{F=5/2}, {m_{\mathrm{F}}=5/2}\right\rangle$$\leftrightarrow$$\left| {^3P_1},{F=7/2}, {m_{\mathrm{F}}=7/2}\right\rangle$. The experimental data for the BSB spectroscopy is shown in Fig.  6. The BSB pulses of $^{27}\hbox {Al}^+$ are applied with constant intensity and duration, corresponding to $\uppi$ pulses on this transition when the laser beam is tuned to the resonance. Each data point is an average of 200 times. The observed linewidth of the transition is Fourier-transform limited with a fitted full width at half maximum (FWHM) of 11.5(4) kHz for an excitation pulse duration of $\hbox {t}_\pi$ = 100 $\upmu$s. The contrast is limited by the fluctuation of the laser power and the detection laser frequency of $^{25}\hbox {Mg}^+$, magnetic field fluctuation of the system, and radial motion Debye-Waller factors.

4.3 The detection of the clock transition

The clock transition for the $^{27}\hbox {Al}^+$ ion clock based on the QLS is carried out with the method mentioned in Sect. 2.1. As discussed in Sect. 2.1, when the $^{27}\hbox {Al}^+$ ion is in $\left| c \right\rangle _{\mathrm{{Al}}}$ ( $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$) state, the QLP will be blocked (opened), and bright (dark) state of $^{25}\hbox {Mg}^+$ will be observed. To improve the detection fidelity, generally the state of the $^{27}\hbox {Al}^+$ ion is repeatedly probed and analyzed. In the following we separate the detected fluorescence result into a subset $\{n_{\mathrm{i}}\}$. To determine the $^{27}\hbox {Al}^+$ ion’s internal state, the probability threshold method and the maximum likelihood method [61] are commonly used.

For the probability threshold method, we set a threshold $n_0$ and probability $p_0$, if the probability of $n_{\mathrm{i}}<n_0$ is larger than $p_0$, we say the $^{27}\hbox {Al}^+$ ion is in the $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$ state and vice versa.

For the maximum likelihood method, we calculate the likelihood $p_{\mathrm{G}}$ that a given subset $\{n_{\mathrm{i}}\}$ arose from a ground-state $^{27}\hbox {Al}^+$ ion, and compare this with the likelihood $p_{\mathrm{C}}$ that $\{n_{\mathrm{i}}\}$ arose from a clock-state $^{27}\hbox {Al}^+$ ion at the start of the detection period. $p_{\mathrm{G}}$ is given by the product of probabilities $p_{\mathrm{G}}=\prod _{\mathrm{i}=1}^N G(n_{\mathrm{i}})$, where $G(n_{\mathrm{i}})$ is the probability of observing $n_{\mathrm{i}}$ counts from a ground-state $^{27}\hbox {Al}^+$ ion case. $p_{\mathrm{C}}=\prod _{\mathrm{i}=1}^N C(n_{\mathrm{i}})$, where $C(n_{\mathrm{i}})$ is the probability of observing $n_{\mathrm{i}}$ counts from a clock-state $^{27}\hbox {Al}^+$ ion case. If $p_{\mathrm{G}} >p_{\mathrm{C}}$ we infer that the ion is at the ground state and vice versa. So the key for the maximum likelihood method is to obtain a suitable photon number probability distribution of the $^{25}\hbox {Mg}^+$ ion’s fluorescence for the clock state ( $\left| c \right\rangle _{\mathrm{{Al}}}$) and the ground state ( $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$).

There are several methods to obtain the probability distribution for two different cases. (a) We can use the measured probability distribution of a dark $^{25}\hbox {Mg}^+$ state ($P_{\mathrm{{D}}}$) for the $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$ and a bright $^{25}\hbox {Mg}^+$ state ($P_{\mathrm{{B}}}$) for the $\left| c \right\rangle _{\mathrm{{Al}}}$ state, as shown in Fig.  7a. This is called B/D discrimination method. b) The transition probability of the $^{3} P_1$ BSB of the $^{27}\hbox {Al}^+$ ion is taken into account. From Fig.  6, it can be seen that the minimum point for the $^{3} P_1$ BSB spectrum ($\eta _0$) is not zero. To take this into account, we still use the histogram of bright $^{25}\hbox {Mg}^+$ state for the $\left| c \right\rangle _{\mathrm{{Al}}}$ state. However, for the $\left| \downarrow \right\rangle _{\mathrm{{Al}}}$ case, the probability distribution has the form of $P{\left( n \right) _ \downarrow } = \left( {1 - {\eta _0}} \right) {P_{\mathrm{D}}} + {\eta _0}{P_{\mathrm{B}}}$, as shown in Fig.  7b. This is called $^{3} P_1$ calculation method. (c) Similar to (b), but the probability distributions of $P_{\mathrm{{D}}}$ and $P_{\mathrm{{B}}}$ are generated by the Poisson distributions where the expected numbers are the same as the experimental results, as shown in Fig.  7c. This is called Poisson calculation method. (d) Similar to (a), but we use the measured probability distributions for the two cases of $^{27}\hbox {Al}^+$, as shown in Fig.  7d. This is called $^{3} P_1$ experiment method.

Fig. 7

a The histogram of the observed photon number for dark state (${\left| \uparrow \right\rangle _{\mathrm{L}}}$) and bright state (${\left| \downarrow \right\rangle _{\mathrm{L}}}$) of $^{25}\hbox {Mg}^+$ ion. b The histogram of the ${\left| \downarrow \right\rangle _{\mathrm{{Al}}}}$ state and ${\left| c \right\rangle _{\mathrm{{Al}}}}$ state of $^{27}\hbox {Al}^+$ ion based on calculation from the $^{3}P_1$ BSB amplitude and the histogram of (a). c The histogram of the ${\left| \downarrow \right\rangle _{\mathrm{{Al}}}}$ state and ${\left| c \right\rangle _{\mathrm{{Al}}}}$ state of $^{27}\hbox {Al}^+$ ion based on calculation from the $^{3} P_1$ BSB amplitude and Poisson distribution of a $^{25}\hbox {Mg}^+$ ion’s dark/bright state. d The histogram of the photon number for the ${\left| \downarrow \right\rangle _{\mathrm{{Al}}}}$ state and ${\left| c \right\rangle _{\mathrm{{Al}}}}$ state of $^{27}\hbox {Al}^+$ ion based on the experimental measurement. The detection time is 150 $\mu$s in the experiments

We use the Monte-Carlo method to simulate the above detection methods. The procedure is as follows. (1) The probability distribution is set up for each method. (2) For a given clock transition probability $p_{\mathrm{{real}}}$, a series of imaginary experiment processes are generated and a series of states of the $^{27}\hbox {Al}^+$ ion are obtained. (3) For each $^{27}\hbox {Al}^+$ ion state, according to the transition probability of the $^{3}P_1$ BSB, a series of imaginary quantum logic detection results (photon numbers of $^{25}\hbox {Mg}^+$ ion) are generated. (4) Based on the probability threshold method or the maximum likelihood method, a clock transition probability $p_{\mathrm{{sim}}}$ can be figured out. The whole sequence can be repeated to obtain the average value and standard deviation of $p_{\mathrm{{sim}}}$.

The simulation results are shown in Fig. 8. It can be seen that both the $^{3}P_1$ experiment method and the probability threshold methods predict the clock transition probabilities very well, while other methods have rather large residual errors in some regions of $p_{\mathrm{{real}}}$. However, it should be noted that in this simulation, when $p_{\mathrm{{real}}} = 0$ and the initial state of the $^{27}\hbox {Al}^+$ ion is ${\left| \downarrow \right\rangle _{\mathrm{{Al}}}}$, only the Poisson calculation method predicts the exact probability of 0, as shown in Fig. 8a. This is very important for people who is searching for the clock transition for the first time or is diagnosing the experimental system, because the potential clock transition probability could be quite low.

Fig. 8

The simulated residual error of threshold and maximum likelihood methods by different clock transition probabilities. Each point is repeated 100 times and the error bar corresponds to one standard deviation. a The initial state of $^{27}\hbox {Al}^+$ ion in each simulation is ${\left| \downarrow \right\rangle _{\mathrm{{Al}}}}$. b The initial state of $^{27}\hbox {Al}^+$ ion in each simulation is ${\left| c \right\rangle _{\mathrm{{Al}}}}$

Fig. 9

The $^{27}\hbox {Al}^+$ ion’s clock transition $\left| {^1S_0},{F=5/2}, {m_{\mathrm{F}}=5/2}\right\rangle$$\leftrightarrow$$\left| {^3P_0},{F=5/2}, {m_{\mathrm{F}}=5/2}\right\rangle$. based on QLS. The data are fit by $\hbox {sinc}^2$ function

At last, we use the Poisson calculation method to observe the clock transition. The clock transition is probed with a 22 ms interrogation time, as shown in Fig. 9. The FWHM of the transition is 38(4) Hz, which is Fourier-transform limited by the interrogation time. The interrogation time is limited by the coherent time of the system which includes the decoherence of the transfer mode and the decoherence of the internal state of the $^{27}\hbox {Al}^+$ and $^{25}\hbox {Mg}^+$. The decoherence effect comes from the fluctuation of the laser power, phase fluctuation of the clock laser, magnetic field fluctuation and radial motion Debye–Waller factors. The contrast of the signal is limited by the coupling coefficient of the wrong state which comes from the transition probability of the BSB for $^{27}\hbox {Al}^+$ ion’s logic transition and RSB for $^{25}\hbox {Mg}^+$ ion’s Raman transition heating rate and the coupling of the motional number from radial directions.

5 Conclusion

In conclusion, we have observed the clock transition of the $^{27}\hbox {Al}^+$ based on QLS. The influence of different logic ions on the time-dilation shift is discussed, and we point out that the choice of $^{25}\hbox {Mg}^+$ could achieve much smaller time-dilation shift uncertainty for the optical clock. As a premise for the QLS, we realize Raman sideband cooling of the $^{27}\hbox {Al}^+$–$^{25}\hbox {Mg}^+$ ion pair to the motional ground state along the axial direction with the mean phonon number of 0.01(1) for the COM mode, and 0.10(1) for the STR mode. The heating rate is evaluated to be 0.23(5) phonons/s for the STR mode and 3.0(7) phonons/s for the COM mode, respectively. To detect the state of the $^{27}\hbox {Al}^+$ ion with high fidelity, we use the Monte-Carlo method to simulate several different detection methods. Based on the simulation result we propose the best method for clock transition diagnosing and the best method for clock transition locking. At last, the clock transition is observed with a FWHM of 38(4) Hz in our system.

In the future, precision measurement of the clock transition frequency will be performed and compare with the clock at NIST [9] and the clock being constructed at the University of Innsbruck [45], WIPM [52] and PTB [48].