1 Introduction

Cold atoms trapped in an optical lattice potential have been considered as one of the promising physical systems for the implementation of a quantum simulator in condensed matter physics and a quantum computer [1]. Global experimental parameters of the system, such as a lattice depth and an interatomic interaction, have been well controlled; however, the measurement and manipulation of atoms with a single-site spatial resolution have been challenging tasks.

So far, the observation of a spatial distribution of ultra-cold atoms in an optical lattice has been realized in two different methods: diffraction-limited optical imaging using an objective with a large numerical aperture [2–4] and magnetic resonance imaging (MRI) with microwave transitions [5–7]. In this paper, we demonstrate MRI with sub-wavelength spatial resolution using a narrow optical transition [8, 9]. The energy level diagram relevant to this work is shown in Fig. 1. To carry out this optical magnetic resonance imaging (OMRI), we chose the 1S0↔3P2 (m≠0) transition of Yb (507 nm) to utilize the non-zero magnetic dipole moment of the 3P2 state. The narrow line width of this transition (the natural line width is about 10 mHz) allows us to achieve a high spatial resolution with a relatively low magnetic field gradient. To measure the excitation spectrum for this weak transition, we repump the excited atoms to a state with a strong cycling transition to enhance the detection sensitivity. A line shape of a measured spectrum of the OMRI is well explained with a theoretical spatial distribution of atoms in a one-dimensional (1D) optical lattice based on the Thomas–Fermi approximation. Spectroscopy of a Bose–Einstein condensate (BEC) of 174Yb is also performed.

Fig. 1
figure 1

Energy levels of an Yb atom. For the laser cooling, the 1S0↔1P1 transition at 399 nm and the 1S0↔3P1 transition at 556 nm are used for the Zeeman slower and the collection-MOT, respectively. The 1S0↔1P1 transition is also used for the absorption imaging and the detection-MOT for the fluorescence measurement. The ultra-narrow 1S0↔3P2 transition at 507 nm is used for the spectroscopy. The 3P2↔3S1 transition at 770 nm and the 3P0↔3S1 transition at 649 nm are used for repumping the 3P2-state atoms back to the 1S0 state

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2 Experimental setup

We developed a vacuum system which has regions for laser cooling to prepare ultra-cold atoms and for high-resolution laser spectroscopy. A schematic description is shown in Fig. 2a. In the spectroscopy area, there is an octagonal thin glass cell which has high optical accessibility and compactness to locate magnetic coils close to an ultra-cold atomic sample. The height and the width of the cell are 14 mm and 70 mm, respectively. The cell is made of 3-mm-thick glass plates by optical contact bondings. For the fluorescence detection, we use the side surface of the cell, which covers a solid angle of about 0.14 steradian. The laser cooling area is the same as previous apparatus [10].

Fig. 2
figure 2

Schematic diagram of the experimental setup. (a) Simplified schematic of the experimental chamber. Inset shows a schematic 3D drawing of the glass cell. (b) The Zeeman slower and the collection-MOT. (c) The atom transferring system

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2.1 Cooling and trapping

We first prepare a sufficient number of thermal Yb atoms in an optical far-off-resonant trap (FORT) at the center of the laser cooling area by the following procedure. An Yb atomic beam from an oven at 375∘C is decelerated in a Zeeman slower. The 1S0↔1P1 transition at 399 nm is used for the slower transition. A typical power and a beam diameter of the slower laser are 250 mW and 14 mm, respectively. The decelerated atoms are captured by a MOT as shown in Fig. 2b (collection-MOT). The intercombination 1 S 0↔3P1 transition at 556 nm is used for the collection-MOT. A typical laser power and a beam diameter of the collection-MOT laser are 25 mW and 25 mm, respectively, for each direction [11]. The trapped atoms are loaded into a single FORT. The wavelength and the beam waist of the FORT laser are 532 nm and 23.5 μm, respectively. The maximum power of the FORT laser is 6.5 W. The number of atoms in the FORT is up to 3.0×106.

2.2 Atom transfer

The atoms in the FORT are moved into the glass cell by translating a lens for the FORT laser (Fig. 2c). The lens is put on a linear air-bearing stage (Aerotech, ABL20030), which controls the axial position of the lens. The motion algorithm of the stage sets a moving distance, a maximum acceleration, and a jerk (derivative of acceleration) to specify the motion trajectory, as in previous work [12].

The results of the transfer experiment in various trajectory settings are shown in Fig. 3. The number of transferred atoms does not depend much on the jerk, presumably due to the sufficient FORT depth. To transfer a larger number of atoms to the cell, a shorter moving time is desirable to suppress atom loss during the moving. In the OMRI experiment, we set the transfer time to 1.7 s.

Fig. 3
figure 3

Transfer efficiency in different motion settings. The number of atoms in the glass cell is plotted versus the maximum acceleration (a) and the moving time (b). Circles, squares, and diamonds correspond to the jerk of 1.0, 2.0, and 3.0×103 mm/s3, respectively. For a comparison, the number of atoms in the FORT after 1.9 s hold time without the transfer is 1.6×106

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2.3 Creating BEC

Another FORT laser beam (532 nm) is introduced vertically to the transferred atoms to make a crossed FORT at the center of the glass cell. The evaporative cooling is performed by continuously decreasing the intensities of the FORT lasers. After evaporative cooling for about 10 s, we obtain an almost pure BEC of 174Yb with up to 3.0×104 atoms.

2.4 Atom detection

The number of atoms in the glass cell is measured by the standard absorption imaging method or by detecting the fluorescence from atoms recaptured by a MOT with 399-nm laser beams (detection-MOT), which is near resonant with the 1S0↔1P1 transition. The beam diameter and power in each detection-MOT laser beam are 3 mm and 0.5 mW, respectively. The magnetic field gradient of the detection-MOT is 100 G/cm. A schematic description is shown in Fig. 4a. In the case of the fluorescence detection, we use an electron-multiplying charge-coupled-device (EMCCD) camera (ANDOR, iXon+885) with small read-out noise and high gain in order to detect weak fluorescence from a small number of atoms. The stray light is suppressed with spatial and interference filters. The CCD exposure settings are controlled depending on the number of atoms in the detection-MOT. A typical CCD exposure time and an EMCCD gain are 300 ms and 300, respectively.

Fig. 4
figure 4

Histograms of the fluorescence intensity from a detection-MOT with and without a small number of atoms. (a) A schematic diagram for the fluorescence detection. The fluorescence from the detection-MOT is guided onto the EMCCD camera through lenses and filters which block stray light. (b) The measurement result without atoms. The solid line denotes a fit to the data with a Gaussian function. (c) The result with atoms loaded. (d) An enlarged view of the whole histogram shown in (c). The scales of the horizontal axis in (b–d) are the same

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Figure 4b–d show results of the fluorescence measurement with and without atoms in the crossed FORT. The number of trapped atoms is controlled by the transfer condition and the loading time of the collection-MOT. Without atoms in the FORT (Fig. 4b), we observe a peak in the histogram, which represents the background stray light at the detection-MOT. On the other hand, a few peaks appear in addition to the large stray light peak in the result with atoms loaded (Fig. 4c and d). Figure 4d, an enlarged view of Fig. 4c, shows a periodic peak structure which corresponds to fluorescence signals from different numbers of atoms. The solid line in Fig. 4d is a sum of six equally separated Gaussian functions, shown to guide the eye. The separation between each Gaussian function corresponds approximately to fluorescence counts from a single atom (n atom=0,1,2,…). The amount of fluorescence signal from a single atom is consistent with the fluorescence estimation made with a large number of atoms. As a result, we have achieved single-atom-level sensitivity by this fluorescence detection system.

3 Spectroscopy with an ultra-narrow optical transition

One of the features of our detection scheme is that the observed spectrum has zero background, that is, the fluctuation of the initial number does not affect the background signal level. The excitation spectrum for the 1S0↔3P2 transition is measured in the following way.

  1. 1.

    A small portion of the 1S0 atoms are excited to the metastable 3P2 state by a laser pulse with a duration of 1–5 ms [13, 14].

  2. 2.

    The remaining atoms in the 1S0 state are removed from the optical trap by irradiating a laser pulse resonant to the strong 1S0↔1P1 transition for 0.5–2 ms.

  3. 3.

    The atoms in the 3P2 state are repumped to the 1S0 state via the 3S1 state by two laser pulses, whose wavelengths are 649 nm for the 3P0↔3S1 transition and 770 nm for the 3P2↔3S1 transition. The pulse duration is 2–10 ms for both repumping lasers. In this period, the magnetic field gradient for the detection-MOT is ramped up.

  4. 4.

    The repumped atoms are captured by the detection-MOT with the strong 1S0↔1P1 transition and the fluorescence from the detection-MOT is detected by the EMCCD camera to measure the number of repumped atoms.

The beam waists of the excitation laser and the repumping lasers are 40 μm, 170 μm (770 nm) and 140 μm (649 nm), respectively. The intensity and the pulse shape of the excitation laser are precisely controlled by a feedback of the monitor laser power, and the typical intensity at the atoms is 40 W/cm. The powers of the repumping lasers with 770 nm and 649 nm are 0.8 mW and 0.7 mW, respectively.

The experimental setup for the spectroscopy is shown in Fig. 5, schematically. In the figure, \(\overrightarrow{{\mathbf{e}}}_{\mathrm{L}}\), \(\overrightarrow{{\mathbf{e}}}_{\mathrm{OT}}\), \(\overrightarrow{{\mathbf{e}}}_{\mathrm{ex}}\), and \(\overrightarrow{{\mathbf{e}}}_{\mathrm{rep}}\) denote the polarization direction of the lattice laser, the optical tweezer laser, the excitation laser, and the repumping laser, respectively. Two coils placed in the anti-Helmholtz configuration, which are used for the detection-MOT and the OMRI, produce a magnetic field gradient of 250 G/cm with a current of 10 A. In addition to the anti-Helmholtz coil pair, a Helmholtz coil pair produces a bias magnetic field of 11 G with a current of 8 A.

Fig. 5
figure 5

Schematic diagram for the spectroscopy of the 1S0↔3P2 (m=−2) transition in the glass cell. The Helmholtz coil pair produces a bias magnetic field. The anti-Helmholtz coil pair is used for the detection-MOT and the OMRI. The polarization of the excitation laser is determined by the selection rule of the 1S0↔3P2 (m=−2) transition

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The polarizabilities of the 3P2 (|m|=0,1,2) state in the FORT laser depend on the FORT laser polarization direction and the direction of the bias magnetic field [13]. Since the light shift of the 3P2 state is different from that of the 1S0 state, the spatial atom distribution in the FORT results in an additional broadening of the excitation spectrum, which is theoretically estimated to be about 0.5 kHz in this experiment. In the following analysis, we also take into account the spectral line width of the excitation laser (∼1 kHz) in addition to this broadening.

3.1 High-resolution laser spectroscopy of a thermal gas and a Bose–Einstein condensate

In this section, we show the demonstration of the 1S0↔3P2 laser spectroscopy of a thermal gas and a BEC with the experimental setup described above. In our previous work [13], it was necessary to excite a large number of atoms in order to observe the excitation spectrum; however, in this work, the system described above enables the detection of even a few atoms. The detection sensitivity enables us to perform the spectroscopy in a weak excitation regime, which clarifies the interpretation of the spectrum, as described below. In the weak excitation limit, the shape of the excitation spectrum of a BEC is given by the following equation under the Thomas–Fermi approximation [15–17]:

$$I(\nu)\propto \frac{15h(\nu-\nu_0)}{4n_0\varDelta U}\sqrt{1-\frac{h(\nu-\nu_0)}{n_0\varDelta U}} ,$$
(1)

where

$$\varDelta U=\frac{4\pi\hbar^2}{m}(a_\mathrm{ge}-a_\mathrm{gg}) .$$
(2)

Here, ν is the laser frequency, ν 0 is the resonance frequency of non-interacting atoms in a trap, n 0 is a peak density, h is Planck’s constant, a gg is the scattering length between two atoms in the 1S0 state, and a ge is the scattering length between atoms in the 1S0 and 3P2 states. The value of a gg has been reported to be 5.53 nm previously [18, 19]. A strong excitation will introduce an additional term, the interaction between excited atoms, in (2). This simple form overwhelmingly clarifies the interpretation of the spectra observed.

The excitation spectra of the 1S0↔3P2 (m=−2) transition at different temperatures are shown in Fig. 6. The atomic samples at different temperatures are prepared in the crossed FORT by adjusting the final trap depth of the evaporative cooling. In the spectroscopy of the BEC, the intensities of the FORT lasers are increased in 100 ms to the same values for the thermal gas after the preparation of the BEC in order to set the same trap condition between the two different temperatures. The temperatures of the two different samples are determined from the time-of-flight images.

Fig. 6
figure 6

Results of high-resolution laser spectroscopy of the 1S0↔3P2 (m=−2) transition of (a) the thermal gas (600 nK) and (b) the BEC (180 nK). The bias magnetic field is 8 G. The solid line in each plot is a convolution of the theoretical calculation (see text) with a Lorentz function (FWHM=1.5 kHz) which models the line width of the excitation laser and the inhomogeneity of the optical trap. The dashed line in (b) denotes the thermal component

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Figure 6a shows the spectrum for a thermal gas. The temperature and the number of atoms are 600 nK and 5.6×104, respectively. The spectrum in Fig. 6a is well fitted by an absorption line shape introduced in Ref. [20] with the temperature of 600 nK. The zero frequency in Fig. 6 is defined by this fit.

The spectrum of a BEC is shown in Fig. 6b. The temperature and the number of atoms are 180 nK and 1.3×104, respectively. The critical temperature of the BEC transition is 270 nK and the condensate fraction is about 70 percent. The maximum number of excited atoms is 1.1×103, which is less than 10 percent of the total atom number of the BEC. The spectrum of the BEC in Fig. 6b is described by a sum of the absorption line shape for the thermal gas and the line shape described in (1). The ratio of the thermal component to the condensate is determined by the temperature and thus the fitting parameters are only the scattering length a ge and a scale of the fluorescence signal intensity. The solid line in Fig. 6b denotes the fitting result and well describes the experimental observation. We determine the scattering length a ge=−21(2) nm from the fit. The error of a ge is mainly due to the uncertainty of the zero-frequency point (±0.5 kHz), which is determined by the fit of the spectrum of the thermal gas.

4 Optical magnetic resonance imaging

High-resolution laser spectroscopy of the 3P2 state in a magnetic field gradient enables magnetic resonance imaging with high spatial resolution. With a magnetic moment of 3 μ B in the 3P2 state of the 174Yb atom and a magnetic field gradient of 100 G/cm, the energy gradient induced by the Zeeman effect is h×42 Hz/nm [9].

We performed the OMRI with a BEC confined in a one-dimensional (1D) optical lattice potential. The BEC prepared in the glass cell is loaded into a vertical 1D optical lattice potential. The optical lattice laser has a wavelength of 532 nm and the intensity is precisely controlled by a feedback of the monitored laser power. The magnetic field gradient is ramped up to the desired value during the evaporative cooling. The current for the magnetic field gradient coils is stabilized by a feedback controller with a current sensor (HIOKI E E Corporation, CT6862).

Figure 7a and b display spectra of the 1S0↔3P2 (m=0) transition without the magnetic field gradient and the 1S0↔3P2 (m=−2) transition with a magnetic field gradient of 250 G/cm, respectively. The uniform applied bias magnetic fields in Fig. 7a and b are 0.7 G and 12 G, respectively. The atom numbers in Fig. 7a and b are 2.0×104 and 1.5×104, respectively. The trap frequencies of external harmonic confinement (ω x ,ω y ,ω z ) are 2π×(77,219,205) Hz and the lattice depth is 15 E R. In the case of the spectroscopy of the 1S0↔3P2 (m=0) transition, the magnetic field is tilted to the horizontal plane, which is required due to the selection rule of the 1S0↔3P2 transition. The widths of the excitation pulse in Fig. 7a and b are 1 ms and 5 ms, respectively. The spectrum of the 1S0↔3P2 (m=−2) transition is significantly broadened by the magnetic field gradient compared with the spectrum of the 1S0↔3P2 (m=0) transition. The broadening of the spectrum in Fig. 7b reflects the spatial distribution of the BEC in the optical lattice and the whole width of the observed spectrum is well explained by a theoretical calculation described below.

Fig. 7
figure 7

OMRI spectra of a BEC trapped in a 1D optical lattice potential. (a) The spectrum of the 1S0↔3P2 (m=0) transition, which is insensitive to the magnetic field. The solid line denotes a fit to the data with a Gaussian function. The error bars denote the standard deviation of individual measurements. (b) The whole OMRI spectrum with the 1S0↔3P2 (m=−2) transition. The position labeled on the top of the x-axis is calculated from the applied magnetic field gradient and the excitation laser frequency detuning. Each data point in (b) corresponds to a single shot of the experiment

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A BEC in a sufficiently deep 1D optical lattice potential can be considered as an array of quasi-2D condensates. The local wavefunction of a quasi-2D condensate in the ith site is assumed to be factorized as [21]

$$\varPhi _i(x,y,z)=f_i(z)\phi_i(x,y).$$
(3)

Here, f i (z) and ϕ i (x,y) are the wavefunctions describing the axial and the radial degrees of freedom in each optical lattice potential well, respectively. Here, we assume the Gaussian function f i (z)=exp(−(z−id)2/2σ 2), where d is the lattice period and σ characterizes the width of the condensate in each lattice well. For a sufficiently deep optical lattice, the value of σ is not affected by the two-body interaction energy, and is calculated as σ=d/(πs 1/4) with s=((ħω z )/(2E R))2 by the harmonic expansion of the optical lattice potential around its minima, where ω z is the trap frequency of the optical lattice well and E R is a recoil energy of the lattice photon.

Due to the weak confinement along the radial direction of the lattice laser, the interaction plays an important role for the shape of the wavefunction in the radial direction. Under the Thomas–Fermi approximation, the wavefunction ϕ i (x,y) in the radial direction is given by

$$\bigl|\phi_i(x,y)\bigr|^2=\frac{m}{2\sqrt{2}\pi\hbar^2a_\mathrm{gg}}\mu_i\biggl(1-\frac{x^2+y^2}{R_i^2}\biggr),$$
(4)

where μ i and \(R_{i}=\sqrt{2\mu_{i}/m\omega_{\mathrm{r}}^{2}}\) are an effective site-dependent chemical potential and the Thomas–Fermi radius of the condensate in the ith site, respectively. Here, ω r is the trap frequency in the radial direction.

We assume that the OMRI spectrum consists of spectra of quasi-2D condensates which are shifted by the Zeeman shift corresponding to the magnetic field at the center of each lattice well. With this assumption, the OMRI spectrum is calculated as follows:

$$I(\nu)=\sum_{k=-k_m}^{k_m} I_k\biggl(\nu-g \mu_\mathrm{B} k d \frac{\partial B}{\partial z}\biggr) ,$$
(5)

where

$$I_k(\nu)\propto \frac{15h(\nu-\nu_0)}{4n_k\varDelta U}\sqrt{1-\frac{h(\nu-\nu_0)}{n_k\varDelta U}}. $$
(6)

Here, g is the Landé g-factor of the excited state, μ B is the Bohr magneton, and n k is a peak density in the kth site. The value of ΔU in (6) is calculated from the value of a ge obtained from the BEC spectrum in the previous section. The solid line in Fig. 7b shows the calculated line shape of (5). The envelope of the solid line is in agreement with the experimental data. One can recognize a relatively large fluctuation of the experimental data in Fig. 7b. To look at this feature in more detail, we show in Fig. 8 a detailed structure of an OMRI spectrum in a slightly different condition from Fig. 7b.

Fig. 8
figure 8

Detailed OMRI spectrum of a BEC in the edge of a 1D optical lattice potential. (a) The observed OMRI spectrum. Each data point corresponds to a single shot of the experiment. The horizontal line shows a threshold to calculate the excitation probability. (b) The excitation probability histogram. The bin width is 7 kHz. The solid lines in (a) and (b) denote the calculated line shapes based on (5) (see text) and are shown to guide the eye

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The number of atoms, the trap frequencies, and the lattice depth are 8.0×103, 2π×(92,235,217) Hz, and 17 E R, respectively. The width of the excitation pulse is 1 ms. The magnetic field gradient and the bias magnetic field are 250 G/cm and 9 G, respectively. The solid lines in Fig. 8a and b are also the calculated line shapes based on (5).

In this experiment, the radial size of the BEC in the optical trap is important. The quadrupole magnetic field produced by the anti-Helmholtz coils has a magnetic field gradient in the radial direction which causes a frequency shift within the same quasi-2D condensate in a lattice well. In order to suppress the frequency shift in the radial direction, a small radial size of the atoms in a lattice well is desirable. To that end, the frequency scan range of the excitation laser is set to the edge of the BEC in the axial direction of the optical lattice. At the edge, however, the number of atoms in each lattice well strongly depends on the initial cloud size of the BEC. As a result, many data points in Fig. 8a show zero fluorescence signal due to the excitation of an empty lattice site. In order to clearly observe the periodic structure induced by the optical lattice, the observed fluorescence signals are binned with a bin width of 7 kHz. Within a bin, we calculated the excitation probability with a threshold of a fluorescence intensity shown in Fig. 8a. The calculated probabilities exhibit equally separated peaks, whose periodicity is consistent with the lattice constant and the applied magnetic field gradient, as shown in Fig. 8b.

4.1 Discussion

In addition to the stabilization of the initial cloud size of the BEC, there are some possibilities to suppress the fluctuation of the OMRI with the single-site resolution of the optical lattice. In particular, the stabilization of the position of the optical lattice retroreflecting mirror and the suppression of the thermal expansion of the magnetic field coils for the magnetic field gradient should provide an improvement of the measurement.

5 Conclusion

In summary, we demonstrated OMRI with the ultra-narrow optical transition 1S0↔3P2 (m=−2) of 174Yb atoms. The experimental setup with separated regions to perform the laser cooling and the spectroscopy has been developed for the OMRI experiment. The spectroscopy scheme with the repumping technique and the fluorescence detection allowed us to perform the spectroscopy in a sufficiently weak excitation regime. The envelope of the observed OMRI spectrum agrees with the theoretical calculation and the periodic structure which reflects the optical lattice is observed in the OMRI spectrum.