1 Introduction

Since its invention by Ramsey [1], separated oscillatory fields have been widely used in physics experiments, especially in the field of atomic clocks. When an atomic transition is excited by separated oscillatory fields, the resonance width of the transition is inversely proportional to the free evolution time between the two excitation fields. Thus, the performance of an atomic clock can be improved by increasing the free evolution time, which is achieved by lengthening the interaction-zone separation or by lowering the atom velocity. The use of a thermal atomic beam in the Ramsey method is mainly restricted by the distance between the two excitation fields. Using fountains of laser-cooled atoms increases the free evolution time by lowering the velocity of the atoms, but this method is mainly limited by free fall due to Earth’s gravitational field and system’s sizes [2, 3]. In efforts to develop a novel atomic frequency standard, another atom source is a laser-cooled continuous atomic beam [4–9]. These beams have been used in an increasing number of applications, including matter-wave interferometer-based quantum sensors [10–13], atom nanolithography [14–16] and in the production of Bose–Einstein condensates (BECs) [17–19].

A laser-cooled atomic beam generates a continuous stream of slow atoms; compared with thermal beams, a laser-cooled atomic beam can shrink the physics package of the instrument it is used in, and it can increase the sensitivity of the sensor because of its longer coherent time. When considering an atomic clock, using a continuous atomic beam can reduce noise caused by intermodulation effects and collision shifts [20, 21] compared with using a pulsed atomic fountain [22] or vapor cell as the atom source [23, 24]. Early research in atomic-beam-based atomic clocks used continuous cold beams produced by laser-cooling a thermal beam generated from an oven [5, 25]; however, this system is usually bulky because of the long distance needed for laser cooling. Other methods for generating continuous cold atomic beams directly from a vapor-cell-based magneto-optical trapping (MOT) have been developed, including a low-velocity intense source (LVIS) of cold atoms [26], used in efforts to develop a novel atomic clock for satellites [27, 28]. Recently, Devenoges et al. [21] demonstrated a stability of \(6 \times 10^{-14}/ \sqrt{\tau }\) with their Cesium fountain clock, which uses a continuous cold atomic beam as the source, and produces its microwave transition in a single microwave cavity.

In traditional atomic clocks, a microwave cavity is used to produce the microwave transition and to achieve two-zone excitation, which produces Ramsey fringes. A laser-cooled fountain clock usually achieves two-zone excitation using a single cavity, while a beam clock uses two cavities, separated a distance along the beam path. In both thermal-beam clocks [29–32] and laser-cooled fountain clocks [33–35], a major cause of systematic error are the frequency shifts caused by the distributed cavity phase and by microwave leakage. One alternative to using a microwave cavity to produce Ramsey fringes is to use optically separated oscillatory fields. Using a stimulated Raman transition as the optical oscillatory field has been studied theoretically by Moler et al. [36]. Optical Raman–Ramsey fringes have been observed in a Na thermal atomic beam [37]; the width of its central fringe of 650 Hz corresponds to an interaction-zone separation of 30 cm. They have also been observed in Cs thermal atomic beams with central fringe widths of 1 kHz [38] and less than 50 Hz [39], corresponding to interaction-zone separations of 15 cm and 2 m, respectively. Recently, Zholnerov et al. [40, 41] proposed a satellite slow-beam atomic clock based on optical oscillatory fields.

In this paper, we used two optical \(\pi /2-\pi \)/2 Raman pulses to drive a laser-cooled continuous \(^{87}\hbox {Rb}\) atomic beam, producing corresponding Raman–Ramsey fringes that we observed. The free evolution time was 3.4 ms, corresponding to a central fringe width of 146 Hz or an interaction-zone separation of 42 mm, and the most probable longitudinal atomic velocity of the atomic beam was 12.4 m/s.

2 Experiment

Energy level structure of the \(^{87}\hbox {Rb}\) atom and the laser frequencies used in the experiment are shown in Fig. 1. We cooled and trapped atoms with cooling lasers tuned to the \(5s^2S_{1/2} F = 2\) to \(5p^2P_{3/2} F^{\prime }=3\) allowed transition with a typical red detuning of \(\delta = 4-5 \varGamma \) (where \(\varGamma = 2\pi \times 6\hbox { MHz}\) is the natural linewidth of \(^{87}\hbox {Rb}\)) and with repumping lasers tuned to the \(5s^2S_{1/2} F =1\) to \(5p^2P_{3/2} F^{\prime }=2\) allowed transitions. In this system, the laser-cooled atomic beam is ejected from a three-dimensional magneto-optical trap (3D-MOT) with a ground state of \(5s^2S_{1/2} F=2\); it is then pumped into a single hyperfine \(5s^2S_{1/2} F=1\) level by the optical pumping laser (1) tuned to the \(F=2 \rightarrow F^{\prime }=2\) transition frequency. Then, another optical pumping laser (2) tuned to \(|F=1, m_F =-1> \rightarrow |F^{\prime } =0, m_F=0>\) of the \(\sigma ^+\) transition or \(|F=1, m_F =+1> \rightarrow |F^{\prime }=0, m_F=0>\) of the \(\sigma ^-\) transition prepares atoms from the \(|F=1, m_F=-1>\) and \(|F=1, m_F=+1>\) states into the magnetically insensitive \(|F=1, m_F=0>\) state. Two phase-coherent Raman lasers with a frequency offset at the \(^{87}\hbox {Rb}\) hyperfine splitting frequency of \(\omega _{\mathrm{hfs}} =\omega _1 -\omega _2 = 2\pi \times 6.835\hbox { GHz}\) are tuned to \(\varDelta \) below the \(F = 1\) and \(F = 2\) to \(F^{\prime }=1\) allowed optical transition. Finally, the atomic signals are detected by the probe laser, tuned to the \(F=2 \rightarrow F^{\prime }=3\) cyclic transition.

Fig. 1
figure 1

Energy level diagram of Rb, showing the laser frequencies used in the experiment

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Figure 2 shows a simplified diagram of the experimental setup used to observe the Raman–Ramsey fringes in the laser-cooled \(^{87}\hbox {Rb}\) atomic beam. The low-velocity intense atomic beam is generated by a vapor-cell 3D-MOT [42]. Two vacuum chambers, used for the 3D-MOT and the Raman–Ramsey zone, are connected only by openings with diameters of 1 mm in the centers of a \(\lambda /4\) plate (QWP1) and a mirror (M1). Six mutually perpendicular cooling beams are focused on the zero center of the quadrupole magnetic field generated by a pair of anti-Helmholtz coils. The cold atomic beam is generated continuously from the unbalanced 3D-MOT induced by a tubular cooling beam along the z axis, reflected by the mirror M1. The direction of the atomic beam is adjusted by \(3.7^{\circ }\) with respect to the horizontal and separated from the leaking cooling laser beam via the gravitational effect. We could vary the most probable velocity of the beam in the range of 10–20 m/s, and we could increase the flux up to \(4 \times 10^9\hbox { atoms/s}\). The launching atoms initiated in the \(F=2\) ground state are then mostly transferred into the magnetically insensitive \(|F=1,m_F=0>\) ground state by a combination of optical pumping beams 1 and 2. The atomic beam can be plugged and switched off quickly by a plug beam tuned to the \(F=2 \rightarrow F^{\prime }=3\) resonance.

Fig. 2
figure 2

Schematic of the experimental layout for Raman–Ramsey interactions in a laser-cooled continuous Rb atomic beam. BS beam splitter, M mirror, HWP half-wave plate, QWP quarter-wave plate, PMF polarization-maintaining fiber, FEOM fiber electro-optic modulator, TA tapered diode laser amplifier, PMT photomultiplier

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Figure 2 also shows the optical apparatus of the Raman lasers. The master laser is a 780-nm external cavity diode laser (ECDL, DL Pro, Toptica, Germany), which is frequency-stabilized to the \(F=1 \rightarrow F^{\prime }=1\) allowed transition with a large red detuning of \(\Delta = -2\pi \times 1.07\hbox { GHz}\) by using a high-sensitivity wavelength meter (WS6-200, Toptica, Germany). Slave laser 1 is directly injection-locked to the master laser, operating at a frequency denoted as \(\omega _1\). Another master laser beam passes through a fiber electro-optical modulator (FEOM, EOspace, USA), driven by an RF signal of \(\omega _{00} = 2\pi \times 6.835\hbox { GHz}\) corresponding to the ground-state hyperfine transition of the\(^{87}\hbox {Rb}\) atoms. Slave laser 2 is frequency-selectively sideband injection-locked to the \(-\)1 sideband of the FEOM output, denoted as \(\omega _2\) [43]. Finally, the 6.835 GHz RF source is referred to a Cs atomic clock; the \(-\)3 dB linewidth of the beat note between \(\omega _1\) and \(\omega _2\) is less than 1 Hz, the resolution of the spectrum analyzer. The beams of slave lasers 1 and 2 are amplified by two semiconductor laser amplifiers (BoosTA, Toptica, Germany), TA1 and TA2 respectively, and then combined into a single-mode polarization-maintaining fiber with crossed polarizations, which can drive co-propagating stimulated Raman transitions for the polarization selection rule. The laser beam is then shaped by a slice and divided into two parallel beam with spacings of L and equal widths of d. The intensity of the two laser beams can be adjusted to get \(\pi /2\) pulses of ~\(6\hbox { mW/cm}^2\) and \(\pi \) pulses of ~\(12\hbox { mW/cm}^2\). To define a quantization axis, a constant magnetic bias field is applied throughout the length of the interaction region, horizontally in the direction of the Raman beams, with a four-conductor magnetic field assembly running inside the vacuum chamber (not shown in Fig. 2). To detect the signals of the stimulated Raman transitions and the Raman–Ramsey fringes, the fluorescence of atoms at the \(F=2\) state, induced by the probe laser beam, is collected by a photomultiplier tube (PMT, H7422-50, Hamamatsu, Japan) in real time while sweeping the RF signal driving the FEOM.

3 Results and discussion

The velocity and flux of the cold atomic beam were measured by the TOF method, realized by plugging the atomic beam using the plug beam and simultaneously recording the falling edge of the signal of the atoms in the \(F=2\) state. The spectrum of the atomic velocity was deduced from the TOF signal, as shown in Fig. 3. We adjusted the most probable longitudinal velocity of the atomic beam to \(v_p =15.0\hbox { m/s}\) with a velocity distribution of \(\delta v =\sim 3.5\hbox { m/s}\) at the full width at half maximum (FWHM); we adjusted it by changing the red detuning range \(\delta \) and the polarization of the cooling lasers. The atomic flux was tuned to ~\(1\times 10^9\hbox { atoms/s}\), which is high enough to directly measure the fluorescence of the atoms of the \(5s^2S_{1/2}\)(\(F=2\)) state.

Fig. 3
figure 3

Velocity distribution of the atomic beam, measured by the time-of-flight (TOF) method

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Figure 4a shows the Raman transition spectra. Applying a constant-bias magnetic field of ~1.2 G along the Raman beams, by detuning \(\omega _2\) we found three Zeeman splitting peaks separated by 1.65 MHz. After being pumped from the \(|F=2>\) state to the \(|F=1>\) state, the atoms were distributed across all \(m_F\) levels in the \(|F=1, m_F=0, \pm 1>\) ground state. However, only the \(|m_F=0>\) ground levels were used in this system. Thus, to redistribute \(|F=1>\) atoms across the \(m_F\) levels, we used another laser beam tuned to the \(\sigma ^+\) transition of \(|F=1, m_F=-1> \rightarrow |F^{\prime }=0, m_F=0>\), pumping 40 % of the atoms in the \(|F=1, m_F=0>\) level, as shown in Fig. 4a. This Zeeman-state optical pumping (ZOP) enhancement was limited by the accumulation of atoms in the \(|F=1, m_F=1>\) level, but it could be improved by introducing an additional laser beam tuned to the \(\sigma ^-\) transition of \(|F=1, m_F=+1> \rightarrow |F^{\prime }=0, m_F=0>\). As shown in Fig. 4b, by adjusting the intensities of the Raman beams of \(\omega _1\) and \(\omega _2\), we can make both Raman pulses A and B work at the \(\pi /2\) or \(\pi \) Rabi phase, letting us observe the spectrum of each Raman transition. The amplitudes and profiles of both of the \(\pi /2\) pulse Raman transitions were adjusted to equal and half the value of the \(\pi \) pulse, respectively. The linewidth of the transitions is \(\delta _{\mathrm{Rabi}}=v_p/d=15\) kHz (FWHM). Thus, the Rabi frequency of the Raman \(\pi \) pulse is \(f_{\pi ,{\mathrm{Rabi}}}=\delta _{\mathrm{Rabi}}/2=7.5\) kHz.

Fig. 4
figure 4

Raman transitions in the co-propagating (Doppler-insensitive) excitation geometry. a Optical pumping is achieved using a laser tuned to the \(|F=2> \rightarrow |F^{\prime }=2>\) resonance, shown by a thin red line. \(|F=1>\) atoms are redistributed across the \(m_F\) levels using another laser beam tuned to the \(\sigma ^+\) transition of \(|F=1, m_F=-1> \rightarrow |F^{\prime } =0, m_F=0>\), shown with a thick blue line. b Raman transitions tuned to \(\pi \) and two \(\pi /2\) Rabi phases

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Figure 5 shows the Raman–Ramsey fringes observed in the detection region. Figure 5a, b shows the fringes with a contrast of ~50 %, with an interaction-zone separation of \(L=19\) mm and an interaction-zone width of \(d=1\) mm. As shown in Fig. 5b, the linewidth of the central fringe is \(\delta _{\mathrm{Ramsey}}= v_p/{(2L)}=390\hbox { Hz}\), corresponding to an interrogation time of 1.3 ms. The cold beam has higher resolution than the thermal atomic beam for the Ramsey fringes because the cold beam has a narrower longitudinal atomic velocity distribution [39]. The fourth order of the interference fringe, where \(n={v_p}/{\Delta v}=4\), can be clearly identified against the velocity averaging effect [36]. As shown in Fig. 5c, we found Ramsey fringes with a linewidth of 146 Hz (FWHM) for the central fringe, when the two beams were separated by \(L=42\) mm and the longitudinal velocity of the cold atomic beam was tuned to \(v_p=12.4\hbox { m/s}\). Correspondingly, the free evolution time of the cold atomic beam was \(T = 3.4\hbox { ms}\).

Fig. 5
figure 5

Raman–Ramsey fringes driven by \(\pi /2-\pi /2\) Raman pulses. In a, b, the length of the separated oscillatory fields is \(L = 19\) mm, and the width of the interaction zone is \(d = 1\) mm. In c, \(L = 42\) mm and \(d = 4\) mm

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In the two-zone stimulated Raman interaction, the AC Stark shifts are important in the cosine function describing the Ramsey fringe [44], and these constitute an important error source for beam-based Raman clocks. These Ramsey-fringe phase shifts cause frequency shifts of the central-fringe minima and produce line-shape asymmetries, as demonstrated in Fig. 5. In our experiments, we found that the \(\frac{\omega _1}{\omega _2}\) intensity ratio is the critical parameter for reducing the relative AC Stark shifts and making the Ramsey fringes symmetrical. Otherwise, the Ramsey fringes would be out of the Raman transition profile and would eventually be washed out because of velocity averaging. In stimulated Raman transitions, it is typical to find that the relative AC Stark shifts depend on the intensity ratio of the two Raman laser beams, an effect which could be used to reduce or even eliminate such frequency shifts induced by AC Stark shifts [45]. Further experiments should explore how AC Stark shifts affect these systems in detail.

At present, the transverse velocity spread of the cold atomic beam is approximately \(\pm 10\hbox { cm/s}\) (FWHM). When the atomic beam reaches a different Raman interaction zone, its transverse dimensions will expand, producing inhomogeneous Rabi frequencies because of the spatially inhomogeneous intensity distribution of a Raman pulse. Consequently, the average effect of this inhomogeneous intensity will decrease the contrast. The transverse thermal expansion of the cold atomic beam can be compressed by further sub-Doppler cooling of atoms.

4 Conclusion

We performed an optical Raman–Ramsey \((\pi /2\)–\(\pi /2)\) experiment by using a continuous LVIS source of cold \(^{87}\hbox {Rb}\) atoms with a most probable longitudinal velocity of 10–20 m/s. We measured a free evolution time 3.4 ms, corresponding to a central fringe width of 146 Hz (FWHM) and a short interaction-zone separation of 42 mm. Compared with a thermal atomic beam [37–39], this cold LVIS atom source shows promise for use in a compact atomic clock and for applications in high-resolution spectroscopy.

Many factors can influence Raman–Ramsey experiments [46]. The major sources of systematic error are a relative AC Stark shift induced by the Raman laser beams and a frequency shift caused by fluorescence leakage from the MOT-based atomic source. This systematic noise must be evaluated and reduced before developing a Raman–Ramsey atomic clock with a MOT-based continuous cold atomic beam. Additionally, the interaction-zone separation must be increased to narrow the central fringe width, which in the present experiment was limited mainly to the size of the vacuum window and essentially to the longitudinal and transverse temperatures of the atomic beam.