We show that the fluorescence emission spectrum of few atoms can be measured by using an optical nanofiber combined with the optical heterodyne and photon correlation spectroscopy. The observed fluorescence spectrum of the atoms near the nanofiber shows negligible effects of the atom-surface interaction and agrees well with the Mollow triplet spectrum of free-space atoms at high excitation intensity.
© 2010 OSA
Resonance fluorescence of atoms has been the subject of interest in quantum optical measurements for many years [1,2]. The resonance fluorescence consists of two main contributions, the elastic scattering and the inelastic scattering components. The elastic contribution comes from the well-known Rayleigh scattering and the inelastic component consists of fluorescence scattering. If the driving field is weak, the elastic scattering part dominates and if the driving field is strong, the inelastic part dominates. The resonance fluorescence is routinely used as a tool to detect atoms, to generate photons and to generate non-classical states of light . One of the widely used methods to investigate the atomic fluorescence is laser-induced fluorescence (LIF) spectroscopy [1–3]. The LIF spectroscopy may be categorized into two methods. One is the excitation spectroscopy and the other is emission spectroscopy. The excitation spectroscopy measures the fluorescence radiation from emitters by tuning the laser wavelength around atomic resonance; on the other hand, the emission spectroscopy measures the spectrum of the emitted fluorescence radiation by fixing the excitation laser wavelength. Although the two methods give complementary information, the excitation spectroscopy is rather extensively used in various applications compared to the emission spectroscopy. This is because the excitation spectroscopy not only provides high resolution, due to the development of single-mode narrow linewidth lasers, but also gives high sensitivity due to the integration of spectrally distributed fluorescence photons.
Regarding the fluorescence emission spectroscopy, two very high resolution methods are well known. One is optical heterodyne (OHD) spectroscopy, and the other is photon correlation (PCR) spectroscopy. The essence of the OHD spectroscopy is to measure the beat signals between a coherent local-oscillator light and the fluorescence light which is emitted into the same single spatial-mode as that of the local oscillator. On the other hand, the PCR-spectroscopy is based on photon correlation measurements with the Hanbury Brown-Twiss (HBT) setup. For single-mode observations, the intensity correlations can be expressed by the first-order correlation function in the many-emitter limit. Hence, one can obtain the emission spectrum through the PCR measurements by Fourier-transforming the obtained first-order correlation functions [1,2]. Both the OHD and PCR methods readily realize very high spectral resolution, higher than the natural linewidth. The OHD method has been used for the fluorescence spectrum measurement of many-atom [4,5] and single-ion systems [6,7]. However, none of the above methods have been applied to measure the fluorescence spectrum of few atoms, which is distributed in a wide frequency range broader than the natural linewidth. This is due to the fact that these methods require single-spatial-mode observations, which demands many atoms to have a measurable fluorescence photon number. Moreover the PCR method is valid only for multi-emitter system. Recently Hong et al. have proposed to combine both the OHD and PCR methods to realize high resolution and high sensitivity. Using such a combined method they have demonstrated the measurement of the spectrum of an extremely weak coherent light . If such a combined method can be extended to measure the few-atom fluorescence spectrum, it would be very beneficial to laser spectroscopy. However, it still requires efficient single mode collection of few-atom emission.
Recently, a novel method which can measure the fluorescence of a small number of atoms under the single-spatial-mode condition has been reported [9–11]. The key point of the method is to incorporate a subwavelength-diameter silica fiber, called an optical nanofiber, in the measurements. The essential physics of the method is the spatial confinement of the field in the guided mode of the nanofiber and the associated quantum electrodynamic (QED) effects . It has been demonstrated that an appreciable amount of fluorescence photons from a small number of atoms can be channeled into a single guided mode of the nanofiber . Single atoms were measured successfully and their photon correlations were investigated theoretically and experimentally [11–14]. It was also demonstrated that the average atom number in the observation region around the nanofiber could be estimated by measuring the correlations for photons through opposite ends of the fiber [13,14]. Moreover, it was demonstrated that atom-surface interaction could be monitored through the fluorescence excitation spectrum [10,11,15].
In the present paper, we extend the optical nanofiber method to measure atomic fluorescence emission spectrum for strongly driven atoms. The emission spectrum has been investigated for both resonant- and off-resonant excitation. We incorporate the combined method of OHD and PCR spectroscopy into the optical nanofiber method. We measure fluorescence emission spectrum for laser cooled cesium atoms around the nanofiber. We demonstrate that fluorescence emission spectrum for small number of atoms can be measured readily with spectral resolution better than natural linewidth.
2. Theoretical outline
We briefly describe the theoretical outline of the present few-atom fluorescence spectrum measurement using nanofiber. The emission spectral density S(ω), can be directly obtained from the Fourier transform of the first-order correlation function of the photon field E(t) as,12–14]2]. In the conventional photon correlation measurements, we measure the intensity correlation and Eq. (3) shows that the intensity correlation can give a direct measure of when the atom number n is large enough. But in our present case, where the number of atoms is small, the intensity correlation is dominated by, and the derivation of is not straightforward. Also, in some special cases it is possible to obtain from: for example, when the spectrum has the Lorentzian line shape, is derived from as. But generally, such derivation of from is not always possible.
As discussed by Hong et al., one can selectively obtain the g (1)(τ) information from the intensity correlations by using the combined method of OHD and PCR spectroscopy . The key point is to mix a coherent local-oscillator (LO) light signal with the weak fluorescence light. The observable intensity correlations can be formulated readily by replacing E(t) with to include the LO field , where ωLO is the LO-frequency. The intensity correlation is then given asEq. (2) and Eq. (3) into Eq. (4), we obtainEq. (1), expression (4) for the second-order correlation function can be rewritten asEquation (6) shows that the second term which contains the fluorescence spectrum S(ω) information is down-shifted by ωLO in the frequency domain. The shifted spectrum appears around the frequency ω 0-ωLO. So, we can obtain the spectral density S(ω) by taking the Fourier transform of the correlation signal . The LO frequency ωLO should be chosen such that there is no overlap between the second term and the other terms of Eq. (6) in the frequency domain and also such that the shifted spectrum frequency falls within the measurable bandwidth of the detector.
3.1 Experimental setup
The schematic diagram of the experimental setup is shown in Fig. 1 . The nanofiber is located at the waist of a tapered fiber, which is produced by heating and pulling a commercial single-mode optical fiber . The diameter of the nanofiber is 400 nm ± 30 nm over the length of about 2 mm. We use laser-cooled Cs atoms for atomic system produced by using a magneto-optical trap (MOT). Experiments are performed by overlapping cold Cs atoms with an optical nanofiber and detecting the fluorescence photons emitted into the guided modes of the nanofiber. Experiments are carried out, by switching off MOT beams for an observation period of 10 μs in every 200 μs and repeating this process for many cycles. For the initial state preparation of the atoms, the MOT repump beam is switched off 200 ns after the cooling beam during each observation period so that any residual atoms in the F=3 state can be pumped into the F=4 state. During the observation period of 10 μs, atoms around the nanofiber are excited by a laser beam irradiated perpendicularly to the nanofiber in the travelling wave configuration.
The excitation beam is elliptically focused to the nanofiber with a waist size of 1 mm along the fiber axis and 0.5 mm perpendicular to the fiber axis. The excitation beam is linearly polarized with polarization axis perpendicular to the fiber axis. Part of the fluorescence light from atoms around the nanofiber is coupled into the guided mode of the nanofiber and propagates through the signal fiber-line via the tapered region. The fluorescence light is detected using single-photon-counting avalanche-photodiodes D1, D2 and D3 (APD, PerkinElmer SPCM-AQR/FC).
The excitation beam is derived by frequency up-shifting an external-cavity diode-laser (ECDL) by 80 MHz using an acousto-optic modulator (AOM) (Fig. 1). The frequency of the excitation laser is phase-locked to a reference laser using a stable RF-frequency generator at 9.2 GHz. The reference laser frequency is locked to Cs D 2 line transition 6S 1/2 F=3 ↔ crossover of 6P 3/2 F=2→F′=3, using saturated absorption spectroscopy. Thus, the frequency of the excitation beam can be varied around the closed cycle transition of Cs D 2 line, 6S 1/2 F=4 ↔ 6P 3/2 F=5, by tuning the RF-frequency generator. The frequency stability of the excitation laser is limited by the line-width of the reference laser, which is better than 1 MHz, and is enough for resolving the natural line-width of Cs atom of 5.2 MHz. The intensity fluctuation of the excitation laser is less than 2%. Part of the ECDL-output is fiber-coupled to the signal fiber-line, as shown in Fig. 1, and is used as the LO-light. Thus, the fluorescence spectrum frequency falls in the frequency range around 80 MHz (=ω 0-ωLO).
Photon correlation measurements are performed using conventional HBT setup. Photons at one end of the signal fiber-line, consisting of both atomic fluorescence and LO light, is split into two using a 50:50 non-polarizing beam splitter (NPBS), and are detected by two APDs, D1 and D2. The LO power (~200 fW) has been kept low to avoid the saturation of the APDs. Arrival times of all the photons are recorded using a time-correlated photon-counter (PicoHarp 300, PicoQuant GmbH) during each observation period of 10 μs with a resolution of 1 ns. The signals are accumulated for 3 minutes, which requires a total experiment time of 1 hour. Photon correlations are derived by analyzing the recorded arrival times.
Regarding the spectral resolution of the present system, OHD-part and PCR-part have their own limitation. Spectral resolution of the OHD-part is limited by the accuracy of the AOM-frequency, which is about 10 kHz. On the other hand, the resolution of the PCR-part is determined by the correlation time. In the present system, the correlation time is effectively limited by the cold-atom dwell-time around the nanofiber. Assuming the dwell time of 1.8 μs , the spectral resolution is estimated to be around 250 kHz. Thus, the spectral resolution of the whole system is estimated to be 250 kHz.
3.2 Settings for the excitation beam frequency
As reported in Ref , for atoms around the nanofiber, excitation spectrum does not show a simple Lorentzian spectrum observed in free space, but shows slightly asymmetric spectral shape shaded in the red side due to van der Waals (vdW) interaction. Figure 2 displays the excitation spectrum (black square dots) of cold Cs atoms measured through the guided mode of the nanofiber using APD, D3. We have fitted the observed curve using a Lorentzian curve (gray solid line). The full width half maximum (FWHM) linewidth of the spectrum is 8.7 MHz ± 0.2 MHz, which is more than the natural linewidth of Cs atom. The increase in the linewidth can be explained by the following mechanisms. Due to QED-effect there is an increase in the linewidth by about 0.6 MHz [9,10]. The effect of the power broadening on the linewidth is around 1.9 MHz. The Doppler broadening of Cs atoms trapped in a MOT at 100 μK is 117 kHz. In addition to the above mechanisms, the vdW potential also leads to an increase in the linewidth by about 1.5 MHz . Including all the above effects, we expect a linewidth of 9.3 MHz FWHM, which shows reasonable agreement with the observed value.
There can be seen slight asymmetry around Δ= −10 to −25 MHz. One can see some modulations on top of the asymmetric part, but these modulations are not reproducible within the present measurement accuracy. The peak of the spectrum is shifted by −1.9 MHz. The shift includes a measurement error of 1 MHz, which is due to the stability of the excitation laser. In the present work, fluorescence emission spectra are measured by setting the excitation beam frequency at four different detuning around the atomic resonance as marked by (a)-(d).
3.3 Estimation of atom number
All fluorescence emission spectra are measured under the constant atom-number condition by fixing MOT-preparation parameters. We estimated the effective number of atoms around the nanofiber using the method reported in Refs [13,14]. The key point of the method is to measure the intensity correlations for fluorescence photons emitted into opposite direction of the nanofiber under the traveling-wave excitation condition. For this scheme, the intensity correlation is simply governed by g (2)(τ) as following [13,14],13].
The opposite-end correlation measured with APDs, D1 and D3, is displayed in Fig. 3 for delay time of ±100 ns. The excitation beam intensity is 40 mW/cm2. We have fitted the observed curve using the above formula to obtain the average atom number n by assuming the theoretical form of g (2)(τ) as given in Refs [1,12]. The fitting has led to an average atom number of n= 14 ± 2.
4.1 Resonant excitation
We first show the LIF emission spectrum for the case where the excitation beam is resonant to the transition 6S 1/2 F=4 ↔ 6P 3/2 F′=5. Figures 4(a) and 4(c) show the normalized coincidences measured for two excitation beam intensities 30 mW/cm2 and 153 mW/cm2 respectively, for time delay τ = ±2 μs. Figures 4(b) and 4(d) show the enlarged view of the center region of Figs. 4(a) and 4(c) respectively, for τ = ±100 ns.
The typical photon counting rate at one end of the nanofiber for parameters shown in Fig. 4(a) is 2.8 ×105 counts/s. Following the equation shown in Ref , , we estimate the average coupling efficiency of fluorescence into the nanofiber guided mode. The parameter np is the fluorescence photon count, n the atom number, R the atomic scattering rate, ηfiber the average coupling efficiency of fluorescence into nanofiber guided mode, T is the effective transmission of the fluorescence photons through the signal fiber-line which includes the nanofiber transmission and the losses associated with various optics and fiber couplers and ηD is the quantum efficiency of the APD’s. T and ηD are 12% and 45% respectively. The number of atoms, n ≈14 and the atomic scattering rate R is estimated as 1.7 ×107 s −1 with a saturation intensity of 2 mW/cm2. For the above calculation we used an effective spontaneous emission rate of 3.6 ×107 s −1 which includes the QED-enhancement factor [9,10]. Thus we obtain the average coupling efficiency into one end of the nanofiber guided mode ηfiber ≈2.3%.
The bunching effect of g (1)(τ) is clearly seen at zero time delay in Figs. 4(a) and 4(c). In Figs. 4(b) and 4(d), one can readily recognize the oscillations with a period of around 12 ns which reflects the difference frequency of ~80 MHz between the fluorescence frequency and the LO frequency. Envelope of the oscillation is mainly given by the first-order correlation function as shown in Eq. (5). By comparing Figs. 4(b) and 4(d), one can see that at low excitation intensity, the envelope which peaks at τ =0 falls-off smoothly. Whereas Fig. 4(d) at high excitation intensity shows a dip at around τ =25 ns.
The Fourier transform of the correlation signals shown in Fig. 4 will give the emission spectrum. The spectrum appears around the frequency of 80 MHz, which has been shifted to 0 MHz in the plot for convenience. Figures 5(a) and 5(b) show the Fourier transform spectra (gray curves) for Figs. 4(a) and 4(c) respectively, for frequency range from −50 to +50 MHz. The spectrum in Fig. 5(a) shows a broader structure than the natural linewidth with shoulders appearing on both sides of the central peak at around ±10 MHz. The spectrum in Fig. 5(b) shows a symmetric three-peak structure. It consists of a central peak and two smaller side peaks separated by around ±24 MHz from the central peak. Also in both the spectra in Figs. 5(a) and 5(b), there is another sharp peak on top of the central peak. The width of the sharp peak is around 300 kHz FWHM, which is limited by the system resolution. We measured the emission spectrum without atoms and obtained almost the same sharp peak as those with atoms.
The spectra are fitted using a equation for fluorescence spectrum of free atoms [16,17]. Fitted results are plotted by black curves. Rabi-frequency (Ω) is the only fitting parameter. The best fit for the data shown in Figs. 5(a) and 5(b), has been obtained for Rabi frequencies, Ω = 9.9 MHz and 24.5 MHz respectively. In Fig. 5(b), the central peak has a width of 5.8 MHz and a height 3.5 times that of the side peaks. The side peaks are located at ± 24.5 MHz from the central peak and have widths of about 9 MHz.
4.2 Off-resonant excitation
Next we show the emission spectrum when the excitation beam is detuned from the atomic resonance. The excitation intensity has been kept fixed at 153 mW/cm2. Figure 6 shows the measured emission spectra for three excitation beam detuning marked in Fig. 2 by (a), (b), and (d). Results are plotted by gray curves. The spectra for off-resonant excitation are found to be shifted from the resonant case by the respective excitation beam detuning. In the observed spectra, the central-peak appears at the excitation frequency and the separation between the side-peaks and the central-peak increases with increase in detuning of the excitation beam. This is due to the increase in the effective Rabi-frequency, with detuning. Also, with increase in detuning, the height of the central-peak decreases with respect to the side-peaks. To see this effect, the vertical-axis scale has been kept identical for all the plots in Fig. 6. The positive and negative side-peaks for all the spectra observed were found to be symmetric. The signal-to-noise ratio of the observed spectrum decreases with increase in detuning. This is because with increase in detuning, the fluorescence photon count reduces. Measured spectra were fitted by adjusting the Rabi frequency (Ω). The best fitting was obtained for a Rabi-frequency of Ω = 18, 16.5, and 15.5 MHz for Figs. 6(a), 6(b), and 6(c) respectively. Fitted results are plotted by black curves.
We have measured the emission spectrum from atoms around the nanofiber, the average number of which is about fourteen. Measurements were done by integrating the fluorescence photons for a time of 3 minutes, which is much less than the time required for free-space measurements . The present high sensitivity is due to the efficient collection of few-atom fluorescence through the guided mode of the nanofiber and the photon-counting based measurements.
In the present work, the excitation beam intensity has been kept much higher than the saturation intensity of the Cs atoms, to investigate the inelastic scattering part of the fluorescence emission spectrum which is distributed in a frequency range broader than the natural linewidth. As exhibited in Figs. 5 and 6, each of the spectra reveals a broad structure, consisting of a central peak along with the two side peaks and a sharp peak on the central peak. The three-peak structure is well explained by the theoretically calculated Mollow triplet spectrum for free atoms, and the sharp peak is assigned to the elastic scattering. We mention here that although the details of the spectrum depends on the multi-level structure of the Cs atom [9,15], the underlying physics can still be explained in the framework of the two-level atom model.
We note here that the Rabi-frequency, calculated from the experimentally measured excitation beam intensity is found to be larger than that obtained from the theoretical fitting, and differs by a factor of around 1.5 to 2. This difference might be due to the position distribution of MOT atoms with respect to the intensity profile of the excitation beam, along the axis of the nanofiber. Also, due to some residual magnetic field in the MOT region the dipole moment orientation of the atoms will have some distribution which may contribute to such deviations in the observed Rabi-frequency.
Regarding the sharp peak due to the elastic scattering, it is understood as scattering from silica nanofiber, since it is observed without atoms. Elastic Rayleigh scatterings from atoms may be included in the sharp peak, but the contribution from the Cs atoms should be much smaller than the bulk-silica nanofiber scatterings.
We did not see any significant difference from the free atom theory, although some signature of the vdW interaction is observed in the red side of the excitation spectrum. It means that the effect of the vdW interaction on the emission spectrum is still small compared to the free-atom contribution under the present conditions. Although some surface effects might be included in the emission spectrum, but with the present signal-to-noise ratio we could not observe any difference between the positive and negative detuning. We note that the signal-to-noise ratio of the present measurements can be increased by improving the transmission of the fluorescence photons through the signal fiber-line.
Moreover, it is reported that the atom-surface interaction can be enhanced by controlling the surface conditions [10,11]. It would be meaningful to extend the present method to a situation where atom-surface interaction is enhanced. Specifically the interaction of atoms with the surface leads to many vibrational levels , and therefore the spectrum of atom with those vibrational levels will be broad and the three-peak structure should no longer be visible in the emission spectrum. Such measurements may give a novel tool to clarify the mechanism of atom-surface interactions.
Finally, we should mention a possible extension of the present method to the dipole-trapped atoms around nanofiber. Recently, a realistic dipole-trapping scheme has been discussed theoretically , and has been demonstrated experimentally . By incorporating such dipole-trapping scheme, the present method may be used to study atom dynamics in the trap potential.
We show that fluorescence emission spectrum of few cold atoms can be investigated using an optical nanofiber. This is because atoms which lie in the vicinity of the optical nanofiber can emit a significant fraction of fluorescence photons into the single guided mode of the nanofiber. The optical nanofiber method is combined with optical heterodyne technique and photon correlation spectroscopy. We have measured the emission spectrum of approximately fourteen Cs atoms around a nanofiber. The present method may give a new tool for investigating atom behaviors in various boundary conditions, such as atom-surface interaction or dynamics in the dipole potential.
M. D. acknowledges the support of a Japanese Government (Monbukagakusho) scholarship. This work was carried out under the 21st century COE program “Innovation in Coherent Optical Science”.
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