Interaction of laser-cooled $^{87}$Rb atoms with higher order modes of an optical nanofiber

Optical nanofibres are used to confine light to subwavelength regions and are very promising tools for the development of optical fibre-based quantum networks using cold, neutral atoms. To date, experimental studies on atoms near nanofibres have focussed on fundamental fibre mode interactions. In this work, we demonstrate the integration of a few-mode optical nanofibre into a magneto-optical trap for $^{87}$Rb atoms. The nanofibre, with a waist diameter of $\sim$700 nm, supports both the fundamental and first group of higher order modes and is used for atomic fluorescence and absorption studies. In general, light propagating in higher order fibre modes has a greater evanescent field extension around the waist in comparison with the fundamental mode. By exploiting this behaviour, we demonstrate that the detected signal of fluorescent photons emitted from a cloud of cold atoms centred at the nanofibre waist is larger ($\sim$6 times) when higher order guided modes are considered as compared to the fundamental mode. Absorption of on-resonance, higher order mode probe light by the laser-cooled atoms is also observed. These advances should facilitate the realisation of atom trapping schemes based on higher order mode interference.


Introduction
Subwavelength diameter optical waveguides, commonly known as "optical nanofibers (ONFs)", are proving to be of immense value for both fundamental and applied research with many different systems being investigated, such as cold atom manipulation and trapping [2,3], colloidal particle manipulation [4,5], sensing [6,7], and microresonator coupling [8,9,10]. ONFs have a high evanescent field extent outside the waist region of the fiber, making them ideal for light-matter interactions studies. The integration of ONFs into atomic systems has been a focus of ever increasing research interest in recent years [11]. Early experimental efforts have focused on (i) the interaction of the light guided in the fundamental fiber mode, HE 11 , with atoms, or (ii) excitation of the fundamental fiber mode through fluorescence coupling from resonantly excited atoms. While the latter experimental technique provides a means of characterizing the atoms near the surface of the optical nanofiber [3,12,13,14], the former permits scenarios whereby atoms can be trapped around the optical nanofiber when far-detuned light is coupled into it [15,16,17]. Such ONFs can be termed as single-mode ONFs (SM-ONFs) as only the fundamental guided mode can be excited efficiently within them. The advantage of such a system is that, in addition to trapping applications, the nanofiber provides an optical interface that may be exploited for quantum communications using ensembles of laser-cooled atoms.
Other atom trapping geometries based on the interference pattern of higher order modes (HOMs) in a nanofiber have been proposed, permitting greater flexibility of atom position relative to the fiber, and relative to other trapped atoms [17,18]. These schemes have the benefit of allowing for selective mode interference, thereby adjusting the trapping parameters according to experimental requirements. Efficient guiding of HOMs in ONFs was a major technical challenge until the recent reporting of low-loss mode propagation in nanofibers fabricated from 80 µm [19] and, more recently, 50 µm [20] diameter silica fiber. We call such fibers few-mode ONFs (FM-ONFs) to distinguish them from the more conventional SM-ONFs. The crucial step in advancing such experiments was the result of a thorough study on parameters for fiber tapering, which revealed that reducing the fiber core:cladding diameter ratio relaxes the adiabatic criteria, thereby promoting efficient guiding of higher order modes [21]. These experimental achievements opened up a plethora of potential atom trapping scenarios, taking advantage of the few mode behavior of the nanofiber. Visualizing the field distribution of the HOMs (Fig.  1), it is evident that the evanescent field is more intense and its tail extends further into the surrounding medium than for the fundamental mode [1]. This has the added advantage of providing stronger coupling between the light field and surrounding atoms compared to when studies are limited to the fundamental mode. Intensity distribution (normalized to the HE 11 mode) of the first four fiber modes (HE 11 , TE 01 , TM 01 , and HE 21 ) along an axis containing the diameter of the nanofiber (radius 350 nm) for 780 nm light. The shaded region denotes the nanofiber. Increased surface intensity and lateral penetration depth for the evanescent fields of the higher order modes are evident.
Here, we report on the first demonstration of higher order mode propagation in an ONF integrated into a cloud of laser-cooled 87 Rb atoms. Higher order fiber modes are maintained despite the integration of the nanofiber into the ultra high vacuum system. The light maintains its modal properties even after passing through a vacuum seal, the nanofiber, and a second vacuum seal as necessary for fiber integration into the ultra high vacuum system. Recent theoretical studies by Masalov and Minogin [1] predicted that the excitation efficiency of higher order modes from fluorescing atoms outside the nanofiber should also be significantly higher than for the fundamental mode. We show initial experimental results demonstrating this effect. Apart from the enhanced pumping of fluorescent light, our results also show substantially enhanced absorption of the HOM evanescent field by the atoms. Aside from the impact this work will have on trap-ping schemes for cold atoms, it is also expected to drive progress in several other significant areas of research including optical communication, neutral-atom based quantum networks, and sensing.

Higher order modes and the optical nanofiber
Most earlier work on ONFs and atoms focused on using a standard single mode fiber (SMF) for the SM-ONF fabrication. In the work reported here a FM-ONF is fabricated using a few mode fiber for 780 nm. 80 µm fiber (SM1250G80, Thorlabs), with a NA of 0.12, was used . Considering the other options, such as 50 µm fiber [20] which supports the LP 11 group of modes, the primary motivation for using 80 µm fiber is due to the fact that it has higher robustness, making it far easier to handle and integrate into a cold atom system. It can also be spliced using standard optical fiber splicers, without having to rely on more cumbersome -and less available -techniques.
Solving Maxwell's equations for an optical fiber [23] yields a waveguide mode parameter, called the V-number, from which the number of modes supported by the fiber can be obtained. The V-number is given as where a is the radius of the core, λ is the wavelength of the light propagating in the fiber, and n core and n clad are the refractive indices of the core and the cladding, respectively. The Vnumber for this specific fiber is 4.3, implying that the fiber can support four linearly polarized mode groups, LP 01 , LP 11 , LP 21 , and LP 02 , for 780 nm light. When a nanofiber is fabricated, the cladding of the initial fiber becomes the core for the nanofiber and the surrounding medium (e.g. air or vacuum) becomes the cladding. The Vnumber plot for the nanofiber is shown in Fig. 2. The corresponding mode cutoff diameter for the degenerate HE 21 true modes is around 660 nm and for the TE 01 and TM 01 true modes it is around 580 nm.
In our experiments, the higher order modes are created by injecting a Laguerre-Gaussian (LG) beam into the fiber [22]. A liquid crystal on silicon spatial light modulator (Holoeye Pluto SLM) is used to create the LG beam. A computer generated vortex hologram is applied to the SLM and a vertically polarized, 780 nm laser beam is launched on to the SLM surface; the reflected beam forms a donut shape at the far field.The LP 11 mode, which contains the TE 01 , TM 01 , and HE 01 modes, can be excited by coupling the LG 01 free space mode into the fiber. This yields a two lobed beam profile at the output, corresponding to the LP 11 fiber modes group. Mode purity of this beam is 95%.
The FM-ONF is fabricated by the heat-and-pull technique, similar to that described in [19]. The fiber pulling rig is a hydrogen-oxygen flame brushed system. The protective jacket is removed from the fiber for around 2 cm length and the two pigtails are clamped on translational stages on the fiber pulling rig. A flame at the central part heats up the fiber near to its phase transition temperature (1550 • C). The flame moves to-and-fro along the fiber for a particular set length (the "hot zone") while the fiber is being simultaneously pulled by the two translational stages moving in opposite directions with a constant speed (the "pulling speed"). The resultant taper profile is very close to exponential in shape. Unlike for propagation of the fundamental mode, the higher order mode propagation is very sensitive to the taper angle, which itself depends on the pulling speed and hot zone. A longer hot zone facilitates shallower exponential tapers, leading to a higher transmission, but this also elongates the taper length. If the fiber is too long, it is not suitable for integration into the cold atom setup. For these reasons, and in order to achieve high transmission of LP 11 through the nanofiber, an optimal hot zone of 7.7 mm and tapering speed of 0.125 mm/s were used. A photodetector and a charged coupled device (CCD) camera were installed at the output of the fiber to monitor the transmission and mode profile during the tapering process.
First, a nanofiber was fabricated for 90 mm pulling length to get a complete transmission profile with mode cutoffs. The transmission graph ( Fig. 3(a)) suggests that the pulling should be stopped anywhere between 62 to 71 mm to obtain a nanofiber supporting the LP 11 family of modes along with the fundamental mode ( [19] gives the details of the process). Since a smaller diameter fiber yields further extension of the evanescent field into the surrounding medium it is preferable to use a longer pulling length. For the experiments reported here the nanofiber is fabricated using 70 mm pulling length, which corresponds to ∼700 nm waist diameter. Monitoring the mode profile until the end of the pulling process ensures that the prepared nanofiber still supports the LP 11 modes. The nanofiber has 30% transmission for the LP 11 mode (Fig. 3). The rest of the power couples to the cladding modes in the down taper and is, therefore, lost as radiation modes. The fiber is highly adiabatic for the fundamental mode with 92% transmission. The nanofiber is mounted on a U-mount and installed vertically in an octagonal vacuum chamber used for the magneto-optical trap (MOT). Each of the fiber pigtails at either end of the nanofiber pass through a Teflon ferrule (0.2 mm diameter) located on the top or bottom flanges of the chamber. The ferrules are rendered vacuum tight via compressing Swagelok connectors [24]. It is crucial that the nanofiber be held straight to ensure minimal bending loss and distortion to the mode profiles of the light passing through it.
2.2. Magneto-optical trapping of atoms 87 Rb atoms are cooled and trapped using a standard magneto-optical trapping technique [25] and the system has been described elsewhere [13,14], though for 85 Rb. Base pressure in the vacuum chamber is 2×10 −9 mbar. A Rb dispenser current of 5 A brings the pressure up to 4×10 −9 mbar and it remains stable during the experiments. A 780 nm laser is locked to the 5 S 1/2 F g = 2 → 5 P 3/2 F e = (2, 3) co crossover peak of 87 Rb to be used as the cooling beam. The frequency is further shifted by an acousto-optical modulator (AOM) in double-pass configuration so that it is 14 MHz red-detuned from the cooling transition, 5 S 1/2 F g = 2 → 5 P 3/2 F e = 3. The cooling beam is split into four beams, two of which are retro-reflected to get three pairs of σ + and σ − beams for the MOT. Another laser, used as the repump, is locked to the 5 S 1/2 F g = 1 → 5 P 3/2 F e = (1, 2) co peak and shifted to the repump transition, 5 S 1/2 F g = 1 → 5 P 3/2 F e = 2, using an AOM. The repump beam is mixed into one of the cooling beams using a beam splitter. The magnetic field for the MOT is created by a pair of coils carrying equal currents of 3.5 A in opposite directions to generate a field gradient of 10 G/cm at the center of the vacuum chamber. Each cooling beam has an intensity of 6 mW/cm 2 and a diameter of 18 mm. The cold atoms are trapped around the waist of the ONF. For precise alignment of the atom cloud with the waist of the nanofiber, a compensation coil is used to generate a small magnetic field in the transverse direction to the MOT coils' axis. The diameter of the cloud is ∼1 mm and there are ∼3×10 6 atoms in the trap. The temperature of the cloud is measured to be ∼150 µK by taking a series of images of the cloud for different expansion times.

Coupling of MOT beams to the nanofiber
As a first test, in the absence of both cold atoms and probe beam through the fiber, an analysis of MOT beam coupling to the nanofiber higher order modes was conducted. Since the MOT beams intersect at the nanofiber waist some light can, in principle, be coupled to either the fundamental or higher order modes. The mode profile of the coupled light can be analyzed at the output of one of the ONF fiber pigtails (see Fig. 4) by focusing it on to a CCD camera. The observed profile (Fig. 5) was donut shaped. This indicates that photons from the MOT beams coupled into the nanofiber excite the higher order modes more efficiently than the fundamental mode.

Fluorescence measurements
Masalov and Minogin predicted that the excitation rate for higher-order modes in a FM-ONF can be 5-10 times higher than for the fundamental mode for 85 Rb [1]. A similar effect should be observable for 87 Rb, the atom used in the experiments reported here. In order to verify this prediction, we looked at the fluorescence coupling into the FM-ONF from resonantly excited atoms. The cold atom cloud is formed around the waist of the FM-ONF and both the output pigtails are connected to single photon counting modules (SPCMs, SPCM-AQRH-14-FC, Excelitas Technologies). In this condition, light coupled into the nanofiber has a contribution from (i) the MOT beams and (ii) the resonantly excited atom fluorescence. Photon counts are monitored on both the SPCMs and the signals are found to be equal, i.e. 50% of total photons coupled to the nanofiber travel in either direction along it. In order to estimate the contribution of the fundamental mode to the total photon count rate, one output pigtail was spliced to a SMF (780HP, Thorlabs). The SMF acts as a filter for any higher order mode propagation (Fig. 6 (a)) and only fundamental mode guiding survives. The SPCM at this output, therefore, only records photons guided by the fundamental mode. The second SPCM records the total number of photons coupling into all the fiber modes, i.e. both the fundamental and higher order, collectively. Both SPCMs are kept on during the experiment. The MOT magnetic field is switched on and off in few second intervals (to have enough time for the cloud to reach steady state) in order to separate the photon count contribution from the atom cloud (as shown in Figs. 6 (b) and (c)) and that arising from the MOT beams. By comparing the total photon signal (fundamental plus higher modes excitation) with that obtained for only the fundamental mode, it is found that 85.4% of the atomic fluorescence coupling to the nanofiber is coupled into the higher order modes. Note that the different FM-ONF transmissions (from the waist to the output) for the fun-damental mode (96%) versus the higher order modes (55%) must be taken into consideration in this calculation.
In order to reconfirm this result, the same experiment was repeated with a different method of filtering out the higher order modes from one side of the detection system. This time a SM-ONF was spliced to the output pigtail of the FM-ONF in place of the mode filtering SMF. Similar results were obtained (data not shown).

Absorption measurements
Following from the HOM fluorescence coupling investigations, the effect of these modes on atom absorption was studied. A fraction of the cooling laser beam was passed through a doublepass AOM setup to generate the probe beam for subsequent coupling into the nanofiber. The frequency of the probe beam can be changed by this AOM as per requirement. The first order output of the AOM was coupled to a fiber and transported to the SLM system, where it was outcoupled and passed through a linear polarizer before reflecting from the phase imprinted SLM to generate the LG 01 beam. This donut-shaped beam was coupled into a 20 m length of fewmode fiber (the same type which is used for fabricating the FM-ONF) to transport it to the cold atom system, where it is used as a probe to be passed through the nanofiber in the LP 11 mode. The computer-controlled SLM can switch the probe into the LP 11 or LP 01 depending on what is needed. The 20 m length of fiber was needed since the SLM setup is in the fiber pulling rig laboratory. We use the same SLM for mode checking during fiber pulling and for generating the probe beam in order to ensure consistency in our measurements. After transporting the probe beam through this distance, the mode profile is checked using a CCD camera at the output of the fiber. This fiber length is then spliced to one of the pigtails of the FM-ONF. Propagation of the higher order mode through the nanofiber is verified by launching the LG 01 mode from the SLM into the few mode "fiber and nanofiber" combination and checking the mode profile with a CCD camera at the other end (Fig. 4). This arrangement excites the three supported modes (TE 01 , TM 01 , and HE 21 ) of the FM-ONF. Note that the Teflon fiber feedthroughs, as described above, still allow for the propagation of light in HOMs into the vacuum chamber and back out again.
The cold atom cloud was formed around the waist of the nanofiber, ensuring that the atoms in the cloud interacted with the guided light via evanescent field coupling. The MOT beams, i.e. the cooling and repump beams, and the photon counter were switched on and off by the timing sequence shown in Fig. 7 in order to check absorption of the probe beam by the cold atoms. The trapping magnetic field and the probe beam remain on at all times; however, the photons guided through the nanofiber were only counted during the 1 ms time when the MOT beams were off. A repetition rate of 5 Hz was used for the experiments to ensure that the atom cloud was fully loaded before proceeding with any measurements. Effectively, this gave sufficient time for the cloud to return to its steady state by recollecting any atoms lost in the 1 ms expansion during the detection phase. The photon counts were collected for 400 runs.   8. Absorption spectra obtained when either a higher order modes (squares) or a fundamental mode (circles) probe beam was coupled into the FM-ONF. Probe power measured at the output end was maintained at ∼ 4.3 pW in both cases by changing the input intensities. Linewidths, as measured by Lorentzian fitting to the spectra, were found to be 19 ± 5 MHz and 17 ± 2 MHz for higher and fundamental mode probes, respectively.
Next, the atom cloud was switched off by turning off the trapping B-field. The same collection sequence (Fig. 7) was repeated 400 times to determine the photon signal arising purely from the probe beam in the absence of the cloud. Using these two sets of data, the percentage absorption of the probe is calculated for a particular detuning with respect to the cooling transition of 87 Rb. The same experiment is repeated using different probe beam detunings in order to obtain the absorption spectrum for the 87 Rb cloud (Fig. 8). A similar experiment was performed for the LG 00 mode as probe instead of LG 01 using the same power level at the output end of the nanofiber. Again, the absorption spectrum of the cloud was obtained. From Fig. 8 it is clearly evident that absorption was significantly more pronounced when higher order modes were used, compared to using the fundamental mode as a probe beam. In all the cases with different detunings or modes, the probe power was managed to maintain 4.3 pW at the output end in cloud off condition. However, when the probe power was increased by up to a factor of three there was no significant change in the percentage of light absorbed by the atoms.
The stronger absorption of the probe beam for higher order modes reflects the fact that more atoms surrounding the waist of the nanofiber interact with the evanescent field, since it extends further away from the fiber surface than when the fundamental mode is used. The measured linewidths (19 ± 5 MHz for higher and 17 ± 2 MHz for fundamental mode probes) are found to be higher than the natural linewidth (6 MHz) of the cooling transition. The reasons behind this could be due to modification of the spontaneous emission rate of atoms due to the presence of the nanofiber [26], surface interaction between atoms and the nanofiber [27], power broadening due to the probe beam, and the presence of the magnetic field at all times during the measurement.

Conclusion
In this work, we have demonstrated for the first time, the propagation of higher order modes (TE 01 , TM 01 , and HE 21 ) in an optical nanofiber integrated into a magneto-optical trap for neu-tral atoms. Significantly, absorption of the higher order modes by laser-cooled 87 Rb is enhanced compared to that of the fundamental mode. This work enables numerous heretofore theoretical atom trapping schemes to finally be realized, such as that relying on modal interference of far detuned higher order modes by selectively exciting different modes in required combinations [18,17]. In addition, we have studied the fluorescence from atoms coupled into the optical nanofiber guided modes. These experiments show that coupling of photons into the higher modes is significantly enhanced compared to coupling into the fundamental mode alone, in line with the theoretical predictions by Masalov and Minogin [1]. Furthermore, by analyzing the polarization of the fluorescence light collected by the nanofiber, coupling of atoms with different modes can be studied. Future work will focus on trapping atoms around the nanofiber using higher order mode interference [18] and will include consideration of alternative complex mode patterns [17]. Note that the nanofiber used in this work is installed vertically in the vacuumchamber; for trapping atoms around the nanofiber waist a horizontal orientation may be preferable in order to remove gravitational field effects. -