Abstract
Unlike photons, which are conveniently handled by mirrors and optical fibres without loss of coherence, atoms lose their coherence via atom–atom and atom–wall interactions. This decoherence of atoms deteriorates the performance of atomic clocks and magnetometers, and also hinders their miniaturization. Here we report a novel platform for precision spectroscopy. Ultracold strontium atoms inside a kagomelattice hollowcore photonic crystal fibre are transversely confined by an optical lattice to prevent atoms from interacting with the fibre wall. By confining at most one atom in each lattice site, to avoid atom–atom interactions and Doppler effect, a 7.8kHzwide spectrum is observed for the ^{1}S_{0}−^{3}P_{1}(m=0) transition. Atoms singly trapped in a magic lattice in hollowcore photonic crystal fibres improve the optical depth while preserving atomic coherence time.
Introduction
Quantum metrology^{1} with atoms relies on the long coherence times of atoms and photons to perform measurements at the quantum limit, which finds broad applications including atomic clocks^{2}, atom interferometers^{3}, magnetometers^{4} and quantum simulators^{5}. In particular, seeking minimally perturbed, opticallydense atomic samples^{6} has been a longstanding endeavour in atomic, molecular and optical physics, as the quantum projection noise (QPN)^{7} in the measurements reduces with the number of atoms as , which improves sensitivities in spectroscopy^{8}, optical magnetometry^{4} and the search for the permanent electric dipole moment (EDM)^{9}. Since the atomic absorption crosssection of a resonant light with wavelength λ is given by σ_{0}=3λ^{2}/2π, a small beam radius is preferable for efficient coupling^{10} between atoms and photons. In a freespace Gaussian beam, however, as the Rayleigh range changes as , an effective interaction volume rapidly decreases as the waist radius . One then is confronted with severe tradeoffs: a strong atom–photon interaction is available at the price of increased atom density n=N_{a}/V, which causes harmful atom–atom interactions. Atoms in a cavity have been used to get rid of this constraint by enabling multiple interaction^{11,12} of photons N_{p} with atoms N_{a} inside a cavity mode volume , with L the length of the cavity.
Atoms coupled to optical fibres may offer an alternative, as their effective interaction length can be arbitrarily longer than the Rayleigh range z_{R}. A strong coupling of spontaneous photons into the guided mode of a nanofibre was predicted and observed^{13}. The evanescent trapping of atoms in the vicinity of the nanofibre allowed for an interface between atoms and guided modes^{14}. A stateinsensitive twocolour dipole trapping around the nanofibre was demonstrated^{15} to reveal the natural linewidth of the 6S_{1/2}, F=4→6P_{3/2}, F′=5 transition of Cs, by cancelling out the light shifts of trapping lasers. In contrast, hollowcore photonic crystal fibres (HCPCFs)^{16} have been proven to confine thermal atoms or molecules together with the guided light inside its core over several metres, enabling the generation of optical nonlinearities at ultralow optical powerlevels, and sensitive spectroscopy on weak transitions^{17}. For example, a 10kHz precision was demonstrated for a saturated absorption spectroscopy of acetylene by extrapolating the zero pressureshift^{18}. To prevent atoms from interacting with the fibre walls, optical dipole trapping of atoms in HCPCFs was applied to guide thermal Rb gas^{19}, Bose condensed Na gas^{20} and lasercooled Rb gas^{21}. Removal of the light shift perturbation during absorption spectroscopy was demonstrated by temporarily turning off a guiding laser^{22}.
However, in all the above configurations, the finest fibrebased spectroscopy that has so far been measured still exhibits a severalmegahertzwide spectral linewidth. This broad linewidth is set by the strong collision of the confined thermal molecules or atoms^{18,23} with the fibre core innerwall, or by the natural linewidth of alkali atoms^{15,22}. As such, the coherence of atoms or molecules longer than tens of nanoseconds in fibres remains a challenge. Targeting ultraprecision laser spectroscopy of atoms at a fractional uncertainty of 10^{−17} and beyond, which is regarded as a goal accuracy for nextgeneration atomic clocks^{24,25}, we considered possible fibrebased configurations. Depending on the atomwall distance, from tens of nanometres to tens of micrometres, the atom fibre–wall interactions change from van der Waals, CasimirPolder and finally to thermalbath regimes^{26}. The van der Waals interaction strongly shifts atomic lines by as much as 10^{−10} of the fractional frequencyshift, for atoms trapped in the evanescent field^{13,14,15} at tens of nm from the nanofibres. To keep the atom–wall interactioninduced fractional frequency shift below 10^{−17}, one has to keep the atomfibre walls distance r_{c}≳20 μm (see Methods).
Here we investigate the ^{1}S_{0}−^{3}P_{1} transition of ^{88}Sr atoms in a 1D optical lattice tuned to the magic condition, which confines atoms near the centre of the HCPCF and in the LambDicke regime without introducing a light shift^{15,24,27}. The moderately narrow linewidth γ_{p}=7.5 kHz of the transition^{28} offers an efficient probe to characterize atom–atom and atom–fibre interactions by its spectral line broadenings and shifts. Wellcharacterized collisional properties^{29} allow investigating the occupancy of atoms in lattices through the reinforced collisional shift by the resonant dipole–dipole interaction, while the total angular momentum J=1 of the upper state probes the fibre birefringence effects via the tensor light shift^{27}. By carefully eliminating collisional shift and birefringenceinduced light shift, we show that the atomic resonance frequency can be unaffected by the fibre within an uncertainty of 0.11 kHz or ≈3 × 10^{−13}. Our investigations provide useful insight for designing fibrebased optical lattice clocks on the mHznarrow ^{1}S_{0}−^{3}P_{0} clock transitions^{30}, where both the collisional and polarizationdependent light shift are expected to be suppressed by more than 3–7 orders of magnitude, depending on the isotopes to be used^{31}.
Results
Experimental setup
Figure 1a shows our experimental setup. ^{88}Sr atoms are lasercooled and trapped at a temperature of a few μK using a narrow line magnetooptical trap (MOT)^{32}. A 32mmlong kagome cladding lattice HCPCF with hypocycloid core shape^{16,33,34} is placed near the MOT. The fibre covers the experimental wavelengths (689–914 nm) with loss figures of <650 dB km^{−1} and guides dominantly in the HE_{11} mode (see Fig. 1ef and Methods). We couple optical lattice lasers at wavelength λ_{L}≈813 nm from both ends of the HCPCF. The potential depth of the optical lattice is about 30 μK at the MOT position (z=−1.6 mm from the entrance end of the HCPCF) and 300 μK inside the fibre. After loading approximately 10^{4} atoms into the optical lattice, the atoms are adiabatically accelerated up to ν_{m}≈53 mm s^{−1} and transported inside the fibre hollow core to a position z. The atom acceleration and positioning is controlled by the frequency difference δν(t)=ν_{2}(t)−ν_{1} of the lattice lasers, as outlined in Fig. 1a,b. For the adiabatic acceleration (deceleration) of atoms, we linearly chirp (≈2 kHz ms^{−1}) the frequency difference δν(t) over 60 ms. The transport velocity ν_{m} is optimized to maximize the number of atoms that pass through the 32mmlong fibre by considering the tradeoffs: while shorter transit time through the fibre reduces the collision losses, larger v_{m} increases the heating loss of atoms, as we discuss below.
Lifetime of atoms in a fibre
The lifetime of the atoms in the fibre is of serious concern to discuss its potential applications, as glancing collisions with residual gases severely limit the coherence time of trapped atoms. The inset of Fig. 2 shows the lifetime τ=347 ms±8 ms of atoms trapped at z=23.4 mm. Figure 2 maps out the positiondependent lifetime of atoms along the fibre. The lifetime of τ=500 ms near the entrance of the fibre, which is close to that measured outside the fibre, decreases to τ=350 ms in the middle of the fibre. Using a glancingcollision model^{35} and taking into account the measured lifetimes and trap depth, we estimate the vacuum pressure in the middle and outside of the fibre to be P_{in}≈1.7 × 10^{−6} Pa and P_{out}≈1 × 10^{−6} Pa. The latter is in good agreement with the measured vacuum pressure, and the increase of the pressure in the fibre is reasonably accounted for by a small core radius r_{c}=17 μm of the fibre and by the outgassing rate q per unit surface area of the fibre wall. By solving the 1D diffusion equation, we obtain the steadystate solution (∂P_{PCF}/∂t=0) of the pressure inside the fibre as P_{PCF}(z,l)=−qz(z−l)/(r_{c}D)+P_{out}, for 0<z<l , with l being the length of the fibre and D the diffusion constant. On the basis of the estimated pressure P_{PCF}(l/2, l) with l=32 mm, the pressure in the middle section of the fibre for an arbitrary length l would scale as P_{PCF}(l/2, l) ≈7 × 10^{−4}l^{2} Pa m^{−2}+P_{out}. We expect that intensive baking of the fibre may reduce the outgassing rate q, thus extending the lifetimes of atoms for a longer fibre for future experiments.
The above result also suggests that there is no extra heating loss of atoms in the fibre as long as atoms are held at the same position. However, we observe larger heating of atoms, as the transport velocity ν_{m} increases. At ν_{m}=53 mm s^{−1}, the heating rate is estimated to be ~300 μK s^{−1} for the moving lattice potential depth of 180 μK. We attribute this to a parametric heating of atoms caused by a residual standingwave field, which is created by a partial reflection (~0.5 %) of the lattice laser by a viewport. This standingwave potential modulates the moving lattice potential by 7% at the frequency f_{m}=2ν_{m}/λ_{L}~130 kHz, as atoms travel every λ_{L}/2. As ν_{m} increases, this frequency becomes closer to the parametric resonance condition^{36} f_{m}=2f_{lattice}/n, where f_{lattice}≈300 kHz is the vibrational frequency of the lattice and n=4. To cope with this heating, we apply laser cooling during transport, which successfully reduces the heating loss of atoms.
Absorption spectroscopy
We perform absorption spectroscopy for atoms trapped at z~4 mm. The ^{1}S_{0}−^{3}P_{1}(m=0) intercombination transition at λ_{p}=689 nm is probed by a laser, whose linewidth and frequency drift per hour are both <1 kHz by referencing a cavity made of ultralow expansion (ULE) glass. We apply a bias field of B_{0}=(0.14 mT)ê_{x} perpendicular to the horizontal plane (see Fig. 1a) to define the quantization axis. The probe laser is linearly polarized with its electric field E_{p} parallel to B_{0} to excite the π transition. The differential light shift for the transition is given by^{27}
where is the differential polarizability, which depends on the lattice laser wavelength λ_{L} and its polarization ?_{L}. The magic condition , to remove the differential light shift, can be satisfied for 690 nm<λ_{L}<915 nm by tuning the tensor contribution of the light shift in the ^{3}P_{1}(m=0) state, which is determined by the angle θ_{L} of the linearlypolarized lattice laser ?_{L}=E_{L}/E_{L} with respect to the quantization axis. It is noteworthy that, despite the fact that the HCPCF guided mode exhibits a small longitudinal component E_{z} (see methods), this is cancelled out in the standingwave configuration.
We couple a probe intensity of I_{p}≈0.15I_{0} into the fibre, with I_{0}=3 μW cm^{−2}, the saturation intensity of the transition. The transmission through atoms in the fibre is coupled to an avalanche photodiode (APD), as shown in Fig. 1a, where the overall photoncounting efficiency is estimated to be 30%. We define the frequencydependent optical depth as
where Δν_{p}=ν_{p}−ν_{0} is the detuning of the probe laser. Here, we approximate the Besselmode profile of the guided mode to a Gaussian one with w_{0}=11.8 μm (see Methods), and assumes an atomic density distribution with w_{a}≈2.0 μm, estimated from the atomic temperature and radialtrapping frequency of ≈1.3 kHz. Here, l=32 mm and r_{c}=17 μm are the length and hollowcore radius of the fibre, respectively. Using the photoncounting rates with and without atoms, Π_{w} and Π_{w/o}, and the background count rate Π_{bk}, the transmittance of the fibre is given by T=(Π_{w}−Π_{bk})/(Π_{w/o}−Π_{bk}), which is used to derive the optical depth as OD(Δν_{p})=−ln T. The number of atoms in the fibre is given by N_{a}≈1200·OD(0). To avoid excess light shifts during spectroscopy, we reduce the lattice intensity by one order of magnitude from that used during the atom transfer. The probing time of the transition is limited to 3 ms to reduce the photonrecoil heating loss of atoms out of the lattice potential.
Collisional shift and its suppression
Figure 3a shows the measured optical depth OD(Δν_{p}) as a function of the probe laser frequency. The LambDicke confinement and the lightshift cancellation allow us to approach the natural linewidth of the transition. However, as shown by the red symbols in Fig. 3b,c, we observe a collisional shift and broadening (see methods) for OD(0)>0.8, which corresponds to the mean atomoccupation of each lattice site . Here, the atom cloud length l_{a} is measured by the laserinduced fluorescence image of atoms, after extraction from the fibre by the moving lattice.
To make the high optical depth compatible with reduced atomic interactions, we expand the atom cloud over the lattice sites in the fibre by temporarily turning off the lattice confinement for t_{f}=60 ms, while maintaining the dipole trapping in the radial direction. The time chart outlined in Fig. 1c allows us to extend the cloud length to , where we use an atomic velocity of , estimated from the Doppler width of 55 kHz. This procedure reduces the mean atom occupation from down to (see the blue circles in Fig. 3a), while preserving an optical depth of OD(0) ≈2.5. The blue symbols in Fig. 3b,c show that the collisional broadening and shift are successfully suppressed by applying this procedure. However, the achieved linewidth of 11 kHz suggests that some unexplained broadening of several kHz still remains.
Light polarizationdependent shift
To elucidate the source of this residual broadening, we investigate the birefringence of the HCPCF. In addition, to improve the spatial resolution in the fibre, we reduce the collisional shift by limiting the number of atoms to N_{a}<1,200, that is, OD(0)<1, instead of expanding atom clouds. Assuming the lattice laser polarization ?_{L} to be parameterized by an angle θ_{L} as defined previously, equation (1) becomes . For the lattice laser wavelength at λ_{L}=813 nm, the differential light shift can be removed by setting θ_{L}=46°. The angle sensitivity of the tensor light shift^{27} makes the light shift an efficient probe for the fibre birefringence.
Figure 4a shows the lattice intensitydependent light shift, where the gradient indicates the effective differential polarizability . The blue filled and empty circles are measured for atoms inside (z_{813}=3.7 mm) and outside (z_{0}=−1.6 mm) the fibre, respectively. While the data confirm that the atomic resonance frequencies are unaffected by being guided in the fibre (as demonstrated by the I_{L}→0 extrapolations that coincide at the same frequency), the change of polarizabilities inside and outside the fibre indicates the presence of fibreinduced birefringence. Assuming an angle sensitivity of the tensor light shift, corresponds to a polarization rotation of δθ_{L}≈0.5° between z_{0} and z_{813}. We investigate the positiondependent birefringence effect throughout the fibre, which is found to be within δθ_{L}≈0.3° and, in particular, nearly constant for 0<z<8 mm. A relatively large deviation is found in the region of fibre support and clamp (see Fig. 1a), which may indicate the presence of pressureinduced fibrebirefringence. In the following measurements, and to be free from stressinduced effects, we focus our attention to the fibre position at around z≈4 mm.
To moderate the fibrebirefringence effect, further experiments are performed at a magic wavelength λ_{L}=914 nm, with θ_{L}=90°, where the angle dependence appears only in second order. The filled and empty red circles in Fig. 4a show the resulting reduction in sensitivity measured at z_{914}=4.3 mm and at z_{0}=−1.6 mm, respectively. The slight change of the position from the measurements at λ_{L}=813 nm results from the lattice wavelength difference, which scales as , since we use the same detuning sequence δν(t) for the moving lattice. In this measurement, we simultaneously record the absorption spectra for five different intensity settings so as to minimize the influence of laser frequency drift in extrapolating the lattice intensity I_{L} → 0. The data points are fit by , where ν_{PCF(FS)} and denote the resonant frequency and the differential polarizability in the PCF or in free space (FS), respectively, and y_{0} assumes an offset frequency chosen to be zero in Fig. 4a. We evaluate the uncertainty of zerointensity intercepts by the uncertainty of y_{0}, which are 0.18 kHz and 0.11 kHz for 813 and 914 nm, respectively, as indicated by error bars at I_{L}=0. The results indicate that the atomic resonance frequency is unaffected by the fibre with an uncertainty of ≈3 × 10^{−13}.
Figure 4b shows a spectrum measured at the magic wavelength λ_{L}=914 nm with the lattice intensity of I_{L}=37 kW cm^{−2} measured at z≈5.3 mm. The linewidth of 7.8(4) kHz agrees well with the saturationbroadened linewidth of 7.8 kHz for the probe laser intensity I_{p}≈0.077I_{0}, demonstrating that there is no significant decoherence of atoms in the fibre at the kHz level. At λ_{L}=914 nm, we investigate the atomic resonance frequencies throughout the fibre, which are found to be within 2 kHz. This variation is partly due to the frequency drift of the probe laser and partly due to the spatial inhomogeneity of the fibre. The detailed investigations of fibredependent inhomogeneity, such as local stress on the fibre, charging effect and formation of patch potential on inner surface of the fibre, are underway.
Discussion
The scheme developed here offers a new and ideal platform for highprecision spectroscopy with enhanced signaltonoise ratio, particularly suitable for the miniaturization of optical lattice clocks operated on the ^{1}S_{0}−^{3}P_{0} clock transitions^{30}. The systematic uncertainties of such clocks are essentially characterized by the nuclear spin I of an interrogated isotope, which at the same time decides its quantum statistical nature. Bosonic isotopes^{31,37,38,39}, for example, ^{88}Sr and ^{174}Yb are highly susceptible to collisional shift; therefore, they certainly demand singly occupied lattices as demonstrated in a 3D optical lattice clock^{31}. Recent observations suggest that, as the uncertainty of the clocks approaches 10^{−17}, collisional interactions become a concern for clocks even with spinpolarized ultracold fermions^{40}, such as ^{87}Sr and ^{171}Yb, where the swave collisions are suppressed. Our demonstrations of a singlyoccupied lattice by expanding atoms in the fibre should be effective for both isotopes to reduce collisional shift while preserving the number of atoms or the QPN limit.
In contrast to free space optics, fibre optics requires special care for the state of light polarization, which is easily affected by mechanical stress or the inhomogeneity of the fibre. As the light polarization affects the light shift for the electronic states that have nonzero angular momentum F≠0, fermionic isotopes with halfinteger nuclear spin become susceptible to fibre birefringence even in the clock states with total electronic angular momentum J=0. However, compared with the ^{3}P_{1} state chosen here as a sensitive probe, the tensor shift in the clock transition of ^{87}Sr is seven orders of magnitude smaller^{41}, as it originates solely from its nuclear spin I=9/2. Our measurements, therefore, suggest that the tensor contribution is safely neglected in achieving 10^{−18} clock uncertainty.
A 32mmlong HCPCF, as employed here, will support as many as 10^{5} lattice sites or N_{a}≈10^{5} atoms free from both collisions and light shifts, allowing to achieve a projection noiselimited stability of , with τ the averaging time. This is in contrast with freespace latticeclock experiments that employ ~10^{3} atoms confined in less than 1mmlong 1D lattices. Further increase in the number of atoms should be possible by extending the fibre length. Moreover, high optical depth and long atomic coherence time allow applying dispersive measurement of atoms^{42}, quantum nondemolition (QND) measurement protocols and spinsqueezing of atoms during clock operation^{6}. By heterodyning or homodyning the transmittance of a probe laser^{8}, a quantum feedback scheme^{43} may be used to steer the probe laser frequency, instead of applying conventional projection measurements^{7}. The strong coupling of atoms to guided modes allows the investigation of collective effects such as collective Lamb shifts^{44} and superradiance^{45}. In particular, superradiant lasing^{46} on the clock transition or generation of narrowline light source via the phasematching effect^{47} may have potential to replace bulky reference cavities^{48,49} required for optical clocks, which will lead to significant miniaturization of optical clocks. Moreover, a fully populated 1D chain of 10^{5} or more qubits sharing an optical bus of the fibreguided mode could be used for quantum computing and simulation^{5} by providing individual spectroscopic access^{50} with a magnetic or electric field gradient.
In summary, we have demonstrated precision spectroscopy of atoms in a HCPCF, investigating possible hurdles intrinsic to fibres, such as collisionlimited lifetime, atom–atom interactions and fibreinduced birefringence. In the present experiment, the coherence time of the system is essentially limited by the natural lifetime of the ^{3}P_{1} state. Further investigation of the coherence time up to a second is possible by interrogating the ^{1}S_{0}−^{3}P_{0} clock transition^{30}, which also reduces the sensitivity to fibre birefringence. The novel platform demonstrated here could have an immense impact on future metrology and quantum information sciences using miniaturized atomic devices.
Methods
Frequency shift due to atom fibre–wall interactions
The CasimirPolder interaction energy^{51} between an atom with polarizability α and an infinite surface at a relatively large distance r_{c}, for which the retardation limit is valid, is given by with ε_{0} the vacuum permittivity. The coefficient Γ depends on the properties of the surface; Γ=1 for ideal metals and Γ<1 for dielectric materials. If one considers a photonic crystal with airfilling fraction P=0.94 as a dielectric with relative permittivity ε=1·p+ε_{FS}·(1−p), then Γ=(ε−1)/(ε+1)≈0.08 is expected, where we assume the static permittivity of fused silica to be ε_{FS}=3.8. The energy shift of an atom inside the HCPCF can be larger than the value given by U_{CP} by a geometric factor G~6, which accounts for the atom interaction with six walls (see Fig. 1e) when each of these walls is approximated by an infinite plane.
The difference in polarizabilities for a Sr atom in the ^{1}S_{0} and ^{3}P_{0} states^{52} is Δα≈4·10^{−39} C m^{2}V^{−1}. For r_{c}=20 μm, the frequency shift is given by , which corresponds to a fractional clock shift Δν/ν_{0}≈1.5 · 10^{−18}. This estimate applies for the zero temperature limit. At room temperature, thermal effects become the same order as the zero fluctuations input. According to ref. 26, this gives a 3–4 times enhancement, and the atom–wall interaction corresponds to a fractional frequency shift of ~10^{−17}.
For the ^{1}S_{0}−^{3}P_{1} transition, the fractional shift can be 20% larger, because of a 20% increase in the differential polarizability Δα. In this transition, however, the resonant dipole–dipole interaction may be more relevant^{53}, because of the significantly larger dipole moment than that of the ^{1}S_{0}−^{3}P_{0} clock transition. As the atomwall distance is much larger than the transition wavelength, r_{c}/(λ_{p}/2π)~180, the retardation effect dominates. Considering the current measurement precision of ~10^{−13}, which is ~10^{−2}γ_{p} with γ_{p}=7.5 kHz the natural linewidth, the atom–wall interactions can be safely neglected.
Coupling light into the hollowcore fibre
The probe and lattice lasers, which are sent through polarizationmaintaining singlemode fibres (PMSMF), are coupled to the HCPCF using aspheric lens pairs. The output of the PMSMF is collimated by a f=4.6 mm lens to pass through a vacuum viewport without aberrations and is then matched to the HE_{11} mode of the HCPCF by a f=18.4 mm lens. Typically 90% of the laser power is transmitted through the 32mmlong fibre. The farfield intensity pattern is nearly Gaussian as shown in Fig. 1f. The spatial mode after the HCPCF is verified by recoupling it to another PMSMF, where we achieve an overall (SMFHCSMF) coupling efficiency of 70%.
HCPCF design and fabrication
The fibre is fabricated using the standard stackanddraw technique. The cladding structure is that of a kagomelattice with a pitch of 14 μm (Fig. 5b) and strut thickness of 196 nm (Fig. 5c). This is the smallest silica strut thickness so far reported for a hypocycloid core HCPCF^{33}. This allows the fibre to guide light with low loss for wavelengths as short as 400 nm (see Fig. 5a), and thus covering the experimental operating wavelengths of 813 and 689 nm with loss figures of 530 and 650 dB km^{−1}, respectively.
HE_{11} mode intensity profile and electric field distribution
The fibre hollow core has a hypocycloid contour with inner radius r_{c}~17 μm (ref. 33). Figure 6 shows the norm of the two polarization degenerate electric fields of the modes along the two axes of symmetry of the fibre core. The presented simulations are performed over a spectrum of 800–830 nm to cover our operating wavelength of 813 nm, where the mode size has a very moderate change with wavelength. The vertical dashed lines indicate the radial position of the field at e^{−1} of its maximum, corresponding to a modefield (MF) radius of the HE_{11} mode of ~12.7 μm. This Bessel intensity transverse profile fits to a Gaussian profile with e^{−2} of the maximum radial position at w_{0}=11.8 μm. The electric and magnetic fields for the fundamental coremode HE_{11} are computed using the finiteelementmethod. Figure 7 shows the components of the electric E (V m^{−1}) and magnetic B (T) fields, when the total optical power contained in the HE_{11} mode is set to 1 W. The results show that the magnitude of the longitudinal component E_{z} is almost 100 times smaller than the transverse components (E_{x}, E_{y}).
HE_{11} birefringence
The fabricated fibre core exhibits a small ellipticity, which results in a residual birefringence Δn_{eff}. Figure 8 shows the spectrum of the birefringence near the lattice laser wavelength 813 nm. The birefringence is found to be 9.6 × 10^{−8} (that is, a beat length of 8.4 m), which is more than one order of magnitude lower than the typical photonic bandgap HCPCF^{17}. It is noteworthy that, in addition to the intrinsic fibre form, the birefringence is also induced by mechanical and/or thermal stress. In the case of a photonic bandgap HCPCF, the lateral pressureinduced birefringence was measured^{54} to be in the range of ∂Δn_{eff}/∂p~10^{−11} Pa^{−1}.
Collision shifts
We evaluate the collisionshift as
where β is the collisionshift coefficient, n_{1}=1/ν is the atom density for a singly occupied lattice site with ν=7.8 × 10^{−13} cm^{3} and assumes the Poisson distribution of atoms with mean occupancy . The red and blue dashed curves in Fig. 3c show , with β=−1 × 10^{−9} Hz cm^{3} to fit the corresponding data points, with red and blue filled circles. This collisionshift coefficient β agrees reasonably well with that measured previously by the JILA group^{29} β_{JILA}= −1.3(3) × 10^{−9} Hz·cm^{3}.
Additional information
How to cite this article: Okaba, S. et al. LambDicke spectroscopy of atoms in a hollowcore photonic crystal fibre. Nat. Commun. 5:4096 doi: 10.1038/ncomms5096 (2014).
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Acknowledgements
This work received support partly from the JSPS through its FIRST Program and from the Photon Frontier Network Program of MEXT, Japan. We thank N. Nemitz for a careful reading of the manuscript. F.B. acknowledges support from ‘Agence Nationale de Recherche’. F.N. is partially supported by the RIKEN iTHES Project, MURI Center for Dynamic MagnetoOptics, JSPSRFBR contract No. 120292100, and GrantinAid for Scientific Research (S).
Author information
Affiliations
Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 731 Bunkyoku, Tokyo 1138656, Japan
 Shoichi Okaba
 , Tetsushi Takano
 & Hidetoshi Katori
Innovative SpaceTime Project, ERATO, Japan Science and Technology Agency, 731 Bunkyoku, Tokyo 1138656, Japan
 Shoichi Okaba
 , Tetsushi Takano
 & Hidetoshi Katori
GPPMM group, Xlim Research Institute, CNRS UMR7252, 123, Albert Thomas, Limoges 87060, France
 Fetah Benabid
 , Tom Bradley
 & Luca Vincetti
Physics department, University of Bath, Claverton Down, Bath BA2 7AY, UK
 Fetah Benabid
 & Tom Bradley
Department of Engineering "Enzo Ferrari", University of Modena and Reggio Emilia, Modena I41125, Italy
 Luca Vincetti
A.Ya. Usikov Institute for Radiophysics and Electronics, National Academy of Science of Ukraine, 12 Acad. Proskura Str., Kharkov 61085, Ukraine
 Zakhar Maizelis
 & Valery Yampol'skii
V.N. Karazin Kharkov National University, 4 Pl. Svobody, Kharkov 61077, Ukraine
 Zakhar Maizelis
 & Valery Yampol'skii
CEMS, RIKEN, 21 Hirosawa, Wako, Saitama 3510198, Japan
 Franco Nori
Physics Department, University of Michigan, Ann Arbor, Michigan 481091120, USA
 Franco Nori
Quantum Metrology Laboratory, RIKEN, 21 Hirosawa, Wako, Saitama 3510198, Japan
 Hidetoshi Katori
RIKEN Center for Advanced Photonics, 21 Hirosawa, Wako, Saitama 3510198, Japan
 Hidetoshi Katori
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Contributions
H.K. envisaged and initiated the experiments. H.K., T.T. and S.O. designed the apparatus and experiments. S.O. and T.T. carried out the experiments and analysed the data. S.O., T.T. and H.K. discussed experimental results and equally contributed to the experiments. F.B. and T.B. designed and fabricated the fibre for the experimental requirements, and L.V. calculated the fibre modal fields. Z.M., V.Y. and F.N. calculated the atom–wall interactions, and all authors participated in discussions and the writing of the text.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Hidetoshi Katori.
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