1. INTRODUCTION

Efficient interaction between light and matter and particularly the faithful coherent mapping between photons and atomic excitations lie at the heart of many quantum optics processes and applications, such as quantum networks. One appealing platform is room-temperature atomic vapor, which is successfully employed in first-generation quantum technologies, including atomic clocks and magnetometers [1,2], and quantum light sources and memories [3,4]. Light–matter interaction can be enhanced by a tight optical mode volume and by a collective coupling of this mode to an ensemble of atoms. While reduced mode volumes are achieved in small optical cavities [5] or with tightly focused beams in free space [6,7], they are typically incompatible with large ensembles of atoms due to the associated short Rayleigh range ${z_{\rm{R}}}$.

An alternative approach employs a tightly confined optical mode that is supported by a low-loss waveguide over an extended length [8–13]. The optical mode guided by a dielectric structure with dimensions below the optical wavelength $\lambda$ extends beyond the structure boundaries as an evanescent field and can interact with the surrounding atomic ensemble. In this case, the field typically decays over a range of ${\sim}\lambda$ away from the structure [9,14,15]. Such tight mode confinement is advantageous for processes requiring high intensity or a steep field gradient (e.g.,  nonlinear optics at ultralow power levels or atomic trapping) [16,17]. However, for coherent light–matter interaction with thermal atomic ensembles, tight confinement may limit the effectiveness of the interaction. Transverse motion of the atoms through the optical mode results in transit-time broadening and in a reduction of the absorption cross-section [9,10,14,18,19]. Additionally, the small mode volume limits the fraction of atoms simultaneously interacting with the mode, and the proximity of these atoms to the dielectric waveguide may lead to atom–surface interactions that impair the coherent interaction with the guided field [12,20]. Finally, for a given uniform atomic density, the collective light–matter coupling strength (or the optical depth) is limited by the fraction of the optical field that resides outside the core material.

In this work, we tackle these challenges by realizing a guided optical mode with an evanescent field part that extends several wavelengths away from the waveguide surface, as illustrated in Fig. 1(a). This unique mode is supported by the extremely thin waist of a tapered optical fiber. Tapered optical fibers, with waist diameters as small as a few hundred nanometers, have been shown to enable single mode operation with high transmission [21]. In the past decade, these nanofibers were used in a multitude of applications from atom trapping [17] and sensing [22] to chiral quantum optics [23] and cavity QED [24,25]. Previous demonstrations of an interface between thermal vapor and a nanofiber have shown several appealing features, such as polarization rotation, electromagnetically induced transparency [18], and nonlinear effects at low power [14]. However, all of these have been limited by transit time broadening, as well as by the interaction between the vapor and the waveguide surface.

 

Fig. 1. Super-extended evanescent field of a nanofiber interacting with atomic vapor. (a) Illustration of the extent of an optical mode surrounding a thin optical fiber. The fiber on the left (fiber diameter $D = 0.9\lambda /n$) has a guided mode with field diameter ${\rm{MFD}} = 1.2\lambda$, comparable to the wavelength $\lambda$. The fiber on the right ($D = 0.37\lambda /n$) has a mode extending to ${\rm{MFD}} = 13\lambda$, guided over a distance of $5000\lambda$. (b) Mode field diameter (in units of $\lambda$) as a function of fiber diameter (in units of wavelength in matter $\lambda /n$). Blue and red circles mark the parameters of the two fibers shown in (a). Dashed black line marks the physical fiber dimensions; the MFD diverges from this line around $nD/\lambda \lt 1$. The dashed purple line shows the fraction of the power residing outside the fiber ${\eta _{{\rm{power}}}}$ and is larger than 99% for the thin fiber. (c) Calculated Doppler-free absorption spectra for Rb vapor in the evanescent field of a $D = 0.9\lambda /n = 500 \;{\rm{nm}}$ (blue) and $D = 0.37\lambda /n = 200 \;{\rm{nm}}$ (red) silica fibers ($n = 1.45$). The effect of transit-time broadening is clearly apparent.

Download Full Size | PDF

Here we show that reducing the diameter of a nanofiber to around $0.37\lambda /n$ (where $n$ is the refractive index of the fiber core) yields a super-extended evanescent mode, and that this mode greatly suppresses the above limitations. In our system, the mode extends to a diameter of $2{w_0} \approx 13\lambda$ and is guided over 5 mm (50 times larger than ${z_{\rm{R}}} = \pi w_0^2/\lambda$), with more than 99% of the optical power residing outside the core material. We interface this fiber with atomic vapor and perform one- and two-photon spectroscopy, as well as saturation and temporal transient measurements. The unique characteristics of the system provide for spectral features much narrower than previously measured and correspondingly longer coherence times, thus establishing its potential for more intricate processes and opening a path to coherent light–matter interactions with thermal vapor in evanescent fields.

2. SUPER-EXTENDED EVANESCENT FIELD

A guided optical mode can be characterized by the mode field diameter (MFD), defining an area containing $(1 - {e^{- 2}})$ of the optical power. For vacuum-clad fibers, the MFD of the fundamental mode ${{\rm{HE}}_{{{11}}}}$ depends on a single parameter: the fiber diameter $D$ in units of wavelength in matter $\lambda /n$. We show calculations of the MFD as a function of this parameter in Fig. 1(b). As the fiber diameter is decreased, the MFD initially follows the diameter of the fiber core. In this regime, a significant fraction of the energy is contained inside the fiber core and its remainder resides in an evanescent field, decaying over ${\sim}\lambda$. When the fiber diameter is further narrowed below $nD/\lambda = 1$, it can no longer support a mode residing predominantly in the core. Yet a single bound solution, with most of the energy residing outside the core, always exists [26].

In this regime, the MFD varies steeply with the fiber diameter. This is demonstrated by comparing two cases, illustrated in Fig. 1(a) and marked by circles in Fig. 1(b). A fiber with $D = 0.9\lambda /n$ has ${\rm{MFD}} = 1.2\lambda$ (blue circle), whereas a narrower fiber with $D = 0.37\lambda /n$ has ${\rm{MFD}} = 13\lambda$ (red circle). The latter, which we realize in this work, is a super-extended mode that contains a remarkably high fraction of ${\gt}99\%$ of the power in the vacuum cladding. This mode is well approximated by a modified Bessel function ${\cal E}(r \gt D/2) = {{\cal E}_0}{K_0}(\kappa r)$ [27], where $\kappa = \sqrt {{\beta ^2} - {{(2\pi /\lambda)}^2}}$ is the transverse component of the wavevector ($\beta$ is the propagation constant; i.e., the longitudinal component of the wavevector). For $\lambda = 780 \;{\rm{nm}}$, a $D = 200 \;{\rm{nm}}$ silica fiber can guide a mode with ${\rm{MFD}} = 10 \;{{\unicode{x00B5}{\rm m}}}$ (${\kappa ^{- 1}} \simeq 0.44{\rm{MFD}} = 4.43 \;{{\unicode{x00B5}{\rm m}}}$), such that the mode area is 2500 times larger than the fiber cross-section. The exact mode structure and polarization are presented in Appendix A. The ability to produce such a waveguide by tapering down a standard optical fiber, with adiabatic following of the fundamental mode, is an exciting capability that lies at the edge of feasibility for waveguide tapering [28,29].

Both fibers highlighted in Fig. 1(a) guide the optical modes over many times the equivalent ${z_{\rm{R}}}$ in free space. However, when interfaced with atomic vapor, the interaction characteristics will strongly depend on mode confinement. Thermal ballistic motion, with thermal velocity ${v_{\rm{T}}}$ along both the longitudinal and the transverse wavevectors, leads to motional broadening. For modes extending to ${\kappa ^{- 1}} \sim \lambda /2$ away from the fiber, the transverse transit-time broadening ${\Gamma _{{\rm{tt}}}} = \sqrt 2 \kappa {v_{\rm{T}}}$ (full width at $1/e$) [30] becomes similar to the longitudinal Doppler broadening $\sigma = \beta {v_{\rm{T}}}$. In contrast, for the super-extended optical mode with ${\kappa ^{- 1}} \gtrsim 5\lambda$, the transverse motional broadening is suppressed by more than one order of magnitude.

We note that while many Doppler-free techniques exist, including compensation by light shift [31,32], transit-time broadening cannot be easily mitigated by purely optical means. The calculated (Doppler-free) absorption spectra for both fibers presented in Fig. 1(c) indeed show a tenfold decrease in spectral width for the thinner fiber and a corresponding enhancement in the atomic absorption cross-section. In this calculation, we average over atoms with different thermal velocities traversing the two-dimensional field distribution and neglect atomic trajectories that hit the nanofiber. This is a valid approximation for the super-extended mode due to the large ratio of mode field area to the fiber cross-section.

3. EXPERIMENTAL SYSTEM

The experimental setup is presented in Fig. 2. A fiber with a super-extended evanescent field is fabricated by tapering a silica fiber down to a nominal diameter $D = 200 \;{\rm{nm}}$ (with variations better than ${\pm}5\%$) in a heat-and-pull method [28]. Figure 2(a) shows a scanning electron microscope image of the nanofiber waist of a tapered fiber that we have pulled using the same parameters and flame trajectory as the tapered fiber used in the subsequent experiments. To fulfill the adiabaticity criterion, the fiber is tapered down from a diameter of 125 µm to the final diameter of 200 nm over a length of 3.3 cm. The fiber is then glued to a custom mount [Fig. 2(b)], which provides three axes of optical access and an ultrahigh vacuum-compatible fiber feedthrough. The mount is installed in a vacuum chamber [Fig. 2(d)], which is wrapped in resistive heating elements and surrounded by a thermally insulating polyurethane enclosure, allowing the stabilization of the system temperature. Rubidium vapor is released into the chamber by breaking a glass capsule containing a metallic rubidium pellet (at natural abundance). We set two different temperature regions in the chamber: The area containing the capsule is kept cooler so that it remains a rubidium reservoir, and its temperature sets the vapor pressure inside the cell.

 

Fig. 2. Experimental system. (a) Scanning electron microscope image of a tapered fiber waist, with a nominal diameter of 200 nm. (b) Custom, vacuum-compatible, fiber mount with external optical access and a fiber feedthrough (f/t). (c) Schematic of the optical setup. Probe and control fields are coupled into the tapered fiber in counterpropagating directions. Fiber-coupled electro-optical modulators (EOMs) shape the temporal intensity of the two fields. Polarizing beam-splitter (PBS) picks out the outgoing probe, which is further filtered by a band-pass interference filter (IF) and sent to a single-photon counting module (SPCM) or to an avalanche photodiode (APD), allowing us to monitor down to pW powers. (d) Vacuum chamber houses the fiber and a natural abundance metallic Rb pellet. A free-space path provides an absorption reference.

Download Full Size | PDF

We use the electronic ladder transitions of rubidium shown in Fig. 3(a). A probe field at 780 nm probes and drives the D2 transition $5{S_{1/2}} \to 5{P_{3/2}}$ in all the experiments. Two-photon spectra, demonstrated in Fig. 3(b), are obtained by adding a control field at 776 nm that drives the transition $5{P_{3/2}} \to 5{D_{5/2}}$. The control and probe lasers are frequency stabilized and fed into the nanofiber in counterpropagating directions. We show the optical setup in Fig. 2(c). The vacuum chamber also has a long free-space path, shown in Fig. 2(d), to measure a reference absorption spectrum. More details on the fiber and setup are given in Appendix A.

 

Fig. 3. Experiment overview. (a) Level structure of rubidium atoms used in the experiment. The ground and intermediate levels are coupled by a probe field ${\cal E}$, detuned by $\Delta$ from the resonance frequency. The intermediate level can be coupled to a doubly excited level $5{D_{5/2}}$ by a control field ${{\cal E}_{\rm{c}}}$. The $5{D_{5/2}}$ level in $^{85}{\rm{Rb}}$ is composed of several tightly spaced hyperfine states. The three states accessible in our experiment are spaced over 18.3 MHz. (b) Typical transmission spectrum of Rb vapor coupled to a super-extended evanescent field guided by a tapered optical fiber. The probe and control fields with propagation constants $\beta$ and ${\beta _{\rm{c}}}$ counterpropagate in the fiber. The two-photon transition is nearly Doppler-free as the propagation constants differ by about 0.5%. Both evanescent fields have a similar decay length ${\kappa ^{- 1}}$ in the transverse direction. The control field is filtered by polarization and narrowband interference filter.

Download Full Size | PDF

The transmission of the fiber at 780 nm is ${\sim}90\%$ after fabrication, and it drops to ${\sim}70\%$ after international transfer, splicing to commercial fibers, and installation in the chamber. However, the transmission may further degrade over time due to rubidium adsorption on the fiber surface. We are able to mitigate this effect and prevent the transmission degradation by heating up the fiber using 1 µW of light at 776 nm that is kept constantly on. For the experiments studying one-photon processes alone, this light is detuned by more than 1 GHz from the $5{P_{3/2}} \to 5{D_{5/2}}$ transition, while for the two-photon spectra it is resonant with this transition (and acting as the control field).

4. SUPER-EXTENDED EVANESCENT FIELD INTERFACED WITH THERMAL VAPOR

We begin by characterizing the super-extended mode by measuring the atomic absorption spectra with only the probe light present. In Fig. 4(a), we show the transmission spectrum of light guided by the tapered fiber and compare it to the free-space transmission spectrum of a large diameter (2 mm) beam measured simultaneously. The spectrum exhibits two transmission dips corresponding to $^{87}{\rm{Rb}}$ and $^{85}{\rm{Rb}}$ isotopes according to their natural abundance. The width of these absorption lines is dominated by Doppler broadening due to longitudinal atomic motion along the fiber axis. Importantly, we do not observe any additional broadening due to the transverse motion across the evanescent mode, when comparing the spectrum (blue line) to the free-space spectrum (dashed red line) and when fitting it to a numerical model (not shown) that accounts for the multilevel structure of rubidium. This is already an indication of an extended mode spanning at least several wavelengths.

 

Fig. 4. Transmission spectra and low-power saturation. (a) Transmission spectrum of light guided in a tapered fiber with super-extended evanescent field surrounded by Rb vapor with natural abundance (blue). Two distinct dips correspond to two Rb isotopes. A free-space absorption spectrum is measured simultaneously as a reference, through a 280 mm long optical path across the chamber (red; transmission below ${10^{- 3}}$ around the $^{85}{\rm{Rb}}$ dip is governed by noise and not shown). (b) Transmission spectra for different probe powers ${P_{{\rm{in}}}}$. (c) Resonant transmission as a function of ${P_{{\rm{in}}}}$. Solid line is a fit to a saturation model. All data in (a)–(c) are normalized by the off-resonance transmission.

Download Full Size | PDF

In Fig. 4(b), we plot absorption spectra for different probe powers. As the probe power increases, the absorption decreases, with the resonant transmission approaching unity already at a few hundreds of nW. We summarize these results in Fig. 4(c) by plotting the transmission at the $^{85}{\rm{Rb}}$ $F = 3 \to F^\prime = 4$ resonance for different probe powers and fitting the data to a saturation model for an inhomogeneously broadened ensemble of two-level systems [33] $T = {\exp}(- {{\rm{OD}}_{{0}}}/\sqrt {1 + {P_{{\rm{in}}}}/{P_{{\rm{sat}}}}})$, where ${{\rm{OD}}_0}$ is the resonant, Doppler-broadened, optical depth of the ensemble, ${P_{{\rm{in}}}}$ is the probe power, and ${P_{{\rm{sat}}}}$ defines the saturation power. We find that the saturation power is ${P_{{\rm{sat}}}} = 35 \pm 7 \;{\rm{nW}}$. Given the imperfect transmission through the bare tapered fiber, we estimate the actual saturation power (reaching the atoms) to be $29 \pm 6 \;{\rm{nW}}$. We note that for an extended evanescent field with decay length ${\kappa ^{- 1}}$, the saturation parameter per atom (often denoted as $s$) increases as the atom–nanofiber distance decreases and, for ${P_{{\rm{in}}}} = {P_{{\rm{sat}}}}$, becomes larger than unity for atoms within a distance of ${\kappa ^{- 1}}$ from the nanofiber axis.

This low saturation power is characteristic of a tightly confined optical mode. Even though the realized mode is larger than typical evanescent fields, the measured saturation power is identical, up to experimental uncertainty, to saturation powers reported across several platforms interfaced with hot vapor [8,9,14]. This universality stems from the trade-off that sets the saturation threshold, as both the Rabi frequency and the transit-time broadening increase linearly with 1/MFD, and the transit-time broadening dominates the relevant relaxation rates (be it within the ground-level manifold or between the excited and ground levels). Consequently, we find that this type of nonlinearity does not benefit from tighter mode confinement and, in fact, super-extended evanescent fields are advantageous as they enable the same saturation level with better coherence times. On the flip side, this trade-off also implies that ${P_{{\rm{sat}}}}$ measurements cannot unambiguously confirm the dimensions of the optical mode.

To determine the transit-time broadening and thus the dimensions of the mode, we perform temporal transient measurements. In essence, we prepare a saturated atomic population and measure the rate at which the excited atoms leave the interaction region. In practice, we monitor the response of the resonant transmission to a sudden change in probe power in a pump-probe-like experiment.

The dynamics is governed by the optical Bloch equations for the excited population ${\rho _{{\rm{ee}}}}$ and the atomic coherence (optical dipole) ${\rho _{{\rm{eg}}}}$, so

Figure 5 shows the results of such an experiment. We start with a probe power of ${\sim}4{P_{{\rm{sat}}}}$ (i.e., above the saturation power), and abruptly attenuate it to $0.6{P_{{\rm{sat}}}}$. Equations (1) and (2) under the condition $\sigma \gg \Gamma ,\Omega$ result in an over-damped solution with a two-step decay process: a fast decay with rate $\sigma$ followed by a slower decay with rate $\Gamma$. Indeed, we observe in Fig. 5(a) an initial short transient, where the atomic coherence follows the fast change in the incoming field, and subsequently a slow relaxation due to equilibration of atomic population.

 

Fig. 5. Temporal dynamics in the nanofiber-vapor system. (a) The transient response of the atomic coherence (optical dipole) ${\rho _{{\rm{eg}}}}$, quantified by the difference $\Delta {\cal E}$ between the incoming and outgoing probe (black dots, see text) after an abrupt reduction of the probe power (dashed blue line). The atomic coherence initially follows the rapidly changing incoming field and then slowly relaxes due to motional exchange of pumped and unpumped atoms and due to radiative decay. (b) Absorption $A(t)$ following the abrupt power reduction, plotted as the difference from the final steady-state value $A({t_{{\rm{end}}}})$ in a semilog scale. Solid line is a fit to a pure exponential decay, also plotted in (a), from which the relaxation time ${\tau _{{\rm{fall}}}}$ is extracted. (c) An abrupt increase in the laser power again initiates a fast following of the field and then a slower decay, determined by the optical pumping rate. (d) Absorption following the abrupt power increase, providing the faster relaxation time ${\tau _{{\rm{rise}}}}$ from which the probe Rabi frequency can be determined.

Download Full Size | PDF

To quantify the slow relaxation rate, we plot in Fig. 5(b) the absorption $A(t) = 1 - {P_{{\rm{out}}}}(t)/{P_{{\rm{in}}}}(t)$. The (pumped) excited atoms are gradually replaced by (fresh) ground-state atoms that enter the interaction region, and correspondingly the medium relaxes from low to high absorption. We measure an exponential relaxation time of ${\tau _{{\rm{fall}}}} = 10.7 \pm 0.5 \;{\rm{ns}}$, corresponding to a depopulation rate of $\Gamma = 14.9 \pm 0.8 \;{\rm{MHz}}$ (hereafter, $1 \;{\rm{MHz}} \equiv {10^6} \cdot 2\pi \;{\rm{rad}}/{\rm{s}}$). After subtracting the radiative decay rate (6 MHz), we attribute the remaining rate of ${\Gamma _{{\rm{tt}}}} = 9 \;{\rm{MHz}}$ to transit time broadening. This is consistent with the numerical calculation of the transit time broadening of ${\Gamma _{{\rm{tt}}}} = \sqrt 2 \kappa {v_{\rm{T}}} = 9 \;{\rm{MHz}}$ for a fiber with a diameter $D = 200 \;{\rm{nm}}$ ($\kappa = 1/4.428 \;{{\unicode{x00B5}}}{{\rm{m}}^{- 1}}$), as plotted in Fig. 1(c).

Upon ramping the strong field back on [Fig. 5(c)], we observe again a fast transient response followed by an exponential relaxation, with an absorption decay time [Fig. 5(d)] of ${\tau _{{\rm{rise}}}} = 6.7 \pm 0.3 \;{\rm{ns}}$. We attribute this time scale to optical pumping to the excited state at a rate of $\Gamma + 4{\Omega ^2}/\sigma = 24 \;{\rm{MHz}}$, from which we infer a conversion of optical power to Rabi frequency of $\Omega ({P_{{\rm{in}}}}) = 2 \;{\rm{MHz}}\sqrt {{P_{{\rm{in}}}}[{\rm{nW}}]}$. This conversion ratio is in good agreement with a weighted average over the calculated field distribution, summing over the relevant atomic transitions; specifically, it characterizes atoms at a distance of 2 µm (${\sim}0.5{\kappa ^{- 1}}$) from the fiber.

We now move on to study a coherent, two-photon process, taking advantage of the long coherence time provided by the super-extended optical mode. We use a counterpropagating configuration that is nearly Doppler free, as the relative mismatch of the propagation constants $({\beta _{\rm{c}}} - \beta)/\beta$ is only about 0.5%. This reduces the longitudinal Doppler broadening to 0.5% of its value for the one-photon transition. We add the counterpropagating control field and observe the appearance of narrow electromagnetically induced transparency (EIT) peaks inside the one-photon absorption lines, as shown in Fig. 3(b). Typically, two-photon processes observed in atomic vapor via a waveguide were limited to cascaded absorption [9,10,14]. Vapor EIT with linewidth larger than 100 MHz was observed in a 300 nm thick nanofiber [18]. In our system, the small transit-time broadening and high signal-to-noise ratio enable the coherent effect of induced transparency, originating from the interference of two possible excitation pathways. Furthermore, we are able to measure EIT with ${P_{{\rm{in}}}} \simeq {\rm{pW}}$ or, equivalently, with probe photons entering the medium at a rate of few photons per µs, such that the atoms interact with less than a single probe photon (on average) during their lifetime in the mode. This is a prerequisite for observing quantum nonlinear optics.

In our configuration, the control field is also guided by the fiber with a similar, super-extended, mode and not applied externally. To leading order, the coherent two-photon process depends on the product of the probe and control fields ${\cal E}(r) \cdot {\cal E}_{\rm{c}}^*(r)$, which varies with time while a given atom traverses the mode. Both evanescent fields are of the form ${K_0}(\kappa r) \approx {e^{- \kappa r}}/\sqrt r$, such that their product is approximately proportional to ${e^{- 2\kappa r}}/r$, yielding an increased two-photon transit time broadening of $2 \times {\Gamma _{{\rm{tt}}}}$.

In Fig. 6(a), we plot the transmission around the EIT resonance for different control powers, normalized by the one-photon absorption (in the absence of a control field). We note that increasing the probe power above a few pW reduces the EIT contrast, and, for probe powers higher than ${P_{{\rm{sat}}}}$, we have observed cascaded absorption.

 

Fig. 6. EIT in a bichromatic super-extended evanescent field. (a) Transmission spectrum of a weak probe in the presence of a control field of varying power ${P_{\rm{c}}}$, counterpropagating in the nanofiber. (b) FWHM of the transparency resonance. Solid line is a fit to a power-broadening model. (c) Contrast of the transparency resonance.

Download Full Size | PDF

We present in Figs. 6(b) and 6(c) the full width at half-maximum (FWHM) of the EIT resonance ${\Gamma _{{\rm{EIT}}}}$ and the EIT contrast for different control powers ${P_{\rm{c}}}$. The width and contrast increase linearly with ${P_{\rm{c}}}$. The solid line in Fig. 6(b) is a fit to an EIT power-broadening model of the form ${\Gamma _{{\rm{EIT}}}} = \alpha {P_{\rm{c}}} + \gamma$. By extrapolating to ${P_{\rm{c}}} = 0$, we find the effective width of the $5{S_{1/2}} \to 5{D_{5/2}}$ transition $\gamma = 40 \pm 1.2 \;{\rm{MHz}}$. The measured EIT line is composed of three transitions to different hyperfine states [$F^\prime = 4 \to F^{\prime \prime} = 5,4,3$, also see Fig. 3(b)] whose transition frequencies are spaced over 18.3 MHz. When summing over these transitions with their corresponding oscillator strengths, we find that the measured width $\gamma$ is consistent when accounting for the contributions of two-photon transit time ($2 \times {\Gamma _{{\rm{tt}}}} = 18 \;{\rm{MHz}}$, as calculated and as measured from transient measurements presented in Fig. 5), residual longitudinal Doppler width ($2\sigma = 1.3 \;{\rm{MHz}}$), laser linewidth (${\sim}0.8 \;{\rm{MHz}}$), and radiative decay rate ($0.66 \;{\rm{MHz}}$). The contribution of different hyperfine states also accounts for the asymmetric lineshape, with a more pronounced tail at higher frequencies, as visible in Fig. 6(a). We expect a narrower EIT line, on the order of 20 MHz, to be obtained if only one hyperfine level (${F^{{\prime \prime}}} = 5$) is addressed, which is possible by spin polarizing the ensemble using optical pumping [4].

5. CONCLUSIONS

We have introduced a new platform to explore light–matter interactions employing a nanofiber-guided mode with a super-extended evanescent field, characterized by low transit-time broadening. This unique mode is reached through adiabatic following of the fundamental mode in a single-mode fiber tapered down to a quarter of a wavelength. Our demonstration, with a high overall transmission, sustaining a $13\lambda$ extended mode over several mm, lies near the asymptotic limit on tapered fiber dimensions capable of such wave guiding. It should be possible to further elongate the fiber waist and to increase the overall transmission by tuning the tapering angles along the fiber profile. In principle, the MFD could be further increased by further reducing the fiber waist diameter. However, this eventually would come at the expense of strongly increased taper losses because meeting the adiabaticity criterion becomes excessively hard when coupling light into a tapered fiber waist with a diameter below a quarter of a wavelength [29]. Moreover, bending losses due to misalignment of the tapered fiber structure would also strongly increase when further decreasing the waist diameter.

Interfaced with atomic vapor, the system balances high local intensity with an extended coherence time. This balance enables saturation of transmission at optical powers that are on par with much tighter confined modes–and at ultralow power levels with ${\lt}{{2}}$ photons present in the interaction region at any given time. We further observe coherent, spectrally narrow, two-photon resonances, owing to the suppression of transit-time broadening. Applying fast in-fiber modulation of the input signal, we observe the intricate dynamics of a nanofiber-vapor coupled system composed of fast modulation of the optical dipole followed by relatively long relaxation times, which are, to the best of our knowledge, one order of magnitude longer than previously measured for a thermal vapor-waveguide interface.

Employing ultracold atoms in this platform is another attractive possibility that can benefit from the super-extended mode, encapsulating a large number of atoms with minimal atom–surface interaction. Naturally, ultracold atoms and especially trapped arrays of atoms would provide for a larger atom–photon coupling strength compared to hot vapor, while the latter offers technical simplicity that can facilitate possible scaling up of such systems. It should be noted, however, that the super-extended evanescent mode, in contrast to tightly confined modes, does not provide a significant enhancement of the atom-waveguide coupling strength. In this respect, it can be considered as a free-space system with weak long-distance guiding that is less suited for typical waveguide-quantum electrodynamics (QED) applications.

The super-extended evanescent mode combines several features that make it particularly well suited for photon–photon interactions and quantum nonlinear optics with Rydberg atoms [36]. First, more than 99% of the guided energy resides outside the material core, as opposed to ${\lesssim} 50\%$ in other evanescent field platforms. In addition, the effective field diameter ${\rm{MFD}} = 10 \;{{\unicode{x00B5}{\rm m}}}$ is suitable for confining photons to below the Rydberg blockade distance. The proximity of Rydberg atoms to dielectric surfaces induces inhomogeneous surface interactions that have been studied, for example, in hollow-core fiber experiments [20]. Due to a favorable ratio of mode volume to fiber surface, we expect such dielectric surface interactions to be suppressed in our extremely tapered fiber [37]. Indeed, Rydberg excitations in a cold atomic cloud near a standard nanofiber were recently observed [38]. In addition to van der Waals and other dipolar interactions, static charges and adsorbed atoms generating stray electric fields pose the major challenge in interfacing a waveguide with Rydberg atoms due to the strong polarizability of the latter [39]. Controlling and removing single charges may, therefore, be necessary as part of the generation of Rydberg excitations in our system. Such control and expulsion of charge down to the single electron level was already achieved in several other platforms [40,41]. We conclude that new and surprising capabilities can emerge from interfacing the platform presented here with various atomic ensembles for waveguide-based quantum optics and sensing applications.

APPENDIX A: TECHNICAL DETAILS

1. Mode Profile and Polarization

Figure 7 shows calculations of the exact field distribution and polarization of the super-extended mode guided by a 200-nm diameter silica fiber at a wavelength of 780 nm. The mode shown is for input light that is linearly polarized along the y axis. While close to the fiber surface, the field magnitude and polarization depend on the azimuthal angle, for the majority of the mode area, the field has a cylindrical symmetry. This is most evident in the radial cross-sections presented in Fig. 7(b), where it can also be seen that the mode is well approximated by the radially symmetric Bessel function ${k_0}(\kappa r)$. In Fig. 7(c) we plot the mode polarization: in-plane polarization is denoted by arrows, and the magnitude of the out-of-plane ellipticity $\epsilon = (i{\mathop{\cal E}\limits^\rightarrow}\times{\mathop{\cal E}\limits^\rightarrow}) / |{\cal E}|^2$ is marked by the color. In contrast to tighter confined fields, where nearly perfect circular polarization with spin-momentum locking of light can be achieved, the weak confinement of the super-extended mode reduces the longitudinal component of the electric field, and thus for the majority of the mode area the polarization is quasilinear with ellipticity $|\epsilon| \le 0.1$.

 

Fig. 7. Super-extended mode structure and polarization. (a) Electric field distribution of the super-extended mode. (b) Radial cross-sections of the field in parallel (red dashed line) and perpendicular (blue solid line) to the input polarization direction. The modified Bessel function ${K_0}(\kappa r)$ is plotted in black dots for comparison. All plots are in semi-log scale. (c) The mode polarization, with arrows denoting the in-plane polarization and color marking the degree of ellipticity out-of-plane.

Download Full Size | PDF

2. Fiber Tapering

We use a Fibercore SM800 fiber, which is tapered gradually along 3.3 cm. The tapering profile consists of three stages: a steep linear taper along 4 mm (taper angle 5 mrad), a moderate linear taper along 18 mm (taper angle 2 mrad), and an exponential taper along the remaining 11 mm. The nanofiber waist is 5 mm long. We image the tapered fibers with a scanning electron microscope [Fig. 2(a)] and, after installation in the vacuum system and while transmitting light through them, with an optical camera. We observe high uniformity along the waist and no light scattering from point defects along the whole fiber. The transmission loss of order 10% can thus be attributed to non-perfect adiabaticity in the tapering region, residual surface roughness, or to slow variations of the waist diameter (on the order of 5% or less).

3. Detailed Experimental Setup

The probe laser is a Vescent Photonics distributed Bragg reflector laser, which is frequency-offset locked to a stable reference laser locked to a cavity. The control laser is a Toptica Photonics external-cavity diode laser, which is side-of-fringe locked to an EIT signal obtained in a vapor cell. Both lasers are inserted into the system via fiber-coupled high bandwidth EOMs (NIR-MX800-LN-10, iXblue). The EOM modulation signal is produced by a SRS DG645 delay generator, equipped with a SRD1 fast rise time module. The probe signal after exiting the fiber is picked off and filtered, by polarization and wavelength, and measured by an avalanche photodiode or by a single-photon counting module (SPCM). To measure the tapered fiber transmission at the intensity level of single photons and with high temporal resolution, we use an Excelitas Technologies NIR-14-FC SPCM whose output is fed to a Fast ComTec MCS6 time tagger. For Fig. 3, we average more than 1024 data traces and use a least-squares digital smoothing filter with a frame length of 19 points, which is equivalent to 9 MHz and is much smaller than the smallest spectroscopic features in this measurement. For Fig. 4, 2.5M repetitions are performed to acquire the data and no extra smoothing is performed. For Fig. 5, 40 K repetitions are performed to acquire each data trace, a moving average filter with frame length of 100 points, or 2.5 MHz, is further used to smooth data.

Funding

H2020 Excellent Science (899275); Ministry of Defense; Israel Science Foundation; International Council for Open Research and Open Education (ICORE); Minerva Foundation; European Research Council (678674); Alexander von Humboldt-Stiftung

Acknowledgment

The authors thank Liron Stern and Uriel Levy for fruitful discussions. B. Dayan is the Dan Lebas and Roth Sonnewend Professorial Chair of Physics.

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. B. Patton, E. Zhivun, D. C. Hovde, and D. Budker, “All-optical vector atomic magnetometer,” Phys. Rev. Lett. 113, 013001 (2014). [CrossRef]  

2. S. Knappe, V. Shah, P. D. Schwindt, L. Hollberg, J. Kitching, L. A. Liew, and J. Moreland, “A microfabricated atomic clock,” Appl. Phys. Lett. 85, 1460–1462 (2004). [CrossRef]  

3. Y.-S. Lee, S. M. Lee, H. Kim, and H. S. Moon, “Highly bright photon-pair generation in Doppler-broadened ladder-type atomic system,” Opt. Express 24, 28083–28091 (2016). [CrossRef]  

4. R. Finkelstein, E. Poem, O. Michel, O. Lahad, and O. Firstenberg, “Fast, noise-free memory for photon synchronization at room temperature,” Sci. Adv. 4, eaap8598 (2018). [CrossRef]  

5. H. Alaeian, R. Ritter, M. Basic, R. Löw, and T. Pfau, “Cavity QED based on room temperature atoms interacting with a photonic crystal cavity: a feasibility study,” Appl. Phys. B 126, 25 (2020). [CrossRef]  

6. A. Maser, B. Gmeiner, T. Utikal, S. Götzinger, and V. Sandoghdar, “Few-photon coherent nonlinear optics with a single molecule,” Nat. Photonics 10, 450–453 (2016). [CrossRef]  

7. N. Jia, N. Schine, A. Georgakopoulos, A. Ryou, L. W. Clark, A. Sommer, and J. Simon, “A strongly interacting polaritonic quantum dot,” Nat. Phys. 14, 550–554 (2018). [CrossRef]  

8. S. M. Spillane, G. S. Pati, K. Salit, M. Hall, P. Kumar, R. G. Beausoleil, and M. S. Shahriar, “Observation of nonlinear optical interactions of ultralow levels of light in a tapered optical nanofiber embedded in a hot rubidium vapor,” Phys. Rev. Lett. 100, 233602 (2008). [CrossRef]  

9. L. Stern, B. Desiatov, I. Goykhman, and U. Levy, “Nanoscale light-matter interactions in atomic cladding waveguides,” Nat. Commun. 4, 1548 (2013). [CrossRef]  

10. A. Skljarow, N. Gruhler, W. Pernice, H. Kübler, T. Pfau, R. Löw, and H. Alaeian, “Integrating two-photon nonlinear spectroscopy of rubidium atoms with silicon photonics,” Opt. Express 28, 19593–19607 (2020). [CrossRef]  

11. K. P. Nayak, P. N. Melentiev, M. Morinaga, F. L. Kien, V. I. Balykin, and K. Hakuta, “Optical nanofiber as an efficient tool for manipulating and probing atomic fluorescence,” Opt. Express 15, 5431–5438 (2007). [CrossRef]  

12. G. Sagué, E. Vetsch, W. Alt, D. Meschede, and A. Rauschenbeutel, “Cold-atom physics using ultrathin optical fibers: light-induced dipole forces and surface interactions,” Phys. Rev. Lett. 99, 163602 (2007). [CrossRef]  

13. M. Bajcsy, S. Hofferberth, V. Balic, T. Peyronel, M. Hafezi, A. S. Zibrov, V. Vuletic, and M. D. Lukin, “Efficient all-optical switching using slow light within a hollow fiber,” Phys. Rev. Lett. 102, 203902 (2009). [CrossRef]  

14. S. M. Hendrickson, M. M. Lai, T. B. Pittman, and J. D. Franson, “Observation of two-photon absorption at low power levels using tapered optical fibers in rubidium vapor,” Phys. Rev. Lett. 105, 173602 (2010). [CrossRef]  

15. R. Ritter, N. Gruhler, W. Pernice, H. Kübler, T. Pfau, and R. Löw, “Atomic vapor spectroscopy in integrated photonic structures,” Appl. Phys. Lett. 107, 041101 (2015). [CrossRef]  

16. M. A. Foster and A. L. Gaeta, “Ultra-low threshold supercontinuum generation in sub-wavelength waveguides,” Opt. Express 12, 3137–3143 (2004). [CrossRef]  

17. E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010). [CrossRef]  

18. D. E. Jones, J. D. Franson, and T. B. Pittman, “Ladder-type electromagnetically induced transparency using nanofiber-guided light in a warm atomic vapor,” Phys. Rev. A 92, 043806 (2015). [CrossRef]  

19. T. Cutler, W. Hamlyn, J. Renger, K. Whittaker, D. Pizzey, I. Hughes, V. Sandoghdar, and C. Adams, “Nanostructured alkali-metal vapor cells,” Phys. Rev. Appl. 14, 034054 (2020). [CrossRef]  

20. G. Epple, K. S. Kleinbach, T. G. Euser, N. Y. Joly, T. Pfau, P. St.J. Russell, and R. Löw, “Rydberg atoms in hollow-core photonic crystal fibres,” Nat. Commun. 5, 4132 (2014). [CrossRef]  

21. L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003). [CrossRef]  

22. S. Korposh, S. W. James, S. W. Lee, and R. P. Tatam, “Tapered optical fibre sensors: current trends and future perspectives,” Sensors 19, 2294 (2019). [CrossRef]  

23. J. Petersen, J. Volz, and A. Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin-orbit interaction of light,” Science 346, 67–71 (2014). [CrossRef]  

24. S. Kato and T. Aoki, “Strong coupling between a trapped single atom and an all-fiber cavity,” Phys. Rev. Lett. 115, 093603 (2015). [CrossRef]  

25. O. Bechler, A. Borne, S. Rosenblum, G. Guendelman, O. E. Mor, M. Netser, T. Ohana, Z. Aqua, N. Drucker, R. Finkelstein, Y. Lovsky, R. Bruch, D. Gurovich, E. Shafir, and B. Dayan, “A passive photon-atom qubit swap operation,” Nat. Phys. 14, 996–1000 (2018). [CrossRef]  

26. A. Snyder and J. Love, Optical Waveguide Theory (Springer, 2012).

27. L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12, 1025–1035 (2004). [CrossRef]  

28. A. Stiebeiner, R. Garcia-Fernandez, and A. Rauschenbeutel, “Design and optimization of broadband tapered optical fibers with a nanofiber waist,” Opt. Express 18, 22677–22685 (2010). [CrossRef]  

29. M. Sumetsky, “How thin can a microfiber be and still guide light?” Opt. Lett. 31, 870–872 (2006). [CrossRef]  

30. C. J. Bordé, J. L. Hall, C. V. Kunasz, and D. G. Hummer, “Saturated absorption line shape: calculation of the transit-time broadening by a perturbation approach,” Phys. Rev. A 14, 236–263 (1976). [CrossRef]  

31. O. Lahad, R. Finkelstein, O. Davidson, O. Michel, E. Poem, and O. Firstenberg, “Recovering the homogeneous absorption of inhomogeneous media,” Phys. Rev. Lett. 123, 173203 (2019). [CrossRef]  

32. R. Finkelstein, O. Lahad, O. Michel, O. Davidson, E. Poem, and O. Firstenberg, “Power narrowing: counteracting Doppler broadening in two-color transitions,” New J. Phys. 21, 103024 (2019). [CrossRef]  

33. V. M. Akulin and N. V. Karlov, Intense Resonant Interactions in Quantum Electronics (Springer, 1992).

34. M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000). [CrossRef]  

35. S. Rosenblum, Y. Lovsky, L. Arazi, F. Vollmer, and B. Dayan, “Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators,” Nat. Commun. 6, 6788 (2015). [CrossRef]  

36. T. Peyronel, O. Firstenberg, Q. Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012). [CrossRef]  

37. E. Stourm, M. Lepers, J. Robert, S. Nic Chormaic, K. Mølmer, and E. Brion, “Spontaneous emission and energy shifts of a Rydberg rubidium atom close to an optical nanofiber,” Phys. Rev. A 101, 052508 (2020). [CrossRef]  

38. K. S. Rajasree, T. Ray, K. Karlsson, J. L. Everett, and S. N. Chormaic, “Generation of cold Rydberg atoms at submicron distances from an optical nanofiber,” Phys. Rev. Res. 2, 012038 (2020). [CrossRef]  

39. G. Epple, N. Y. Joly, T. G. Euser, P. St.J. Russell, and R. Löw, “Effect of stray fields on Rydberg states in hollow-core PCF probed by higher-order modes,” Opt. Lett. 42, 3271–3274 (2017). [CrossRef]  

40. M. Frimmer, K. Luszcz, S. Ferreiro, V. Jain, E. Hebestreit, and L. Novotny, “Controlling the net charge on a nanoparticle optically levitated in vacuum,” Phys. Rev. A 95, 061801 (2017). [CrossRef]  

41. D. C. Moore, A. D. Rider, and G. Gratta, “Search for millicharged particles using optically levitated microspheres,” Phys. Rev. Lett. 113, 251801 (2014). [CrossRef]