Over the past years, significant research effort has been devoted to interfacing quantum emitters, such as molecules, quantum dots, color centers, and neutral atoms, with fiber-guided light fields. Suitable light–matter interfaces are considered to be key elements for future quantum networks [1]. One way to realize such an interface consists of coupling emitters to the evanescent fields surrounding the nanofiber waist of a tapered optical fiber, that is, a fiber section with sub-wavelength diameter. Such systems already provide absorption probabilities for single fiber-guided photons of about 25% for a single emitter located on the nanofiber surface [2] and about 5% at a distance of 200 nm, typical for cold atoms in nanofiber-based optical dipole traps [3,4].

One way to further enhance the light–matter coupling strength is to increase the number of emitters in the evanescent field and to take advantage of the collective coupling [5]. Another option relies on confining the light in an optical resonator, which allows one to even reach the regime of strong coupling in the sense of cavity quantum electrodynamics (CQED) [6]. There, the coherent emitter-light interaction strength dominates over the incoherent decay channels. In this context, optical nanofibers are a versatile platform, as they allow one to combine both approaches and, in this way, to reach very strong light–matter coupling in a fiber-integrated environment.

Different nanofiber-based Fabry–Perot resonator schemes have been developed, for example, based on Bragg structures created using ion beam milling [7,8] or laser ablation [9] of the fiber waist. A nanofiber placed on an optical grating has been demonstrated, and Purcell enhancement of single quantum emitters coupled to this system has been observed [10]. Tapered optical fibers have been combined with conventional fiber Bragg gratings to form a Fabry–Perot resonator [11] for which strong coupling has been demonstrated with Cesium (Cs) atoms trapped close to the nanofiber surface [12]. Running-wave type resonators, such as nanofiber knot and loop resonators [13] and a nanofiber-segment tapered fiber closed to a ring via a 50:50 beam splitter [14], have also been demonstrated, albeit so far with smaller finesse than the Fabry–Perot counterparts.

Here, we demonstrate a tapered fiber-based ring resonator with optical characteristics that are compatible with entering the regime of single-atom strong coupling. We experimentally reach a finesse of $F=75\pm 1$, which corresponds to a single-atom cooperativity of $C\approx 1$. Our implementation offers easy tuning of the resonator eigenpolarizations and out-coupling rate as well as straightforward adjustment of the free spectral range of the resonator. Our system is compatible with established nanofiber-based schemes for the optical trapping of laser-cooled atoms. This would enable a controlled coupling of large ensembles of atoms to the resonator field which then gives rise to extremely large collective atom-resonator interaction strengths. Furthermore, the evanescent fields around the nanofiber exhibit an inherent link between the local polarization and the propagation direction of the guided light [15]. This gives rise to a direction-dependent light-emitter coupling [16,17] which renders this resonator distinct from traditional Fabry–Perot or ring resonators. In particular, this allows one to implement chiral quantum optics effects [18]. Finally, the optical path length of such a tapered fiber ring (TFR) resonator can be significantly increased without reducing cooperativity. This makes this system a prime candidate for experimentally exploring the regime of optical multimode strong coupling where the atoms simultaneously couple strongly to many longitudinal resonator modes [19,20].

Our ring resonator consists of a tapered optical fiber with a nanofiber waist connected to an adjustable fiber beam splitter (Newport F-CPL-830-N-FA), see Fig. 1(a), whose splitting ratio can be continuously adjusted between 0% and 100%. The tapered fiber, shown schematically in Fig. 1(b), is fabricated from a standard step-index silica fiber (Fiber Core SM 800) using a heat-and-pull process [21,22]. The fiber waist has a length of 5 mm and a diameter of 500 nm. Light guided in such a subwavelength-diameter optical fiber exhibits a pronounced evanescent field [Fig. 1(c)] and enables efficient, homogeneous coupling of optical emitters to the guided mode. The standard fiber ends of the tapered fiber are spliced to the fiber beam splitter, yielding a TFR resonator with a total resonator length of $l=2.35\pm 0.01\mathrm{m}$. The remaining two outputs of the fiber beam splitter then constitute the coupling fiber that allows us to interface the resonator.

 

Fig. 1. (a) Sketch of the experimental setup: a tapered optical fiber with a nanofiber waist forms a fiber ring resonator. An adjustable fiber beam splitter allows one to couple light into and out of the resonator. The characterization of the system is carried out by recording transmission spectra through the coupling fiber around a center wavelength of about 850 nm. The polarization of the input light field can be adjusted using a half and a quarter wave plate. (b) Schematic of the tapered optical fiber section with a nanofiber waist. A waist with a diameter of 500 nm and a length of 5 mm was used in the experiment. (c) Normalized intensity profile of a quasi-linearly polarized nanofiber-guided light field. The pronounced evanescent part of the guided light allows efficient interfacing with optical emitters (parameters: nanofiber radius 250 nm; vacuum optical wavelength 852 nm; and fiber refractive index 1.45).

Download Full Size | PDF

We optically characterize the TFR resonator by recording transmission spectra through the coupling fiber. To this end, we send light from a tunable, narrow-band diode laser into the coupling fiber and align the polarization of the probe light with one of the principle polarization axes of the resonator using wave plates (see below). The transmitted power is recorded with a photodiode as a function of the laser–resonator detuning. The wavelength of the light is about 850 nm, close to the Cs D2 line. By changing the splitting ratio of the beam splitter, we can continuously adjust the fiber–resonator coupling rate ${\kappa }_{\mathrm{ext}}$ from the undercoupled regime (${\kappa }_{\mathrm{ext}}<{\kappa }_0$) to critical coupling (${\kappa }_{\mathrm{ext}}={\kappa }_0$) and to the overcoupled regime (${\kappa }_{\mathrm{ext}}>{\kappa }_0$). Here, ${\kappa }_0$ is the unloaded field decay rate of the uncoupled TFR resonator. Figure 2 shows example spectra for the three different coupling regimes. In each case, the laser is scanned over three consecutive resonances. Narrow transmission dips are clearly apparent with a free spectral range (FSR) of ${\nu }_{\mathrm{FSR}}=87.5\pm 0.4\mathrm{MHz}$, demonstrating a high optical finesse and indicating a small internal resonator field decay rate ${\kappa }_0$.

 

Fig. 2. Measured coupling fiber transmission (gray) as a function of the laser detuning for the (a) undercoupled regime (${\kappa }_{\mathrm{ext}}<{\kappa }_0$), (b) critically coupling (${\kappa }_{\mathrm{ext}}={\kappa }_0$), and (c) overcoupled regime (${\kappa }_{\mathrm{ext}}>{\kappa }_0$). The red dashed lines are Lorentzian fits to the data.

Download Full Size | PDF

The resonator is produced from a non-polarization maintaining optical fiber and is subject to birefringence. The TFR resonator has two eigenpolarizations and the corresponding eigenfrequencies are, in general, different from each other. The eigenpolarizations are these two (orthogonal) polarization states that exactly reproduce themselves after one resonator round trip. In the experiment, we added an intra-resonator fiber paddle polarization controller (Thorlabs FPC) into the resonator fiber. This controller adds additional stress-induced birefringence to the fiber which allows us to adjust the resonator eigenpolarizations. In this way, we were able to fully tune the TFR resonator polarization eigenmodes from the point of degeneracy (same eigenfrequencies) to the point of maximum birefringence (eigenfrequencies of the two polarization eigenmodes are separated by half a FSR). For the data presented in Fig. 2, the resonator was set such that the two modes were clearly separated. The polarization of the probe light field was always adjusted to couple to only one of the resonator polarization modes by optimizing the wave plate settings in the coupling beam path.

Close to a cavity resonance, the field transmission $t$ through the coupling fiber as a function of the frequency of the probe light is given by the Lorentzian [23]:

The intrinsic decay rate ${\kappa }_0$ in our experiment corresponds to round trip power losses of $\mathcal{L}=\exp (2{\kappa }_{0}nl/c)\approx 7.9\%$, where $n$ is the silica refractive index and $c$ the vacuum speed of light. For our resonator length, these losses have two main contributions: one originates from the insertion loss of the fiber beam splitter, specified by the manufacturer to be below 0.1 dB, which corresponds to a loss of 2.3%. The remaining loss stems from the finite tapered fiber transmission, which we estimate to be about 94%. This is slightly lower than typical transmissions of our home-made tapered fibers (about 98%), presumably due to dust that accumulated in the fiber waist during the experiment.

Based on our system parameters and the results of the spectral characterization of the resonator, we now estimate the potential for the realization of a light–matter interface and for experiments in the realm of CQED. We discuss the typical example of interfacing an ensemble of laser-cooled ${}^{133}\mathrm{Cs}$ atoms to the evanescent field present around the nanofiber section of the TFR resonator [3]. The closed optical transition $|F=4,m_F=4⟩\leftrightarrow |F^{\prime }=5,m_{F^{\prime }}=5⟩$ from the $6S_{1/2}$ ground to the $6P_{3/2}$ excited state (wavelength $\lambda =852\mathrm{nm}$, dipole decay rate $\gamma =2\pi \times 2.6\mathrm{MHz}$) is ${\sigma }^{+}$-polarized. For a suitably chosen quantization axis, this optical transition has almost unit polarization overlap with the nanofiber-guided light [16]. In order to calculate the coupling strength $g$, we recall that this quantity corresponds to the single-photon optical Rabi frequency of the driven atom inside the resonator. Using the explicit solution for the evanescent fields of the nanofiber [15], we can calculate the local electric field at the position of the atom. From the electric field associated with the optical power of one single photon circulating the resonator and the atomic dipole moment of the above-mentioned transition, we then obtain a coupling strength of $g=2\pi \times 1.5\mathrm{MHz}$ for an atom located 200 nm away from the nanofiber surface, that is, at a typical atom–surface separation realized with nanofiber-based traps [3]. This corresponds to a single-atom cooperativity $C_0=g^2/(2\kappa \gamma )=0.74$, close to the regime of strong coupling. For comparison, for an atom at the surface, we find a coupling strength of $g_{\mathrm{surf}}=2\pi \times 5\mathrm{MHz}$, which is high enough to enter the strong coupling regime, that is, $C_{0,\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}\approx 4.4>1$. The increase of $\gamma$ due to the close proximity of the atomic dipole to the surface is taken into account in the calculation [25]. By employing established techniques, it should be feasible to optically trap about $N=2000$ atoms close to the nanofiber [3]. This would realize a large collective interaction between the atomic ensemble and the light in the resonator, yielding a collective cooperativity of $C_{\mathrm{coll}}={\mathit{NC}}_0\approx 1500$, which sets the system far in the strong coupling regime.

We now estimate how the resonator and coupling parameters change with the resonator length. Diffraction effects make it challenging to maintain a small mode area (and, thus, large coupling strength) for long resonators build from free-space optical components. However, for both Fabry–Perot and ring-type fiber-based resonators, the light is guided and the mode cross section is independent of the resonator length. Then, the atom-resonator coupling strength scales as $g\propto l^{-1/2}$ with the length $l$ of the resonator. At the same time, the cumulative propagation losses in the standard fiber part are typically much smaller than the losses that occur when the light propagates through the tapered section. In this case, the unloaded resonator decay rate ${\kappa }_0$ depends on the length as ${\kappa }_0\propto l^{-1}$. Consequently, the cooperativity is then independent of the resonator length. Remarkably, it is thus possible to maintain large cooperativities also for large resonator lengths. For our experimental system, Fig. 3 shows the predicted behavior of the collective cooperativity $C_{\mathrm{coll}}$ as well as collective coupling strength $g_{\mathrm{coll}}=\sqrt{N}g$ and FSR ${\nu }_{\mathrm{FSR}}$ as a function of the geometrical resonator length of the TFR resonator. The calculation takes fiber propagation losses into account. For fiber lengths l of more than 4 m, the collective coupling strength $g_{\mathrm{coll}}$ exceeds the FSR, and the system enters the regime of multimode strong coupling [19]. In this regime of CQED, the atomic ensemble is simultaneously strongly coupled to several non-degenerate longitudinal resonator modes. So far, this has only been realized in the microwave domain [26]. Multimode strong coupling enables atom-mediated interactions between different optical modes and results in a nonlinear quantum dynamics that is not present in the single-mode regime [20]. Atom-mediated mode coupling has recently also been considered for implementing photonic quantum simulation protocols [27].

 

Fig. 3. (a) Transmission through the coupling fiber as a function of the total field decay rate of the resonator $\kappa ={\kappa }_{\mathrm{ext}}+{\kappa }_0$. From small to large values of $\kappa$, the regimes of undercoupling, critical coupling, and overcoupling are observed. The curve is a fit using Eq. (1). Error bars, indicating the standard deviation of the $\{\kappa ,T_r\}$ fit values, are small and hardly visible. (b) Expected collective coupling strength $g_{\mathrm{coll}}$ (blue dashed line), free spectral range ${\nu }_{\mathrm{FSR}}$ (red dashed-dotted line), and cooperativity $C_{\mathrm{coll}}$ (black solid line) of the TFR resonator as a function of the resonator length. The plot is calculated for fiber losses of 1.5 dB/km for optical fibers at a wavelength of about 850 nm [24] and for the case of a state-of-the-art nanofiber trap loaded with 2000 atoms [3]. The dashed vertical line indicates the length at which the $g_{\mathrm{coll}}$ exceeds the free spectral range, and the atom-resonator system enters the regime of multimode strong coupling.

Download Full Size | PDF

Apart from the large possible collective coupling strength and the intrinsic fiber integration, a TFR resonator differs from other resonator types because of the extremely tight confinement of the light in the nanofiber section. This confinement gives rise to an inherent link between the local polarization and the propagation direction of the light. In conjunction with suitable quantum emitters, such as spin-polarized atoms, NV-centers or quantum dots, which can exhibit polarization dependent scattering cross-sections, this gives rise to strong direction-dependent light–matter interaction strengths [16,17], similar to the case of whispering-gallery-mode resonators [28]. This renders the TFR resonator a conceptually novel type of optical resonator for which collectively enhanced chiral light–matter interaction [18] provides novel quantum functionalities [29] and that can, for example, be employed to realize optical nonreciprocal devices with significantly higher efficiency that also work for light levels that are higher than in previous implementations [30,31]. Moreover, it enables the study of chiral interactions and the consequences of collective effects such as sub- and super-radiance in the regime of multimode strong coupling, that is, where the dynamics of the system is neither that expected for a conventional CQED system nor that of an open waveguide.

In summary, we realized a TFR resonator that incorporates a nanofiber section, and identified two key areas of application. The high optical finesse, the tight confinement of the light in the nanofiber section, and the compatibility with homogeneously coupling to large ensembles of optical emitters render this system well suited for protocols that require strong light–matter interaction. By taking additional precautions against the pollution of the nanofiber waist, we expect to be able to reduce the transmission losses in this section and, thus, to further enhance the finesse and cooperativity of the system. Our experimental approach also offers practical advantages such as a simple tuning of key system parameters. Both, the chiral light–matter interactions present in our resonator and the option to enter the regime of multimode strong coupling, make TFR resonators a powerful platform for future experiments in classical and quantum photonics.