Coupling of ultrathin tapered fibers with high-Q microsphere resonators at cryogenic temperatures and observation of phase-shift transition from undercoupling to overcoupling

We cooled ultrathin tapered fibers to cryogenic temperatures and controllably coupled them with high-Q microsphere resonators at a wavelength close to the optical transition of diamond nitrogen vacancy centers. The 310-nm-diameter tapered fibers were stably nanopositioned close to the microspheres with a positioning stability of approximately 10 nm over a temperature range of 7-28 K. A cavity-induced phase shift was observed in this temperature range, demonstrating a discrete transition from undercoupling to overcoupling.

. 33. The overall transmittance of the tapered fibers (including the fiber coupling loss, the fiber connection loss, and the scattering loss at the UV adhesive) was 0.13 at room temperature. It decreased to 0.03 at 7 K, probably due to temperature-induced deformation of the UV adhesive. This reduction in transmittance does not essentially affect the present fiber-microsphere coupling experiment. 34. Note that the microspheres fabricated from the silica fibers (S630-HP, Thorlabs) usually show Q factor of ∼ 10 7 due to impurities doped in the silica fibers. By using high-purity and low-OH silica for the starting material, we confirmed that the microspheres have Q factor of greater than 10 8 . Note also that the cryogenic experiments do not affect the Q-factor of the microsphere as we already experimentally confirmed previously [24]. 35. H. Konishi, H. Fujiwara, S. Takeuchi, and K. Sasaki, "Polarization-discriminated spectra of a fiber-microsphere system," Appl. Phys. Lett. 89, 121107 (2006). 36. K. Totsuka and M. Tomita, "Slow and fast light in a microsphere-optical fiber system," J. Opt. Soc. Am. B 23, 2194-2199 (2006). 37. The taper-microsphere distance was calibrated by the data of room-temperature coupling experiments using a 330-nm-diameter tapered fiber and a 125-µm-diameter microsphere with the quality factor of Q = 1.0 × 10 7 .

Introduction
Coupling diamond nitrogen vacancy (NV) centers with nanophotonic cavities such as microresonators and photonic crystals is currently of great importance for quantum information science [1][2][3][4][5][6][7][8][9][10]. Such NV-coupled cavity quantum electrodynamics (cavity QED) systems can be used to develop coherent photonic quantum devices including quantum phase gates [11,12] and quantum memories [13,14]. These applications exploit the narrow zero-phonon optical transitions of NV centers located at around λ = 637 nm, which appear only at cryogenic temperatures [6]. To develop coherent quantum devices that employ NV centers, it is thus essential to couple NV optical transitions with these nanophotonic cavities in a cryogenic environment.
To satisfy the first requirement, ultrathin tapered fibers (typically 300-400 nm in diameter) have to be used to realize efficient coupling with microresonators at about λ = 637 nm. The distance between these tapered fibers and microresonators needs to be controlled with an accuracy of 10 nm. However, it is difficult to cool ultrathin tapered fibers to cryogenic temperatures and to realize such precise positioning in a cryogenic environment due to fiber fragility and the mechanical vibrations generated by cryogenic systems [3]. In our previous study at cryogenic temperatures [24], the coupling efficiency between tapered fibers and microsphere cavities could not be controlled due to the large taper diameters used (∼ 1.0 µm diameter) and the positioning instability.
Satisfaction of the second requirement can be demonstrated by investigating the cavityinduced phase-shift spectra. We recently obtained phase-shift spectra of fiber-coupled microsphere resonators at room temperature by using a polarization analysis technique [27]. However, the phase-shift spectra of such fiber-coupled microresonators have not been measured at cryogenic temperatures until now. Phase-shift spectrum measurements at cryogenic temperatures are particularly important because they directly demonstrate the potential of applying these fiber-coupled microresonators to coherent quantum information devices.
Here, we demonstrate cooling of 310-nm-diameter ultrathin tapered fibers down to 7 K and their fully controllable coupling with high-Q microsphere resonators at about λ = 637 nm. The tapered fibers were stably nanopositioned close to the microspheres with a positioning stability of less than 13 nm over at least 7 min over a temperature range of 7-28 K. A cavity-  induced phase shift was observed. These observations clearly show a discrete transition from undercoupling to overcoupling [27,28]. In addition, 0.8-GHz frequency tuning of the cavity resonance was realized by increasing the temperature from 7 to 28 K while preserving nearcritical coupling throughout this temperature range. The present cavity system has high-Q factor of 1.9 × 10 7 (potentially up to 10 9 [29]) in the visible wavelength region together with the full controllability of waveguide-cavity coupling condition, which makes it distinct from other cryogenic fiber-coupled microcavity systems recently reported [18][19][20]30] in view that the present system can provide a testbed for quantum optics using stable solid-state quantum nanoemitters like diamond NV centers. The present demonstration is thus especially important for developing coherent quantum devices using those solid-state quantum nanoemitters.

Experiments
Ultrathin tapered fibers with diameters of ∼ 300 nm were fabricated from conventional singlemode optical fibers (Thorlabs, 630HP) by the method described elsewhere [31,32]. Adiabatic tapering was confirmed as the transmittance was larger than 0.9 during fabrication. UV adhesives were used to attach the tapered fibers to a silica substrate [33]. The tapered fibers used in the present experiments do not exhibit birefringence or depolarization in tapered region [35], which ensures the polarization selectivity of cavity modes in fiber-microsphere coupling experiments. Microspheres were fabricated by melting the tips of silica fibers (S630-HP, Thorlabs) using a CO 2 laser [34].
Using these ultrathin tapered fibers and microspheres, we performed transmittance and phase-shift measurements based on polarization analysis of the light in the fiber-coupled microsphere cavities. Figure 1 shows the experimental setup. The output of a tunable laser diode (λ = 638.8 nm) was coupled to a single-mode fiber, which was connected with a tapered fiber in the cryostat. Polarization controllers were inserted before and after the tapered fiber to compensate the fiber birefringence. The output light from the fiber was then sent to the polarization analysis setup that consisted of a half-wave plate, a quarter-wave plate, a polarizing beam splitter, and two photodetectors. The tapered fiber used in this experiment was 310 nm in diameter. The microsphere was 120 µm in diameter and its stem was 20 µm in diameter. It was mounted on a three-axis piezo stage [24]. After cooling the cryostat, the microsphere was brought close to the fiber to couple the guided light and the cavity mode. The polarization analysis provided four Stokes parameters from S 0 to S 3 , which can be used to calculate the phase shift (∆φ ) due to microsphere coupling [∆φ = arctan (S 3 /S 2 )]. The experimental details of the phase-shift measurement are described in Ref. [27,38]. Figures 2(a) and (b) show normalized transmittance (η) spectra of the cavity resonance measured at 10 K for taper-microsphere distances (D) of 390 and 250 nm, respectively. Here, η was normalized to the off-resonance transmittance of the cavity mode. The transmittance minimum and linewidth in Fig. 2(a) are respectively η = 0.152 and 77 MHz (Q = 6.1 × 10 6 ) and are respectively η = 0.395 and 220 MHz (Q = 2.1 × 10 6 ) in Fig. 2(b). The higher quality factors were obtained as the microsphere went away from the tapered fiber; e.g. Q = 1.2 × 10 7 (linewidth: 40 MHz) was observed at D = 1030 nm [see Fig. 2

Results and Discussions
These two transmittance spectra exhibit different coupling conditions, namely undercoupling and overcoupling [27, 28, 36]. In Fig. 2(a), the microsphere is located far from the fiber and consequently the coupling strength is smaller than the cavity loss (the undercoupling regime). In contrast, the microsphere is near the fiber in Fig. 2(b) and the coupling strength is greater than the cavity loss (the overcoupling regime). Between these two regimes there is critical coupling for which the coupling strength equals the cavity loss. Critical coupling gives the highest coupling efficiency.
The striking difference between these two regimes can be clearly observed in their phase-shift spectra. The phase shift (∆φ ) exhibits a discrete transition between these two regimes [27,36]. In the undercoupling regime, the phase shift caused by cavity resonance is less than π and the phase shift does not change between the lower frequency side of the cavity resonance and the higher side. In contrast, a phase shift of 2π is always obtained in the overcoupling regime and the phase shift changes by 2π as the frequency increases across the cavity resonance [36]. The present observation of a discrete transition from undercoupling to overcoupling demonstrates the full controllability of taper-microsphere coupling in a cryogenic environment at λ ∼ 637 nm.
In addition to observing these drastic changes, we have controlled the coupling condition from undercoupling to overcoupling by varying the taper-microsphere distance. Figures 2(e) and (f) show plots of the transmittance minimum and the linewidth as a function of the tapermicrosphere distance, respectively [37]. As the microsphere approaches the fiber, the transmittance minimum decreases until D = 360 nm. Below D = 360 nm, the transmittance minimum increases until η ∼ 0.5 at D = 0 nm. At the same time, the linewidth of the cavity resonance dip increases rapidly below D = 360 nm. The positioning stability of the tapered fiber relative to the microsphere was estimated to be less than 13 nm over at least 7 minutes by calibrating the fluctuations in the transmittance minimum observed in 7 minutes with the distance dependence of the transmittance depicted in Fig. 2(e). Such coupling controllability is crucial for developing cavity QED systems [16]. Thus, the present system can be effectively used for cavity QED studies using NV centers. Note that the nonzero transmittance minimum for critical coupling in Fig. 2(e) originates from the phase mismatch between the waveguide and cavity modes. This mismatch can be eliminated by optimizing the taper diameter to the microsphere coupling [28,39]. This coupling controllability based on stable nanopositioning of the microsphere allows their efficient coupling over the temperature range of 7-28 K. Figure 3(a) shows the frequency shift of the cavity resonance as a function of the temperature from 7 to 28 K and Fig. 3(b) shows the corresponding several transmittance spectra that were measured at which the microsphere was positioned to the critical coupling position. As the temperature increases from 7 K, the central frequency of the cavity mode initially increases, but it starts to decrease at 20 K. Importantly, the transmittance minimum of the resonance dip remains constant over the entire temperature range, as shown in Fig. 3(b). This indicates that near-critical coupling was obtained over the whole temperature range.
The frequency shift of the resonance peak showed nonlinear temperature dependence in contrast to the linear frequency shift at room temperature (-2.6 GHz / K at room temperature [40]). As the temperature increased from 8 K the center frequency showed positive shift up to 20 K, and from there it switched to negative shift. It is known that the cavity resonance frequency shift dT as a function of temperature from 7 to 28 K, which are indicated by black rectangle, red circle, blue triangle, respectively. The data of α is taken from Ref. [43,44]. In (b), cavity mode A has the Q factor of 7.3 × 10 6 and cavity mode B 1.9 × 10 7 .
is governed by thermal expansion coefficient of silica (α) and the effective refractive index of the cavity mode (n eff ) at cryogenic temperatures, which can be described as follows [41,42], Using the experimental values for 1 ν 0 dν 0 dT and the literature data of α [43,44], the variation in the effective refractive index ( 1 n eff dn eff dT ) is determined using this equation. Fig. 3(c) shows the plot of these three components. At room temperature, the n eff term is known to be one order of magnitude greater than α, so that 1 ν 0 dν 0 dT depends mostly on the n eff term [40,41]. However, the n eff term rapidly decreases as the temperature goes down to LHe temperature regime, thereby competing with the thermal coefficient α. The resultant frequency shift of the cavity resonance peak in Fig. 3(c) hence shows such a nonlinear behavior (reversal of the frequency shift).
This nonlinear temperature dependence of the frequency shift in the fiber-coupled microsphere agrees well with the results obtained in Ref. [42], where the temperature dependence of the microsphere cavity resonance was measured by far-field excitation and scattering detection. This good agreement between the results of the previous study for far-field excitation and the present results for near-field excitation indicates that cavity QED experiments using NVcoupled microsphere resonators by far-field excitation, such as NV-photon strong coupling [3], can be more effectively performed using the present near-field excitation system.
The coupling of NV centers with microcavities at cryogenic temperatures has been the central issue in the recent solid-state cavity QED studies. Until now, several works have reported the coupling of single NV centers with microcavities (cavity only) at cryogenic temperature, demonstrating the NV-photon strong coupling [3] or Purcell factor of F ∼ 70 [8]. However, these microcavities still do not have efficient optical access like tapered fibers. Forthcoming step to implement these NV-coupled cavity systems to the coherent quantum information devices is to provide the cavities with an efficient optical access. Thus our present demonstration of the fiber-coupled microsphere system at cryogenic temperatures is an important step towards the realization of coherent quantum devices using diamond NV centers.
The present fiber-coupled microsphere system is also distinct from other cryogenic fibercoupled microcavity systems recently reported [18][19][20]30] in view that the system is optimized for the visible wavelength range necessary for the coherent excitation of the stable solid-state quantum nanoemitters, including diamond NV centers or CdSe/ZnS quantum dots, and has the full controllability of the waveguide-cavity coupling condition with high-Q factors. The cavity modes that we have shown above have the Q factor of 0.6-1.2 ×10 7 in the undercoupling regime [see Fig. 3(b) and cavity mode A in Fig. 3(b)]. We were also able to observe cavity modes that have Q factor of 10 7 in many cases; the cavity mode B in Fig. 3(b) showed Q = 1.9 × 10 7 , for example. Furthermore, the microsphere-based cavity system can potentially provide Q factors up to 10 9 [29]. The present system is therefore especially important for quantum optics experiments employing those promising solid-state nanoemitters.

Summary
We have demonstrated cooling of 310-nm-diameter ultrathin tapered fibers down to 7 K and fully controllable coupling with high-Q microsphere resonators at λ ∼ 637 nm in the vicinity of the zero-phonon optical transitions of diamond NV centers. Stable nanopositioning of the 310nm-diameter tapered fibers in close proximity to the microsphere with a positioning stability of less than 13 nm over at least 7 min has been realized over the temperature range 7-28 K. A cavity-induced phase shift has been successfully obtained. The results clearly showed a discrete transition from undercoupling to overcoupling. In addition, 0.8-GHz frequency tuning of the cavity resonance by increasing the temperature from 7 to 28 K has been realized while preserving near-critical coupling over this temperature range. The present cavity systems have high-Q factor of Q = 1.9 × 10 7 in the visible wavelength region and therefore can be used for developing coherent quantum devices using diamond NV centers.