Design of a fiber cavity ion trap for a high-efficiency and high-rate quantum network node

Graphical abstract Scheme and parameters of the designed fiber cavity ion trap.


Introduction
Quantum nodes, which can act as either a quantum repeater or a terminal quantum node, can be created easily using coupled systems of cold atoms (ions) and optical resonant cavities [1][2][3][4] . In this scheme, the ions trapped in the ion trap can be used as carriers for stationary qubits, and the fluorescent photons emitted by the ions can be used as carriers for flying qubits. The optical resonant cavity can be used as a quantum interface for the interaction of qubits between the ions and the photons.
An ion trap was first invented in the 1960 s by Paul and Dehmelt [5,6] , who were awarded the Nobel Prize in 1989. This Paul trap uses an alternating electric field to form a saddle potential that alternates at RF frequency, and ions in this alternating potential are eventually trapped in the middle of the saddle. In recent years, scientists have invented various iontrap structures [7] , such as quadrupole traps [8] , blade traps [9,10] , and two-dimensional surface traps [11,12] . Given the current state of development, ion-trap systems are among the best quantum processors. Ion traps can stably trap cold ions with long coherence times for long periods. These ions also provide rich interactions through Coulomb forces, thus enabling various complex quantum logic gates [3,9,13,14] . Ions trapped in an ion trap are excellent physical materials for constructing quantum networks [1] , and one of the keys to employing this structure as a quantum node is to achieve an efficient quantum interface between ions and photons. One of the most straightforward methods is to collect fluorescent photons through a high numerical aperture objective; however, the collection efficiency of this method is limited to less than and is difficult to be further improved. Another possible method is to collect fluorescent photons using a parabolic ion trap, which can theoretically reach a collection efficiency close to ; however, the ions in such ion traps can only receive external modulated light in the direction of the parabolic plane, which make it difficult to be modulated [15,16] . The most common method is to enhance the collection efficiency by the Purcell effect with a Purcell factor greater than . This scheme effectively collects fluorescent photons into the resonant cavity mode, achieving a photon collection efficiency greater than . An optical microcavity with a small mode volume ensures that a single photon generates a strong electric field in the cavity, thus achieving a strong coupling of photons to atoms (ions) [17] . A fiber Fabry-Pérot cavity (FFPC) developed internationally in the last decade [18] , is an ideal optical microcavity in the field of cold atoms (ions). This microcavity has a small mode volume and provides a direct fiberoptic interface, which allows a strong and efficient coupling between single photons and trapped cold atoms.
The long trapping time of the ion trap system provides its critical advantage in developing quantum network research, especially in the case of requiring long storage time of quantum information, such as quantum repeaters. A high-efficiency interface between trapped ion quantum nodes has been realized in a classical optical cavity with mode volume in the range of mm 3 [19,20] , the entangle rate is limited by the coupling factor provided by the millimeter size of the cavity. To further develop the system, minimizing the cavity mode volume is essential, whereas integrating a FFPC into a trapped ion system is a quite realistic approach. The FFPC has been verified as a powerful tool in neutral cold atom area, which has been successfully applied to basic quantum cavity electronic dynamic research [21][22][23] , efficiently preparing entangled multiphoton graph states from a single cold atom [23] . However, integrating a FFPC with an ion trap is still challenging, e. g. Tracy's group tried to place a FFPC in a linear trap [24] , but since the fiber end-face is a dielectric material and UV light is required to manipulate the ion energy level, the photoelectric effect when UV light acting on the dielectric material causes the fiber to be charged, which changed the confining potential of the ion trap and eventually leads to the inability of ions to move to the middle of the FFPC; Ballance et al. tried to fabricate an ultraviolet FFPC and an ion trap for strong ionphoton interaction [25] , the finesse of the FFPC in the ultraviolet band slowly deteriorates with time in vacuum, and eventually no effective coupling between the cavity and the ions is achieved; Moonjoo Lee et al. designed a micromechanicalbased fiber cavity system [26] , Takahashi proposed a new singleion fiber-optic cavity scheme [27] , and Teller integrated a fiber cavity into a wheel trap for strong ion-cavity coupling [28] . Although using microcavities to achieve efficient interfaces is one of the feasible ways, it is still very challenging to achieve highly reliable and efficient ion-cavity coupled systems. Experiments in this area have been led by foreign research groups in recent years, and experimental research in this area is also emerging in China.

System design
We design the whole experimental setup according to the schematic diagram shown in Fig. 1. In this setup, the fiber cavity ion trap (FCIT) is placed in the middle of the vacuum chamber. The core parts of the FCIT are the fiber electrodes. These fiber electrodes consist of bare optical fibers with metalcoated surfaces, which are used to transport both charges and photons. Our emitter in this diagram is a single ion in the center of the fiber electrodes. The fiber cavity in this trap enhances photon emission on the electronic dipole transition. Because the wavelength is in the visible light band, it is well suited for transition and modulation.
138 Ba + To stably trap a ion and efficiently collect the emitted fluorescent single photons, we constructed a FCIT system. This system is shown in Fig. 2.
The core part of the system is mainly composed of three symmetrical optical fibers with metal-coated surfaces. The optical fiber in the middle is metal-coated optical fiber that can pass light. And the end face of this fiber is properly coated with gold, which can shield the surface charges on the dielectric high-reflection film. Therefore, the fibers in the middle can be used to transmit both light and electric charges. To isolate the external vibration, we use springs to connect the two parts of the stainless steel base, the vacuum chamber is mainly connected with the lower part, and the whole system is assembled on the upper part.
In the following sections, we describe the design requirements of each part in Fig. 2 in detail.

Fiber electrode
In addition to being used as the electrode of the ion trap, the fiber cavity electrode is also used to form the fiber cavity. The basic steps for fabricating fiber electrodes are as follows: CO 2 (1) Use a laser to create a pit with the desired radius of curvature on the cut fiber end face.
(2) Use negative photoresist, through steps such as exposure and development, to create a protective area of appropriate size on the pit.
(3) Fabricate a metal layer with a thickness of about a dozen nanometers on the entire fiber by evaporation, sputter- ing, etc.
(4) Thick the entire metal layer to a few tens of microns by electroplating.
(5) Use organics such as acetone to remove the photoresist. The entire flow chart is presented in Fig. 3. The last structure in Fig. 3 is the required fabricated fiber cavity electrode. The central area of the fiber cavity electrode can be used to pass light, and the outer metal layer can be used to pass electricity.
When the cavity surface of the fiber cavity is very close to the location of the ions, the dielectric high-reflection film on the cavity surface is one of the noise sources affecting the ion confinement. The possible effects of the dielectric high-reflection film on ions contain the following aspects: 1. The induced electric field that varies with ion motion affects ion cooling [29,30] ; 2. The charging effect of the laser on the dielectric high-reflection film changes the electric field [24] ; 3. The cavity working for a long time is contaminated with ions, which makes extra surface charges exist on the surface of the cavity. To eliminate or weaken these effects, as shown in Fig. 3, the fiber surface is wrapped inside a metal layer, and the metal layer is connected to an external power source. By this design, the potential on the dielectric high-reflection film is consistent with the potential of the fiber electrode, i.e., shielding of the charges on the dielectric high-reflection film is achieved. In our evaluation, since the area of the non-conductive region (the region with high resistance) is one order of magnitude smaller, the conductive resistance is one order of magnitude smaller, and the charge is more easily conducted away. In addition, the area receiving light is also one order of magnitude smaller, which makes the area for charging even smaller. Therefore, the charging voltage in our evaluation is reduced by at least two orders of magnitude.

Ion trap
In our design, the ion trap is mainly composed of six opposing fiber electrodes, as shown in Fig. 1. The two electrodes in the middle propagate the RF electric field, and the four side electrodes propagate the DC electric field. To achieve high light-passing and high Purcell enhancement, the spacing between the RF electrodes is approximately , and the DC electrodes are placed at an angle of to the center, as shown in Fig. 4. The DC electrodes in the figure are grounded by default. In practice, a small voltage can be applied to the DC electrodes to adjust the position of the ions.
All electrodes are placed on borosilicate glass. To install these six fiber electrodes on the same plane and make the opposite fibers coaxial, the fibers are glued in the three scribed grooves of the smoothed high borosilicate glass.

Support structure
In addition to the core part, the FCIT has a stable support structure to support the core part and a circuit system for powering the fiber electrodes. In Fig. 2, these support structures consist of three parts: base, circuit support plate, and fiber electrode support plate.
The base is made of stainless steel and is fixed to the vacuum chamber with screws and grippers. An external wire pressing device on the base can hold all input wires. These wires are connected to the circuit support plate by conductive elastic wires.
The circuit support plate is constructed of stainless steel and has an aluminum nitride ceramic circuit board screwed onto its surface. The ceramic circuit board implements the circuit system required by the core part. The ceramic circuit board is printed with the power supply circuit, and other conductive separate parts are glued to the ceramic circuit board by conductive silver glue.
The fiber electrode support plate is formed by stacking three parts: high borosilicate glass, shear piezoelectric ceram- ics, and aluminum nitride ceramic base. The shear piezoelectric ceramics is used for locking the fiber cavity with a travel of about a few microns, which will not affect the whole RF and DC electric fields. After these parts are glued together with AB glue, a grinding process is used to make the two borosilicate glass surfaces overlap and flat, and a scribing process is used to create fiber electrode mounting grooves on the surface.

Simulation and performance
3.1 the optimal fiber cavity To achieve Purcell enhancement, the structure of the fiber cavity needs to be optimized to match the ion energy level. Referring to Fig. 1 and Fig. 5, the parameters to be considered for the optimal fiber cavity are the fiber type parameters ( , , , ), the cavity structure parameters ( , , , , ) and the coating parameters ( , , , , , , which satisfy ). These parameters are adjusted to maximize the probability of cavity photons emitted by the ions output to the fibers.
Although the smaller the cavity length , the easier it is to achieve strong coupling, the cavity length is often limited by the size of the fiber and the overall structure. In this design, an optical fiber with a cladding diameter of is used. Fibers with larger sizes are difficult to use for fabricating fiber electrodes and also limit external laser transmission. In this design, the cavity length is fixed to . Due to the large length of this cavity, a single-sided cavity structure is considered, i. e., photons are emitted from only one side of the fiber cavity. By default, the left fiber is the output side. Since the design is a single-sided cavity structure, there is no need to consider the type of fiber on the right side. For convenience, the fiber types on both sides are set to be the same, i.e., and . Using the latest coating technology in China, we set the reflectivity at for the right fiber and set the coating loss for the left fiber. As the reflectivity of the required dielectric film increases, the number of layers of the dielectric in the film will also increase, which will increase the absorption loss. Therefore it is difficult to raise the re-  Design of a fiber-cavity ion trap for a high-efficiency and high-rate quantum network node Bao et al. flectivity of the dielectric film to an extremely high level. However, for the end face with low reflectivity, the coating loss comes mainly from scattering from the coating surface.
The reflectivity of the left fiber is a very important parameter.
affects the finesse of the fiber cavity and the percentage of transmission in total channels. Using the definition in Appendix 5, the percentage of transmission in total channels is given by (1) The above equation decreases slowly in the interval and decreases sharply when . when , but when . A too small leads to a low cavity finesse and thus reduces the coupling of ions to the cavity photons, but a too large affects the photon output. Therefore, the reflectivity of the left fiber can be estimated to be in the interval .
Considering the flat-concave cavity structure, when the clipping loss of the concave surface is small enough (the cavity diameter satisfies or the clipping loss is less than ), the cavity diameter of the concave surface for a flat-concave cavity structure needs to satisfy μm for the S460-HP single-mode fiber, while for the LMA-10 photonic crystal fiber. Preference is given to the LMA-10 photonic crystal fiber for constructing the fiber cavity because the diameter cavity surface can be easily fabricated in experiments. The large mode field and single-mode characteristics of LMA-10 photonic crystal fiber facilitate the fabrication of our system.
In addition to the parameters analyzed above, the only parameters that have large tunability are and , which change the mode of the cavity significantly. When the cavity structure is close to a flat-concave cavity ( ), the optimal mode matching efficiency can be approximated to .
However, in this case, the ion is not on the waist of the cavity mode, and the single-atom cooperativity parameter between the ion and the cavity photon is small, which makes the efficiency of photon emission into the cavity small. On the other hand, when the cavity structure is close to a symmetric cavity ( ), the ion is on the beam waist of the cavity, and is close to , but the mode matching efficiency between the fiber and cavity will be significantly low. The specific effects of both are shown in Fig. 6. The overall efficiency that can be achieved by flat-concave cavities shown in Fig. 6 is generally larger than that achieved by symmetric cavities. However, the ROC of the concave surface of the flat concave cavity is large. Since the fiber cavity fabrication technology can easily fabricate fiber cavity surfaces with ROC in the range of , we prefer to fabricate a concave cavity when the overall efficiency improvement is not significant.
Considering the above constraints, we optimize the parameters of the fiber cavity by numerical optimization methods 5. Schematic diagram of the interaction between the fiber cavity and the ion. In the presence of an external electric field , the single ion interacts with the cavity. In cavity quantum electrodynamics, this interaction can be described using . There is only a fundamental mode in the cavity and the distribution is affected by the structure of the cavity. In the figure, is the radius of the mode field at each location, is the radius of the waist, is the radius of curvature (ROC), is the refractive index, and is the cavity length. At the fiber end face, only the mode field within diameter can be reflected by the fiber.  Table 1. These parameters are achievable using current technology in our laboratory. When this cavity is coupled to a single ion, the optimal cavity can be described . For this optimal fiber cavity, and . Therefore, the quantum efficiency is , the matching efficiency is , the transmission efficiency is and the overall efficiency is .

10000
In our design, the overall efficiency is mainly limited by the coating technology. If the finesse can be improved while reducing coating loss, the optimized overall efficiency will be higher under the same conditions. Compared with other research groups in which the finesse of the resonant cavity is larger than [19,26,27] , our design has a lot of room for improvement. Compared to a collection device consisting of an objective lens with a high numerical aperture, a collection device consisting of a resonant cavity can ensure the collection of photons in opposite directions with significantly higher collection efficiency. In addition, the small-mode fiber cavity has a more significant improvement in overall collection efficiency compared to the resonant cavity implemented with optical lenses, although the fiber cavity introduces an addi-tional mode matching efficiency.

Performance of the ion trap
The total potential energy in the ion trap is the sum of the RF pseudopotential and the DC potential, which satisfies m Ω where represent the mass of a single ion, is the RF frequency, is the electric potential term independent of the RF frequency, and is the electric potential of the DC field. As shown in Fig. 4 Fig. 7. The ions will be trapped in a region with a radius of approximately in the center of the fiber electrodes. Although the depth of the ion trap is shallow, it is sufficient to trap a single cold ion. To achieve deeper potential well, the drive voltage can be increased to V without affecting ion trapping. These electrical parameters are consistent with those commonly used in various ion traps, allowing the use of off-the-shelf auxiliary devices such as voltage sources, amplifiers, and helical resonators to be assembled in the FCIT, reducing the additional expense required to purchase and deploy auxiliary devices.
A system composed of a single ion and a resonant cavity is an ideal and easy-to-implement solution for realizing a quantum node. When the number of ions in the ion trap is larger than one, the design of the whole system needs to include the correspondence between photons and individual ions in the resonant cavity, which undoubtedly increases the diffi-  Design of a fiber-cavity ion trap for a high-efficiency and high-rate quantum network node Bao et al. culty of system design and fabrication. Quantum nodes formed by multiple-ion ion traps can be realized by expanding quantum nodes formed by single-ion ion traps. Therefore, the FCIT is mainly used to trap a single ion. However, the potential energy shown in Fig. 7 does not have a flat region, which makes the FCIT capture only a single ion with high probability.

Stability of baking
Since the complete system needs to operate in ultra-high vacuum, the system needs to be baked and evacuated at high temperatures. The higher the baking temperature, the higher the vacuum degree that can be achieved. In this system, the maximum baking temperature is determined by the maximum operating temperature of the shear piezoelectric ceramic. We use the shear piezoelectric ceramic CSAP03 from Noliac, which can reach a maximum operating temperature of 200 ℃.
It is worth noting that, during the heating and cooling process, the fiber cavity may shift due to the different expansion coefficients of different materials, which makes the overall coupling efficiency lower. To solve this problem, a very thin AB glue is needed, and the part connected to the fiber cavity is required to have a similar thermal expansion coefficient. In our design, the fiber electrodes are glued in the thin grooves of the fiber mounting base shown in Fig. 2. Therefore, the fiber electrode can be considered as a tight fit to the groove, and the offset of the fiber cavity can be regarded as the offset of the relative positions of the two glass grooves. The designed fiber mounting base is composed of high borosilicate glass with , shear piezoelectric ceramics with , and aluminum nitride with , where is the linear thermal expansion coefficient of each material. And the fiber mounting base has a width of mm and a height of 6.5 mm as shown in Fig. 2. Therefore, by simulating the heating and cooling processes and assuming that the contact surfaces of each material simply slide relative to each other, eventually, in the range of this fiber mounting base, the possible radial offset after heating can be up to 4 μm. This offset size is consistent with the radius of the cavity mode field on the fiber cavity surface and therefore can very significantly reduce the coupling strength between the fiber cavity and the ion. However, the symmetrically mounted high borosilicate glasses are about mm relative to each other, so that the offset of the fiber cavity is only about 400 nm in this range.
In practice, during the baking process, the materials are held together by the internal stress and hardly slide relative to each other. In order to minimize relative slip, we need to use homogeneous materials and thin uniform strong layer of AB glue. Current manufacturing techniques allow for very small differences in components composed of the same material. Therefore, the main inhomogeneity of the fiber mounting base lies in the difference in gluing thickness between the components. This can be overcome by creating a mounting positioning device that is complementary to the fiber mounting base. By positioning the components in the mounting positioning device and applying pressure, the AB glue can be evenly distributed between the components. In experiments, the bonding strength of the AB glue is sufficient to provide strong internal stresses and eventually a small offset of the fiber cavity is achieved as long as the operation is correct.

Influence of vibration
Vibration is an important factor affecting the coupling of this system. To isolate the external vibration, we use springs to connect the core part of the FCIT to the external vacuum chamber. This elastic connection is achieved by a two-part stainless steel base, as shown in Fig. 8. The designed FCIT is symmetric, which means that the translational resonant mode does not affect the coupling of this system. As shown in Fig.  8, the stainless steel base uses a total of ten springs to achieve the connection, and the springs in the same direction have the same properties. In the figure, the side springs can be mounted using spring mounting holes and secured using screws.
To evaluate the vibration of this system, we mainly use COMSOL to analyze the modes of this system at resonance. In our evaluation, the core parts mounted on the upper half of the stainless steel base are seen as a whole, i.e. the relative displacement of the contact surfaces of each part is the same. The core parts are in contact with the vacuum chamber only at the individual spring connections. The stiffness coefficient of all springs is set to be , which can be easily implemented in experiments. Based on this model, we analyzed the resonant frequencies and modes of the system using COMSOL.
The basic resonant modes of this upper part in Fig. 8 are shown in Fig. 9, and the resonant frequencies of this part are between 22 Hz and 39 Hz. Since it is very difficult to install damping devices in vacuum, the damping of the system in vacuum is mainly provided by the springs themselves. The springs made of stainless steel in the system have a damping factor of about 0.003. The magnitude of the transfer function of the spring system in any single direction is given by where is the ratio of the excitation frequency to the undamped intrinsic frequency in a single direction, and is the damping factor. Therefore, this spring system is able to suppress high frequency vibrations above 100 Hz. The typical resonant frequency of the CSAP03 piezoelectric ceramics used in the system for locking the fiber cavity can reach several kilohertz and higher. Therefore, the resonant frequencies of the spring system are in the operating frequency range of the piezoelectric ceramics. We use piezoelectric ceramics to build a servo to actively suppress vibrations at frequencies below 100 Hz.

Discussion
This designed coupled system can be used as a quantum node in a quantum network. This design overcomes the difficult problem of assembling the fiber cavity with the ion trap and solves the problem of shielding the charges in the dielectric high-reflection film on the end face of the fiber cavity [24][25][26][27]31] . After our initial validation, our scheme can be easily implemented in the laboratory. The remarkable feature of this designedsystem is the coupling between the ion trap and the  Design of a fiber-cavity ion trap for a high-efficiency and high-rate quantum network node Bao et al. fiber cavity using electrodes made of optical fibers. This reduces the complexity of the system and makes it easy to manufacture in volume. Multiple such systems can be connected using optical fibers, and such extended system can be used in complex quantum networks. 138 Ba + 6 2 P 1/2 ↔ 6 2 S 1/2 6 2 P 1/2 ↔ 5 2 D 3/2 138 Ba + In this paper, we adopt a single as the trapped ion. The transitions and in can be used to achieve deterministic photon emission. The corresponding transition wavelengths are 493.5 nm and 650 nm, both in the visible wavelength band, which can pass through single-mode fibers with low loss. However, the spontaneous radiation rate of is relatively large, which corresponds to a relatively short lifetime. Therefore, a large coupling factor is required to achieve deterministic photon emission, which is difficult to achieve in a resonant cavity composed of optical lenses. If an ion with a smaller spontaneous radiation rate , such as , is used, the overall coupling efficiency corresponding to the same designed coupled system will increase accordingly.

Conclusions
53% 138 Ba + We design and investigate an ion-cavity coupled system that can be implemented using current technology. The ion-cavity is implemented by fabricating a metal layer on the surface and end face of the fiber. The fabricated fiber electrodes are mounted on a fiber mounting base composed of materials with similar coefficients of linear thermal expansion and are connected to the vacuum chamber using springs. The resonant frequencies are approximately a few tens of hertz. In addition, the cavity length of the fiber cavity is set to approximately 250 μm, and the cavity structure is optimized to form a strong trapping potential for ion-cavity coupling. This overall coupling efficiency after optimization is about for a single ion, mainly limited by the recent coating technology. This design constitutes a promising approach to constructing scalable quantum networks.

Coupling between ion and fiber cavity
(g, κ, γ) g κ 2κ γ γ A common ion-cavity system can be described by the parameters , where is the coherent interaction rate between the particle and cavity, is the HWHM (half-width half-maximum) of the cavity transmission line (intracavity field decay at a rate ), and is the FWHM (full-width halfmaximum) of the spontaneous line (spontaneous decay at a rate ).
Note that where is the FWHM frequency of the cavity transmission line, is the single-pass effective cavity loss, is the cavity length, is the refractive index of the cavity, and and are the reflectivity of the left and right cavity mirrors, respectively. In this equation, represents an approximate round-trip loss, and this approximation only holds when the cavity loss is very small and the reflectivity of the mirror or is very large.
It is worth noting that the intensity of a single reflection is affected by the clipping loss because the size of the cavity surface of the fiber cavity is finite. The clipping loss is defined as w m where is the radius of the cavity mode at the position of D the fiber surface, and is the diameter of the effective reflective surface.
g The coupling coefficient is evaluated by µ d E where is the electric dipole moment of the particle and is the electric field strength of a single photon at the position of the particle. In general, the single-photon electric field intensity is is the modal volume, is the waist radius, is the position of the ion and is the normalized cavity mode distribution where . Using the Gaussian mode to approximate the cavity mode, the size of the beam waist satisfies , where is the Rayleigh length and is the wavelength.
Define the single-atom cooperativity parameter . Then, the quantum efficiency of single-photon sources in the bad cavity regime ( ) is given by g ≫ κ, γ η q In the strong regime ( ), the quantum efficiency of single-photon sources is given by g κ By changing the structure of the cavity, we can change the values of and and thus achieve Purcell enhancement. κ in the above equation contains many additional transmission channels and loss channels, while the available single photons are only transmitted in a specific transmission channel. Therefore, it is necessary to consider the percentage of transmission in the total channels, that is, T o where is the transmissivity of the output channel.
η q η f c In addition to , the matching between cavity modes and fiber modes also limits the overall coupling efficiency. This matching efficiency is given by where is the refractive index of the output fiber, is the waist radius of the corresponding fiber mode, is the radius of the cavity mode at the position of the mirror of the output fiber and is the radius of curvature of this mirror.

L
(R, T , L) In this system, the tunable parameters are the cavity length , coating parameters , structure of the cavity, and type of fiber.