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Abstract
We present a compact optical design for a scalable trapped ion quantum processor employing a single high numerical aperture lens for the excitation of ions and collection of photons, both of which are essential for remote entanglement generation. We verified the design by performing a quantum interference experiment between two photons generated by two sets of the proposed design and observed a 82(3) % suppression of coincidence within 8.13 ns time window when the two photons became indistinguishable. This design can be extended for the simultaneous generation of multiple pairs of entangled qubits with existing fiber-array devices.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A trapped-ion system is considered as one of the most promising platforms for quantum information technology. Ion qubits, which are units of quantum information, are inherently identical, and the coherence of stored quantum information in a trapped-ion system is ensured over long time scales relative to other approaches [1]. High-fidelity quantum logic gates have been implemented for both single-qubit and two-qubit gates [2–4]. However, the number of qubits that can be entangled in a single ion chain cannot be increased indefinitely [5].
To overcome this limitation, qubit ions can be interfered using photonic qubits with optical wavelengths. The photons entangled with ions can be transported over long distances via optical fibers, and remote ion-ion entanglement can be generated by the projection measurement of two photons in the Bell basis. Previous proof-of-principle experiments have demonstrated the generation of remote quantum entanglement between distant ion qubits [6–10]. The remote entanglement can also be a key ingredient of a quantum network or quantum key distributions [11,12].
With the photonic interface and the remote entanglement generation technique, it is possible to scale up ion trap quantum processors [13]. Each processor has several data qubits, which are connected by motional modes. Network qubits can be remotely entangled with other qubits in distant processors. This system design allows the quantum teleportation of data qubits or even quantum gate teleportation between distant processors [14,15]. To continuously create entangled states between network qubits, quantum states of network qubits must be manipulated without disturbing the quantum information stored in data qubits, which requires tightly focused optical beams formed by a high numerical aperture (NA) objective lens [16,17].
Here, we present a compact optical design for the generation of entangled states between two distant ions trapped in separate chambers, which is compatible with both shuttling and individual addressing that is necessary to build a scalable quantum processor based on the ion trap. The same high-NA objective lens is used for both excitation of a network ion and the collection of photons emitted from the same ion, hence simplifying the optical path entering the vacuum chamber. Single ions were excited with a tightly focused beam, and the quantum interference between two indistinguishable single photons emitted from two independent ions trapped in two separate vacuum chambers were observed. This system design can easily be combined with currently existing fiber-array technologies and generate multiple pairs of heralded entangled ions, as shown in Fig. 1.
Fig. 1. Schematic diagram of proposed compact optical design for generation of multiple entangled ion pairs. In our experimental setup, a single high NA lens is used for both excitation of an ion and collection of a single photon emitted from the corresponding ion. The combination of polarizing beam splitter (PBS) and quarter wave plate (QWP) directs the incoming excitation pulse and outgoing single photons respectively. The collected photons are coupled into a single-mode fiber for mode cleaning and polarization filtering [18]. This setup for a single trapped ion (network qubit) can be easily extended to a simultaneous entanglement generation scheme by trapping multiple network qubits (and data qubits in a separate region for a scalable quantum processor), and replacing single fibers with fiber arrays as shown in the insets.
2. Setup
Compared to other approaches [6,9,10], our experimental setup uses a single optical path for both the selective excitation of an ion and the collection of photons emitted from the same ion. A high-NA objective lens (NA = 0.6, Photon Gear, 15470-S) is placed in the optical path, which allows maximization of the collection efficiency of ion fluorescence up to 10 %, and enables tight focus of an excitation laser beam.
Figure 1 illustrates the optical design of our experimental setup. We loaded single $^{174}$Yb$^{+}$ ions into micro-fabricated surface traps in two separate vacuum chambers. The excitation pulse launched from an optical fiber tip passes through the high-NA lens and is focused onto a target ion. The ion excited by the pulse spontaneously emits a photon and the objective lens couples this photon into another single-mode fiber. Note that the excitation pulse and the emitted photon can be directed along two separate beam paths with the combination of a polarizing beam splitter and a quarter wave plate, similar to the typical reflection measurement setup for an optical cavity [19]. In Fig. 1, if we define the direction of the magnetic field B as positive z-axis, the propagation vector of the incoming laser is along the negative z-axis. When the atom absorbs or emits photons, the selection rule of electric dipole transition only depends on the handedness of the polarization rotation with respect to B-field, and it is independent of the propagation direction of the light. Therefore, when the excitation laser interacts with the atom, the handedness of circular polarization should be considered with respect to the positive z-axis (B-field direction) and, even though the incoming light is right-circularly-polarized (RCP) with respect to the negative z-axis (laser propagation direction), $\sigma ^{-}$-transition can be driven as shown in Fig. 3(b). During the emission process, $\sigma ^{-}$-transition is accompanied by a single photon whose polarization rotates in the same direction as the incoming laser, but because the propagation direction of the emitted photon is reversed with respect to the excitation pulse, the polarization should be interpreted as left-circularly-polarized (LCP). To preserve the polarizations in both paths, polarization-maintaining (PM) single-mode fibers are used.
To achieve the maximum fiber coupling efficiency, we used an objective lens which was specially designed to match the UV fiber mode with NA of 0.07 [20,21]. Figure 2 shows the procedure adopted to optimize fiber coupling. First, to determine the extent of image aberration, multiple ion images were acquired by changing the electron-multiplying charge-coupled device (EMCCD) position through the focus. Then, to reduce the aberration, the tilt angle and three-dimensional position of both the objective lens and fiber were carefully aligned.
Fig. 2. Ion fluorescence-fiber coupling procedure. (a) Under initial alignment, multiple images of a trapped ion are acquired as the camera moves through the image plane of the lens. The image aberration is estimated by comparison with the Zemax simulation and fixed by adjusting the objective lens. (b) Images of the ion acquired by the high-resolution CCD camera after reducing the aberrations by fine alignment. (c) Highest-resolution image of an ion obtained by 2D scanning of the fiber tip. The data in this figure are interpolated for better visualization. The upper (right) plot represents the cross-section of the contour plot along the x-axis (y-axis). Red dots represent the acquired data via fiber tip scanning, and the blue line plots depict the Gaussian.
Subsequently, we switched the camera to a charge-coupled device (CCD), which provides higher resolution than the EMCCD. Figure 2(b) shows the CCD images upon aberration minimization with careful alignment. The residual astigmatism may be caused by uneven thickness of the chamber viewport. We added a cylindrical lens (f = 500 mm) near the image plane to compensate for the astigmatism [21]. To control the fiber tip position, we used a motorized three-dimensional stage. Long-term drift was compensated by scanning the fiber tip in a transverse plane. Therefore, we could maintain maximum coupling during a long experiment sequence.
Figure 2(c) shows the image of the ion after the optimization was completed. Because the fluorescence image of the ion is too small to be captured with the CCD camera, we obtained the high-resolution image by scanning the fiber tip and measuring the amount of ion fluorescence coupled into the fiber. The data agree well with the 2D Gaussian fit, and the full widths at $1/e^{2}$ obtained by the curve fitting are 4.72(6) and 6.72(2) µm in x- and y-axis, respectively, as shown in Fig. 2(c). These results are comparable with the mode field diameter (3.5(5) µm) of the fiber mode. The emission pattern of the photons is in principle not necessarily a Gaussian distribution; however, because we can only count the photons whose electric field component is projected into a near-Gaussian mode of the single mode fiber, we assume that the final image in Fig. 2(c) follows a Gaussian profile.
To maximize the single photon collection efficiency, we tested three different types of fibers (Table 1). All fibers are step-index fused silica single mode fibers, and we assume that a small deviation of the mode field diameter among different fibers caused the differences in the fiber coupling efficiency. Based on our measurement, we chose Corning PM 400 fibers to collect fluorescence with the optimum fiber coupling efficiency ($\eta _{\textrm{FC}}$) of 18.6 %. Although the cutoff wavelength of the fiber is comparable with that of atomic transition, the fiber coupling into high-order modes was negligible.
Table 1. Various fibers and collection efficiency
The level structure of $^{174}$Yb$^{+}$ is shown in Fig. 3(a) and we employ $^{2}$S$_{1/2}$ and $^{2}$P$_{1/2}$ electronic levels of the ion as ground and excited states, respectively. To deterministically excite the ion into $^{2}$P$_{1/2}$ $|{m=-1/2}\rangle$, an ultra-short pulse is required, as its excited state has a very short lifetime ($\tau$ = 8.12 ns) [22]. We used the Ti-Sapphire mode-locked pico-second laser (MIRA-900, Coherent), followed by a second harmonic generation stage, to generate a pico-second pulse of 369.5 nm light. Based on the auto-correlation measurement, the pulse width was 2.44 ps, which is significantly shorter than the excited state lifetime. A pulse picker on the beam path transmits a single pulse triggered by the digital pulse generated from a sequence controller.
Fig. 3. Simplified diagrams of energy levels and relevant atomic transitions of $^{174}$Yb$^{+}$ ion. (a) The ion is Doppler-cooled with 369.5 nm dipole transition, and a 935 nm repumping laser is used to close the cycling transition for Doppler cooling. (b) A pico-second laser beam is circularly polarized to selectively excite the Zeeman state. (c) The branching ratio of two different decay channels ($\sigma ^{-}$ and $\pi$) is 2:1, while only the circularly polarized photon from $\sigma ^{-}$ transition can be coupled to the single mode fiber due to its spatial distribution [18].
The pulsed laser beam is delivered to an ion through the optical fiber. The excitation pulse becomes circularly polarized by the quarter wave plate after the polarizing beam splitter (PBS) to exclusively excite the $|{m=+1/2}\rangle$ Zeeman sublevel of $^{2}$S$_{1/2}$ ground state to the $|{m=-1/2}\rangle$ state of the $^{2}$P$_{1/2}$ excited state, while other sublevels remain unaffected. Considering demagnification of the objective lens and the mode-field diameter of the optical fiber, the mode waist of the pulse at the ion position is expected to be 190(30) nm [23]. This waist is significantly smaller compared to typical ion chain spacing of 5 µm, such that the cross-talk to the neighboring ions is negligible.
3. Theory
Quantum interference is a fundamental property of quantum mechanics. The indistinguishability principle of quantum mechanics enables quantum interference between two identical particles, which cannot be explained in classical theory.
Hong-Ou-Mandel interference is a well-known quantum interference phenomenon [24]. When two distinguishable photons simultaneously enter a 50:50 beam splitter, there are four possible outcomes with equal probabilities of occurrence at of 25 % as shown in Fig. 4(b)-(e), Therefore, coincidences are expected to be observed between two single photon detectors at output ports 3 and 4 with a probability of 50 %. However, when two photons are identical in terms of all degrees of freedom, such as polarization, frequency, arrival time, and spatial modes, they are in principle indistinguishable. Therefore, the probability amplitudes of simultaneous triggering of both detectors is canceled out, as indicated by the following equation:
Quantum interference of two photons is also often utilized to distill a remote ion-ion entanglement out of a couple of entangled ion-photon pairs by entanglement swapping. To achieve proper quantum interference, temporal and spatial modes of two incoming photons should be matched, which is also a prerequisite of Hong-Ou-Mandel interference. Therefore, it is a good tool to optimize an optical interferometer independent of the quantum states of the ions before a full generation of the entanglement will be attempted [25].
Fig. 4. Simplified schematics of our Hong-Ou-Mandel experiment. (a)-(d) Four possible outcomes for two simultaneously injected photons. When two photons are distinguishable, i.e., different with respect to polarization and/or frequency, each case has an equal probability of occurrence. However, when two photons are identical and hence indistinguishable, only two bunching cases (a) and (d) can occur. (e) A single trapped ion is excited by a pico-second laser pulse and spontaneously emits a photon. The photon collected by objective lens couples to a single mode fiber. Two photons from distant emitters are mixed by a 50:50 non-polarizing beam splitter, and its outputs are measured by photon counting modules.
4. Experiment
We verify our design by demonstrating single photon generation with a trapped ion. We apply a single excitation pulse to a trapped ion and wait for 75 ns until the excited state of the ion decays to the ground state, accompanied by the emission of a single photon. This photon is coupled into a single-mode fiber and eventually detected by the photo-multiplier tube (PMT).
To determine optimal conditions to generate and collect a single photon emitted by the trapped ion, we varied the laser pulse power or the center frequency of the pulse laser and measured the corresponding success rates. Figure 5(a) shows the Rabi oscillation as a function of the laser pulse power. Only 7.2 fJ of the pulse energy was required to drive a full Rabi flop of the ion owing to the tightly focused excitation beam. From this observation, the beam waist is estimated to be 180 nm, which agrees well with our expected value of 190(30) nm, as discussed in section 2.
Fig. 5. Rabi flopping by a single pico-second pulse. (a) Rabi oscillation with respect to electric field strength. (b) Single photon generation rate as a function of laser frequency detuning.
We observe the loss of visibility of the Rabi oscillation in Fig. 5(a), which can be partially explained by considering a very small beam waist of the excitation pulse. Because the spot size of the excitation pulse is extremely small, even a miniscule deviation of the ion position from the potential minimum can lead to a significant change in the pulse intensity. We assumed that the motional state of the ion will be in a thermal state and the expected success rate is calculated based on the position distribution of the ion inside the excitation pulse using Gaussian distribution according to the Ref. [26]. In our calculation, we assumed that the Rabi angle of each experiment is determined by the laser intensity at the ion position, and we could confirm that this simple model can partially explain the damping of the Rabi oscillation amplitude as the pulse power is increased (solid line in Fig. 5).
Because of the geometry of emission distribution, the fiber blocks photons from the $\pi$-transition, such that only $\sigma ^{-}$-photons are collected during the experiment [18]. The estimated success rate of single photon generation is given by the equation below:
The time-resolved trace of single photons is shown in Fig. 6(a). We built two separate ion traps (setup A and B) and recorded the traces from each setup. The probability of detecting the exponential photon decays and their measured decay time of 8.0(3) ns agree with the atomic natural lifetime of 8.12 ns within the measurement resolution. Despite the presence of the polarization filter with PBS and QWP, a non-negligible amount of back-scattering from ion trap structures was recorded with the PMTs. However, those back-scattering events are caused by instant reflection, such that they can be distinguished from ion fluorescence, which follows an exponential decay curve characterized by the lifetime of the excited state. By discarding immediate events within a short period of time right after the excitation, we could exclude false-triggered events generated by the back-scattering of the pulse while sacrificing the success rate of the single photon generation. We also recorded a trace in the absence of an ion and calculated the false count ratio $FCR = N_0/N_1$ where $N_1$ ($N_0$) is sum of all counts detected after the cutoff time with (without) an ion. As shown in Fig. 6(b), if we discard events up to 2.5 ns after the pulse arrival time, we can ensure that 95 % of count events arise from the single photon emission from the ion. The resulting single photon generation success rates of setup A and B are 0.20 % and 0.14 %, respectively, which correspond to the overall single photon generation success probability of 0.27 % and 0.21 % estimated by extrapolation of the exponential decay fitting curves (dashed line in Fig. 6(b)). The success rate we achieved is comparable to that of previous study [9], although one could achieve higher efficiency by changing ion species with visible wavelength transition that can provide higher fiber coupling efficiency and high detector quantum efficiency [10].
Fig. 6. (a) Time-resolved scattering of a pico-second laser pulse from a $^{174}$Yb$^{+}$ single trapped ion. In the absence of an ion (diamonds), strong back reflection is observed near the temporal origin. Solid lines depict the exponential decay fit to the ion fluorescence data, with a fitted decay time of 8.0(3) ns. (b) Effective success probability and true count ratio as a function of cut-off time. With the cut-off time of 2.5 ns (dashed vertical line), single photon generation success probability is approximately 0.2 % while the false count ratio is 5 %.
After the characterization of each setup, we integrated the two setups (A and B) to test the Hong-Ou-Mandel-type interference. Figure 7 shows the measurement result. The single photons generated from two distinct ions are combined by a non-polarizing beam splitter and the interference between them was observed by measuring the two output modes with the PMTs as shown in Fig. 4(e). If the two independent photons are indistinguishable, the bosonic nature of photons will bind two photons into the same output mode, suich that no coincidence between the two PMTs is observed. Notably, an additional HWP at the left arm of the experimental setup in Fig. 4(e) allows us to alter the polarization of one input photon to render the two single photons entering the interferometer distinguishable.
Fig. 7. Hong-Ou-Mandel interference of single photons generated from two distant ions. The FPGA-based pulse timestamper recorded the photon arrival time differences between the left and right outputs of the beam splitter, and the arrival time differences are plotted for both distinguishable (circle) and indistinguishable photons (triangle). Coincidence events were strongly suppressed (by 82(3) % within detection time difference of 8.13 ns (dotted line)) for the indistinguishable photon case. The dashed line depicts the exponential decay fit to the data for the distinguishable case.
We conducted 500 million trials for each condition and recorded the timestamps of PMT trigger pulses. After each 10 million trials, the fiber coupling had to be re-optimized by the motorized stages to compensate for the thermal drift of the input fiber tips. We normalized the measured coincidence counts with the expected count number under the classical random-decision assumption. To calibrate the effect of the fiber tip drift, which leads to the reduced efficiency of fiber coupling, we also counted the event in which at least one of two PMTs are triggered, and considered it as a single photon generation probability during the experiment. Consequently, while the photons are distinguishable, i.e., in terms of their orthogonal polarization, a normalized coincidence count of 0.98(6) was observed, which is a good agreement with the classical random-decision assumption. The histogram of arrival time differences follows an exponential decay with a characteristic time of 7.89(6) ns. In contrast, when the two photons become indistinguishable with identical polarization, we observed the normalized coincidence count of 0.18(3) when the detection time difference is 8.13 ns, which means that 82(3) % of coincidences were suppressed by destructive interference. The coincidence counts were non-zero because of imperfect mode matching at the beam splitter and the residual false counts caused by back-reflection leakage.
5. Discussion & Conclusion
We proposed a compact optical design that can be used for both the excitation of individual ions and collection of the emitted individual photons. We experimentally verified our proposed design by demonstrating the Hong-Ou-Mandel interference between two emitted photons and showed that the false counts stemming from direct reflection can be suppressed by implementing tight timing requirements for PMT triggers.
Our proposal involves the use of a tightly focused excitation beam, whose spot size is very small compared to the usual ion spacing. However, our analysis shows that the excitation probability is affected by this small beam spot, if the motional state of the ion is in thermal distribution. In contrast, a typical design for a quantum computer based on the ion trap system requires tight focusing for individual addressing, and our proposed design can be easily integrated to add the remote entanglement capability to build a scalable quantum computer. In particular, ions in quantum computers are generally cooled down to their motional ground state, and therefore the reduction of the excitation probability due to the focused excitation beam can be avoided.
To extend this experimental setup to realize actual remote ion-ion entanglement, one should replace $^{174}$Yb$^{+}$ with $^{171}$Yb$^{+}$ isotope that has one-half nuclear spin and hyperfine splitting in the ground state as shown in Fig. 8. Among these hyperfine states, two hyperfine states $|{F=0, m=0}\rangle$ and $|{F=1, m=0}\rangle$ form so-called a clock transition and they are typically defined as qubit states $|{0}\rangle$ and $|{1}\rangle$ accordingly. Once the state of the qubit is initialized into $|{0}\rangle$, a single pulse with a circular polarization can excite the ions via $\sigma$-transition. Since the photons emitted by $\pi$-transition cannot be coupled to the fiber mode [18], only circularly-polarized photons from two $\sigma$-transitions with different frequencies can be coupled to the fiber, which creates entanglement between ion qubit states and the photonic frequency qubits. Finally, with the same interferometer setup used in our experiment, one can discriminate $|{\Psi ^{-}}\rangle =(|{\omega _0}\rangle |{\omega _1}\rangle -|{\omega _1}\rangle |{\omega _0}\rangle )/\sqrt {2}$, one of the four photonic Bell states, by monitoring coincidence between two PMTs, leading to a remote ion-ion entanglement in a heralded way [6,27]. Even though the explanation was based on the $^{171}$Yb$^{+}$, the same argument can be applied to any other ion species with one-half nuclear spin (Fig. 8) such as $^{133}$Ba$^{+}$.
Fig. 8. Simplified energy levels of $^{171}$Yb$^{+}$ ion. (a) The $\sigma ^{-}$-polarized pulsed laser selectively excites the initialized ion state $^{2}$S$_{1/2}$ $|{0, 0}\rangle$ to $^{2}$P$_{1/2}$ $|{1, -1}\rangle$. (b) The possible transitions from $^{2}$P$_{1/2}$ $|{1, -1}\rangle$ state to ground states. The $\pi$-polarized photons are filtered out by the geometry of the optical setup.
Finally, we emphasize that our proposed design can be easily extended to attempt simultaneous generation of entangled pairs using a fiber array to push the current limit of the entangled pair generation rate. The optical design shown in Fig. 1 shows selective excitation of network qubits only, even when they are trapped adjacent to the data qubits in a linear chain of ions. Although there is a possibility that a scattered photon from one ion can excite the other ions in the same chain of ions, this kind of cross-talk can be avoided either by using different species of ions for data qubits and network qubits [17] or just by increasing the interval between ions [28]. Therefore, our design could be a useful component in a scalable quantum computer with quantum network capability [13].
Funding
Research Resettlement Fund for the new faculty of Seoul National University; Samsung Research Funding & Incubation Center of Samsung Electronics (SRFC-IT1901-09); SK Telecom (0418-20170002, 0418-20170062, and 0418-20180033).
Acknowledgments
Raw experimental data were obtained while T. K. belonged to SK Telecom. All analysis and simulations were performed while J.K. belong to Duke University. Other than these, all other authors belonged to SNU for all work. English proof reading service was financed by Brain Korea 21 Plus Project funded by the National Research Foundation of Korea (NRF).
Disclosures
The authors declare no conflicts of interest.
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