Doubly-ionized lanthanum as a qubit candidate for quantum networks

Quantum information has the potential to revolutionize computation and communication. Trapped atomic ions are a leading platform for applications in quantum information, due in part to their long coherence times [1, 2, 3, 4], and precise control and detection of the atomic quantum states using microwave and laser radiation [2, 5, 6, 7]. These pristine quantum systems continue to extend the frontiers of quantum computation [8, 9, 10, 11, 12], quantum simulation [13, 14, 15], and quantum sensing [16, 17].

Trapped atomic ions are also a candidate for quantum networks [18, 19, 20, 21, 22, 23]. Pioneering experiments have demonstrated the ability to entangle ions remotely, over considerable distance, using their emitted photons [24, 25, 26, 27]. In general, though, the ultraviolet and visible wavelengths of photons directly produced by strong transitions between low-lying levels in atomic ions [9] are not conducive to long-distance transmission through optical fibers [28, 29]. Nevertheless, recent experiments demonstrated the ability to convert the short-wavelength photons emitted by trapped ions to the infrared frequencies more amenable to long-distance communication [30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. These experiments represent a promising approach to quantum networks with trapped ions, even though the frequency-conversion stage adds noise and loss, as well as additional complexity to an already intricate system.

Here, doubly-ionized lanthanum (La²⁺) is proposed as an alternative approach to quantum networks. Transitions between the lowest-lying levels in La²⁺ are in the infrared, with low-loss transmission in standard optical fiber, and may be used for laser cooling. The rich hyperfine structure of the ion allow for magnetic-field insensitive qubit states, as well as state detection fidelities comparable to other atomic ions, and methods for directly producing ion-photon entanglement at wavelengths amenable to long-distance transmission for the realization of quantum networks.

The lowest levels in doubly-ionized lanthanum are ${}^{2}D_{3/2}$ and ${}^{2}D_{5/2}$, followed by ${}^{2}F^{o}_{5/2}$ and ${}^{2}F^{o}_{7/2}$, as shown in Fig. 1. The ${}^{2}F^{o}_{5/2}$ and ${}^{2}F^{o}_{7/2}$ levels are predicted to have lifetimes of about 4.6 $\mu$s and 4.4 $\mu$s, respectively, while the metastable ${}^{2}D_{5/2}$ is predicted to have a lifetime of about 15 s [40]. All four low-lying levels have substantial hyperfine structure, as both naturally occurring isotopes have relatively large nuclear spin: the ¹³⁹La isotope has a natural abundance of 99.91% and nuclear spin $I=7/2$; the ¹³⁸La isotope has a natural abundance of 0.09% and nuclear spin $I=5$ [41]. The large nuclear spin combined with the substantial $J$-values of the electronic levels results in a rich hyperfine structure that has recently been measured [42] and calculated [43] for the ¹³⁹La isotope. Despite multiple hyperfine manifolds, the ion is tractable for laser cooling.

Trapped La²⁺ ions could be laser-cooled by using either a 3-level ($\Lambda$-type) or 4-level scheme. A possible 3-level cooling scheme involves driving the ${}^{2}D_{3/2}\leftrightarrow{}^{2}F^{o}_{5/2}$ transition near 1389.4 nm (air), along with the ${}^{2}D_{3/2}\leftrightarrow{}^{2}F^{o}_{5/2}$ transition near 1787.8 nm (air), as shown in Fig. 1(a). This 3-level cooling scheme would be similar to the approach often used for cooling other trapped ions with a low-lying $D$-level, including Ca⁺ [45], Sr⁺ [46], and Ba⁺ [47], and is susceptible to coherent population trapping [48]. Alternatively, a possible 4-level cooling scheme involves driving the ${}^{2}D_{3/2}\leftrightarrow{}^{2}F^{o}_{5/2}$ transition near 1389.4 nm (air) and the ${}^{2}D_{5/2}\leftrightarrow{}^{2}F^{o}_{7/2}$ transition near 1409.6 nm (air), as shown in Fig. 1(b), which would be similar to the approach often used for cooling Yb⁺ [49, 50]. In either case, all necessary frequencies for laser cooling can be generated by just two lasers, with wavelengths accessible using semiconductor diode lasers.

The 4-level cooling scheme has potential advantages. First, this option has the important technical advantage that the two required laser wavelengths are near enough to each other that a common set of fibers and optics may be used. Second, in this case there exists a true cycling transition between the ${}^{2}D_{5/2}$ $|F=6\rangle$ and ${}^{2}F^{o}_{7/2}$ $|F=7\rangle$ levels, despite the substantial hyperfine structure of ¹³⁹La²⁺. Population accumulated in other hyperfine levels, through off-resonant scattering and other processes, can be optically pumped to this cycling transition by driving each ${}^{2}D$ $|F=n\rangle$ to ${}^{2}F^{o}$ $|F=n+1\rangle$ transition, as illustrated in Fig. 1(c). The additional required frequencies may be produced by wide-bandwidth fiber electro-optic modulators (standard components at these wavelengths) driven at multiple frequencies to generate sidebands on the laser light for depopulating all hyperfine levels in ${}^{2}D_{3/2}$ and ${}^{2}D_{5/2}$. Therefore, two lasers and two fiber electro-optic modulators can produce all required frequencies for laser cooling.

The aforementioned lifetime of the ${}^{2}F^{o}_{5/2}$ and ${}^{2}F^{o}_{7/2}$ levels [40] results in a Doppler-cooling temperature limit of less than 1 $\mu$K, which might make La²⁺ ions attractive for setups involving sympathetic cooling. On the other hand, these relatively long lifetimes also limit the photon scattering rate from La²⁺ ions, which is important for fluorescence detection of trapped ions and state detection of a potential La²⁺ hyperfine qubit.

Qubits stored in the hyperfine levels of a trapped ion can exhibit simple state preparation, long coherence times, and high-fidelity state detection (for a review of different qubit realizations in trapped atomic ions, see e.g. [51, 9]). We consider a hyperfine qubit in ¹³⁹La²⁺ stored in states in the ${}^{2}D_{5/2}$ $|F=5\rangle$ and $|F=6\rangle$ hyperfine manifolds.

Initialization of a qubit into a defined pure state is the first step in most quantum information protocols. Doubly-ionized lanthanum could be prepared in the ${}^{2}D_{5/2}$ $|F=6,m_{F}=+6\rangle$ or $|F=6,m_{F}=-6\rangle$ state by optical pumping with $\sigma^{+}$- or $\sigma^{-}$-polarized light, respectively. Optical pumping with $\sigma^{-}$-polarized light to $|F=6,m_{F}=-6\rangle$ may be preferred to red-detune the incident laser light from all ${}^{2}D_{5/2}$ $|F=5,6\rangle$ $\leftrightarrow$ ${}^{2}F^{o}_{7/2}$ $|F=6,7\rangle$ transitions while optimizing the scattering rate on the cycling transition. Although state preparation fidelity based on optical pumping in hyperfine qubits with nuclear spin greater than 1/2 is typically limited by the polarization purity of the laser light, it can be improved by combining optical pumping with additional optical or microwave transitions [52, 2, 5]. A subsequent series of microwave transitions could then move population to any state in the hyperfine manifold, including pairs of states that are first-order insensitive to magnetic field fluctuations.

Qubits that are first-order insensitive to magnetic fields are important for achieving long coherence times. We calculated the energy shift of states in the low-lying levels of ¹³⁹La²⁺ by numerically diagonalizing the Hamiltonian for the hyperfine interaction in the presence of an external magnetic field [53]:

where $A_{\text{hfs}}$ is the magnetic dipole hyperfine coefficient, $B_{\text{hfs}}$ is the electric quadrupole hyperfine coefficient, $I$ is the nuclear spin and $I_{z}$ its projection along the magnetic field, $J$ is the total electron spin and $J_{z}$ its projection along the magnetic field, $\mu_{B}$ is the Bohr magneton, $B$ is the magnetic field defined to be in the $z$-direction, $g_{J}$ is the Landé g-factor, and $g_{I}$ is the nuclear g-factor using the convention of Ref. [53]. We repeated the calculation for both the experimentally [42] and theoretically [43] determined values for the hyperfine coefficients, and searched for pairs of states in the ${}^{2}D_{5/2}$ $|F=5\rangle$ and $|F=6\rangle$ hyperfine manifolds where the first-order derivative of the frequency difference vanishes at finite magnetic field (see Fig. 2). Several options for a first-order magnetic field-insensitive qubit are listed in Table 1.

A qubit stored in ${}^{2}D_{5/2}$ $|F=5,m_{F}=1\rangle$ and $|F=6,m_{F}=1\rangle$, calculated to be first-order magnetic field-insensitive at an external field of about 0.18 (0.64) mT using the experimentally (theoretically) determined hyperfine coefficients, is of particular interest. In this case, the small external magnetic field required is easily obtained in the lab. Additionally, at this magnetic field the induced shift in transition frequencies from ${}^{2}D_{5/2}$ $|F=6\rangle$ to ${}^{2}F^{o}_{7/2}$ $|F=7\rangle$ for $\sigma$-polarized light span about 2 (8) MHz across all states, making the manifold addressable by a single laser frequency of reasonable ($\sim 1$ MHz) linewidth. As noted in Table 1, the splitting between this pair of states is calculated to have a second-order magnetic field dependence of about 1.5 Hz/$\mu\text{T}^{2}$ (0.43 Hz/$\mu\text{T}^{2}$), comparable to the sensitivity of other qubits with demonstrated long coherence times [1].

The qubit state could be read out using standard state-dependent fluorescence [54, 51, 9]. In this case, near-resonant, $\sigma^{-}$-polarized, laser light could drive the ${}^{2}D_{5/2}$ $|F=6\rangle$ to ${}^{2}F^{o}_{7/2}$ $|F=7\rangle$ transition, so that population in ${}^{2}D_{5/2}$ $|F=6,m_{F}=1\rangle$ qubit state is quickly driven back to the cycling transition, scattering many photons and appearing “bright” to a detector. The light used for detection would be detuned from a transition involving the ${}^{2}D_{5/2}$ $|F=5,m_{F}=1\rangle$ qubit state by the difference between the ${}^{2}F^{o}_{7/2}$ $|F=6-7\rangle$ hyperfine splitting and the ${}^{2}D_{5/2}$ $|F=5-6\rangle$ hyperfine splitting, and therefore appear “dark” to a detector. Modeling the system similar to Ref. [55, 56], we find that errors in state detection due to off-resonant coupling to other states should be small, due to the relatively large detuning with respect to linewidth for both the bright and dark states. Even when considering the expected linewidth of the detection laser (1 MHz), the detuning to unintended states is approximately $10^{3}$ linewidths, which is larger than the detuning in several other ions with demonstrated high-fidelity state detection of hyperfine qubits (e.g. ¹¹¹Cd⁺ with off-resonant coupling from the dark state detuned by about $2\times 10^{2}$ linewidths [55]; ¹⁷¹Yb⁺ with off-resonant coupling from the bright state detuned by about $10^{2}$ linewidths [50, 57]). Leakage out of the bright and dark state manifold is further suppressed by small electric dipole transition probabilities between the off-resonantly coupled states, relative to the cycling transition. Error in bright state detection is also suppressed due to the dependence on polarization errors; either $\pi$- or $\sigma^{+}$-polarized light is required to couple ${}^{2}D_{5/2}$ $|F=6,m_{F}=-6\rangle$ to off-resonant states instead of driving the cycling transition. However, decay of the ${}^{2}D_{5/2}$ level appears as an additional error in bright state detection. Another important consideration in our model is the error contribution from background counts, given the small photon scattering rate resulting from the relatively long lifetime of the ${}^{2}F^{o}_{7/2}$ level. Nevertheless, given the high detection efficiency and low dark count rate of available superconducting nanowire single-photon detectors (SNSPDs), high state detection fidelities may still be achieved. Altogether, assuming a simple threshold detection method, modest light collection optics ($NA=0.28$), a wide range of laser intensities ($I/I_{sat}\lesssim 15$), and detection parameters readily achieved with an SNSPD ($QE\approx 0.8$, dark count rate $\leq 2$ s^-1, fiber coupling $\approx 0.2$), our model already predicts a state detection fidelity greater than $99.9\%$. The fidelity increases by considering larger light collection [55, 50, 58, 57, 59, 60, 61, 62, 38, 63, 64, 65, 66], larger fiber coupling [27], or a smaller laser linewidth (e.g. comparable to the natural linewidth of the transition).

The potential for first-order magnetic field-insensitive qubit states with high-fidelity state preparation and readout supports ¹³⁹La²⁺ as an excellent quantum memory candidate, with possible applications in photon-mediated entanglement operations.

Entanglement between an atomic ion and an emitted photon can serve as the foundation for a quantum network, where linking ions over a long-distance is accomplished by the interference and detection of spontaneously emitted photons [18, 19, 20, 21, 22, 23]. Doubly-ionized lanthanum is an excellent candidate for this architecture, as it could be entangled with the polarization or frequency mode of a photon with emitted wavelengths directly amenable to long-distance transmission.

A possible protocol for entangling ¹³⁹La²⁺ with the polarization of an emitted photon is illustrated in Fig. 3(a). The ion would be initially prepared in the ${}^{2}D_{5/2}$ $|F=6,m_{F}=-6\rangle$ state by optical pumping with $\sigma^{-}$-polarized light. A 10 ns, $\pi$-polarized laser pulse then could excite the atom to the ${}^{2}F^{o}_{7/2}$ $|F=7,m_{F}=-6\rangle$ state. Notably, an excitation pulse of 10 ns would be much shorter than the lifetime of the excited state (inhibiting multiple excitation-emission processes), yet result in a pulse linewidth significantly smaller than the excited state hyperfine splitting (inhibiting off-resonant coupling to other states). We estimate the optimal pulse duration is about 10 ns (though this value will depend on the time-bandwidth product of the pulse), with estimated errors due to double-excitation and off-resonant scattering each less than $3\times 10^{-3}$. Following excitation, the subsequent atomic decay and photon emission ideally produces the ion-photon entangled state $|\Psi\rangle=\sqrt{\frac{1}{7}}|6,-6\rangle|\pi\rangle+\sqrt{\frac{6}{7}}|6,-5\rangle|\sigma\rangle$, where the atomic state is denoted by the final $F$ and $m_{F}$ values in the ${}^{2}D_{5/2}$ level, the photonic state is denoted by the polarization $\pi$ or $\sigma$, and the coefficients arise from the dipole transition matrix element for each decay channel. Collecting emitted photons in a direction perpendicular to a quantization axis defined by an external magnetic field, where $\pi$ and $\sigma$ polarizations are linearly polarized and orthogonal, and the intensity of radiation of the $\pi$-decay is twice as strong as the $\sigma$-decay, would result in an ion-photon entangled state $|\Psi\rangle=\frac{1}{2}|6,-6\rangle|V\rangle+\frac{\sqrt{3}}{2}|6,-5\rangle|H\rangle$, where we now denote the two (linear, orthogonal) polarizations as $V$ and $H$ [67, 68]. This protocol has the advantage of simple state preparation and straight-forward manipulation of the photonic state using waveplates and polarizers, although it does not directly connect to a magnetic-field insensitive pair of states for the ion.

One option for entangling a first-order magnetic field-insensitive atomic qubit with the frequency of an emitted photon is shown in Fig. 3(b). In this case, following optical pumping to ${}^{2}D_{5/2}$ $|F=6,m_{F}=-6\rangle$, a series of microwave transitions could move the population to ${}^{2}D_{5/2}$ $|F=5,m_{F}=1\rangle$. A 10 ns, $\sigma^{-}$-polarized laser pulse then could excite the atom to the ${}^{2}F^{o}_{7/2}$ $|F=6,m_{F}=0\rangle$ state, with estimated errors due to double-excitation and off-resonant scattering nearly the same as the polarization-encoded protocol. Collecting photons in a direction along a quantization axis defined by an external magnetic field, the dipole radiation pattern of $\pi$-polarized light does not couple into a single-mode optical fiber, and $\sigma^{+}$- and $\sigma^{-}$-transitions produce light with orthogonal circular polarizations that may be converted to orthogonal linear polarizations using quarter-wave plates [68, 69, 70, 61]. The atomic decay and photon emission, combined with appropriate filtering to select only the $\sigma^{-}$-polarized photons, then ideally produces the ion-photon entangled state $|\Psi\rangle=\sqrt{\frac{25}{36}}|5,1\rangle|\nu_{blue}\rangle+\sqrt{\frac{11}{36}}|6,1\rangle|\nu_{red}\rangle$, where the atomic state is again denoted by the final $F$ and $m_{F}$ values in the ${}^{2}D_{5/2}$ level and the photonic state is denoted by the frequency $\nu_{blue}$ or $\nu_{red}$, which approximately differ by the hyperfine splitting of $F=5$ and $F=6$ in ${}^{2}D_{5/2}$ ($\nu_{blue}-\nu_{red}=0.20$ $(0.27)$ GHz) and are well-resolved due to the narrow linewidth of the transition ($\gamma=2\pi\times 36$ kHz). These resulting atomic states can be first-order magnetic field-insensitive (see Table 1), and the frequency-encoded photonic states should be resilient against dispersion and birefringence [71, 68].

An alternative frequency-encoded photonic qubit protocol is presented in Fig.3(c), where the ion is prepared in a superposition of ${}^{2}D_{5/2}$ $|F=5,m_{F}=0\rangle$ and $|F=6,m_{F}=0\rangle$, then excited to ${}^{2}F^{o}_{7/2}$ using $\pi$-polarized light. Driving the ion with two frequencies, and noting the elimination of nearest-state off-resonant coupling due to selection rules, a few-nanosecond laser pulse could simultaneouly drive the ${}^{2}D_{5/2}$ $|F=5,m_{F}=0\rangle$ to ${}^{2}F^{o}_{7/2}$ $|F=6,m_{F}=0\rangle$ and ${}^{2}D_{5/2}$ $|F=6,m_{F}=0\rangle$ to ${}^{2}F^{o}_{7/2}$ $|F=7,m_{F}=0\rangle$ transitions with estimated errors due to double-excitation and off-resonant scattering each less than $1\times 10^{-3}$. Collecting photons in a direction perpendicular to a quantization axis defined by an external magnetic field allows for polarizaton filtering to select only photons from $\pi$-transitions. Then due to selection rules, the resulting ion-photon entangled state is ideally $|\Psi\rangle=a_{5}c_{5}|5,0\rangle|\nu_{5}\rangle+a_{6}c_{6}|6,0\rangle|\nu_{6}\rangle$, where the atomic state is again denoted by the final $F$ and $m_{F}$ values in the ${}^{2}D_{5/2}$ level, the photonic state is denoted by the resolved frequency $\nu_{5}$ or $\nu_{6}$ such that $\nu_{6}-\nu_{5}=0.96$ (0.89) GHz (the difference between the ${}^{2}F^{o}_{7/2}$ $F=6\leftrightarrow 7$ and ${}^{2}D_{5/2}$ $F=5\leftrightarrow 6$ hyperfine splittings), the initial amplitudes of ${}^{2}D_{5/2}$ $|F=5,m_{F}=0\rangle$ and $|F=6,m_{F}=0\rangle$ are denoted by $c_{5}$ and $c_{6}$, and the effect of the excitation laser pulse and dipole transition matrix element are incorporated into $a_{5}$ and $a_{6}$. Notably, it should be possible to tailor the excitation laser pulse properties such that $a_{5}$ and $a_{6}$ can be assigned arbitrary relative values, including $a_{5}=a_{6}=1$. In that case, the ion-photon entangled state becomes $|\Psi\rangle=c_{5}|5,0\rangle|\nu_{5}\rangle+c_{6}|6,0\rangle|\nu_{6}\rangle$, and may facilitate a photon-mediated quantum gate between remote atoms [71, 25, 72].

Additional ion-photon entanglement protocols can be devised to connect to other potential atomic qubits states in Table 1 (and more beyond these). Moreover, other photonic qubits can be considered, such as using two successive excitations of the cycling transition to produce a time-bin encoded photonic qubit [73, 74, 69, 75]. In all cases, photons at this wavelength are expected to experience $\leq 0.32$ dB/km attenuation in standard optical fiber [76, 77].

The rate of ion-photon entanglement is an also important consideration for large-scale quantum networks. Although the per atom repetition rate for producing ion-photon entanglement is bounded by the relatively long lifetime of the ${}^{2}F^{o}_{7/2}$ excited state in ¹³⁹La²⁺ ($\tau=4.4$ $\mu$s [40]; $\gamma=2\pi\times 36$ kHz), for distances greater than about 5 km between network nodes, the limitation to the per atom repetition rate will be the propagation delay for the photon and (classical) detection signal [78, 79]. As with any trapped ion system facing this distance-limited rate, the probability of successfully registering entanglement over long distances can be improved by incorporating high numerical aperture optics [57, 59, 60, 61, 62] or an optical cavity [38, 63, 64, 65, 66] for efficient photon collection, and the effective rate can be increased by multiplexing a system with multiple communication ions [80, 81, 82, 38, 83]. Therefore, doubly-ionized lanthanum should be able to provide ion-photon entanglement rates over long distances comparable to other ions, and may have a distinct advantage by directly producing the telecom-compatible photons amenable to transmission over these distances.

Doubly-ionized lanthanum may allow for efficient laser cooling using infrared diode lasers, a range first-order magnetic-field insensitive qubit states with high-fidelity state detection, and viable protocols for directly entangling trapped ions with telecom-compatible photons for long-distance transmission in optical fiber. The unique features of doubly-ionized lanthanum may also make it a candidate for other quantum information applications, including as a refrigerant ion [84, 85, 86, 87] (given the small Doppler-cooling temperature limit and large difference in wavelength compared to other trapped ions) or multi-state qudit candidate [88, 89, 90, 91, 92] (given the rich hyperfine structure). While the results presented here are promising, more precise measurements of the hyperfine structure of doubly-ionized lanthanum, as well as additional investigations into producing and trapping multiply-charged ions, may be needed to realize the potential of doubly-ionized lanthanum for applications in quantum information and quantum networks.