Eliminating qubit type cross-talk in the $omg$ protocol

The highest fidelity state-preparation and measurement (SPAM) operations [1] as well as single- and two-qubit gates [2, 3] have all been achieved using trapped atomic ions. This performance is realized in part thanks to the high degree of isolation of trapped ion qubits from their environment as compared to other technologies. This isolation necessitates that the entropic (or open-channel) operations used to prepare a qubit, i.e. motional cooling and state preparation, are introduced deliberately, typically via laser light that is resonant with atomic electronic transitions. As light from these lasers can be extinguished as needed, they provide a strong but severable link to the environmental bath.

However, if cooling is needed as part of a given quantum operation, due to, e.g., heating during a long algorithm [4] or due to shuttling in the QCCD architecture [5], laser cooling cannot be performed directly since the strong coupling of the process to the atomic internal states scrambles any quantum information hosted in the atom. To overcome this limitation, some ion-based quantum processors simultaneously use two species of atoms [5]. One species, the logic ion, is used to host and process the quantum information, while the other ion, the coolant ion, is used only for cooling.

This ‘dual species’ approach is powerful, providing capabilities like mitigation of heating during transport and long algorithms, as well as facilitating mid-circuit measurement [6]. It is not, however, without complication. Beyond requiring significantly more complicated laser and optical systems , the mass difference of the two species, coupled with the pondermotive nature of an ion trap, leads to deleterious effects such as mode-decoupling [7] and increased heating during transport [8, 9]. These necessitate complicated shuttling protocols, specific ion chain arrangements, and significant recooling time after transport [5].

A recent proposal, dubbed the omg protocol, has described how to achieve dual-species functionality using a single atomic species [10]. The omg protocol leverages the three types of qubits; $o$ptical, $m$etastable, and $g$round, available in certain atomic ions species to host quantum information in different parts of the atomic Hilbert space. Allowing operations to be performed on some ions, while other ions, occupying different part of the Hilbert space, are protected from the operation. As such, omg appears to offer many advantages over the dual-species approach including reducing the number of trapped ions needed for a given algorithm, a reduction in laser and optical complexity, the elimination of the effects associated with mass mismatch, and the ability to flexibly define an ion’s role.

However, while some of the basic components of the omg protocol have been demonstrated, such as qubit initialization and conversion [11], many of its promised benefits remain unproven. One open question is the degree to which operations can be performed on one part of the Hilbert space without affecting others. Recent work has shown that laser cooling on the $g$-space does not noticeably affect coherent operations on the $m$-space [11]. However, laser cooling requires relatively low laser intensity as compared to the lasers used for single- and two- qubit gates. Here, we show that the high-intensity operations necessary for laser-based gates in the $m$-space of ¹³³Ba⁺ do indeed lead to detrimental effects, as the intensity instability of the gate lasers causes uncontrolled differential light shifts for qubits stored in the $g$-manifold. However, we find this effect can be completely mitigated by exploiting magnetic-field-induced hyperfine mixing to realize a non-zero, vector light shift between the clock state qubits that cancels the scalar and tensor shifts [12]. Therefore, with the correct laser polarization, which we dub the ‘magic polarization’, the differential light shift of $g$-type qubits by the gate lasers is nulled, protecting information stored in $g$-type qubits from $m$-type qubit laser-based gate operations.

Implementation of the $omg$ protocol in ¹³³Ba⁺ utilizes the $\{g,m,g\}$ architecture – here, the ordered triplet denotes the qubit space used for $\{\text{cooling, gates, storage}\}$ [10]. In this architecture, information is stored in the $g$ subspace and qubits are ‘activated’ to the $m$ subspace for gates, as shown in Fig. 1(a). If the activation resolves single ions, the laser light used for gates can be applied globally, providing a dramatic simplification in systematic complexity. In this work, we define the $g$-type qubit as the $F=0$ and $F=1$ hyperfine zero-field clock (i.e. $m_{F}=0$) states of the ¹³³Ba^+ 2S_1/2 manifold, while the $m$-type qubit is defined on the the $F=2$ and $F=3$ hyperfine zero-field clock states of the ²D_5/2 manifold. Laser cooling, state preparation, and state readout are performed via the $g$ subspace as detailed in Ref. [13]. After high fidelity state preparation of the $F=1,m_{F}=0$ $g$-qubit state, transfer to the $m$-type qubit is accomplished via a heralded coherent operation on the $o$-qubit transition at 1762 nm to the $F=3,m_{F}=0$ state of the ²D_5/2 manifold (see SI). Readout of the $m$ qubit is performed by a shelving operation that involves the simultaneous application of lasers at 1762 nm and 493 nm to transfer the target $m$-state population to the ²D_3/2 manifold, where it can be detected via resonant fluorescence (see SI). A continuous wave laser at 532 nm is directed through acousto-optical modulators to generate the beams for stimulated Raman $m$-type qubit gates. An electro-optical modulator in the beam path, allows the same laser to also perform operations on the $g$-type qubit; while not required by the $\{g,m,g\}$ architecture the ability to use the same laser path to perform $g$ and $m$ qubit gates adds significant flexibility to the system.

Because high-resolution spectroscopy of ¹³³Ba⁺ is not yet complete, before performing $m$ qubit gates it is necessary to first measure the hyperfine splitting of the $5d6s$ ²D_5/2 state that hosts the $m$ qubit. We perform this measurement via a stimulated Raman spectroscopy on the $F=3,m_{F}=0\leftrightarrow F=2,m_{F}=0$ transition at $B=5.036(1)$ G, as shown in Fig. 1(b,c). From this measurement, we report the first high-resolution measurement of the ¹³³Ba⁺ $5d6s$ ²D_5/2 hyperfine constant as $A_{5/2}=29.7565(1)_{sys}$ MHz, with a statistical uncertainty of 40 Hz. Using this measurement, we use a single $m$-qubit rotation to measure the intensity stability limited, $m$-qubit coherence time to be $2.6(9)$ ms (see SI).

To ascertain the effect of the light used for these $m$-type gates on a qubit stored in the $g$ subspace, we perform a detuned Ramsey sequence on a $g$ qubit, using microwaves, with and without illumination by the laser beams used in the $m$-qubit gate. As seen in Fig. 2 for $\sigma^{-}$ polarization and a magnetic field of $B=5.00(1)$ G, a fit of a simple decaying oscillation is shown as a guide to the eye and suggests a coherence time of approximately 30 ms.

While this effect could be somewhat mitigated with technical improvements and dynamical decoupling, it would be beneficial if the differential light shift of the $g$-qubit states could simply be eliminated. To understand how this is possible, it is useful to write the light shift of an atomic state due to a laser at frequency $\omega_{L}$ and polarization $\hat{\epsilon}_{L}$ as $\Delta E_{F,m_{F}}\propto\alpha_{F}(m_{F},\omega_{L},\hat{\epsilon}_{L})I_{L}$, where $\alpha_{F}$ is the dynamic polarizability and $I_{L}$ is the laser intensity.

The dynamic polarizability can be decomposed as [14]:

where the $m_{F}$-independent $\alpha^{s}_{F}$, $\alpha^{a}_{F}$, and $\alpha^{t}_{F}$ are the scalar, vector (axial), and tensor dynamic polarizabilities, respectively, $A\equiv-i(\hat{\epsilon}_{L}^{\ast}\times\hat{\epsilon}_{L})\cdot\hat{z}$ is the helicity projection of circular polarization and $\hat{z}$ is the quantization axis. At first glance, since the vector and tensor polarizabilities are non-zero only for $F\geq 1/2$ and $F\geq 1$, respectively, and the $g$-qubit states have $m_{F}=0$ it would appear the only opportunity for nulling the differential light shift between the qubit states is via controlling the direction of linear polarization relative to the magnetic field to tune the tensor shift contribution. However, the magnitude of the tensor contribution is not large enough to allow matching the light shift of the two qubit states. Fortunately, in a non-zero magnetic field Eq. (1) is only approximate as the magnetic field breaks rotational invariance and $F$ is no longer a good quantum number. The result is a non-zero vector light shift contribution even for $m_{F}=0$ [12], which can be used to match the dynamic polarizabilities of the two clock states and null the differential light shift.

This can be intuitively understood by considering the light shift of the $g$-qubit states under illumination by a laser with $\sigma_{+}$ polarization. The two states couple primarily to the $F=m_{F}=1$ state of the ${}^{2}\mathrm{P}_{1/2}$ manifold (see e.g. Fig. 1(b)). At zero magnetic field, the $F=0$ $g$ qubit state experiences a larger light shift than the $F=1$ $g$ qubit state because it is closer to the ${}^{2}\mathrm{P}_{1/2}$ manifold. Therefore, the $g$ qubit has a negative differential light shift. In a non-zero magnetic field, the two $g$-qubit states are mixed and the coupling is modified such that the light shift now depends on the degree of helicity projection.

To quantify this effect, we measure the differential light shift of the $g$ qubit using a Ramsey sequence (see SI) as a function of $A$ and $B\hat{z}$, under the condition that the angle between $\hat{k}$ and $\hat{z}$ is $\theta_{kz}=180.0(8)^{\circ}$. As seen in Fig. 3, the $g$ qubit exhibits a clear, magnetic-field dependent vector light shift. Interestingly, above a certain magnetic field this vector light shift is large enough to completely counteract the difference in the scalar light shifts. By using gate lasers operating at a polarization where the differential light shift is zero, which we dub a magic polarization, the $g$-qubit splitting becomes completely insensitive to the $m$-qubit gate lasers. The magnetic field required for a given magic polarization is found by finding the zero crossing of the fitted line in Fig. 3(a) and plotted in Fig. 3(b) as black points, where the error bars are 68.5% confidence intervals from the fit. The minimum magnetic field necessary for cancellation of the $g$-qubit light shift, which we dub the critical magnetic field $B_{c}$, is achieved for $A=1$ ($\sigma_{+}$ polarization). However, due to coherent population trapping effects [15], our Doppler cooling efficiency suffers below a magnetic field of $\sim 2$ G, and therefore we cannot probe the critical magnetic field directly. Instead, we fit the expected theoretical behavior to the data to extract $B_{c}$.

The dependence of the magic polarization on magnetic field is found by diagonalizing the optical Hamiltonian leading to Eq. (1) in the presence of a magnetic field $B$ [16]. The result is a vector light shift term that is proportional to $B$ leading to the relation:

where we distinguish between the polarizabilities of the two ($F^{\prime}=0$ and $F=1$) qubit states separated by energy $\hbar\omega_{q}$. Fitting this expression to the data in Fig. 3(b)(the grey shaded region), yields a critical magnetic field for ¹³³Ba⁺ $g$ qubits of $B_{c}=1.15(3)$ G at 532 nm.

Calculation of the expected behavior requires a third-order hyperfine-interaction (HFI) mediated polarizabilities treatment [16] be used in Eq. (2) as the two scalar polarizabilities cancel out otherwise. This partially stems from the fact that the total electron angular momentum $J$ is mixed by the atomic hyperfine interaction, leading to differing transition dipole moments for the hyperfine qubit states to the same fine-structure state. Using a combination of empirical data for low-lying electronic states and ab initio relativistic RPA+BO many-body method [17, 16, 18], we numerical evaluate the polarizabilities in Eq. (2). The results of the numerical calculation are plotted as the dashed blue line in Fig. 3(b), and correspond to a predicted critical magnetic field of at 532 nm of $B_{c}=1.17$ G in agreement with the measurement.

The HFI mediated polarizability calculation has also been performed as a function of gate laser wavelength and the predicted $B_{c}$ is plotted in Fig. 3(c) alongside the data. The rise in the $B_{c}$ with large detunings is due to interference from the counter-rotating terms, which work to lessen the induced vector light shift.This difference can be substantial and clearly shows the need for high-level calculations.

To demonstrate the utility of magic polarization for $omg$ operation, we repeat the measurements of Fig. 2 at $B=5.00(1)$ G, but with the polarization chosen to be magic ($A=0.212$). As can be seen in Fig. 2, the decoherence due to the light shift of the $g$ qubit by the $m$-qubit gate laser is mitigated, leading to a $\sim$100-fold increase in coherence time. In fact, the $g$-qubit coherence time of ($\tau=3.6(2.5)$ s) is consistent with the performance observed when the ion is not illuminated by the gate laser (Fig. 2) despite being exposed to a laser field with an intensity of $\sim 100$ MW/m².

For single and two-qubit gates, $A=1$ polarization is preferred as it yields the largest Rabi rate and therefore minimizes spontaneous Raman scattering error [19, 20]. Thus, to harness the benefits of this magic condition, the applied magnetic field should be at the critical point. However, at 532 nm the critical magnetic field for ¹³³Ba⁺ is well below 2 G, which is the minimum magnetic field we require for efficient Doppler cooling. Luckily, as the gate laser is further detuned to longer wavelengths the critical field grows, due to interference from the emission first term in the light shift, as shown in Fig. 3(c).

The use of a longer wavelength gate laser, while requiring moderately more power, has the added benefit of a lower Raman scattering rate. For example, at 1130 nm $B_{c}\approx 2$ G and the two-qubit gate infidelity due to Raman scattering is expected to be $<10^{-5}$  [20].

In summary, we demonstrate the first preparation, operation, and readout of the $m$ qubit in ¹³³Ba⁺ and report the most accurate measurement to date of the $5d6s$ ²D_5/2 hyperfine splitting of $A_{5/2}=29.75650(10)_{sys}(4)_{stat}$ MHz. Using these tools, we examine the cross-talk of $omg$ laser-based gates between qubit subspaces and find that gate operations on $m$ qubits lead to decoherence of $g$ qubits. This decoherence results from the $g$ qubit differential light shift by the gate laser. However, utilizing a magnetic-field-induced vector light shift we show how the light shift between these zero-field clock state qubits can be matched using a magic polarization. We measure the critical magnetic field required to realize a magic polarization in ¹³³Ba⁺ and compare it to high-level atomic structure calculations and find good agreement. Finally, using the magic polarization to null the differential light shift of the $g$-qubit states, we demonstrate protection of quantum information stored in the $g$ qubit from $m$-qubit laser-based operations.