Rydberg quantum computation with nuclear spins in two-electron neutral atoms

Abstract

Alkaline-earth-like (AEL) atoms with two valence electrons and a nonzero nuclear spin can be excited to Rydberg state for quantum computing. Typical AEL ground states possess no hyperfine splitting, but unfortunately a GHz-scale splitting seems necessary for Rydberg excitation. Though strong magnetic fields can induce a GHz-scale splitting, weak fields are desirable to avoid noise in experiments. Here, we provide two solutions to this outstanding challenge with realistic data of well-studied AEL isotopes. In the first theory, the two nuclear spin qubit states |0〉 and |1〉 are excited to Rydberg states |r〉 with detuning ∆ and 0, respectively, where a MHz-scale detuning ∆ arises from a weak magnetic field on the order of 1 G. With a proper ratio between ∆ and Ω, the qubit state |1〉 can be fully excited to the Rydberg state while |0〉 remains there. In the second theory, we show that by choosing appropriate intermediate states a two-photon Rydberg excitation can proceed with only one nuclear spin qubit state. The second theory is applicable whatever the magnitude of the magnetic field is. These theories bring a versatile means for quantum computation by combining the broad applicability of Rydberg blockade and the incomparable advantages of nuclear-spin quantum memory in two-electron neutral atoms.

Graphical abstract

Keywords

alkaline-earth atom / Rydberg state / quantum computation / neutral atom

1 Introduction

It is widely believed that quantum information processing can carry out fantastic tasks beyond the capability of classical technology [1, 2], although unfortunately a universal quantum computer has not yet been realized especially because of the fast dissipation on the timescale of each system useful for realizing a quantum information processor [3–7]. Recently, neutral atoms have been recognized as a new promising platform for large-scale quantum computing [8–13] because quantum entanglement based on the dipole blockade between neutral Rydberg atoms can be generated rapidly in a large qubit array [14–25]. However, models for reliable and practical quantum computing need entangling gates with a fidelity beyond, for example, 0.999 [26], while the best experimental fidelity of two-qubit entangling gates with alkali-metal atoms is below 0.98 [22, 23].

A new route toward neutral-atom quantum computation is by using atoms with two valence electrons and a nonzero nuclear spin, i.e., some alkaline-earth-metal or lanthanide atoms, which we call alkaline-earth-like (AEL) atoms. Compared to alkali-metal atoms, well-studied AEL atoms such as strontium and ytterbium can be more easily cooled to very low temperatures [25, 27–30] or to the vibrational ground state [31], their nuclear spin states are insensitive to background magnetic noise and can be preserved in the process of cooling [32], and long-lived trapping of both the ground and Rydberg states is realizable [33]. These properties compare favorably to those of alkali-metal atoms concerning quantum control for entanglement generation. A recent Rydberg blockade experiment [25] with alkaline-earth-metal atoms reported a two-atom entanglement fidelity 99.5%, while the best number with alkali-metal atoms is 97.4% [20, 22]. However, the entanglement in Ref. [25] is between a Rydberg state and a metastable state (5s5p)³P₀ of the nuclear-spin-free strontium-88. In principle, by Rydberg blockade one can entangle nuclear spin qubit states in the ground-state space, but its realization faces an outstanding challenge, namely, that there is no hyperfine splitting in the ground state of well-known AEL fermions like strontium-87 and ytterbium-171 and it is unlikely to leave one nuclear spin qubit state unaffected when rotating the other qubit state back and forth to Rydberg state with a megahertz rate.

In this work, two theories, Theory 1 and Theory 2, are given to selectively and rapidly transfer one nuclear spin qubit state from the ground state to a Rydberg state of an AEL atom. For Theory 1 shown in Fig. 1(a), a magnetic field B on the order of 0.1 mT for specifying the quantization axis splits the degeneracy of the two transitions |0〉↔|r〉 and |1〉↔|r〉 with a frequency mismatch ∆ on the order of megahertz. A megahertz-rate Rabi flopping between ground and Rydberg states is required to avoid decoherence in the Rydberg states, so the Rabi frequency Ω for the rotation |1〉↔|r〉 should be on the order of megahertz, leading to an undesired detuned Rabi oscillation between |0〉 and |r〉. The generalized Rabi frequency \overline{\text{Ω}} for the detuned Rabi oscillation of |0〉 is \sqrt{{\text{Ω}}^2+{\text{Δ}}^2} [34], and, remarkably, the condition \overline{\text{Ω}}/\text{Ω}=2N can lead to N cycles for the detuned Rabi rotations between |0〉 and |r〉 when |1〉 is excited to |r〉, where N is an integer. Although there is a phase accumulation in |0〉, it can be canceled by single-qubit phase gates for designing a controlled-Z (C_Z) gate as shown later. For Theory 2 shown in Fig. 1(b), a two-photon Rydberg excitation |1〉→|r〉 is achieved via circularly polarized laser fields. We show by realistic data that it is possible to choose appropriate intermediate states and Rydberg states so that we can completely suppress the Rydberg excitation of |0〉 when |1〉 is excited to Rydberg states. Theory 2 does not rely on a MHz-scale ∆ and, in fact, any value of ∆ is fine. Theory 2 can also be executed by pulse-shaping methods such as stimulated Raman adiabatic passage (STIRAP) [35–37] that are useful for Rydberg atom quantum entanglement [8, 22, 38–64]. In comparison, Theory 1 cannot be implemented with STIRAP, but it can be used with one-photon excitation of p-orbital Rydberg states [18, 65].

After reviewing the present stage of AEL-based quantum computing in Section 2, we detail our theories with ytterbium in Section 3 and briefly study strontium in Section 4 because these two elements are well studied in experiments, as shown in Refs. [27, 28, 33] and [25, 29–31, 66], respectively. Discussions and conclusions are given in Section 5 and 6, respectively. Although Section 3 contains necessary details, more are grouped in the appendixes to avoid blurring the essence of the theories.

2 Nuclear spin states as qubit states

2.1 A brief review of previous methods

It will be useful to briefly review previous gate proposals with AEL atoms. The motivation is that they have stable nuclear spin states which can be used to store information, and there can be collisional interaction that enables entanglement [67–69]. Moreover, for elements with nonzero nuclear spins, an electric singlet-triplet clock transition from the ground state can be decoupled from the nuclear spin. As a consequence, the atoms can be cooled without changing the quantum states of the nuclear spin, which is an awesome benefit to store the information in the state space of the nuclear spin [32]. This in principle means that when qubits heat up after many cycles of quantum control as required in quantum computing, they can be cooled again without changing the information stored in the quantum system. Although there is only one ground level ¹S₀ for well studied AEL atoms, their first excited ³P₀ state has a long lifetime τ, which can be used for shelving one of the nuclear spin qubit states for information readout; for the case of ytterbium studied in this work, τ is about 26 s [70] which can exceed the time scale of relevant quantum control by many times. Besides, coding information in the nuclear spin states can avoid environmental noise such as fluctuation of magnetic fields. These several advantages motivated proposals for entangling atoms with two valence electrons by merging them from different sites for the induction of a blockade or exchange interaction [69, 71], where the latter has been studied in experiments [72–74]. Till now, several theories have been put forward for showing that high-fidelity quantum gates should be feasible via the collisional interaction [75–79].

Another method is to use the optical clock transition ¹S₀ ↔³ P₀ of AEL neutral atoms [80], where a qubit is defined by the electronic ground and long-lived clock states. This method easily offers frequency-selective excitation to Rydberg states. As in the experiment [25], the ³P₀ clock state was coherently and rapidly excited to a Rydberg state, leading to high-fidelity two-atom entanglement between the Rydberg and optical-clock states.

2.2 Difficulties in selective Rydberg excitation of one nuclear spin qubit state

In this work, we show that entanglement between nuclear spins in two-electron atoms can be efficiently generated by Rydberg interactions. Compared to the collisional gates [71, 75–79], our Rydberg gate is implemented on a much shorter microsecond timescale. But there are difficulties to achieve our goal.

The first difficulty is that Rydberg states are not stable. Different isotopes have different properties. We take ¹⁷¹Yb as an example to briefly show the difficulties one by one while more details are grouped in Appendixes AE. The qubit is defined by two nuclear spin states, |0(1)〉≡ |(6s²)¹S₀, m_I = ±1/2〉, among which one should be excited to a Rydberg state |r〉≡|(6sns)³S₁, F = I + 1〉 for entanglement generation. As estimated in Appendix C, the lifetime of |r〉 is about τ = 330 µs for n = 70 at room temperatures. Rydberg excitation should take place within a time much shorter than τ.

The second difficulty is that for frequently studied AEL atoms, there is no hyperfine splitting in the ground states. When laser fields are sent to qubits for exciting |1〉 to Rydberg states, the other qubit state |0〉 can also be excited. Moreover, because the dipole matrix elements between the ground state and a Rydberg state is rather tiny, a two-photon Rydberg excitation is more favorable. A standard way to achieve this is to use a largely detuned intermediate p-orbital state |p〉 for Rydberg excitation. To avoid using ultraviolet light of too short wavelengths, the lowest ¹P₁ or ³P_0,1,2 states can be used as the intermediate state. But for the lowest ¹P₁ state whose excitation from the ground state is spin and dipole allowed, it is difficult to find available laser powers to compensate the fast dissipation of ¹P₁. For the lowest ³P_0,1,2 manifold, it is unclear whether the transition can be fast enough with available technology.

The third difficulty lies in that weak magnetic fields are preferred in experiments when Rydberg states need to be coherently excited. Although strong magnetic fields can in principle lead to a GHz-scale ∆, they are incompatible with Rydberg atom quantum science. First, Rydberg excitation of neutral atoms requires sending laser fields to the vacuum chamber in various directions that cause trouble in setting up devices for generating strong magnetic fields. Second, qubits are often left in free flight during Rydberg excitation so that an inhomogeneous distribution of B leads to an extra dephasing [81, 82], and the gradient of B along the motional direction of the qubits can be larger when stronger fields are employed. The magnitudes of B in Rydberg gate experiments by alkali-metal atoms were 9, 11.5, 3.7, 1.5, 4.8, 3, 1.5, 7.5, 8.5, and 6 G in Ref. [14–23], respectively. In a recent experiment with AEL qubits, a 710 G field was used for state initialization [25], but B was switched to 71 G during Rydberg excitation [25].

3 Selective Rydberg excitation of a nuclear spin state of ¹⁷¹Yb

3.1 Theory 1

Theory 1 is shown in Fig. 1(a) with π polarized laser fields where |1〉 is resonantly excited. The complicated level scheme of AEL requires extra efforts to design the scheme to excite a nuclear spin state.

For ¹⁷¹Yb, there is no quadrupole interaction, and the Hamiltonian for the Zeeman interaction in the presence of a magnetic field B is

\begin{array}{}\text{(1)}& \widehat{H}=A\widehat{\mathit{I}}\cdot \widehat{\mathit{J}}+\mathit{B}\cdot (g_J{\mu }_B\widehat{\mathit{J}}-g_I{\mu }_n\widehat{\mathit{I}})\mathrm{,}\end{array}

where A is the nuclear magnetic dipole constant, \widehat{I} and \widehat{J} are the nuclear and electron spin operators, respectively, and g_I(J) is the nuclear (electron) g factor. Here, µ_B is the Bohr magneton and µ_n is the nuclear magneton, which is equal to µ_n = 0.4919µ_N for ¹⁷¹Yb [83] and µ_N ≈ µ_B/1836. Because different electron orbits have different contacts with the nucleus, different states have different values of A [84–87]. For the ground state the total electron angular momentum is zero, and the only term left in Eq. (1) is the Zeeman splitting g_I µ_nB_z between |0〉 and |1〉. Larger fields can be used for optical pumping during the state initialization, afterward the field can be lowered, as in Ref. [25]. A/(2π) is 3.958 GHz [88] for the (6s6p)³P₁ state, and is−0.21 GHz [87] for the (6s6p)¹P₁ state (the sign of A is delicate in that some previous measurement suggested a positive A [84]). For a highly excited Rydberg electron, its contact with the nucleus is negligible, and the hyperfine interaction of the Rydberg atom is mainly between the nucleus and the inner valence electron, with A/(2π) ≈ 12.6 GHz according to the measurement in Ref. [89]. With n ≈ 70, the analysis in Appendix B shows that A\widehat{I}\cdot \widehat{J} couples the |(6sns)¹S₀, F = I〉 and |(6sns)³S₁ F = I〉 states (when F = 1/2) to form two new eigenstates |λ±〉 while |λ₀〉 = |(6sns)³S₁ F = I + 1〉 remains uncoupled. This leads to three states {|λ₀〉, |λ₊〉, |λ−〉} with energies 2π ×(6.3, 4.8,−6.1) GHz in reference to the unperturbed energy of the ¹⁷⁴Yb |(6sns)¹S₀〉 state. For n∼70, Appendix B shows that when we choose |λ₊〉 as |r〉 a MHz-scale Rydberg Rabi frequency can safely avoid the leakage to other nearby states.

An appropriate intermediate state must have two properties. First, it should possess balanced dipole matrix elements to the ground and Rydberg states. Second, large enough laser power at the transition wavelengths should be available to realize coherent excitations. Several choices are at hand. The (6s6p)¹P₁ state, which is 2π × 751.5 THz over the ground state, can have a strong coupling with the ground state, but it has a short lifetime about 5 ns [90]. To avoid dissipation from the intermediate state, it should be largely detuned, which makes it challenging to achieve a MHz-scale two-photon Rydberg Rabi frequency. A similar issue exists for (6s7p)¹P₁, which is 2π × 1216.1 THz over the ground state and has a short lifetime about 10 ns. The (6s6p)³P₁ state is 2π × 539.4 THz over the ground state and has a lifetime about 800 ns [90]. This makes it a useful choice because a smaller detuning at the intermediate state is enough, so that a MHz-scale two-photon Rabi frequency becomes possible as shown in Appendix D. In principle, the clock state cannot be coupled with the ground state, but due to mixing with the (6s6p)¹P_1^0 by spin–orbit interaction [91] that results in (see Appendix A)

\begin{array}{}\text{(2)}& |(6s6p{)}^3{P}_1\rangle =a|(6s6p{)}^3{P}_1^0\rangle +b|(6s6p{)}^1{P}_1^0\rangle \mathrm{,}\end{array}

it becomes possible to be coupled to the ground state, where the superscript 0 denotes pure Russell-Saunders coupling and (a, b) =(0.991, –0.133) [92].

The spin–orbit coupled state in Eq. (2) is the key feature that enables our theories. The dipole matrix element between the ground (Rydberg, with n = 70) and the intermediate states is on the order of 0.1 (10⁻³)ea₀, as shown in Appendixes B, C, and D. So, fast enough Rabi rotation between the qubit state and |(6s6p)³P₁〉 is feasible even though |b| is only 0.133. On the other hand, a is near 1, and the dipole coupling between |(6s6p)³P₁〉 and |r〉 can reach a large value for n = 70. The laser wavelengths of about 556 and 308 nm for the lower and upper transitions in Fig. 1 are available with commercial or homemade lasers, where the 556 nm light source has been widely used in spectroscopy [88, 93] or atomic trapping and cooling [28, 31, 94]. As for the UV lasers [14, 65], laser systems with wavelenghs tunable in the range 304– 309 nm [95] were frequently used for exciting Rydberg states of a strontium ion [95–97]. Detailed analysis in Appendix D shows that with technically available resources, the effective Rabi frequency between |1〉 and |r〉 can be Ω_1r/(2π) = 1.4 MHz, and meanwhile the Rabi frequency for the transition between |0〉 and |r′〉 = (6s70s)³S₁ F = 3/2, m_F = 1/2〉 is−Ω_1r.

A π pulse with duration t_π can complete the transition |1〉→−i|r〉. To avoid exciting |0〉 to the Rydberg state, it is desirable to realize a generalized Rabi frequency \overline{\text{Ω}}= \sqrt{{\text{Ω}}_{1r}^2+{\text{Δ}}^2} which is 2N times Ω_1r, where N is an integer. As can be easily proved [34, 98], with a duration t_π of the Rydberg lasers, the input state |0〉 undergoes N detuned Rabi cycles, evolving to

\begin{array}{}\text{(3)}& |0\rangle \to {\text{e}}^{\text{i}\alpha }|0\rangle \mathrm{,}\end{array}

where α≡−[N +∆/(2Ω_1r)]π. Note that to implement the Rydberg gate with Rydberg interaction V, there is a phase shift β = π|Ω_1r |/(2V) to the input state |11〉 because of the blockade error as studied in Refs. [99, 100], extra phase shifts to the laser fields are required to recover a controlled-Z (C_Z) gate. More than that, because when the control qubit state |1〉 is excited to |r〉, the off-resonant excitation of |0〉 to |r′〉 in the target qubit results in a similar phase shift β′ = π|Ω_1r |/[2(V +∆)] for the two-qubit input state |10〉. So, based on the Rydberg blockade gate and following Ref. [99], we modify the standard π − 2π−π pulse sequence of Ref. [8] by replacing the 2π pulse on the target qubit with two π pulses, where the latter π pulse has a phase shift 2α relative to the the first one. Then, we realize the state transform

\begin{array}{}\text{(4)}& \begin{array}{ll}|00\rangle \hfill & \to {\text{e}}^{\text{i}4\alpha }|00\rangle \mathrm{,}\hfill \\ |01\rangle \hfill & \to -{\text{e}}^{\text{i}4\alpha }|01\rangle \mathrm{,}\hfill \\ |10\rangle \hfill & \to -{\text{e}}^{\text{i}{\beta }^{\prime }}|10\rangle \mathrm{,}\hfill \\ |11\rangle \hfill & \to -{\text{e}}^{\text{i}\beta }|11\rangle \mathrm{.}\hfill \end{array}\end{array}

Afterwards, single-qubit phase gates can be applied to the control qubit for |0〉→e−i4α|0〉 and |1〉→e−iβ |1〉 so that a quasi controlled-Z (C_Z) gate is realized with the gate matrix U =diag{1, −1,−e^{i(β′−β)},−1}, which differs a little bit from the standard Rydberg C_Z gate \mathcal{U} =diag{1,−1,−1,−1}. In the standard Rydberg blockade gate [10] studied here, we need V≫Ω_1r, so β′−β is a tiny number because ∆∼Ω_1r. Averaging equally over the four possible input states |00〉, |01〉, 10〉, and |11〉 and using the fidelity formula in Ref. [101],

\begin{array}{}\text{(5)}& \frac{1}{20}\left[|\text{Tr}(U^{†}\mathcal{U}{)|}^2+\text{Tr}(U^{†}\mathcal{U}{\mathcal{U}}^{†}U)\right]\mathrm{,}\end{array}

the error due to different phases accumulated in |10〉 and |11〉 is

\begin{array}{}\text{(6)}& \begin{array}{ll}{E}_{\text{ph}}\hfill & =0.3[1-\cos ({\beta }^{\prime }-\beta )]\hfill \\ \hfill & \approx 0.15({\beta }^{\prime }-\beta {)}^2\mathrm{.}\hfill \end{array}\end{array}

The error in Eq. (6) is less than 10−5 for the parameters shown later.

Theory 1 can be realized with small magnetic fields. To have, for example, N = 2, a magnetic field B ≈ 3.9 G (we use g_J = g_L for brevity; the g factor depends on the choice of |r〉 shown in Appendix B) can be used to induce a Zeeman shift \text{Δ}=\sqrt{15}{\text{Ω}}_{1r} , so that the generalized Rabi frequency \overline{\text{Ω}}=\sqrt{{\text{Ω}}_{1r}^2+{\text{Δ}}^2}=4{\text{Ω}}_{1r} is realized. The smallness of B is of vital importance since during Rydberg excitation the qubit is released, experiencing free flight and, hence, different magnetic fields. The smaller the magnetic field generated, the smaller its fluctuation.

3.2 Theory 2

Theory 2 is shown in Fig. 1(b) where the same symbol | r〉 is used though m_F is different from that in Fig. 1(a). The two-photon transition |1〉→|r〉 is resonant with right-hand polarized fields. As a consequence there is no way to couple |0〉 to (6s70s)³S₁ since it does not have a m_F = 5/2 state. As shown in Appendix E, the angular coupling coefficients for |1〉→|r〉 are \sqrt{3} times those in Section 3.1. This basically means that if the field amplitudes are the same for the corresponding lasers used in Figs. 1(a) and (b), the achievable Rabi frequency for exciting |1〉 to |r〉 can be Ω_1r/(2π) = 2.4 MHz, leading to a Rydberg-state decay error 7π/(4Ω_1r) ≈ 1.1 × 10⁻³. Because there is a phase shift to |11〉 due to the imperfect blockade, we follow Ref. [99] to modify the standard π−2π−π pulse sequence of Ref. [8] by replacing the 2π pulse on the target qubit with two π pulses, where the second pulse has a phase shift β relative to the the first one. Moreover, as shown later, because the qubit state |0〉 undergoes highly off-resonant excitation to the intermediate state, there is a certain phase accumulation θ in |0〉 for each π pulse used for |1〉↔|r〉. Then, Theory 2 can realize the gate

\begin{array}{}\text{(7)}& \begin{array}{ll}\hfill & |00\rangle \to {\text{e}}^{\text{i}4\theta }|00\rangle \mathrm{,}\hfill \\ \hfill & |01\rangle \to -{\text{e}}^{\text{i}(2\theta +\beta )}|01\rangle \mathrm{,}\hfill \\ \hfill & |10\rangle \to -{\text{e}}^{\text{i}2\theta }|10\rangle \mathrm{,}\hfill \\ \hfill & |11\rangle \to -{\text{e}}^{\text{i}\beta }|11\rangle \mathrm{,}\hfill \end{array}\end{array}

where θ ≈ 0.084π as shown later in Section 3.3 and Fig. 3(b). Different from Eq. (4), the state |10〉 does not have a blockade-induced phase shift because the qubit state |0〉 can only be excited to the largely detuned intermediate state. Single-qubit phase gates can be applied to both qubits for the transform |0〉→e^−i2θ |0〉, and an extra single qubit gate |1→e^−iβ |1〉 can be applied to the target qubit to recover a C_Z gate. The gate by Theory 2 has no phase error as in Eq. (6).

Theory 2 differs from Theory 1 in that there is a highly off-resonant excitation from |0〉 to the intermediate ³P ₁ state. With the estimate in Appendix E, the ratio between the single-photon Rabi frequency and the detuning to the intermediate state is about 0.01, which leads to an error less than 5 × 10⁻⁵ due to the scattering at ³P₁.

The gates in Eqs. (4) and (7) rely on the Rydberg blockade mechanism. To estimate the van der Waals interaction between AEL Rydberg atoms [102], we use quantum defects in Ref. [103] and radial integration of Ref. [104], and calculate [105] C₆/(2π) = 32 GHz·µm⁶ for two (6s70s)¹S ₀ atoms lying along the quantization axis. The interaction for atoms in the ³S₁ states should be much larger because they have nine dipole-dipole transition channels for each set of principal quantum numbers, while two atoms in ¹S₀ only have one, i.e., it couples with {|}^1{P}_1^1{P}_1\rangle . But the quantum defects for the ³P_0,1,2 Rydberg states of ytterbium are not available. To have a conservative estimate (also because we can choose either (6s70s)³S ₁ or |λ₊〉 as |r〉, see Appendix B), we suppose that the C₆ coefficient of |r〉 is only six times that of (6s70s)¹S₀, and, thus, with a qubit spacing 4 µm, the interaction is V = 2π × 47 MHz, which can validate the blockade mechanism with Ω_1r/(2π) = 1.4 (2.4) MHz in Eqs. (4) [(7)] with a blockade error [81]

\begin{array}{}\text{(8)}& \begin{array}{l}{E}_{\text{bl}}=({\text{Ω}}_{1r}^2/8)[1/V^2+1/(V+\text{Δ}{)}^2]\mathrm{,}\hfill \end{array}\end{array}

for Theory 1, and

\begin{array}{}\text{(9)}& E_{\text{bl}}={\text{Ω}}_{1r}^2/(8V^2)\mathrm{,}\end{array}

for Theory 2. That there is an extra blockade error in Eq. (8) is because for Theory 1, the input state |10〉 also experiences blocked Rydberg excitation, but with a detuning V +∆. Then, we find that E_bl is about 2.0 (3.3)×10⁻⁴ for Theory 1 (2).

3.3 Numerical test

The Hamiltonian for the input state |1〉 in Theory 1 is

\begin{array}{}\text{(10)}& {\widehat{H}}_1=\frac{1}{2}(\begin{array}{cccc}2{\mathbb{D}}_r& {\text{Ω}}_3^{(1)}& {\text{Ω}}_1^{(1)}& 0\\ {\text{Ω}}_3^{(1)}& 2{\delta }_1+2{\delta }_2& 0& {\omega }_3^{(1)}\\ {\text{Ω}}_1^{(1)}& 0& 2{\delta }_1& {\omega }_1^{(1)}\\ 0& {\omega }_3^{(1)}& {\omega }_1^{(1)}& 2{\mathbb{D}}_1\end{array})\end{array}

written in the basis {|r〉, |3/2〉, |1/2〉, |1〉} where |r〉≡ |(6s70s)³S₁, F = 3/2, m_F = −1/2〉, and |3(1)/2〉 are the shorthand notations for the states |(6s6p)³P₁, F = 3(1)/2, m_F = −1/2〉. Here, δ₁ is the detuning given by the frequency of the lower laser field minus the frequency of the atomic transition and−δ₂ is the hyperfine gap of ³P₁. The pulse duration is t_π = π/Ω_1r, where {\text{Ω}}_{1r}=\frac{{\omega }_1^{(1)}{\text{Ω}}_1^{(1)}}{2{\delta }_1}+\frac{{\omega }_3^{(1)}{\text{Ω}}_3^{(1)}}{2({\delta }_1+{\delta }_2)} . With experimentally achievable laser field strengths as specified in Appendix D, we can have ({\omega }_1^{(1)}\mathrm{,}{\text{Ω}}_1^{(1)}\mathrm{,}{\delta }_1)/(2\pi ) = (−39.9, 103, 2970) MHz, δ₂ = −2δ₁, {\omega }_3^{(1)}=\sqrt{2}{\omega }_1^{(1)}\mathrm{,}{\text{Ω}}_3^{(1)}=-{\text{Ω}}_1^{(1)}/\sqrt{2} . Because the adiabatic elimination of the intermediate states, there will be Stark shifts for the states |1〉 and |r〉. To compensate them, we assume that careful calibration is made by, e.g., adding extra highly off-resonant fields [17, 106], to induce effective shifts {\mathbb{D}}_1=-{({\omega }_1^{(1)})}^2/(4{\delta }_1) and {\mathbb{D}}_r= {({\text{Ω}}_1^{(1)})}^2/(8{\delta }_1) .

The Hamiltonian for the input state |0〉 is

\begin{array}{}\text{(11)}& {\widehat{H}}_0=\frac{1}{2}(\begin{array}{cccc}2({\mathbb{D}}_r+\text{Δ})& |{\text{Ω}}_3^{(1)}|& |{\text{Ω}}_1^{(1)}|& 0\\ |{\text{Ω}}_3^{(1)}|& 2{\delta }_1+2{\delta }_2& 0& |{\omega }_3^{(1)}|\\ |{\text{Ω}}_1^{(1)}|& 0& 2{\delta }_1& -|{\omega }_1^{(1)}|\\ 0& |{\omega }_3^{(1)}|& -|{\omega }_1^{(1)}|& 2{\mathbb{D}}_1\end{array})\end{array}

in the basis {|r′〉, |3/2〉, |1/2〉, |1〉}. Here, \text{Δ}=\sqrt{15}{\text{Ω}}_{1r} denotes the Zeeman shift between the states |r′〉 and | r〉 from a magnetic field about 3.9 G; the tiny Zeeman shift between |0〉 and |1〉 is absorbed in ∆ in the rotating frame transformation, and the Zeeman shifts in the intermediate states are small compared to the hyperfine gap, and, thus, are neglected. Details for deriving Eqs. (10) and (11) are given in Appendix D.

Figure 2 shows the simulation results with Eqs. (10) and (11). With a pulse duration t_π, the population transfer is not as accurate as expected because the detunings at the intermediate states |1(3)/2〉 are finite. But with a pulse duration t_o = 0.9994t_π, the population transfer from |1〉 to |r〉 has a fidelity 0.99976, and meanwhile the population in |0〉 remains there with a fidelity 0.99982. After the excitation process shown in Fig. 2, another pulse of the same duration t_o can be used so that the initial states are restored with fidelities 0.99961 and 0.99984 for |0〉 and |1〉, respectively. The bizarre fact that the population error in |1〉 is not twice of the error in the transition |1〉→|r〉 results from a peculiar interference between the different transition pathways. The average Rydberg superposition time is (π/4)(7/Ω_1r + 3Ω_1r/{\overline{\text{Ω}}}^2 ), leading to a decay error about 1.9 × 10⁻³ with Ω_1r/(2π) = 1.4 MHz and τ = 330 µs. With the phase error in Eq. (6), blockade error in Eq. (8), intrinsic rotation error, and the Rydberg-state decay, the infidelity of the C_Z gate is 2.4 × 10⁻³ which is dominated by the Rydberg-state decay. To have a fidelity beyond 0.999, our results suggest that to host qubits in cryogenic chambers is necessary.

In Theory 2, the Hamiltonian for the input state |1〉 is still Eq. (10) but with different parameters because circularly polarized fields are used in Fig 1(b) as detailed in Appendix E. With similar strengths of laser fields used in the simulation for Fig. 2, we have ({\omega }_1^{(1)}\mathrm{,}{\text{Ω}}_1^{(1)}\mathrm{,}{\delta }_1)/(2\pi ) = (56.4, 126, 2970) MHz, δ₂ = −2δ₁, {\omega }_3^{(1)}={\omega }_1^{(1)}/\sqrt{2}\mathrm{,}{\text{Ω}}_3^{(1)}=-\sqrt{2}{\text{Ω}}_1^{(1)} . The required extra detunings at the ground and Rydberg states are {\mathbb{D}}_1={({\omega }_1^{(1)})}^2/(8{\delta }_1) and {\mathbb{\text{D}}}_r=-{({\text{Ω}}_1^{(1)})}^2/(4{\delta }_1) , respectively. For the input state |0〉, it is excited to |3/2〉 with detuning δ₁ + δ₂ and Rabi frequency {\omega }_3^{(1)} = 2π × 49 MHz. By these parameters, the population dynamics for the two input states |0(1)〉 are shown in Fig. 3. As in Fig. 2, we find that a pulse duration t_o = 0.9994t_π, instead of t_π, can lead to a more accurate Rydberg excitation. In sharp contrast to Theory 1, the phase accumulation in the input state |0〉 cannot be neglected. At the end of the pulse we find arg〈0|Ψ〉/π = −0.084. This phase is accumulated due to the detuned Rabi oscillation between the ground state and the intermediate state |3/2〉 with a generalized Rabi frequency \overline{\omega }\equiv \sqrt{{({\omega }_3^{(1)})}^2+{({\delta }_1+{\delta }_2)}^2} [34, 98]. The phase is accumulated in a way similar to that in Eq. (3), but with a much larger \overline{\omega } . In each generalized Rabi cycle, there is a phase accumulation -\pi [1+({\delta }_1+{\delta }_2)/\overline{\omega }] , leading to \pi ({\delta }_1+{\delta }_2+\overline{\omega })/|{\text{Ω}}_{1r}|\approx -0.084\pi for a pulse duration π/|Ω_1r |. This extra phase in |0〉 can be effectively eliminated by the single-qubit phase gates as shown below Eq. (7). So, including the blockade error in Eq. (9), the intrinsic rotation error, and the Rydberg-state decay, the infidelity of the C_Z gate by Theory 2 is 2.0 × 10⁻³ that is again dominated by the Rydberg-state decay.

3.4 Gate error due to the fluctuation of magnetic fields

The fluctuation of magnetic fields can cause frequency jitter in the atomic transitions between the ground, intermediate, and Rydberg states. Both the two qubit states have zero electron angular momentum, and the g factor is simply that of the nuclear spin. Because the nuclear magneton is about 1836 times smaller than the Bohr magneton, the level shifts in the qubit states can be neglected when estimating the gate error from the field fluctuation.

When the magnetic field is exactly equal to the value we desire, the Hamiltonians for the states |1〉 and |0〉 are described by Eqs. (10) and (11), respectively. With a fluctuation σ_B of the magnetic field, Eqs. (10) and (11) become

\begin{array}{}\text{(12)}& \text{Eq}\mathrm{.}(10)-\text{diag}\left\{g_R\mathrm{,}{g}_i\mathrm{,}{g}_i\mathrm{,}{g}_g\right\}{\sigma }_B{\text{μ}}_B/\mathrm{2,}\end{array}

and

\begin{array}{}\text{(13)}& \text{Eq}\mathrm{.}(11)+\text{diag}\left\{g_R\mathrm{,}{g}_i\mathrm{,}{g}_i\mathrm{,}{g}_g\right\}{\sigma }_B{\text{μ}}_B/\mathrm{2,}\end{array}

respectively, where the subscripts R, i, and g distinguish the Rydberg, intermediate, and ground states, and the minus and plus signs in the two lines above originate from the two different Zeeman levels associated with the two qubit states. As discussed above, we set g_g = 0 here since the nuclear spin Zeeman splitting is tiny compared to those of the electrons. The Rydberg g factor g_R depends on the state we choose, shown in Appendix B. For brevity, we take g_R = 1 as an example. The g factors for the two intermediate states should be calculated according to the mixing between the singlet and triplet states shown in Eq. (2), g_p = a²g(³P₁) + b²g(¹P₁) ≈ 1.49 [107–109], where g(³P₁) and g(¹P₁) are 1.5 and 1, respectively.

To quantify the error from the magnetic field fluctuation, we use Eq. (5) by defining \mathcal{U} as the ideal gate matrix, and calculate U for each nonzero σ_B. In the Rydberg blockade gate [8, 10], the blockade error is estimated by averaging gate errors with interactions varying around the desired V. This is because the blockade error can accidentally vanish [58] for a certain V that depends on the impossible condition of absolutely static atoms. Thus, we use Eq. (8) for a useful estimate of the blockade error but assume large enough V here to estimate the error solely from σ_B.

Numerical results with σ_B up to 50 mG are shown in Fig. 4, where the round (rectangular) symbols denote results for Theory 1 (2). One can see that when no field fluctuation occurs, the rotation error is about 5 × 10−4. This is because as shown in Figs. 2 and 3, there are intrinsic rotation errors in the two-photon Rydberg excitations. Figure 4 shows that with a field fluctuation around 10 mG, the rotation error is nearly the same as when there is no field fluctuation. As shown in the Supplemental Materials of Ref. [23], σ_B < 10 mG is achievable with B = 6 G that is of similar magnitude as used here. Even with a large fluctuation σ_B = 50 mG, Fig. 4 shows that our method enables a fidelity over 0.996 for both Theory 1 and Theory 2 when including the phase error in Eq. (6), the blockade error in Eq. (8), and the Rydberg-state decay.

4 Rydberg excitation of nuclear spin qubits with ⁸⁷Sr

It is useful to extend our theories to the case of ⁸⁷Sr which was widely studied in experiments [25, 29–31, 66, 110–114] and theories [91, 102, 115–120]. Because of the different quantum defects in ytterbium and strontium [116], we calculate that the C₆ coefficient of strontium is 20 times that of ytterbium for a certain ¹S₀ Rydberg state. Thus it is possible to set a larger qubit spacing if strontium atoms are used for Rydberg blockade so as to have less crosstalk in the Rydberg lasers. Indeed, the Rydberg blockade interaction of both cold and hot strontium atoms have been experimentally studied [25, 121–123], demonstrating the applicability of strontium in Rydberg atom quantum science.

The key to the shown theories in Section 3 is the hyperfine interactions and spin–orbit couplings. However, such interactions are strikingly different for different AEL atoms. For ⁸⁷Sr with I = 9/2, the qubit states to encode information can be chosen from the states m_I = 7/2 and 9/2, whose degeneracy can be split by a field of the order of 1 G that leads to a kHz-scale Zeeman shift with a nuclear magnetic moment−1.0936µ_N [124]. To coherently excite the nuclear spin states to Rydberg states, however, we find that the lowest ³P₁ state is no longer the best choice for the intermediate state. This is because for strontium, Eq. (2) becomes

\begin{array}{}\text{(14)}& |(5s5p{)}^3{P}_1\rangle =a|(5s5p{)}^3{P}_1^0\rangle +b|(5s5p{)}^1{P}_1^0\rangle \mathrm{,}\end{array}

with b = −0.0286 [91], which is about five times smaller than that for ytterbium. This leads to a rather small Rabi frequency if laser fields of similar magnitudes to those in Section 3 are used.

One solution to the above issue is to use the state |(5s6p)¹P₁〉 as the intermediate state which can be coupled to the ¹S₀ Rydberg states. As detailed in Appendix F, |(5s6p)¹P₁〉 has a linewidth Γ = 2π × 0.3 MHz and it can be coupled to the ground state with a dipole matrix element that is about half of |〈(6s6p)³P₁||er||(6s²)¹S₀〉| of ytterbium. If we choose the |(5s70s)¹S₀〉 state as |r〉 to entangle the qubits, the achievable Rabi frequencies for |1〉→|r〉 are 0.8 times those in Fig. 1 if the same values δ₁₍₂₎ ∼2π × 3 GHz are used. In principle, larger Rydberg Rabi frequencies can be achieved by using smaller detunings. More details are shown in Appendix F. Finally, we note that although the ¹S₀ Rydberg states do not have large Zeeman shifts, they are coupled to the ³S ₁ Rydberg states via the hyperfine interaction. Such a coupling induces a large energy separation on the order of the hyperfine constant, so that we can choose one state as |r〉. The hyperfine induced state mixing was studied in detail in Ref. [114]; see also Appendix B. In this case, a small magnetic field can lead to a MHz-scale ∆ by the electron Zeeman shift although there can be a weak diamagnetic effect in the ¹S₀ state [125].

We calculate that the van der Waals interaction coefficient for the strontium (5s70s)¹S₀ state is C₆/(2π) = −710 GHz·µm⁶ with the quantum defects used in Ref. [116, 119]. Note that if stronger interactions are desired, one can choose the state near n = 63 where Förster resonance occurs [119]. With a qubit spacing of 5 µm the interaction is |V |/(2π) = 45 MHz, which is large enough to validate the blockade gate in the form of Eq. (7). Compared to the ytterbium case studied at the end of Section 3.2, here the qubit spacing is 1 µm larger while the value of V is almost the same. So, using ⁸⁷Sr is more favorable to avoid crosstalk compared to ¹⁷¹Yb.

5 Discussion

We have studied exciting nuclear spin qubit states to Rydberg states for entanglement generation. As for single-qubit gates, their implementation with nuclear spins in AEL atoms has been studied in, e.g., Refs. [71, 78, 80]. It is also an open question whether using Rydberg excitation as introduced in this work can we have fast and accurate single-qubit gates. However, a straightforward extension of Theory 1 to single-qubit gate operation is at hand. In the clock transition from the ground ¹S₀ state to the lowest ³P₀ state, the transition is possible because there is a small mixing of the {}^1{P}_0^0 component in the ³P ₀ state. A detailed analysis in Ref. [91] shows that with a magnetic field applied there will be a differential shift for the transition |¹S₀, m_I〉→|³P₀, m_I〉 considering different m_I. For the case of ⁸⁷Sr, the shift is 110×m_I Hz/G. This basically means that by applying an appropriate B-field we can excite the two nuclear spin qubit states from the ground state to the ³P₀ state, one resonantly with a kHz-scale Rabi frequency Ω₀ while the other with a generalized

Rabi frequency \overline{\text{Ω}} because it is detuned. The detuning can be specifically set so that Ω is N times Ω, where N is an integer. Then, it is possible to induce single-qubit gates based on the exotic selective excitation of nuclear spin qubits as in Theory 1.

The gates in Eq. (4) for Theory 1 and Eq. (7) for Theory 2 belong to the standard Rydberg blockade gate introduced in Ref. [8]. Based on the Rydberg blockade mechanism, there have been various protocols. For example, simultaneous Rydberg excitation of both qubits can lead to entangling gates in the blockade regime. A special advantage of Theory 1 is that both qubit states experience Rydberg excitation, and different ∆ can be easily obtained by different B-fields. It is an open question whether by choosing appropriate ratio between ∆ and the Rydberg Rabi frequencies can we generate a C_Z gate with one step. This should be possible since for alkali-metal atoms, fast C_Z -like gates [22] or CNOT gates [106] can be generated with effective one-pulse sequences.

We have based our discussions on the Rydberg blockade mechanism. It is also of interest to extend our theories to Rydberg gates by the antiblockade method [126, 127] as reviewed in Ref. [128]. Although the antiblockade regime is sensitive to the fluctuation of the Rydberg interactions, the simultaneous excitation of both qubit states to Rydberg state, as shown in Theory 1, can be useful for designing exotic logic gates [98]. With short enough Rydberg superposition times, the error from fluctuation of interactions can be small.

6 Conclusions

We provide two theories to realize Rydberg excitation of one of the two nuclear spin ground states of an AEL atom. We perform a detailed study with ytterbium and briefly study strontium when using our theories. The first theory needs an external magnetic field on the order of 1 G to work, while the second method can work also with larger magnetic fields. We have shown methods to realize C_Z gates based on these two theories and found that fidelities over 0.997 are achievable for both methods with feasible resources. The key advantage of our theories is that strong magnetic fields are not required so that fast dephasing from the noises in the magnetic field can be avoided in practical experiments. Because AEL atoms like ytterbium can be easily cooled, can be cooled without altering the information encoded in the nuclear spin space, and their nuclear spins are insensitive to magnetic fluctuations in the environment, our theories can lead to new opportunities for Rydberg atom quantum science and technology.

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