Fast nuclear-spin gates and electrons−nuclei entanglement of neutral atoms in weak magnetic fields

Abstract

We present a novel class of Rydberg-mediated nuclear-spin entanglement in divalent atoms with global laser pulses. First, we show a fast nuclear-spin controlled phase gate of an arbitrary phase realizable either with two laser pulses when assisted by Stark shifts, or with three pulses. Second, we propose to create an electrons−nuclei-entangled state, which is named a super bell state (SBS) for it mimics a large Bell state incorporating three small Bell states. Third, we show a protocol to create a three-atom electrons-nuclei entangled state which contains the three-body W and Greenberger−Horne−Zeilinger (GHZ) states simultaneously. These protocols possess high intrinsic fidelities, do not require single-site Rydberg addressing, and can be executed with large Rydberg Rabi frequencies in a weak, Gauss-scale magnetic field. The latter two protocols can enable measurement-based preparation of Bell, hyperentangled, and GHZ states, and, specifically, SBS can enable quantum dense coding where one can share three classical bits of information by sending one particle.

Graphical abstract

Keywords

nuclear-spin qubit / electrons−nuclei entanglement / super Bell state / Greenberger−Horne−Zeilinger state / Rydberg-mediated entanglement / quantum dense coding

1 Introduction

Rydberg-mediated entanglement [1, 2] with individual alkali-metal atoms was extensively explored experimentally in the last fourteen years [3–18]. Recently, people demonstrated Rydberg-mediated entanglement in alkaline-earth-like (AEL) atoms, such as ytterbium and strontium [19–23]. Compared to alkali-metal atoms, divalent AEL atoms of the alkaline-earth-metal, lanthanide, or transition-metal species can be easily cooled to low temperatures [19, 24–28] and long-lived trapping of both the ground and Rydberg states is realizable [29]. A special advantage in AEL is that for isotopes with nuclear spins, quantum information stored in nuclear spin states [30, 31] is insensitive to magnetic noise, can be preserved during laser cooling [32, 33], and one neutral atom can host multiple, stable, and controllable nuclear-spin states useful for coding information [31, 34, 35]. But concerning entanglement, no protocols were designed specifically for nuclear spins that exhibit a privilege for nuclear-spin qubits in AEL atoms.

In this paper, we study a class of nuclear-spin entanglement enabled by a novel mechanism for creating unit Rydberg excitation in multiple atoms solely applicable for AEL atoms with nuclear spins. We show protocols to generate two-atom nuclear-spin entanglement that can be executed exceedingly fast, and present protocols for creating two- and three-atom electrons−nuclei entanglement. We further show that the electrons−nuclei entangled state can be used for measurement-based creation of W and Greenberger−Horne−Zeilinger (GHZ) states, and can enable quantum dense coding in which three classical bits of information can be securely shared when only one particle is sent in a public channel. These reveal nuclear-spin-possessing AEL atoms as a versatile platform for exploring functional quantum devices.

1.1 Fast nuclear-spin entanglement

In AEL atoms with nuclear spins, the g-factors of the ground and clock states are mainly of nuclear-spin character [36] which can lead to simultaneous Rydberg excitation of both nuclear-spin qubit states [37]. A useful protocol of entanglement generation is simultaneously exciting one out of the two qubit states in each atom to Rydberg states [11, 18, 38], which means that the other qubit state in each atom is not Rydberg excited. To employ this method with nuclear-spin qubits, one can choose a Rydberg Rabi frequency \mathrm{\Omega } small compared to the Zeeman splitting {\mathrm{\Delta }}_{\mathrm{Z}} in the Rydberg states [20, 23, 35], or use selection rules [37]. Nonetheless, for nuclear spin qubits defined in the clock state, sizable UV-laser Rydberg Rabi frequencies \mathrm{\Omega } can be realized [19, 22, 35] which seems an advantage, but the selection-rule method of [37] cannot work for the clock-Rydberg transition, and using \mathrm{\Omega }\ll {\mathrm{\Delta }}_{\mathrm{Z}} (e.g., the experiments in Refs. [20, 23] had ) would either lengthen the gate duration with Gauss-scale B-fields or bring extra magnetic noise when strong B-fields are used.

Here, we present nuclear-spin entangling gates realizable with a large \mathrm{\Omega } in a weak magnetic field when \mathrm{\Omega } /{\mathrm{\Delta }}_{\mathrm{Z}}⪅1. By exciting the four nuclear spin qubit states in two atoms simultaneously from the ground or clock state to Rydberg states, a two-qubit controlled phase gate of an arbitrary phase is realizable with three laser pulses of total duration about 5.05 \pi / {\mathrm{\Omega }}_{\mathrm{m}}, or with two pulses of total duration about 2.59 \pi / {\mathrm{\Omega }}_{\mathrm{m}} when assisted by Stark shift, where {\mathrm{\Omega }}_{\mathrm{m}} is the maximal Rabi frequency in the pulses. With {\mathrm{\Omega }}_{\mathrm{m}} over 2\pi \times 6 MHz [19, 22, 35], the gate duration can be less than 0.42(0.22) μs for the three (two)-pulse gate, or even shorter for clock-Rydberg Rabi frequencies over 2\pi \times 10 MHz are realizable [19]. Importantly, the gates are compatible with Gauss-scale magnetic fields as in recent nuclear-spin-qubit experiments [20, 23, 39, 40].

1.2 A super Bell state

The nuclear-spin states in both the ground and metastable clock states of AEL atoms can enable a novel entanglement, namely, a simultaneous entanglement between two atoms in the external space, and entanglement between the electrons and nuclei. However, it was not reported how such states can be prepared.

In this paper, we present a three-pulse protocol to realize the following state entangled between the electrons (e) and nuclear spins (n) in two atoms via Rydberg blockade,

(1)| SB⟩\equiv {\displaystyle \frac{1}{\sqrt{2}}}(|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}+ |\mathrm{\Phi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}}),

where

(2)\begin{array}{rl}& |\mathrm{\Phi }{⟩}_{\mathrm{n}}=\frac{|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{2}},\\ & |\mathrm{\Psi }{⟩}_{\mathrm{n}}=\frac{{\mathrm{e}}^{\mathrm{i}\theta }|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\theta }^{\prime }}|\downarrow \downarrow {⟩}_{\mathrm{n}}}{\sqrt{2}}\end{array}

are two orthogonal Bell states entangled in the two nuclear spin states \uparrow and \downarrow, \theta and {\theta }^{\prime } are two angles, and

|\mathrm{\Phi }{⟩}_{\mathrm{e}}= {\displaystyle \frac{|\text{cg}{⟩}_{\mathrm{e}}+|\text{gc}{⟩}_{\mathrm{e}}}{\sqrt{2}}}

is a Bell state entangled in the electronic ground (g) and clock (c) states. Because of the similar structure, |\mathrm{\Phi }{⟩}_{\mathrm{n}} and |\mathrm{\Phi }{⟩}_{\mathrm{e}} are labeled by the same Greek letter. Looking at it as a whole, Eq. (1) is entangled between the two-atom electronic states (|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}, |\mathrm{\Phi }{⟩}_{\mathrm{e}}) and the two-atom nuclear spin states (|\mathrm{\Phi }{⟩}_{\mathrm{n}}, |\mathrm{\Psi }{⟩}_{\mathrm{n}}). For want of a better term, Eq. (1) is called a supper Bell state (SBS) because it is like a large Bell state including three “small” Bell states. To our knowledge, no such exotic two-particle entangled state containing three Bell states was reported.

SBS is prepared by three laser pulses, two UV laser pulses for the clock-Rydberg transition and one laser pulse for the ground-Rydberg transition. The three pulses have a total duration 3.4\pi /{\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}} (or 7.7\pi /\mathrm{\Omega }), where {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}} is the Rabi frequency for the ground-Rydberg transition and \mathrm{\Omega } is the largest UV laser Rabi frequency among the two UV laser pulses for the clock-Rydberg transition.

SBS can enable quantum dense coding [41]. By sending one of the two particles in the state of SBS, three bits of information can be shared as shown in Section 7.

1.3 A three-atom state including W and GHZ states

Our theory can be used to realize the following state

(3)|▴⟩={\displaystyle \frac{1}{2}}[( \sqrt{3}|\mathrm{c}\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes \mathit{\text{₩}}{⟩}_{\mathrm{n}}+|W{⟩}_{\mathrm{e}}\otimes |\mathrm{G}\mathrm{H}\mathrm{Z}{⟩}_{\mathrm{n}})],

which has rich entanglement in three atoms, where

\begin{array}{rl}|\mathit{\text{₩}}{⟩}_{\mathrm{n}}=& \frac{1}{\sqrt{6}}[|\uparrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\uparrow \downarrow \uparrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}\mathrm{\Theta }}(|\uparrow \downarrow \downarrow {⟩}_{\mathrm{n}}\\ & +|\downarrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow \uparrow {⟩}_{\mathrm{n}})]\end{array}

is the sum of two different nuclear-spin W states with a relative phase \mathrm{\Theta },

|W{⟩}_{\mathrm{e}}= {\displaystyle \frac{1}{\sqrt{3}}}\left(|\text{gcc}{⟩}_{\mathrm{e}}+|\text{cgc}{⟩}_{\mathrm{e}}+|\text{ccg}{⟩}_{\mathrm{e}}\right)

is an electronic W state [42, 43] which is maximally entangled in the ground-clock state space, and

|\mathrm{G}\mathrm{H}\mathrm{Z} {⟩}_{\mathrm{n}}= {\displaystyle \frac{1}{\sqrt{2}}}\left(|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}}\right)

is a Greenberger−Horne−Zeilinger (GHZ) state [44] which is maximally entangled in the nuclear-spin state space.

Like SBS, the state in Eq. (3) can be prepared by two UV laser pulses for the clock-Rydberg transition and one laser pulse for the ground-Rydberg transition with a total duration 3.5\pi /{\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}} (or 11\pi /\mathrm{\Omega }). Because {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}} is in general small [37], the speed for creating |SB⟩ and |▴⟩ is bottlenecked by the available laser powers for realizing the ground-Rydberg transition.

The remainder of this paper is organized as follows. In Section 2, we study a three-pulse protocol to realize a quantum gate in the nuclear spins of the clock or ground states. In Section 3, we study a similar nuclear-spin quantum gate realized by two laser pulses when assisted by Stark shift. In Section 4, we show a three-pulse protocol to create SBS. In Section 5, we show a three-pulse protocol to create a three-atom state which contains W and GHZ states simultaneously. In Section 6, we discuss creation of Bell, hyperentangled, and GHZ states by measuring the states studied in Sections 4 and 5. In Section 7, we show quantum dense coding with SBS. Sections 8 and 9 give discussions and conclusions, respectively.

2 Fast nuclear-spin quantum gates

2.1 A controlled-phase gate of any desired phase

We first show a sequence to realize a nuclear-spin quantum gate of the form,

(4)\begin{array}{rl}& |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\rightarrowtail {\mathrm{e}}^{-\mathrm{i}\alpha }|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩,\\ & |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩\rightarrowtail {\mathrm{e}}^{-\mathrm{i}\beta /2}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩,\\ & |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩\rightarrowtail {\mathrm{e}}^{-\mathrm{i}\beta /2}|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩,\\ & |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\rightarrowtail {\mathrm{e}}^{\mathrm{i}\alpha }|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩,\end{array}

where the first (second) {\mathrm{c}}_{\uparrow (\downarrow )} represents the state of the first (second) atom, \alpha and \beta are angles where \beta is determined by a global laser phase, and

\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\equiv |\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\uparrow {⟩}_{\mathrm{n}},\\ |{\mathrm{c}}_{\downarrow }⟩\equiv |\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\downarrow {⟩}_{\mathrm{n}},\\ |{\text{g}}_{\uparrow }⟩\equiv |\text{g}{⟩}_{\mathrm{e}}\otimes |\uparrow {⟩}_{\mathrm{n}},\\ |{\text{g}}_{\downarrow }⟩\equiv |\text{g}{⟩}_{\mathrm{e}}\otimes |\downarrow {⟩}_{\mathrm{n}},\end{array}

where e (n) denotes the electronic (nuclear spin) state of the atom, \otimes is used because to a good approximation, the electron and nuclear spin are decoupled in both the ground (g) and clock (c) states [35, 37, 45]. The gate in Eq. (4) is a controlled-phase gate because by using the single-qubit phase gates

\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\rightarrowtail {\mathrm{e}}^{\mathrm{i}\alpha /2}|{\mathrm{c}}_{\uparrow }⟩,\\ |{\mathrm{c}}_{\downarrow }⟩\rightarrowtail {\mathrm{e}}^{\mathrm{i}(\beta -\alpha )/2}|{\mathrm{c}}_{\downarrow }⟩,\end{array}

in both atoms [37, 45], the gate in Eq. (4) becomes

(5)\begin{array}{l}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\rightarrowtail |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩,\\ |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩\rightarrowtail |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩,\\ |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩\rightarrowtail |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩,\\ |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\rightarrowtail {\mathrm{e}}^{\mathrm{i}\beta }|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩,\end{array}

where \beta is adjustable by varying laser phases. The case \beta = \pi corresponds to the canonical CZ gate as realized in the Rydberg quantum gate [46]. In principle, protocols used in alkali-metal atoms [47] can also be used for creating the gate in Eq. (5). But for nuclear spins in AEL, two nearby nuclear spin states in either the ground or clock states are nearly degenerate in a weak B-field, so that both nuclear spin qubit states can be excited to Rydberg states [37]. To use protocols tested with alkali-metal atomic hyperfine qubits for entangling AEL nuclear-spin qubits, one may use strong magnetic fields to suppress the Rydberg excitation of the nontarget nuclear spin states [35]. The benefit of using strong magnetic fields is that the polarization of the laser fields can fluctuate without decreasing the gate fidelity too much [35]. On the other hand, strong magnetic fields can lead to large field fluctuation in an array of atoms. A compromise is to use quite small Rydberg Rabi frequency when a weak magnetic field is employed, as in the experiments reported in Refs. [20, 23]. This raises a question whether fast nuclear-spin quantum gates can be created in a weak magnetic field. Below, we first show the physical possibility for our gate protocols and then show a gate realizable with large Rydberg Rabi frequencies in a weak magnetic field.

2.2 Laser excitation of the ground-Rydberg and clock-Rydberg transitions

The gate protocols in this paper depend on exciting the ground and clock states to Rydberg states with MHz-scale Rabi frequencies, for which we analyze in detail below.

The ground state can be excited to a (6s6n){}^3{S}_1 Rydberg state via the largely detuned (6s6 p){}^3{P}_1 state as theoretically analyzed [37] and experimentally verified [20, 48]. The hyperfine interaction can mix the singlet and triplet states [49], but for the F=I+1=3 /2 state there is no mixing [50]. In this paper, we consider exciting the ground or clock states to the F=I+1 manifold of (6s6 n){}^3{S}_1 state with n\sim 70, for which the nearby Rydberg states (of different F) are separated by more than 1 GHz [37], which is orders of magnitude larger than the Rydberg Rabi frequency \mathrm{\Omega } considered in this paper. So, we can ignore the excitation of the nearby Rydberg states. Reference [37] showed that a ground-Rydberg Rabi frequency 2\pi \times 1.4 MHz can be realized for a Rydberg state of principal quantum 70. As for the clock state, the experiment of Ref. [19] excited the clock state of strontium to a Rydberg state of n=61 with a Rabi frequency up to 2\pi \times 13 MHz, and large Rabi frequencies were possible as shown in detailed analyses [35].

We consider a Gauss-scale magnetic field for specifying the quantization axis, so that the two Zeeman substates |\uparrow ⟩\equiv |m_I=1 /2 ⟩ and |\downarrow ⟩\equiv |m_I=- 1/ 2⟩ in the ground (or clock) state can be assumed degenerate [37]. In the F=3/2 level of the (6s6 n){}^3{S}_1 Rydberg state, there is a frequency separation 2\mathrm{\Delta }\approx 2\pi \times 1.9 B MHz/G [35] between the two hyperfine substates |r_{\pm }⟩\equiv (6s6n){}^3{S}_1 |F= 3/ 2,m_F=\pm 1/2⟩, where B is the magnetic field in units of Gauss.

According to the angular momentum selection rule, the Rabi frequencies {\mathrm{\Omega }}_{\mathrm{g}\uparrow (\downarrow )} for the ground-Rydberg transition via the (6s6 p){}^3{P}_1 state, and the Rabi frequencies {\mathrm{\Omega }}_{\mathrm{c}\mathrm{r}\uparrow (\downarrow )} for the clock-Rydberg transition satisfy the condition [37]

(6)\begin{array}{rl}& {\mathrm{\Omega }}_{\mathrm{g}\uparrow }=-{\mathrm{\Omega }}_{\mathrm{g}\downarrow },\\ & {\mathrm{\Omega }}_{\mathrm{c}\uparrow }={\mathrm{\Omega }}_{\mathrm{c}\downarrow },\end{array}

where the ground-Rydberg transition is used in the creation of |SB⟩ and |▴⟩. Note that Eq. (6) does not mean that the four entanglement protocols are limited to the forms shown. The nuclear-spin quantum gates in Sections 2.3 and 3 can also be executed by ground-Rydberg transitions for nuclear spins in the ground state, and other forms of |SB⟩ and |▴⟩ can be prepared with the “g” and “c” states exchanged in the equations defining them.

For the nuclear-spin gates and the first two pulses of the entanglement protocols in Sections 4 and 5, the laser is tuned to the middle of the gap between the two Rydberg states |r_{+}⟩ and |r_{-}⟩ as shown in Fig.1, i.e., the “detuning” of the Rydberg lasers are \mathrm{\Delta } and -\mathrm{\Delta } for |r_{+} ⟩ and |r_{-}⟩, respectively, where the “detuning” is given by the transition frequency deducted by the laser frequency (which has a sign difference compared to the usual definition of detuning for the sake of convenience). The magnetic field is fixed, while the laser fields can be tuned so that Rydberg Rabi frequencies can be varied between pulses.

2.3 A three-pulse sequence

We consider a controlled-phase gate with three sequential laser pulses sent to the two atoms. The Hamiltonians for the gate are shown in Appendix A. According to Refs. [51, 52] and the Rydberg blockade condition [1], by applying a pulse of duration {\mathbb{T}}_1=2 \pi / \sqrt{{\mathrm{\Delta }}^2+2{\mathrm{\Omega }}_x^2}, where {\mathrm{\Omega }}_{\mathrm{c}\uparrow }= {\mathrm{\Omega }}_{\mathrm{c}\downarrow }\equiv {\mathrm{\Omega }}_x [according to Eq. (6)] is the Rabi frequency in the x-th pulse, x=1,2, or 3, we have the state map

\begin{array}{rl}& |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\stackrel{}{\to }{\mathrm{e}}^{-\mathrm{i}\pi (1+\mathrm{\Delta }/\sqrt{{\mathrm{\Delta }}^2+2{\mathrm{\Omega }}_x^2})}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩,\\ & |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\stackrel{}{\to }{\mathrm{e}}^{-\mathrm{i}\pi (1-\mathrm{\Delta }/\sqrt{{\mathrm{\Delta }}^2+2{\mathrm{\Omega }}_x^2})}|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩.\end{array}

For the first pulse, we find that when

\begin{array}{l}{\mathrm{\Omega }}_1/\mathrm{\Delta }=1.6088,\end{array}

the states |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩ and |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩ are excited to states with unit Rydberg excitation. For example, driven by the Hamiltonian ({\mathrm{\Omega }}_{\mathrm{c}\uparrow }|r_{+} ⟩⟨{\mathrm{c}}_{\uparrow }|+{\mathrm{\Omega }}_{\mathrm{c}\downarrow }|r_{-}⟩ ⟨{\mathrm{c}}_{\downarrow }| +\text{H.c.}) /2 +\mathrm{\Delta }(| r_{+}⟩⟨ r_{+}|-|r_{-}⟩⟨r_{-}|) for each atom, |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩ evolves to

(7)({\mathrm{e}}^{\mathrm{i}{\phi }_{+}}|r_{+} {\mathrm{c}}_{\downarrow }⟩+{\mathrm{e}}^{\mathrm{i}{\phi }_{-}}|{\mathrm{c}}_{\uparrow }{r}_{-}⟩)/\sqrt{2},

where {\phi }_{+}=2.645 rad and {\phi }_{-}=0.497 rad. To map the state down to ground states, we find that it is necessary to change the phase of the Rydberg state, which is why we apply a second pulse in the condition

\begin{array}{l}{\mathrm{\Omega }}_2/\mathrm{\Delta }=0.5932{\mathrm{e}}^{\mathrm{i}3\pi /4}.\end{array}

After the second pulse, the state in Eq. (7) becomes {\mathrm{e}}^{-\mathrm{i}{\phi }_{+}}|r_{+} {\mathrm{c}}_{\downarrow }⟩+{\mathrm{e}}^{-\mathrm{i}{\phi }_{-}}|{\mathrm{c}}_{\uparrow }{r}_{-}⟩)/\sqrt{2}. Then, with a third pulse of condition

\begin{array}{l}{\mathrm{\Omega }}_3={\mathrm{\Omega }}_1{\mathrm{e}}^{\mathrm{i}\beta /2},\end{array}

a gate in Eq. (4) is realized with \alpha =1.793 rad in the ideal blockade condition. If \beta =\pm \pi is chosen, a CZ gate is realized, for which a numerical simulation is shown in Fig.2 with a finite V. The total duration of the gate is \sum _{x=1}^3{\mathbb{T}}_x\approx \frac{5.054\pi }{{\mathrm{\Omega }}_1}, which is in terms of the maximal Rabi frequency {\mathrm{\Omega }}_1 (or {\mathrm{\Omega }}_3) during the three pulses. If we assume |{\mathrm{\Omega }}_1|/(2\pi )=3.25 MHz (corresponding to \mathrm{\Delta }/(2 \pi )=2.02 MHz in a B-field of 2.1 G), the gate duration would be 0.78 μs. We use |{\mathrm{\Omega }}_x|\le 2\pi \times 3.25 MHz in the numerical example for it is equal to the maximal Rabi frequencies used later on in creating |SB⟩ and |▴⟩ in Sections 3 and 4, but in experiments much larger UV Rydberg Rabi frequencies are realizable since strong UV fields are available; for example, Rydberg Rabi frequencies up to 2\pi \times 13 MHz were realized for exciting a Rydberg state of n=61 with optimization of laser system and beam path [19].

2.4 Numerical analyses

Here, we numerically study the fidelity of the nuclear-spin gate. Because a CZ gate has the maximal entangling power among the two-qubit gates [53], we study the case with \beta = -\pi. Three main factors limit the intrinsic fidelity, the Rydberg state decay, the finiteness of V, and the fluctuation of V. The error due to Rydberg-state decay is [54]

(8)\begin{array}{l}{E}_{\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}}=t_{\mathrm{R}\mathrm{y}\mathrm{d}}/\tau ,\end{array}

where t_{\mathrm{R}\mathrm{y}\mathrm{d}} is the Rydberg superposition time and \tau is the lifetime of the Rydberg state. We numerically found t_{\mathrm{R}\mathrm{y}\mathrm{d}}\approx \frac{2.7\pi }{{\mathrm{\Omega }}_1}. If the maximal Rabi frequency |{\mathrm{\Omega }}_1| is 2\pi \times 3.25 MHz, the decay-induced error is E_{\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}}= 4.15\times {10}^{-3} with \tau =100 μs estimated by using the lifetime of a lower state of n=59 [23].

The analyses in Section 2.3 assumed infinite Rydberg interactions, but a blockade error will arise with finite V [52, 54]. The experiment of Ref. [20] estimated that the C_6 coefficient for the |(6 sns{)}^3{S}_1,F=I+1⟩ state with n=50 can be up to 2\pi \times 15 GHz·\text{μ}{\mathrm{m}}^6. With the n^{11} scaling of van der Waals interaction, this would imply C_6/(2\pi )=15\cdot (70/50{)}^{11}\approx 607 GHz·\text{μ}{\mathrm{m}}^6 for the state we consider. To have a more conserved estimate about the gate fidelity in this paper, we use a smaller C_6/(2\pi )=192 GHz·\text{μ}{\mathrm{m}}^6 from the analyses of Ref. [37]. We consider an atomic separation of 3 μm (this short atomic separation is possible since a value 2 μm was used in the experiment of Ref. [18]), for which we have V_0/(2\pi ) \approx 260 MHz. The gate protocols in this paper do not need single-site addressing since all the pulses simultaneously excite all atoms. So, the small atomic separation does not bring extra difficulty. There is nearly no population loss to the states caused by the finiteness of V as shown in Fig.2. With the finite V, the final phases for the four different input states differ from the desired ones. We numerically found that the protocol maps the states \{| {\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩ ,|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩,|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩ ,|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\} to \{{\mathrm{e}}^{-\mathrm{i}{\alpha }^{\prime }}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩ ,{\mathrm{e}}^{-\mathrm{i}{\beta }^{\prime }/2}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩,{\mathrm{e}}^{-\mathrm{i}{\beta }^{\prime }/2}|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩ ,{\mathrm{e}}^{\mathrm{i}{\alpha }^{″}}|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\} when the interaction is V=V_0, where \{{\alpha }^{\prime },{\beta }^{\prime }/2,{\alpha }^{″}\}\approx \{1.771,-1.587,1.816\} rad. This means that by using |{\mathrm{c}}_{\uparrow }⟩\rightarrowtail {\mathrm{e}}^{\mathrm{i}{\alpha }^{\prime }/2}|{\mathrm{c}}_{\uparrow }⟩ ,|{\mathrm{c}}_{\downarrow }⟩ \rightarrowtail {\mathrm{e}}^{\mathrm{i}(-{\alpha }^{\prime }+{\beta }^{\prime })/2}| {\mathrm{c}}_{\downarrow }⟩, a gate of map diag\{1,1,1,{\mathrm{e}}^{\mathrm{i}({\alpha }^{″}-{\alpha }^{\prime }+{\beta }^{\prime })}\} is realized, where {\alpha }^{″}-{\alpha }^{\prime }+{\beta }^{\prime }\approx -3.128 rad, which differs from -\pi by about 0.0135 rad. To quantify the gate fidelity, we define

(9)\begin{array}{l}\hat{U}=\text{diag}\{{\mathrm{e}}^{\mathrm{i}(\theta -{\alpha }^{\prime })},{\mathrm{e}}^{-\mathrm{i}({\beta }^{\prime }/2+\theta )},{\mathrm{e}}^{-\mathrm{i}({\beta }^{\prime }/2+\theta )},{\mathrm{e}}^{\mathrm{i}({\alpha }^{″}+\theta )}\}\end{array}

as the target gate map, where \theta =-(\pi +{\alpha }^{″}-{\alpha }^{\prime }+{\beta }^{\prime })/4. Then, using the single-qubit phase gates

\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\rightarrowtail {\mathrm{e}}^{\mathrm{i}({\alpha }^{\prime }-\theta )/2}|{\mathrm{c}}_{\uparrow }⟩,\\ |{\mathrm{c}}_{\downarrow }⟩\rightarrowtail {\mathrm{e}}^{-\mathrm{i}({\alpha }^{\prime }-\theta )/2+\mathrm{i}({\beta }^{\prime }/2+\theta )}|{\mathrm{c}}_{\downarrow }⟩,\end{array}

in both atoms one can transform Eq. (9) to the canonical CZ gate.

Beside that the finiteness of V can cause error, the fluctuation of V results in error, too. We define the rotation error by [55]

(10)E_{\mathrm{r}\mathrm{o}}= 1-{\displaystyle \frac{1}{20}}\left[|\text{Tr}({\hat{U}}^{†}\hat{\mathcal{U}}){|}^2+\text{Tr}({\hat{U}}^{†}\hat{\mathcal{U}}{\hat{\mathcal{U}}}^{†}\hat{U})\right],

where \hat{\mathcal{U}}=diag\{{\mathrm{e}}^{\mathrm{i}{\eta }_1},{\mathrm{e}}^{\mathrm{i}{\eta }_2},{\mathrm{e}}^{\mathrm{i}{\eta }_3},{\mathrm{e}}^{\mathrm{i}{\eta }_4}\} is the state transform matrix with a fluctuating V, i.e., a V not equal to V_0. In order to evaluate the effect of fluctuation, we consider the average

(11){\overline{E}}_{\mathrm{r}\mathrm{o}}={\displaystyle \frac{\int E_{\mathrm{r}\mathrm{o}}(V)\mathrm{d}V}{\int \mathrm{d}V}},

where the integration is over V\in [(1-ϵ )V_0,(1+ϵ )V_0]; we use a uniform distribution for it can lead to a larger gate error compared to a Gaussian distribution so that we can estimate the lower bound for the gate fidelity. The results shown in Fig.3 indicate that the fluctuation of V does not result in small fidelities. Even when V tends to deviate from the desired interaction with ϵ=0.8, we numerically found that the fidelity is 1-{\overline{E}}_{\mathrm{r}\mathrm{o}}-E_{\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}}\approx 0.996.

3 Faster gates assisted by Stark shifts

Here, we show that it is possible to realize the nuclear-spin gate in Section 2 with only two global pulses when assisted by Stark shifts. To be consistent with Section 2, we suppose that during the first pulse of the gate we use a rotating frame that transfers a Hamiltonian \hat{\mathbb{H}} to {\mathrm{e}}^{\mathrm{i}t\hat{R}}\hat{\mathbb{H}}{\mathrm{e}}^{-\mathrm{i}t\hat{R}}- \hat{R}\equiv \hat{H} with

\begin{array}{l}\hat{R}=\omega (|r_{+}⟩⟨r_{+}|+|r_{-}⟩⟨r_{-}|),\end{array}

where \omega \pm \mathrm{\Delta }=E_{\pm } is the energy separation (divided by the reduced Planck constant) between the clock state and |r_{\pm }⟩, the energy is measured in reference to that of the clock state, and we ignore the energy separation between |{\mathrm{c}}_{\uparrow }⟩ and |{\mathrm{c}}_{\downarrow }⟩ since it is orders of magnitude smaller than the MHz-scale Rabi frequencies in a Gauss-scale magnetic field. In the second pulse, we assume that there are Stark shifts {\delta }_{\pm } in |r_{\pm }⟩ with the condition {\delta }_{–}-{\delta }_{+}=4 \mathrm{\Delta }, and another rotating frame

\begin{array}{l}{\hat{R}}^{\prime }=(E_{+}+{\delta }_{+}+\mathrm{\Delta })|r_{+}⟩⟨r_{+}|+(E_{-}+{\delta }_{–}-\mathrm{\Delta })|r_{-}⟩⟨r_{-}|\end{array}

is used. In this new frame, the state in Eq. (7) becomes ( {\mathrm{e}}^{\mathrm{i}{\phi }_{+}^{\prime }}|r_{+}{\mathrm{c}}_{\downarrow }⟩ +{\mathrm{e}}^{\mathrm{i}{\phi }_{-}^{\prime }}|{\mathrm{c}}_{\uparrow }{r}_{-}⟩)/\sqrt{2}, where {\phi }_{+}^{\prime }={\phi }_{+}+{\mathbb{T}}_1 ({\delta }_{+}+2\mathrm{\Delta } ) and {\phi }_{-}^{\prime }={\phi }_{-}+{\mathbb{T}}_1( {\delta }_{–}- 2\mathrm{\Delta }). With the condition {\delta }_{–}-{\delta }_{+}=4\mathrm{\Delta }, one can see that {\phi }_{+}^{\prime }- {\phi }_{+}= {\phi }_{-}^{\prime }-{\phi }_{-} which is equal to \kappa = {\mathbb{T}}_1( {\delta }_{+}+2\mathrm{\Delta }). In other words, the frame transfer changes the state in Eq. (7) to

(12)\begin{array}{l}{\mathrm{e}}^{\mathrm{i}\kappa }({\mathrm{e}}^{\mathrm{i}{\phi }_{+}}|r_{+}{\mathrm{c}}_{\downarrow }⟩+{\mathrm{e}}^{\mathrm{i}{\phi }_{-}}|{\mathrm{c}}_{\uparrow }{r}_{-}⟩)/\sqrt{2},\end{array}

while the four two-atom Rydberg states |r_{\pm }{r}_{\pm }⟩ (which appear during the pulse sequence when V is finite) get an extra phase 2\kappa which is accounted for in the numerical simulation. By tuning the laser frequency to be at the middle between |r_{-}⟩ and |r_{+}⟩ and adding a phase \pi +\beta / 2+\kappa to the laser field, namely,

\begin{array}{l}{\mathrm{\Omega }}_2=-{\mathrm{e}}^{\mathrm{i}(\beta /2+\kappa )}{\mathrm{\Omega }}_1,\end{array}

the second pulse is with a Hamiltonian [{\mathrm{\Omega }}_2|r_{+}⟩⟨{\mathrm{c}}_{\uparrow }| +{\mathrm{\Omega }}_2|r_{-}⟩ ⟨{\mathrm{c}}_{\downarrow }| +\text{H.c.}] /2 -\mathrm{\Delta }(| r_{+}⟩⟨ r_{+}|-|r_{-}⟩⟨r_{-}|) after dipole and rotating wave approximations. After the second pulse which has the same duration as the first one, we realize Eq. (4) with \alpha = 0, and a CZ gate is realizable if \beta = \pm \pi is chosen in the laser field. The total gate duration is 2{\mathbb{T}}_1\approx 2.589 \pi / {\mathrm{\Omega }}_1. A numerical simulation of the state dynamics for this two-pulse quantum gate is shown in Fig.4 by assuming {\delta }_{-}= 4\mathrm{\Delta } and {\delta }_{+}=0 (a different choice for the Stark shift does not bring difference to the target gate map).

We turn to the analyses of the gate fidelity with finite Rydberg interactions. With V=V_0, numerical simulation shows that the gate map is diag\{{\mathrm{e}}^{\mathrm{i}\nu }, {\mathrm{e}}^{\mathrm{i}\mu },{\mathrm{e}}^{\mathrm{i}\mu },{\mathrm{e}}^{\mathrm{i}\nu }\}, where (\mu ,\nu )\approx (-1.5583,0.02133 ) rad when \beta = \pi and the state dynamics is shown in Fig.4. The Rydberg superposition time is t_{\mathrm{R}\mathrm{y}\mathrm{d}}\approx 1.47\pi / {\mathrm{\Omega }}_1 which leads to an error E_{\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}}=2.26\times {10}^{-3} with {\mathrm{\Omega }}_1=2\pi \times 3.25 MHz and \tau = 100 μs. To characterize the error from the finiteness and fluctuation of V, we define

(13)\begin{array}{l}\hat{\mathbb{U}}=\text{diag}\{{\mathrm{e}}^{\mathrm{i}(\nu +\vartheta )},{\mathrm{e}}^{\mathrm{i}(\mu -\vartheta )},{\mathrm{e}}^{\mathrm{i}(\mu -\vartheta )},{\mathrm{e}}^{\mathrm{i}(\nu +\vartheta )}\}\end{array}

as the target gate, where \vartheta =\pi /4-(\nu -\mu )/2\approx -0.004 rad. By using the phase gates

\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\rightarrowtail {\mathrm{e}}^{-\mathrm{i}(\nu +\vartheta )/2}|{\mathrm{c}}_{\uparrow }⟩,\\ |{\mathrm{c}}_{\downarrow }⟩\rightarrowtail {\mathrm{e}}^{\mathrm{i}(\nu -2\mu +3\vartheta )/2}|{\mathrm{c}}_{\downarrow }⟩,\end{array}

the map in Eq. (13) can be transformed to the CZ gate. We define the rotation error via Eq. (10) with \hat{U} replaced by the matrix in Eq. (13), and evaluate the average infidelity as in Eq. (11), with the results shown in Fig.5 which shows that with a large relative fluctuation ϵ=0.8 of V, the gate still has a large fidelity 0.997 which is still limited by the Rydberg-state-decay error.

4 Super Bell states

The SBS in Eq. (1) can be prepared starting from the following two-atom state

(14)|\psi (0) ⟩=|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes (|\uparrow \uparrow {⟩}_{\mathrm{n}}+|\uparrow \downarrow {⟩}_{\mathrm{n}}+ |\downarrow \uparrow {⟩}_{\mathrm{n}}+ |\downarrow \downarrow {⟩}_{\mathrm{n}})/2,

where the two-atom state is separable in that for each atom, the electronic state is in the optical clock state and the nuclear spin is in (|\uparrow {⟩}_{\mathrm{n}}+ |\downarrow {⟩}_{\mathrm{n}})/\sqrt{2}. Here, \otimes is used because to a good approximation, the electron and nuclear spin are decoupled in the ground (g) and clock (c) states [35, 37, 45]. In Eq. (1), (\theta ,{\theta }^{\prime }) are dependent on the Rydberg interactions and they approach (0,\pi ) in the ideal blockade condition. Because {}_{\mathrm{e}}⟨\mathrm{c}\mathrm{c}|\mathrm{\Phi }{⟩}_{\mathrm{e}}= {}_{\mathrm{n}}⟨ \mathrm{\Psi }|\mathrm{\Phi }{⟩}_{\mathrm{n}}= 0, one can define a pair of orthogonal two-atom electronic states

\begin{array}{l}|+{⟩}_{\mathrm{e}}\equiv |\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}},\\ |-{⟩}_{\mathrm{e}}\equiv |\mathrm{\Phi }{⟩}_{\mathrm{n}},\end{array}

and a pair of orthogonal two-atom nuclear-spin states

\begin{array}{l}|+{⟩}_{\mathrm{n}}\equiv |\mathrm{\Phi }{⟩}_{\mathrm{n}},\\ |-{⟩}_{\mathrm{n}}\equiv |\mathrm{\Psi }{⟩}_{\mathrm{n}},\end{array}

so that Eq. (1) can also be written as

(15)\begin{array}{l}|SB⟩\equiv {\displaystyle \frac{1}{\sqrt{2}}}(|+{⟩}_{\mathrm{e}}\otimes |+{⟩}_{\mathrm{n}}+|-{⟩}_{\mathrm{e}}\otimes |-{⟩}_{\mathrm{n}}),\end{array}

which is a Bell state entangled between the electronic and nuclear spin states in two atoms. Combining Eqs. (2), (3), and (15), one can find that SBS is a “large” Bell state formed with three “smaller” Bell states and a product state |\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}. As shown in Section 7, SBS can enable novel quantum dense coding [41].

4.1 A three-pulse protocol

We describe a three-pulse sequence for creating SBS.

4.1.1 The first pulse

First, with an ultra-violet (UV) laser excitation of pulse duration {\mathbb{T}}_{\mathrm{p}1}^{(\text{{S}})},

\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\underset{\text{detuned by}\mathrm{\Delta }}{\overset{{\mathrm{\Omega }}_{\mathrm{c}\uparrow }={\mathrm{\Omega }}^{(\mathrm{S})}}{\to }}|r_{+}⟩,\\ |{\mathrm{c}}_{\downarrow }⟩\underset{\text{detuned by}-\mathrm{\Delta }}{\overset{{\mathrm{\Omega }}_{\mathrm{c}\downarrow }={\mathrm{\Omega }}^{(\mathrm{S})}}{\to }}|r_{-}⟩,\end{array}

the following state transform is realized,

(16)\begin{array}{l}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\stackrel{}{\to }{\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(\text{{S}})}}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩,\\ |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\stackrel{}{\to }{\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(\text{{S}})}}|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩,\\ |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩\stackrel{}{\to }|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩,\\ |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩\stackrel{}{\to }|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩,\end{array}

where \hat{H} is the Hamiltonian shown in Appendix A. In the blockade regime, one can easily find, according to the picture of generalized Rabi oscillation [38, 51, 56, 57], that the last two transitions in Eq. (16) can be realized with a pulse duration {\mathbb{T}}_{\mathrm{p}1}^{(\text{{S}})}=2\pi /\sqrt{{\mathrm{\Delta }}^2+0.5[{\mathrm{\Omega }}^{(\mathrm{S})}{]}^2}, where the ratio between \mathrm{\Delta } and \mathrm{\Omega } is determined in Sec. 4.1.2 below.

4.1.2 The second pulse

Second, with a two-photon Rydberg excitation of pulse duration {\mathbb{T}}_{\mathrm{p}2}^{(\text{{S}})}=2\pi /\sqrt{{\mathrm{\Delta }}^2+0.5{\eta }^2[{\mathrm{\Omega }}^{(\mathrm{S})}{]}^2},

\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\underset{\text{detuned by}\mathrm{\Delta }}{\overset{{\mathrm{\Omega }}_{\mathrm{c}\uparrow }=\eta {\mathrm{\Omega }}^{(\mathrm{S})}}{\to }}|r_{+}⟩,\\ |{\mathrm{c}}_{\downarrow }⟩\underset{\text{detuned by}-\mathrm{\Delta }}{\overset{{\mathrm{\Omega }}_{\mathrm{c}\downarrow }=\eta {\mathrm{\Omega }}^{(\mathrm{S})}}{\to }}|r_{-}⟩,\end{array}

the following state transform is realized,

(17)\begin{array}{l}{\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(\text{{S}})}}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\stackrel{}{\to }{\mathrm{e}}^{\mathrm{i}\alpha }{\displaystyle \frac{|r_{+}{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{r}_{+}⟩}{\sqrt{2}}},\\ {\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(\text{{S}})}}|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\stackrel{}{\to }{\mathrm{e}}^{\mathrm{i}(\pi -\alpha )}{\displaystyle \frac{|r_{-}{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{r}_{-}⟩}{\sqrt{2}}},\\ |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩\stackrel{}{\to }|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩,\\ |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩\stackrel{}{\to }|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩,\end{array}

where the Hamiltonian is still in the form as in Appendix A. We find that with

(18)\begin{array}{rl}& |{\mathrm{\Omega }}^{(\mathrm{S})}|/\mathrm{\Delta }=1.608,\\ & \eta =-0.4606,\end{array}

the transitions in Eqs. (16) and (17) can be realized perfectly when V /\mathrm{\Delta } is infinite, where \alpha =-0.8502 (2.291) rad when {\mathrm{\Omega }}^{(\mathrm{S})} is positive (negative). Errors due to finite blockade will be analyzed later.

4.1.3 The third pulse

Before going to the third pulse, it is useful to point out that the description from Section 4.1.1 to Section 4.1.2 is in one rotating frame, and a different rotating frame is used in the third pulse. Details for this change of rotating frames is given in Appendix B, which shows that the frame transform changes the phase \alpha in the first two equations of Eq. (17) to

\begin{array}{l}{\alpha }^{\prime }=\alpha +\mathrm{\Delta }\left[{\mathbb{T}}_{\mathrm{p}1}^{(\text{{S}})}+{\mathbb{T}}_{\mathrm{p}2}^{(\text{{S}})}\right].\end{array}

The third pulse is realized by two-photon Rydberg laser excitation of the transitions

\begin{array}{l}|{\text{g}}_{\uparrow }⟩\stackrel{{\mathrm{\Omega }}_{\mathrm{g}\uparrow }={\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}(1+{\mathrm{e}}^{2\mathrm{i}t\mathrm{\Delta }})}{\to }|r_{+}⟩,\\ |{\text{g}}_{\downarrow }⟩\stackrel{{\mathrm{\Omega }}_{\mathrm{g}\downarrow }=-{\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}(1+{\mathrm{e}}^{-2\mathrm{i}t\mathrm{\Delta }})}{\to }|r_{-}⟩,\end{array}

i.e., two sets of laser fields of equal strength are sent, one resonant with |{\text{g}}_{\uparrow }⟩\leftrightarrow |r_{+}⟩, the other resonant with |{\text{g}}_{\downarrow }⟩\leftrightarrow |r_{-}⟩, each with a Rabi frequency {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}} where eff denotes that it is derived by adiabatic elimination of the intermediate states [37]. According to Eq. (17), there is no Rydberg population in the states evolved from the input states |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩ and |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩, and only the states evolved from the two input states |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩ and |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩ respond to the third pulse. For the input state |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩ which is {\mathrm{e}}^{\mathrm{i}{\alpha }^{\prime }}\frac{|r_{+}{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{r}_{+}⟩}{\sqrt{2}} at the beginning of the third pulse, the Hamiltonian is

\begin{array}{l}{\hat{H}}_{\uparrow \uparrow }={\displaystyle \frac{{\mathrm{\Omega }}_{\mathrm{g}\uparrow }}{2}}{\displaystyle \frac{|r_{+}{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{r}_{+}⟩}{\sqrt{2}}}{\displaystyle \frac{⟨{\text{g}}_{\uparrow }{\mathrm{c}}_{\uparrow }|+⟨{\mathrm{c}}_{\uparrow }{\text{g}}_{\uparrow }|}{\sqrt{2}}}+\text{H.c.}\end{array}

For the input state |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩ which is {\mathrm{e}}^{\mathrm{i}(\pi -{\alpha }^{\prime })}\frac{|r_{-}{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{r}_{-}⟩}{\sqrt{2}}, we have

\begin{array}{l}{\hat{H}}_{\downarrow \downarrow }={\displaystyle \frac{{\mathrm{\Omega }}_{\mathrm{g}\downarrow }}{2}}{\displaystyle \frac{|r_{-}{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{r}_{-}⟩}{\sqrt{2}}}{\displaystyle \frac{⟨{\text{g}}_{\downarrow }{\mathrm{c}}_{\downarrow }|+⟨{\mathrm{c}}_{\downarrow }{\text{g}}_{\downarrow }|}{\sqrt{2}}}+\text{H.c.}\end{array}

We have also shown the above Hamiltonians without using the collective entangled basis states in Appendix C. We find that with

(19)\begin{array}{l}|{\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}|/\mathrm{\Delta }=0.7064,\end{array}

the third pulse leads to the state transform

(20)\begin{array}{rl}& {\mathrm{e}}^{\mathrm{i}{\alpha }^{\prime }}\frac{|r_{+}{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{r}_{+}⟩}{\sqrt{2}}\stackrel{}{\to }{\mathrm{e}}^{\mathrm{i}({\alpha }^{\prime }+\beta )}\frac{|{\text{g}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{\text{g}}_{\uparrow }⟩}{\sqrt{2}},\\ & {\mathrm{e}}^{\mathrm{i}(\pi -{\alpha }^{\prime })}\frac{|r_{-}{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{r}_{-}⟩}{\sqrt{2}}\stackrel{}{\to }{\mathrm{e}}^{\mathrm{i}(\pi -{\alpha }^{\prime }-\beta )}\frac{|{\text{g}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{\text{g}}_{\downarrow }⟩}{\sqrt{2}},\end{array}

with a pulse duration

\begin{array}{l}{\mathbb{T}}_{\mathrm{p}3}^{(\text{{S}})}={\displaystyle \frac{1.688\pi }{\mathrm{\Delta }}},\end{array}

while the states |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩ and |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩ do not evolve. We have \beta =-1.575 rad when {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}} is positive, and \beta =1.567 when {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}} is negative.

By defining nuclear spin up and down states with

(21)\begin{array}{rl}& \text{new}|\uparrow {⟩}_{\mathrm{n}}={\mathrm{e}}^{\mathrm{i}({\alpha }^{\prime }+\beta )/2+\mathrm{i}\pi /4}|\uparrow {⟩}_{\mathrm{n}},\\ & \text{new}|\downarrow {⟩}_{\mathrm{n}}={\mathrm{e}}^{-\mathrm{i}({\alpha }^{\prime }+\beta )/2+3\mathrm{i}\pi /4}|\downarrow {⟩}_{\mathrm{n}},\\ & \text{new}|\mathrm{c}{⟩}_{\mathrm{e}}=-\mathrm{i}|\mathrm{c}{⟩}_{\mathrm{e}},\end{array}

the two final states in Eq. (20) are written as

\begin{array}{l}{\displaystyle \frac{|{\text{g}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{\text{g}}_{\uparrow }⟩}{\sqrt{2}}},\\ {\displaystyle \frac{|{\text{g}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{\text{g}}_{\downarrow }⟩}{\sqrt{2}}}.\end{array}

Looking through Eqs. (16), (17), (20), and the basis definition in Eq. (21), one can see that

(22)\begin{array}{rl}& |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\rightarrowtail \frac{|{\text{g}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{\text{g}}_{\uparrow }⟩}{\sqrt{2}}=\frac{|\text{cg}⟩+|\text{gc}⟩}{\sqrt{2}}\otimes |\uparrow \uparrow ⟩,\\ & |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\rightarrowtail \frac{|{\text{g}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{\text{g}}_{\downarrow }⟩}{\sqrt{2}}=\frac{|\text{cg}⟩+|\text{gc}⟩}{\sqrt{2}}\otimes |\downarrow \downarrow ⟩,\\ & |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩\rightarrowtail |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩=|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\uparrow \downarrow ⟩,\\ & |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩\rightarrowtail |{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\uparrow }⟩=|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\downarrow \uparrow ⟩,\end{array}

is realized with a total duration \sum _{x=1}^3{\mathbb{T}}_{\mathrm{p}\mathrm{x}}^{(\text{{S}})} which is about 4.781\pi /\mathrm{\Delta }, or 7.687\pi /|{\mathrm{\Omega }}^{(\mathrm{S})}|, or 3.377\pi /|{\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}| with the relations shown in Eqs. (18) and (19). If we start from the state in Eq. (14), the transform in Eq. (22) leads to SBS in Eq. (1). A numerical simulation with a practical V is shown in Fig.6.

4.1.4 Another SBS

Using a similar strategy as in Sections 4.1.1-4.1.3, one can find that starting from

\begin{array}{l}|\psi (0)⟩=|\mathrm{g}\mathrm{g}{⟩}_{\mathrm{e}}\otimes (|\uparrow \uparrow {⟩}_{\mathrm{n}}+|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow {⟩}_{\mathrm{n}})/2,\end{array}

the following SBS

(23)\begin{array}{l}|SB{⟩}^{\prime }\equiv {\displaystyle \frac{1}{\sqrt{2}}}(|\mathrm{g}\mathrm{g}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}+|\mathrm{\Phi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}})\end{array}

can be prepared by following the three pulses in Sections 4.1.1, 4.1.2, and 4.1.3 with the UV laser excitation and the two-photon ground-Rydberg excitation exchanged.

The state in Eq. (23) can also be prepared from the SBS in Eq. (1) (with a trivial overall \pi phase). The method is simple, by using the ground-clock state transition

(24)\begin{array}{l}|\text{g}{⟩}_{\mathrm{e}}\leftrightarrow |\mathrm{c}{⟩}_{\mathrm{e}}\end{array}

with a duration \pi / {\mathrm{\Omega }}_{\mathrm{g}\mathrm{c}}, where {\mathrm{\Omega }}_{\mathrm{g}\mathrm{c}} is the Rabi frequency for the ground-clock transition, |\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\to -|\mathrm{g}\mathrm{g}{⟩}_{\mathrm{e}} and |\mathrm{\Phi }{⟩}_{\mathrm{e}}\to -|\mathrm{\Phi }{⟩}_{\mathrm{e}} are realized [56], leading to |SB⟩\to -|SB{⟩}^{\prime }. As analyzed in Ref. [35], {\mathrm{\Omega }}_{\mathrm{g}\mathrm{c}}= 2\pi \times 0.2 MHz can be realized, with which Eq. (23) can be formed from Eq. (1) with a pulse duration 2.5 μs.

4.2 Numerical analyses

Here, we numerically study the fidelity for creating SBS with the pulses of Sections 4.1.1, 4.1.2, and 4.1.3. Three main factors limit the fidelity, the Rydberg state decay, the finiteness of V, and the fluctuation of V.

The Rydberg state decay is inversely proportional to the Rabi frequencies in the protocol. The ground-Rydberg transition requires two-photon excitation, for which the Rabi frequency can be small. As analyzed in Ref. [37], {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}/(2\pi )=1.4 MHz can be realized for exciting a Rydberg state with principal quantum number n\sim 70. During the third pulse of Section 4.1.3, two sets of two-photon transitions shall be used, one resonant with |r_{+}⟩, and the other resonant with |r_{-}⟩. To realize such transitions with one laser source for the ground-intermediate state transition, and another laser source for the intermediate-Rydberg state transition, the transition from the intermediate state to |r_{+}⟩ and |r_{-}⟩ can be realized by dividing the laser beam to two halves for which the frequency of one half shall be shifted by 2\mathrm{\Delta } via pulse pickers. This is experimentally feasible as shown in Ref. [39], where one laser source was used to drive a two-photon Raman transition assisted by several electro−optic modulators and crossed acousto−optic deflectors. This means that if we employ two lasers of similar powers in the analyses of Ref. [37], the available {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}} will be reduced by a factor of \sqrt{2}. For this reason, we assume {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}/(2\pi ) =1 MHz, which corresponds to 2\mathrm{\Delta }/(2 \pi )=2.83 MHz (realized with a B-field of 1.5 G) according to Eq. (19). By Eq (18), the UV laser Rabi frequencies are {\mathrm{\Omega }}^{(\mathrm{S})}/(2\pi )\approx 2.28 MHz for the first pulse, and |\eta |{\mathrm{\Omega }}^{(\mathrm{S})}/(2\pi ) \approx 1.05 MHz for the second pulse. Concerning that the UV laser Rabi frequency can be quite large [35], the above analyses show that the speed for creating SBS is limited by the smaller two-photon ground-Rydberg Rabi frequency. Note, however, that the analyses in Ref. [37] were quite conservative, and in principle higher ground-Rydberg Rabi frequencies should be achievable. The error due to Rydberg-state decay is given by Eq. (8), where t_{\mathrm{R}\mathrm{y}\mathrm{d}} is the Rydberg superposition time. We numerically find t_{\mathrm{R}\mathrm{y}\mathrm{d}}=0.55{\mathbb{T}}_{\mathrm{p}1}+0.57{\mathbb{T}}_{\mathrm{p}2}+0.23{\mathbb{T}}_{\mathrm{p}3}. With {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}/(2\pi )=1 MHz, and {\mathrm{\Omega }}^{(\mathrm{S})} and \mathrm{\Delta } given by Eqs. (18) and (19), we find t_{\mathrm{R}\mathrm{y}\mathrm{d}}= 0.7506 μs, leading to E_{\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}}= 7.51\times {10}^{-3} with \tau =100 μs.

The analyses in Section 4.2 assumed infinite Rydberg interactions, but a finite V leads to population loss to the computational basis states. Moreover, it leads to some population of the two-atom Rydberg states |r_{+}{r}_{+} ⟩,|r_{+}{r}_{-} ⟩,|r_{-}{r}_{+} ⟩, and |r_{-}{r}_{-}⟩ during the laser excitation, which alters the dynamics a little bit and finally leads to phases in the computational states different compared to those in Eq. (20). In other words, the final SBS with a certain V is no longer

\begin{array}{rl}& \frac{1}{2}[\frac{|\text{cg}⟩+|\text{gc}⟩}{\sqrt{2}}\otimes \left({\mathrm{e}}^{\mathrm{i}({\alpha }^{\prime }+\beta )}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}(\pi -{\alpha }^{\prime }-\beta )}|\downarrow \downarrow {⟩}_{\mathrm{n}}\right)\\ & +|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes \left(|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}\right)],\end{array}

but

\begin{array}{rl}& \frac{1}{2}[\frac{|\text{cg}⟩+|\text{gc}⟩}{\sqrt{2}}\otimes \left({\chi }_1{\mathrm{e}}^{\mathrm{i}{\theta }_1}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\chi }_2{\mathrm{e}}^{\mathrm{i}{\theta }_2}|\downarrow \downarrow {⟩}_{\mathrm{n}}\right)\\ & +{\chi }_3{\mathrm{e}}^{\mathrm{i}{\theta }_3}|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes \left(|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}\right)],\end{array}

where {\theta }_1,{\theta }_2, and {\theta }_3 are dependent on V, and {\chi }_1,{\chi }_2, {\chi }_3 should be 1 in the ideal case, but they are smaller than 1 due to the population loss when V is finite. To investigate the rotation error with finite V, we define

(25)\begin{array}{rl}|{SB}_{\theta }⟩=& \frac{1}{2}[\frac{|\text{cg}⟩+|\text{gc}⟩}{\sqrt{2}}\otimes \left({\mathrm{e}}^{\mathrm{i}{\theta }_1}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\theta }_2}|\downarrow \downarrow {⟩}_{\mathrm{n}}\right)\\ & +{\mathrm{e}}^{\mathrm{i}{\theta }_3}|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes \left(|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}\right)],\end{array}

as the target SBS. Since a global phase is trivial, Eq. (25) can be written as Eq. (1) where \theta ={\theta }_1-{\theta }_3 and {\theta }^{\prime }= {\theta }_2- {\theta }_3 in Eq. (2). The reason for the phase {\theta }_1 of \frac{|\text{cg}⟩+|\text{gc}⟩}{\sqrt{2}}\otimes |\uparrow \uparrow ⟩ to differ from {\theta }_2 of \frac{|\text{cg}⟩+|\text{gc}⟩}{\sqrt{2}}\otimes |\downarrow \downarrow ⟩ is that the detunings V\pm 2\mathrm{\Delta } for them are different, shown in Eqs. (A1) and (A2), which result in different state dynamics. With V_0=2\pi \times 260 MHz, we find

\begin{array}{rl}& {\theta }_1=-2.406+\mathrm{\Delta }({\mathbb{T}}_{\mathrm{p}1}+{\mathbb{T}}_{\mathrm{p}2}),\\ & {\theta }_2=-0.697-\mathrm{\Delta }({\mathbb{T}}_{\mathrm{p}1}+{\mathbb{T}}_{\mathrm{p}2}),\\ & {\theta }_3=\mathrm{0.009.}\end{array}

By defining new nuclear-spin and clock states (or, using single-qubit phase gates),

\begin{array}{rl}& \text{new}|\uparrow {⟩}_{\mathrm{n}}={\mathrm{e}}^{-\mathrm{i}(2{\theta }_3-3{\theta }_1-{\theta }_2)/4}|\uparrow {⟩}_{\mathrm{n}},\\ & \text{new}|\downarrow {⟩}_{\mathrm{n}}={\mathrm{e}}^{-\mathrm{i}(2{\theta }_3-{\theta }_1-3{\theta }_2)/4}|\downarrow {⟩}_{\mathrm{n}},\\ & \text{new}|\mathrm{c}{⟩}_{\mathrm{e}}={\mathrm{e}}^{-\mathrm{i}({\theta }_1+{\theta }_2-2{\theta }_3)/2}|\mathrm{c}{⟩}_{\mathrm{e}},\end{array}

Eq. (25) can be formally written as

\begin{array}{l}{\displaystyle \frac{|\text{cg}{⟩}_{\mathrm{e}}+|\text{gc}{⟩}_{\mathrm{e}}}{2\sqrt{2}}}\otimes \left(|\uparrow \uparrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow {⟩}_{\mathrm{n}}\right)+{\displaystyle \frac{|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}}{2}}\otimes \left(|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}\right).\end{array}

However, the target state is still Eq. (25), in which the angles {\theta }_k, k=1,2 ,3 fluctuate when V fluctuates due to the position fluctuation of the two atoms, leading to a rotation error,

(26)\begin{array}{l}{E}_{\mathrm{r}\mathrm{o}}=1-⟨{SB}_{\theta }|\hat{\rho }|{SB}_{\theta }⟩,\end{array}

where \hat{\rho } is the density matrix of the actual state realized with the three-pulse protocol. Because the Rydberg-state decay is analytically approximated above so as to get an upper bound for the error [54], we use unitary dynamics to simulate the blockade error so that Eq. (26) reduces to 1-|⟨\psi | SB⟩{|}^2 where |\psi ⟩ is the final state evolved from Eq. (14). To evaluate the rotation error, we consider the average {\overline{E}}_{\mathrm{r}\mathrm{o}}=\int E_{\mathrm{r}\mathrm{o}}(V) \mathrm{d}V/\int \mathrm{d}V, where the integration is over V\in [(1-ϵ )V_0,(1+ϵ )V_0]. To have a lower bound for the fidelity, we consider a uniform distribution of V instead of a Gaussian distribution. The numerical results for different ϵ are shown in Fig.7. We find that for , the average rotation error {\overline{E}}_{\mathrm{r}\mathrm{o}} is smaller than {10}^{-4}. Remarkably, even with ϵ=0.8, {\overline{E}}_{\mathrm{r}\mathrm{o}} is 2.14 \times {10}^{-4}. Different from entanglement protocols that require matching the Rydberg interaction with laser detuning [58, 59], the robustness here benefits from the intrinsic property of the blockade mechanism [47]. The fidelity \mathcal{F}=1-{\overline{E}}_{\mathrm{r}\mathrm{o}}- E_{\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}} for creating SBS is about 99.2% for ϵ up to 0.8, which is dominated by the Rydberg-state decay. This fidelity should be possible to be improved if larger ground-Rydberg Rabi frequencies are available.

5 A three-atom state including W and GHZ states

5.1 A three-pulse protocol

Here, we show a three-pulse protocol to excite the initial product state

(27)\begin{array}{rl}|\psi (0)⟩=& |\mathrm{c}\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes (|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+|\uparrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\uparrow \downarrow \uparrow {⟩}_{\mathrm{n}}\\ & +|\downarrow \uparrow \uparrow {⟩}_{\mathrm{n}}+|\uparrow \downarrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow \uparrow {⟩}_{\mathrm{n}}\\ & +|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}})/(2\sqrt{2})\end{array}

to realize |▴⟩ in Eq. (3). We will do it briefly for the pulses induce state transform similar to that in Section 4.1.

In the first pulse, a UV laser with a pulse duration {\mathbb{T}}_{\mathrm{p}1}^{(▴)} for the transitions

\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\underset{\text{detuned by}\mathrm{\Delta }}{\overset{{\mathrm{\Omega }}_{\mathrm{c}\uparrow }={\mathrm{\Omega }}^{(▴)}}{\to }}|r_{+}⟩,\\ |{\mathrm{c}}_{\downarrow }⟩\underset{\text{detuned by}-\mathrm{\Delta }}{\overset{{\mathrm{\Omega }}_{\mathrm{c}\downarrow }={\mathrm{\Omega }}^{(▴)}}{\to }}|r_{-}⟩\end{array}

evolves the the input states as |{\mathrm{c}}_x {\mathrm{c}}_y{\mathrm{c}}_z⟩\stackrel{}{\to }{\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(▴)}} |{\mathrm{c}}_x {\mathrm{c}}_y{\mathrm{c}}_z⟩ where x,y,z \in \{\uparrow ,\downarrow \}. Unlike the case for creating SBS, each input state here is in superposition of the clock and Rydberg states after the first pulse. The second pulse for the excitation

\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\underset{\text{detuned by}\mathrm{\Delta }}{\overset{{\mathrm{\Omega }}_{\mathrm{c}\uparrow }={\eta }^{(▴)}{\mathrm{\Omega }}^{(▴)}}{\to }}|r_{+}⟩,\\ |{\mathrm{c}}_{\downarrow }⟩\underset{\text{detuned by}-\mathrm{\Delta }}{\overset{{\mathrm{\Omega }}_{\mathrm{c}\downarrow }={\eta }^{(▴)}{\mathrm{\Omega }}^{(▴)}}{\to }}|r_{-}⟩,\end{array}

of duration {\mathbb{T}}_{\mathrm{p}2}^{(▴)} leads to

(28)\begin{array}{rl}& {\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(▴)}}|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\stackrel{}{\to }\frac{{\mathrm{e}}^{\mathrm{i}{\alpha }^{(▴)}}}{\sqrt{3}}(|r_{+}{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{r}_{+}{\mathrm{c}}_{\uparrow }⟩\\ & +|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }{r}_{+}⟩),\\ & {\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(▴)}}|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩\stackrel{}{\to }\frac{{\mathrm{e}}^{\mathrm{i}(\pi -{\alpha }^{(▴)})}}{\sqrt{3}}(|r_{-}{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{r}_{-}{\mathrm{c}}_{\downarrow }⟩\\ & +|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }{r}_{-}⟩),\end{array}

and

(29)\begin{array}{l}{\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(▴)}}|{\mathrm{c}}_x{\mathrm{c}}_y{\mathrm{c}}_z⟩\stackrel{}{\to }{\mathrm{e}}^{\mathrm{i}{\zeta }^{(▴)}}|{\mathrm{c}}_x{\mathrm{c}}_y{\mathrm{c}}_z⟩,\end{array}

when one of x,y, and z is \downarrow while the other two are \uparrow, and

(30)\begin{array}{l}{\mathrm{e}}^{-\mathrm{i}\hat{H}{\mathbb{T}}_{\mathrm{p}1}^{(▴)}}|{\mathrm{c}}_x{\mathrm{c}}_y{\mathrm{c}}_z⟩\stackrel{}{\to }{\mathrm{e}}^{-\mathrm{i}{\zeta }^{(▴)}}|{\mathrm{c}}_x{\mathrm{c}}_y{\mathrm{c}}_z⟩,\end{array}

when one of x,y, and z is \uparrow while the other two are \downarrow. We find that when

(31)\begin{array}{rl}& |{\mathrm{\Omega }}^{(▴)}|=1.976\mathrm{\Delta },\\ & {\eta }^{(▴)}=-0.3735,\\ & {\mathbb{T}}_{\mathrm{p}1}^{(▴)}=5.933\pi /|{\mathrm{\Omega }}^{(▴)}|,\\ & {\mathbb{T}}_{\mathrm{p}2}^{(▴)}=1.902\pi /|{\mathrm{\Omega }}^{(▴)}|,\end{array}

the transitions in Eqs. (28), (29) and (30) can be realized with a fidelity over 0.995 for each input state. For infinite V/\mathrm{\Delta }, {\alpha }^{(▴)} is -1.036 (2.106) rad when {\mathrm{\Omega }}^{(▴)} is positive (negative), and {\zeta }^{(▴)}=-0.8717 rad. After the frame transform as in Appendix B, the angle {\alpha }^{(▴)} in the state on the right side of Eq. (28) becomes

\begin{array}{l}{\alpha }^{(▴{)}^{\prime }}={\alpha }^{(▴)}+\mathrm{\Delta }({\mathbb{T}}_{\mathrm{p}1}^{(▴)}+{\mathbb{T}}_{\mathrm{p}2}^{(▴)}),\end{array}

while {\zeta }^{(▴)} in Eqs. (29) and (30) remains the same. The third pulse induces a ground-Rydberg transition as in Section 4.1.3. We find that with the condition

(32)\begin{array}{rl}& |{\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{(▴)}|=0.6072\mathrm{\Delta },\\ & {\mathbb{T}}_{\mathrm{p}3}^{(▴)}=1.789\pi /\mathrm{\Delta },\end{array}

the third pulse can restore the population in the states of Eq. (28) back to the ground states, i.e., the states in Eq. (28) evolve as

\begin{array}{rl}& \frac{{\mathrm{e}}^{\mathrm{i}{\alpha }^{(▴{)}^{\prime }}}}{\sqrt{3}}(|r_{+}{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{r}_{+}{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }{r}_{+}⟩)\\ & \stackrel{}{\to }\frac{{\mathrm{e}}^{\mathrm{i}[{\alpha }^{(▴{)}^{\prime }}+{\beta }^{(▴)}]}}{\sqrt{3}}(|{\text{g}}_{\uparrow }{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{\text{g}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }{\text{g}}_{\uparrow }⟩)\\ & ={\mathrm{e}}^{\mathrm{i}[{\alpha }^{(▴{)}^{\prime }}+{\beta }^{(▴)}]}|W{⟩}_{\mathrm{e}}\otimes |\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}},\\ & -\frac{{\mathrm{e}}^{-\mathrm{i}{\alpha }^{(▴{)}^{\prime }}}}{\sqrt{3}}(|r_{-}{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{r}_{-}{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }{r}_{-}⟩)\\ & \stackrel{}{\to }-\frac{{\mathrm{e}}^{\mathrm{i}[-{\alpha }^{(▴{)}^{\prime }}-{\beta }^{(▴)}]}}{\sqrt{3}}(|{\text{g}}_{\downarrow }{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{\text{g}}_{\downarrow }{\mathrm{c}}_{\downarrow }⟩+|{\mathrm{c}}_{\downarrow }{\mathrm{c}}_{\downarrow }{\text{g}}_{\downarrow }⟩)\\ & =-{\mathrm{e}}^{\mathrm{i}[-{\alpha }^{(▴{)}^{\prime }}-{\beta }^{(▴)}]}|W{⟩}_{\mathrm{e}}\otimes |\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}},\end{array}

where {\beta }^{(▴)} =-1.607 (1.534) rad when {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{(▴)} is positive (negative), while the states in Eqs. (29) and (30) remain the same because the third pulse excites the ground-Rydberg transitions. So, starting from the state of Eq. (27), the above pulses realize the state

\begin{array}{rl}& \frac{1}{2}\{\frac{\sqrt{6}}{2}|\mathrm{c}\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes [{\mathrm{e}}^{\mathrm{i}{\zeta }^{(▴)}}\frac{|\uparrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\uparrow \downarrow \uparrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{3}}\\ & +{\mathrm{e}}^{-\mathrm{i}{\zeta }^{(▴)}}\frac{|\uparrow \downarrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{3}}]\\ & +|W{⟩}_{\mathrm{e}}\otimes \frac{{\mathrm{e}}^{\mathrm{i}[{\alpha }^{(▴{)}^{\prime }}+\beta ]}|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}-{\mathrm{e}}^{-\mathrm{i}[{\alpha }^{(▴{)}^{\prime }}+\beta ]}|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}}}{\sqrt{2}})\},\end{array}

in which the relative phase in the nuclear-spin GHZ state can be removed by redefining the basis states. The total duration is about 11.43\pi /|{\mathrm{\Omega }}^{(▴)}| for creating |▴⟩. A numerical test of the above protocol is shown in Fig.8 with relevant Hamiltonians shown in Appendix D.

5.2 Numerical simulations

We consider three atoms forming a triangle in the x−y plane when the laser fields are sent along \mathit{z} [12, 17], and the atoms are in a configuration where the three distances between atom traps are equal as in the experiment [18]. To have equal laser-field strength for the three atoms, the laser is focused at the center of the triangle. When there is no fluctuation of the atomic positions, the interactions between any two Rydberg atoms are equal to the desired value V_0. Even so, the final state has some population loss in the target state mainly due to the finiteness of V, as shown in Fig.8.

The Rydberg-state decay induces an error E_{\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{y}}\approx 2.51\pi /(\tau \mathrm{\Delta }), which is about 7.62\times {10}^{-3} when {\mathrm{\Omega }}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{(▴)}/(2\pi ) =1 MHz and \tau = 100 μs. The fluctuation of V and its finiteness result in gate errors analyzed as follows. We define the state with state phases realized when V=V_0 as the target state

(33)\begin{array}{rl}|{▴}_{\theta }⟩=& \frac{1}{2}\{\frac{\sqrt{6}}{2}|\mathrm{c}\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes [{\mathrm{e}}^{\mathrm{i}{\vartheta }_1}\frac{|\uparrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\uparrow \downarrow \uparrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{3}}\\ & +{\mathrm{e}}^{\mathrm{i}{\vartheta }_2}\frac{|\uparrow \downarrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{3}}]\\ & +|W{⟩}_{\mathrm{e}}\otimes \frac{{\mathrm{e}}^{\mathrm{i}{\vartheta }_3}|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\vartheta }_4}|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}}}{\sqrt{2}}\},\end{array}

where the four angles {\vartheta }_k,k= 1–4 are given in the caption of Fig.8. If we define new nuclear-spin states by absorbing phases to them as

(34)\begin{array}{rl}& \text{new}|\text{g}{⟩}_{\mathrm{e}}={\mathrm{e}}^{\mathrm{i}(2{\vartheta }_4+7{\vartheta }_3-6{\vartheta }_1)/9}|\text{g}{⟩}_{\mathrm{e}},\\ & \text{new}|\mathrm{c}{⟩}_{\mathrm{e}}={\mathrm{e}}^{\mathrm{i}(3{\vartheta }_1+{\vartheta }_3-{\vartheta }_4)/9}|\mathrm{c}{⟩}_{\mathrm{e}},\\ & \text{new}|\downarrow {⟩}_{\mathrm{n}}={\mathrm{e}}^{\mathrm{i}({\vartheta }_4-{\vartheta }_3)/3}|\downarrow {⟩}_{\mathrm{n}},\end{array}

the four phases {\vartheta }_1–{\vartheta }_4 of the state in Eq. (33) become 0,{\vartheta }_2-{\vartheta }_1+({\vartheta }_3-{\vartheta }_4)/3, 0, and 0, respectively. The basis transform is not necessary since in real experiments the interesting part of the state is the electrons−nuclei entanglement. So, the state in Eq. (33) is the target |▴⟩ state. To quantify the error from the fluctuation of V, we define E_{\mathrm{r}\mathrm{o}}=1- ⟨{▴}_{\theta }|\hat{\rho }|{▴}_{\theta }⟩ as the rotation error, where \hat{\rho } is the density matrix of the actual state realized with the three-pulse protocol. With unitary dynamics, the rotation error becomes 1- |⟨\psi |{▴}_{\theta }⟩{|}^2 where |\psi ⟩ is the final state evolved from Eq. (14). To evaluate the rotation error, we consider the average {\overline{E}}_{\mathrm{r}\mathrm{o}}=\int E_{\mathrm{r}\mathrm{o}}(V) \mathrm{d}V/\int \mathrm{d}V, where the integration is over V\in [(1-ϵ )V_0,(1+ϵ )V_0]. With a uniform distribution of V, the numerical results for different ϵ are shown in Fig.9. The average rotation error {\overline{E}}_{\mathrm{r}\mathrm{o}} increases from 2 to 4.5\times {10}^{-3} when ϵ grows from 0 to 0.8. The fidelity can be about 0.988 if ϵ is as large as 0.8.

Note that the numerical example shown above can be optimized so as to have a faster preparation of the state. For example, Appendix E shows another set of parameters to create |▴⟩ with a faster speed. The purpose of this paper is to reveal the possibility to create exotic multi-atom entanglement between electrons and nuclear spins.

6 Realizing lower-dimensional entanglement by measurement

SBS and |▴⟩ are highly entangled. Here, we show how to use them to create other types of entangled states. Taking SBS as an example, SBS incorporates three small Bell states as discussed in Section 3 and it is a Bell-like state by itself, shown in Eq. (15). This means that it is in principle possible to extract the “smaller” Bell states from it.

6.1 Measurement of one atom at a time in SBS

Below, we show that by measuring one or two atoms, the SBS in Eq. (25) can be projected to different entangled states depending on the measurement results. In other words, it is possible to create lower-dimensional entanglement by measurement of SBS.

By measuring one of the two atoms, one can determine whether the atom is in the ground state or not by detecting the light scattered via the transition between the ground state and (6s6p{)}^1{P}_1) [40]. However, we need to ensure that the measurement does not change the nuclear-spin states which demands efforts to quench the hyperfine-interaction-induced spin mixing [32, 33]. If the first atom is measured, and the result is that no light is detected, the state in Eq. (25) becomes

(35)\begin{array}{rl}|{\xi }_1⟩=& \frac{1}{\sqrt{6}}[|\text{cg}{⟩}_{\mathrm{e}}\otimes \left({\mathrm{e}}^{\mathrm{i}{\theta }_1}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\theta }_2}|\downarrow \downarrow {⟩}_{\mathrm{n}}\right)\\ & +{\mathrm{e}}^{\mathrm{i}{\theta }_3}\sqrt{2}|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes \left(|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}\right)].\end{array}

Then, measuring the second atom in the state of Eq. (35) has two outcomes. If no light is detected, the state in Eq. (35) collapses to

(36)\begin{array}{l}|{\xi }_{1a}⟩={\displaystyle \frac{1}{\sqrt{2}}}|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes \left(|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}\right),\end{array}

but if light is detected, the state in Eq. (35) collapses to

(37)\begin{array}{l}|{\xi }_{1b}⟩={\displaystyle \frac{1}{\sqrt{2}}}|\text{cg}{⟩}_{\mathrm{e}}\otimes \left({\mathrm{e}}^{\mathrm{i}{\theta }_1}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\theta }_2}|\downarrow \downarrow {⟩}_{\mathrm{n}}\right).\end{array}

When the measurement of the first atom results in light detected, meaning that the atom is in |\text{g}⟩, the state in Eq. (25) becomes

(38)\begin{array}{l}|{\xi }_2⟩={\displaystyle \frac{1}{\sqrt{2}}}|\text{gc}{⟩}_{\mathrm{e}}\otimes \left({\mathrm{e}}^{\mathrm{i}{\theta }_1}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\theta }_2}|\downarrow \downarrow {⟩}_{\mathrm{n}}\right).\end{array}

The state in Eq. (36) is an entangled nuclear-spin state in the clock state. The states in Eqs. (37) and (38) can be converted to

\begin{array}{l}|{\xi }_{1b}^{\prime }⟩=|\mathrm{g}\mathrm{g}{⟩}_{\mathrm{e}}\otimes \left({\mathrm{e}}^{\mathrm{i}{\theta }_1}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\theta }_2}|\downarrow \downarrow {⟩}_{\mathrm{n}}\right),\end{array}

and

\begin{array}{l}|{\xi }_2^{\prime }⟩=|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes \left({\mathrm{e}}^{\mathrm{i}{\theta }_1}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\theta }_2}|\downarrow \downarrow {⟩}_{\mathrm{n}}\right),\end{array}

respectively by using a \pi pulse of ground-clock transition in the first atom (with an extra phase determined by the Rabi frequency) as shown around Eq. (24).

The above discussions are based on detecting ground-state atoms without losing the atoms. It was reported in Ref. [28] that high-fidelity (over 0.9999) measurement of strontium atoms can proceed with a survival probability over 0.999. Imaging AEL atoms with nuclear spins would require extra efforts. Recently, Ref. [40] demonstrated imaging of the ground-state ¹⁷¹Yb atoms in tweezers with a fidelity about 0.997 and a loss probability around 2%–3% (near-unit fidelity can be achieved with larger atom loss probability).

6.2 Measurement of two atoms at a time in SBS

Measurement of two atoms can also lead to interesting states. For example, if no light is detected when measuring the two atoms initially in the state of Eq. (25), the state collapses to

\begin{array}{l}|{\mathrm{\Xi }}_1⟩={\displaystyle \frac{1}{\sqrt{2}}}|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes \left(|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}\right).\end{array}

But if light is detected, and if it is possible to collect light scattered from the two nearby atoms when neither any device nor the environment knows which atom scatters the light, the state collapses to

\begin{array}{l}|{\mathrm{\Xi }}_2⟩={\displaystyle \frac{1}{2}}(|\text{cg}{⟩}_{\mathrm{e}}+|{\text{gc}}_{\mathrm{e}}⟩)\otimes ({\mathrm{e}}^{\mathrm{i}{\theta }_1}|\uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\theta }_2}|\downarrow \downarrow {⟩}_{\mathrm{n}}),\end{array}

which is a hyperentangled state where both the electronic states and the nuclear-spin states of the two atoms are maximally entangled [45]. Here, we need to emphasize that though it was not tested with AEL atoms for collecting light scattered from two atoms without distinguishing the light pathway, it is possible since the physical principle is similar to that of the Young’s double-split experiment.

6.3 Measurement of |▴⟩

The three-atom state |▴⟩ in the form of Eq. (33) can also be used for creating interesting entangled states. Because of the descriptions shown above about SBS, we only briefly discuss the measurement of |▴⟩.

We first discuss the outcome of measuring one atom. If we measure the first atom and find light scattered, then Eq. (33) collapses to |\text{gcc}{⟩}_{\mathrm{e}}\otimes ( {\mathrm{e}}^{\mathrm{i}{\vartheta }_3}|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\vartheta }_4}|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}})/\sqrt{2}. This state can be further converted to |\mathrm{c}\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes ({\mathrm{e}}^{\mathrm{i}{\vartheta }_3}|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\vartheta }_4}|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}})/\sqrt{2} via the ground-clock transition in the first atom, which is the canonical GHZ state considering that the relative phase in it can be absorbed by defining new spin basis states as in Eq. (34). But if no light is scattered when measuring the first atom, the state collapses to

(39)\begin{array}{rl}& \frac{1}{2}\{\frac{\sqrt{6}}{2}|\mathrm{c}\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes [{\mathrm{e}}^{\mathrm{i}{\vartheta }_1}\frac{|\uparrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\uparrow \downarrow \uparrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{3}}\\ & +{\mathrm{e}}^{\mathrm{i}{\vartheta }_2}\frac{|\uparrow \downarrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{3}}]\\ & +\frac{|\text{cgc}{⟩}_{\mathrm{e}}+|\text{ccg}{⟩}_{\mathrm{e}}}{\sqrt{2}}\otimes \frac{{\mathrm{e}}^{\mathrm{i}{\vartheta }_3}|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\vartheta }_4}|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}}}{\sqrt{2}}\}.\end{array}

Measurement of the second atom in the state of Eq. (39) can lead to two outcomes. (i) If no light is detected, then Eq. (39) collapses to the state where \frac{1}{\sqrt{2}}(| \text{cgc}{⟩}_{\mathrm{e}}+|\text{ccg} {⟩}_{\mathrm{e}}) in the last line of Eq. (39) is replaced by |\text{ccg} {⟩}_{\mathrm{e}}, and if we continue to measure the third atom, and detect light, then the state finally collapses to |\text{ccg} {⟩}_{\mathrm{e}}\otimes ({\mathrm{e}}^{\mathrm{i}{\vartheta }_3}|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+ {\mathrm{e}}^{\mathrm{i}{\vartheta }_4}|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}})/\sqrt{2}. (ii) If light is detected, Eq. (39) collapses to |\text{cgc}{⟩}_{\mathrm{e}}\otimes ({\mathrm{e}}^{\mathrm{i}{\vartheta }_3}|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\vartheta }_4}|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}})/\sqrt{2}, which can be converted to the canonical GHZ state as discussed above.

We then discuss measuring three atoms at a time. We consider the case that we can collect light scattered from the atoms when neither any device nor the environment knows which atom scatters the light. If we find light scattered, Eq. (33) collapses to

\begin{array}{l}{\displaystyle \frac{1}{\sqrt{2}}}|W{⟩}_{\mathrm{e}}\otimes \left({\mathrm{e}}^{\mathrm{i}{\vartheta }_3}|\uparrow \uparrow \uparrow {⟩}_{\mathrm{n}}+{\mathrm{e}}^{\mathrm{i}{\vartheta }_4}|\downarrow \downarrow \downarrow {⟩}_{\mathrm{n}}\right),\end{array}

which is a hyerentangled state where the electronic state of the three atoms is in the maximally entangled W state, and the nuclear-spin state is in the maximally entangled GHZ state. In principle, it is possible to convert the electronic W state to the electronic GHZ state [60] so that the electronic GHZ state and the nuclear-spin GHZ state can simultaneously exist in three atoms.

7 Quantum dense coding with SBS

7.1 Dense coding model of Ref. [41]

It is known that Rydberg atoms are useful in quantum computing (for a review, see, e.g., Refs. [46, 47, 61]), quantum simulation [62], and quantum optics [63]. In this section, we show a possible application of Rydberg atoms in the context of SBS in quantum dense coding [41]. Before presenting our quantum dense coding with SBS, it is useful to first review the original method of [41]. Consider two qubits, each with two qubit states |0⟩ and |1⟩, then we can have four Bell states in the two qubits,

(40)\begin{array}{rl}& |{\mathbb{B}}_1⟩=(|10⟩-|01⟩)/\sqrt{2},\\ & |{\mathbb{B}}_2⟩=(|10⟩+|01⟩)/\sqrt{2},\\ & |{\mathbb{B}}_3⟩=(|11⟩-|00⟩)/\sqrt{2},\\ & |{\mathbb{B}}_4⟩=(|11⟩+|00⟩)/\sqrt{2}.\end{array}

We consider single-qubit gates represented by \hat{I},\hat{X}, \hat{Y},\hat{Z}, where \hat{I} is the identity, \hat{X} and \hat{Z} are the first and third Pauli matrices, and \hat{Y}=\mathrm{i}{\sigma }_y. By \hat{X}\{ |0⟩,|1⟩\} =\{|1⟩,|0⟩\}, \hat{Y}\{| 0⟩, |1⟩\}=\{| 1⟩, -|0⟩\}, and \hat{Z}\{|0⟩,|1⟩\} =\{- |0⟩,|1⟩\}, one can examine that by applying these gates to the first qubit in | {\mathbb{B}}_1⟩, the four states in Eq. (40) transform to |{\mathbb{B}}_1⟩,-|{\mathbb{B}}_3 ⟩,- |{\mathbb{B}}_4⟩, and |{\mathbb{B}}_2⟩, respectively. Dense coding proceeds with four steps. First, somebody prepares, e.g., |{\mathbb{B}}_1⟩, and sends the two qubits to two persons named, e.g., Alice and Bob. Second, Bob applies one of the four single-qubit gates on his bit, where he does nothing if he chooses the identity. Third, Bob sends his qubit to Alice via a public channel. Fourth, Alice measures the state of the two qubits to determine which of the four single-qubit gates Bob used to rotate the qubit Alice just received. Because the four Bell states in Eq. (40) are orthogonal to each other, it is possible for Alice to distinguish the four Bell states. If Alice and Bob already agreed with each other before the above mentioned four steps that the four operations \hat{I},\hat{X}, \hat{Y},\hat{Z} represent binary series 00, 01, 10, and 11, respectively, then the above procedure is one example for Bob to send information of two bits when actually one qubit is sent to Alice. Note that the nature of quantum mechanics guarantees that when an eavesdropper tries to gain knowledge about the qubit Bob sends to Alice via the public channel, Alice can find it out and then can discard that qubit from Bob.

7.2 Dense coding with SBS

Before presenting the quantum dense coding with SBS, we note that by using single-qubit gates or redefinition of basis states as in Eq. (21), we can prepare the following SBS

(41)\begin{array}{l}|SB1⟩\equiv {\displaystyle \frac{1}{\sqrt{2}}}(|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}+|\mathrm{\Phi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}})\end{array}

via the protocol of Section 4, where

(42)\begin{array}{rl}& |\mathrm{\Phi }{⟩}_{\mathrm{e}}=\frac{|\text{cg}{⟩}_{\mathrm{e}}+|\text{gc}{⟩}_{\mathrm{e}}}{\sqrt{2}},\\ & |\mathrm{\Phi }{⟩}_{\mathrm{n}}=\frac{|\uparrow \downarrow {⟩}_{\mathrm{n}}+|\downarrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{2}},\\ & |\mathrm{\Psi }{⟩}_{\mathrm{n}}=\frac{|\uparrow \uparrow {⟩}_{\mathrm{n}}+|\downarrow \downarrow {⟩}_{\mathrm{n}}}{\sqrt{2}}.\end{array}

By using single-qubit nuclear-spin gates \{{\hat{I}}_{\mathrm{n}}, {\hat{X}}_{\mathrm{n}}, {\hat{Y}}_{\mathrm{n}}, {\hat{Z}}_{\mathrm{n}}\} for the first atom, |\mathrm{\Phi }{⟩}_{\mathrm{n}} becomes \{|\mathrm{\Phi }{⟩}_{\mathrm{n}},|\mathrm{\Psi }{⟩}_{\mathrm{n}},|\mathrm{\Psi } {⟩}_{\mathrm{n}}^{\prime },| \mathrm{\Phi }{⟩}_{\mathrm{n}}^{\prime }\}, respectively, where

(43)\begin{array}{rl}& |\mathrm{\Phi }{⟩}_{\mathrm{n}}^{\prime }=\frac{|\uparrow \downarrow {⟩}_{\mathrm{n}}-|\downarrow \uparrow {⟩}_{\mathrm{n}}}{\sqrt{2}},\\ & |\mathrm{\Psi }{⟩}_{\mathrm{n}}^{\prime }=\frac{|\uparrow \uparrow {⟩}_{\mathrm{n}}-|\downarrow \downarrow {⟩}_{\mathrm{n}}}{\sqrt{2}},\end{array}

while |\mathrm{\Psi }{⟩}_{\mathrm{n}} becomes \{|\mathrm{\Psi }{⟩}_{\mathrm{n}},|\mathrm{\Phi }{⟩}_{\mathrm{n}},|\mathrm{\Phi } {⟩}_{\mathrm{n}}^{\prime },| \mathrm{\Psi }{⟩}_{\mathrm{n}}^{\prime }\}, respectively. Bob can also apply one among the two possible operations \{{\hat{I}}_{\mathrm{e}},{\hat{X}}_{\mathrm{e}}\} on the electronic state space in the first atom. When {\hat{X}}_{\mathrm{e}} is used, |\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}} and |\mathrm{\Phi }{⟩}_{\mathrm{e}} become |\text{gc}{⟩}_{\mathrm{e}} and |\mathrm{\Psi }{⟩}_{\mathrm{e}}= \frac{|\mathrm{g}\mathrm{g}{⟩}_{\mathrm{e}}+|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}}{\sqrt{2}}, respectively. For brevity, we list the operations used for the first atom and the resulting super Bell states,

(44)\begin{array}{rl}& {\hat{I}}_{\mathrm{e}}{\hat{I}}_{\mathrm{n}}:|SB1⟩,\\ & {\hat{I}}_{\mathrm{e}}{\hat{X}}_{\mathrm{n}}:|SB2⟩\equiv \frac{1}{\sqrt{2}}(|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}}+|\mathrm{\Phi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}),\\ & {\hat{I}}_{\mathrm{e}}{\hat{Y}}_{\mathrm{n}}:|SB3⟩\equiv \frac{1}{\sqrt{2}}(|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}}^{\prime }+|\mathrm{\Phi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}^{\prime }),\\ & {\hat{I}}_{\mathrm{e}}{\hat{Z}}_{\mathrm{n}}:|SB4⟩\equiv \frac{1}{\sqrt{2}}(|\mathrm{c}\mathrm{c}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}^{\prime }+|\mathrm{\Phi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}}^{\prime }),\\ & {\hat{X}}_{\mathrm{e}}{\hat{I}}_{\mathrm{n}}:|SB5⟩\equiv \frac{1}{\sqrt{2}}(|\text{gc}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}+|\mathrm{\Psi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}}),\\ & {\hat{X}}_{\mathrm{e}}{\hat{X}}_{\mathrm{n}}:|SB6⟩\equiv \frac{1}{\sqrt{2}}(|\text{gc}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}}+|\mathrm{\Psi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}),\\ & {\hat{X}}_{\mathrm{e}}{\hat{Y}}_{\mathrm{n}}:|SB7⟩\equiv \frac{1}{\sqrt{2}}(|\text{gc}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}}^{\prime }+|\mathrm{\Psi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}^{\prime }),\\ & {\hat{X}}_{\mathrm{e}}{\hat{Z}}_{\mathrm{n}}:|SB8⟩\equiv \frac{1}{\sqrt{2}}(|\text{gc}{⟩}_{\mathrm{e}}\otimes |\mathrm{\Phi }{⟩}_{\mathrm{n}}^{\prime }+|\mathrm{\Psi }{⟩}_{\mathrm{e}}\otimes |\mathrm{\Psi }{⟩}_{\mathrm{n}}^{\prime }).\end{array}

The eight super Bell states are orthogonal to each other, so they can be distinguished in experiments in principle and can represent the binary numbers \{000,001 ,010,011,100,101, 110,111\}. Like Section 7.1, quantum dense coding with SBS also needs four steps as described there. The difference with SBS lies in that Alice and Bob agree beforehand that the eight different operations correspond to the eight binary numbers. Then, by sending only one atom to Alice, Bob conveys three bits of classical information.

As light can carry quantum information while flying a long distance, we will discuss the possibility to map the entanglement from atoms to photons so as to realize dense coding for long-distance communication.

7.3 Mapping atomic entanglement to photons

Sending information via photons can enable long-distance quantum communication. If SBS of the atomic state is mapped to a super Bell state of photons, then the dense coding of Section 7.2 can proceed as follows. First, map the entanglement from the two atoms to two photons, where the “c” and “g” electronic qubit states of atoms are mapped to the time-bin qubit states of photons, and the up and down nuclear-spin qubit states of atoms are mapped to the polarization-encoded qubit states of photons. Second, the two photons are sent to Alice and Bob, respectively. Third, Bob performs one of the eight operations on his photon in a way similar to those for atoms shown in Eq. (44). Fourth, Bob sends his photon to Alice. Fifth, Alice performs measurement to check whether the two-photon state is one of the eight super Bell states. If not, an eavesdropper ever tried to observe the photonic state Bob sent to her, so Alice can discard the photons; if yes, Alice can determine the three-digit binary number Bob meant to send according to her measurement result. By this, Alice can receive three bits of classical information when only one photon is sent by Bob.

We consider the mapping outlined in Fig.10. (i) Move the two entangled atoms into two separate cavities, each of which is coupled to a fibre with a high efficiency. Note that the coherent transport of atoms while preserving the internal states was experimentally possible [16]. (ii) Use two sets of two-photon laser fields to realize

(45)\begin{array}{l}|{\text{g}}_{\uparrow }⟩\underset{}{\overset{}{\to }}|[6s5d{}^3{D}_1]F=3/2,m_F=3/2⟩,\end{array}

(46)\begin{array}{l}|{\text{g}}_{\downarrow }⟩\underset{}{\overset{}{\to }}|[6s5d{}^3{D}_1]F=3/2,m_F=-3/2⟩,\end{array}

where both sets of fields are via the intermediate state 6s6p{}^3{P}_1 but with different detunings at it so that the two fields for the transition of Eq. (45) and the two fields for the transition of Eq. (46) are independent with each other. The transitions 6s6p{}^1{S}_0\to 6s6 p{}^3{P}_1 and 6s6p{}^3{P}_1\to 6s5 d{}^3{D}_1 are with dipole matrix elements about 0.2a_{0}e (which includes a factor due to the spin-orbit coupling [37]) and 1.6a_{0}e [64], respectively. This makes it possible to realize large Rabi frequencies for Eqs. (45) and (46). However, we do not need to have large Rabi frequencies, and one can tune the fields to be near to [6s6 p{}^3{P}_1]F=3 /2 as in Ref. [20]. In this case, the [6s6 p{}^3{P}_1]F=1 /2 hyperfine level, which is about 6 GHz away from the F=3/2 level [37], does not participate in the transitions of Eqs. (45) and (46). (iii) The cavity field is near resonant with the transition [6s5d{}^3{D}_1]F =3/2\leftrightarrow [6s6 s{}^3{P}_1]F=1 /2. As a result, the state in Eq. (45) can lead to an emission of {\sigma }^{+} polarized photon, while the state of Eq. (46) can lead to an emission of {\sigma }^{-} polarized photon. The photons are coupled to optical fibres and sent to Alice and Bob, respectively. (iv) Use two sets of UV laser fields to realize

(47)\begin{array}{l}|{\mathrm{c}}_{\uparrow }⟩\underset{}{\overset{}{\to }}|[6s5d{}^3{D}_1]F=3/2,m_F=3/2⟩,\end{array}

(48)\begin{array}{l}|{\mathrm{c}}_{\downarrow }⟩\underset{}{\overset{}{\to }}|[6s5d{}^3{D}_1]F=3/2,m_F=-3/2⟩.\end{array}

(v) As in step (iii), the two states in Eqs. (47) and (48) lead to emission of cavity photons of different polarizations, which are further coupled to fibres and sent to Alice and Bob, respectively. Because there is time difference between steps (iii) and (v), the “c”−“g” electronic degree of freedom is mapped to the time-bin degree of freedom. Here, we only show the basic process for mapping the atomic SBS to photonic SBS. In practice, one can use adiabatic passage process to realize the mapping as experimentally realized in Ref. [65].

We note that high-fidelity transport of neutral atoms while preserving the atoms and their internal state is not trivial [16], and the collection of photons emitted from the cavities with high efficiency also needs efforts. Therefore it is a demanding task to realize a photonic SBS with a high fidelity. On the other hand, the basic elements to realize the novel quantum dense coding can be tested with the atomic SBS with the single-qubit operations of Eq. (44).

8 Discussion

Here, we summarize the mechanism introduced in this paper for the novel entanglement protocols, and compare the theory shown here and the previous results published about the quantum control over nuclear-spin qubits in divalent neutral atoms. We also discuss the gate speed, magnetic field used in the protocols, and connections with experiments in hope to stimulate further explorations.

8.1 Creation of unit Rydberg excitation in detuned Rabi oscillations

The fast nuclear-spin gates presented in Sections 2 and 3, and the exotic electrons−nuclei entanglement in Sections 4 and 5 rely on two methods to create unit Rydberg excitation among two atoms when there are both laser detuning and Rydberg blockade.

For the nuclear-spin gates in Sections 2 and 3, the novel mechanism used is that when a common laser field is sent to two nearby atoms in the initial state, as an example, |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩, with the laser tuned to the middle between the two clock-Rydberg transitions of Fig.1, we have the following transition in the strong blockade limit,

(49)\begin{array}{l}|r_{+}{\mathrm{c}}_{\downarrow }⟩\stackrel{\text{detuned by}\mathrm{\Delta }}{\leftarrow }|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩\stackrel{\text{detuned by}-\mathrm{\Delta }}{\to }|{\mathrm{c}}_{\uparrow }{r}_{-}⟩.\end{array}

If the left transition in Eq. (49) is absent, one can prove [51] that there can never be unit Rydberg population, i.e., the state |{\mathrm{c}}_{\uparrow }{r}_{-}⟩ will not be fully populated. Likewise, if the right transition in Eq. (49) is absent,the state |r_{+}{\mathrm{c}}_{\downarrow }⟩ can not be fully populated either. A novel mechanism used here is that, it is possible to realize a unit Rydberg excitation starting from the state |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\downarrow }⟩ when Eq. (49) is realized in the blockade condition. In other words, it is possible to create one Rydberg excitation even though neither |r_{+}{\mathrm{c}}_{\downarrow }⟩ nor |{\mathrm{c}}_{\uparrow }{r}_{-}⟩ is resonantly excited. This surprising mechanism is in sharp contrast to the usually employed creation of unit Rydberg excitation via resonant laser excitation of two nearby atoms [3, 19]. By the symbols used in this paper, the method of Refs. [3, 19] relies on exciting

(50)\begin{array}{l}|r_{+}{\mathrm{c}}_{\uparrow }⟩\stackrel{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{t}}{\leftarrow }|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\stackrel{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{t}}{\to }|{\mathrm{c}}_{\uparrow }{r}_{+}⟩\end{array}

in the strong blockade condition so that the state (|r_{+}{\mathrm{c}}_{\uparrow }⟩+|{\mathrm{c}}_{\uparrow }{r}_{+}⟩)/\sqrt{2} can be created, i.e., unit Rydberg excitation among two atoms is achieved. Unit Rydberg excitation was useful for creating maximal entanglement between atoms [3, 19], which is why in this paper we explore the creation of unit Rydberg excitation via the novel method of Eq. (49).

The electrons−nuclei entanglement in Sections 4 and 5 is realized via another method for creating unit Rydberg excitation in detuned transitions with Rydberg blockade. In practice, it was known that by

(51)\begin{array}{l}|r_{+}{\mathrm{c}}_{\uparrow }⟩\stackrel{\text{detuned by}\mathrm{\Delta }}{\leftarrow }|{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩\stackrel{\text{detuned by}\mathrm{\Delta }}{\to }|{\mathrm{c}}_{\uparrow }{r}_{+}⟩\end{array}

with a constant Rabi frequency, there can be no unit Rydberg excitation [38, 51]. But Sections 4 and 5 show that if we change the Rabi frequency though the laser frequency is constant, it is possible to create unit Rydberg excitation among the two atoms. The simplest case is to use two laser pulses where the Rabi frequency in the second pulse differs from that in the first pulse in a certain way as in Sections 4 and 5. By this method, it is possible to create the state (|r_{+}{\mathrm{c}}_{\uparrow }⟩ +|{\mathrm{c}}_{\uparrow }{r}_{+} ⟩)/ \sqrt{2} from |{\mathrm{c}}_{\uparrow }{\mathrm{c}}_{\uparrow }⟩ even though the lasers are detuned during the excitation process. Importantly, the same unit excitation can be realized if \mathrm{\Delta } is replaced by -\mathrm{\Delta }, which means that it fits perfectly for the nuclear-spin qubits studied in this paper.

8.2 Comparison with other entanglement of divalent atoms via Rydberg blockade

It is useful to compare the theory in this paper with other Rydberg-mediated entanglement in electronic state space [19, 21, 22] or nuclear-spin state space [20, 23, 35, 37, 66] of divalent atoms.

First, this paper studies entanglement between electrons and nuclear spin states in divalent atoms, while other papers studied entanglement between nuclear spins [20, 23, 35, 37], or entanglement between electrons [19, 21, 22]. Reference [45] studied simultaneous entanglement between electrons and entanglement between nuclear spins of two atoms, but the hyperentangled state there doesn’t contain entanglement between electrons and nuclear spins.

Second, for the entanglement protocols relying on unit Rydberg population [19, 37, 45], a resonant ground-Rydberg [37, 45] or clock-Rydberg [19, 45] transition was employed in the blockade condition. The theory in this paper relies on off-resonant ground- or clock-Rydberg transition. To the best of our knowledge, this novel mechanism was not reported.

Third, for nuclear-spin entanglement with experimental demonstration [20, 23] or theoretical analysis [66], either the gate of [11] via dynamical phases in detuned Rabi cycles [38] was employed, or the gate via optimal control was used [23]. Theses methods, however, fit well for hyperfine qubits in alkali-metal atoms or optical clock qubits of AEL atoms [47]. In AEL atoms with nuclear spins, the frequency separation between the nuclear-spin qubit states in ground or clock state is negligible compared to a MHz-scale Rydberg Rabi frequency in a Gauss-scale magnetic field as frequently used in experiments. To use the methods designed for hyperfine qubits or optical clock qubits [47] with nuclear-spin qubits, one can enlarge the ratio between the Zeeman splitting of nearby Rydberg Zeeman substates and the Rydberg Rabi frequency so as to avoid the Rydberg excitation of the nontarget nuclear spin state as done in Refs. [20, 23]. This, however, strongly limits the gate speed which may explain why using similar optimal-control protocols, the gate of Ref. [18] with hyperfine qubits has a higher fidelity than that of Ref. [23] with nuclear-spin qubits. These methods differ from the theory in this paper in two aspects. (i) The theory here does not purely rely on dynamical phases for entanglement though it uses detuned Rabi cycles. For example, the nuclear-spin gate in Sections 2 and 3 realizes a controlled-\beta gate when the last laser pulse has a relative phase \beta, while the gate in Section 3 realizes a controlled-\beta gate when the last laser pulse has a relative phase \beta plus another phase due to a rotating frame transform. These are in sharp contrast to the method of Refs. [11, 38], where it was shown in Ref. [47] that the protocol of Ref. [11] can be used to realize a controlled-\beta gate with an arbitrary \beta, but there is no linear relation between \beta and the laser phases. (ii) The theory here is designed for nuclear-spin qubits in divalent atoms with nuclear spins, so it works well for nuclear-spin qubits while not so for hyperfine qubits of alkali metals or optical clock qubits of AEL atoms [47]. Our theory relies on exciting the two qubit states to Rydberg states off-resonantly, which fits perfectly the nuclear-spin qubits in a Gauss-scale magnetic field. As a result, the Rydberg Rabi frequency is comparable to the Zeeman splitting of two nearby Rydberg Zeeman substates, which ensures fast gate speeds. In other words, when fixing the magnetic field, the achievable gate speed with the theory in this paper will be larger than those in [20, 23] for realizing nuclear-spin entanglement in divalent atoms.

8.3 Gate speed

The speed of our nuclear-spin quantum gates is large. To discuss the speed of a Rydberg-mediated quantum gate, it is useful to denote the gate duration in terms of \pi / {\mathrm{\Omega }}_{\mathrm{m}} where {\mathrm{\Omega }}_{\mathrm{m}} is the maximal Rabi frequency during the protocol. To our knowledge, the fastest Rydberg-blockade-based entangling gate that can be transferred to the CZ gate via single-qubit gates needed two laser pulses of total duration about 2.732\pi /{\mathrm{\Omega }}_{\mathrm{m}} [11] without using optimal control, or about 2.414\pi /{\mathrm{\Omega }}_{\mathrm{m}} [18] by optimal control. The gates of [11, 18] depend on exciting only one of the two qubit states to Rydberg states, which means that it fits qubits defined either by the hyperfine substates as used in the experiments of Refs. [11, 16, 17] or by the optical clock qubits (there was an experiment [21] with the optical clock qubits via another gate protocol [67]). Of course, one can try to use it for nuclear spin qubits as experimentally done in Refs. [20, 23] with a {\mathrm{\Omega }}_{\mathrm{m}} small compared to the Zeeman splitting between Rydberg states. The small {\mathrm{\Omega }}_{\mathrm{m}} results in relatively long gate duration, which was possibly why the gate fidelity of Ref. [20] was smaller than that of Ref. [11], and that of Ref. [23] was smaller than that of Ref. [18] though in principle AEL atoms can allow high-fidelity entanglement generation [19, 22] due to the advantages possessed by them as introduced in Section 1. In this paper, the two or three-pulse gate has a total duration 2.589\pi /{\mathrm{\Omega }}_{\mathrm{m}} or 5.054\pi /{\mathrm{\Omega }}_{\mathrm{m}}. With {\mathrm{\Omega }}_{\mathrm{m}}/(2\pi )=13 MHz realizable [19], even the slower three-pulse gate in this paper would have a short duration 0.19 μs. This gives hope to realize fast and high-fidelity nuclear-spin entanglement.

8.4 Magnetic field

The theory shown in this paper is based on using a magnetic field below 10 G for specifying the quantization axis of the atoms. The numerical simulations for the nuclear-spin quantum gates shown in Fig.2, Fig.3, Fig.4, and Fig.5 assume 2\mathrm{\Delta }/(2\pi ) \approx 4 MHz, which corresponds to a B-field of about 2 G. For the data of SBS shown in Fig.6 and Fig.7, the Zeeman splitting is 2\mathrm{\Delta }/(2\pi ) \approx 2.8 MHz which corresponds to B\approx 1.5 G. For the data about |▴⟩ shown in Fig.8 and Fig.9, the field B\approx 1.7 G is assumed. If Rabi frequencies used in them are increased by three times which can lead to much faster entanglement generations with higher fidelities, B-fields below 10 G are still sufficient. So, the protocols in this paper are compatible with recent experimental facilities studying nuclear spins, where a B-field of 4.11 G [20], 5 G [23], 11 G [39], or a value in the range (0,18] G [40] was used with ⁸⁷Sr [39] or ¹⁷¹Yb [20, 23, 40]. These bring hope to realize functional quantum devices with nuclear-spin quantum memories [30, 31, 34, 35, 45, 66].

8.5 Application with other atomic species

The numerical simulations in this paper are performed with ¹⁷¹Yb for it can simplify the analysis due to that it has the simplest nuclear spin I=1/2. Extending the theory to isotopes with larger I is possible. For example, for nuclear-spin qubits defined with 1-I and -I in ⁸⁷Sr, the ground-Rydberg transition was demonstrated [39], and the clock-Rydberg transition was demonstrated with ⁸⁸Sr [19, 22]. Moreover, when the Rydberg Rabi frequencies for the two nuclear spin states do not have the same magnitude as in ¹⁷¹Yb shown in Eq. (6), application of the theory in this paper may lead to more exotic entanglement because different Rydberg excitation rates in the two qubit states of each atom can result in different dynamics useful for quantum control [57]. In general, we expect the entanglement protocols to be useful for divalent atoms with nuclear spins and metastable clock states, such as calcium, barium, zinc, cadmium, and mercury [33].

8.6 Connections with experiments

The protocols introduced in this paper rely on techniques intimately related with recent experiments.

The fast clock-Rydberg transition assumed throughout this paper can be experimentally realized. In Ref. [19], Rabi frequencies about 2\pi \times 6.8 MHz were used to excite the clock state to 5s61s{}^3{S}_1 of ⁸⁸Sr. The two atoms to be entangled were separated by about 3.6 μm and experienced the same laser fields in Ref. [19]. In Ref. [21], time-dependent Rabi frequencies up to 2\pi \times 18 MHz were used for the transition from the clock state to 5s40d{}^3{D}_1 of ⁸⁸Sr, and short atomic separation 1.2 μm was employed. This setting with very short atomic separation fits well for our theory since each Rydberg excitation should occur for the two or three atoms simultaneously. For the state |▴⟩ where three atoms should be close enough, the experiment of Ref. [18] placed three atoms with an atom−atom separations of only 2 μm in triangular gate sites, where the triangular configuration enabled strong, symmetric interactions between three qubits. These point to the experimental feasibility of the protocols here.

The ground-Rydberg transition via a low-lying intermediate state was possible. In Ref. [20], the Rydberg state 6s74s{}^3{S}_1 was excited from the ground state of ¹⁷¹Yb for entangling nuclear spin states in the ground state. The Rydberg Rabi frequency was 2\pi \times 0.763 MHz so as to avoid the excitation of the nontarget nuclear spin state as required by the protocol used in [20]. However, it was analyzed in Ref. [37] that Rabi frequencies over 2\pi \times 1 MHz for exciting states with principal number around 70 is feasible.

The fast shifting of energy of Rydberg states required in Section 3 is experimentally feasible. There are two possible approaches to shifting the transition energy between the Rydberg state and the clock or ground state. First, one can shift the energy of the atomic states by exciting the outer valence electron. In Ref. [39], two nuclear spin qubit states with m_I=-I, 1-I in ⁸⁷Sr were isolated via Stark shifting other nuclear spin states away from the resonant beams. In Ref. [40], light shifts in excited states were used to coherently control the nuclear spin states in ¹⁷¹Yb. Second, the excitation of the inner valence electron of Rydberg states can be used. For example, energy shifts over 2\pi \times 10 MHz were demonstrated for both the (6s6 n{)}^3{S}_1 [48] and the (6s6n{)}^1{S}_0 [68] Rydberg states via the excitation of the Yb{}^{+}6s\to 6p_{1/2} transition of the inner valence electron in ¹⁷⁴Yb. Extending the methods of Refs. [48, 68] to ¹⁷¹Yb would require calibrating polarization dependence of light shifts as in [39, 40].

9 Conclusions

We present fast and high-fidelity nuclear-spin controlled-phase gates of an arbitrary phase in AEL atoms. By exciting two AEL atoms with global laser fields where both nuclear spin qubit states of each atom are Rydberg excited, a controlled-phase gate of an arbitrary phase can be realized via three laser pulses of total duration 5.05\pi /{\mathrm{\Omega }}_{\mathrm{m}}, where {\mathrm{\Omega }}_{\mathrm{m}} is the largest Rydberg Rabi frequency during the pulses. We also show that if spin-dependent Stark shift (\sim {\mathrm{\Omega }}_{\mathrm{m}}) is realizable, the same gate can be realized with two laser pulses of total duration 2.59\pi /{\mathrm{\Omega }}_{\mathrm{m}}. With large {\mathrm{\Omega }}_{\mathrm{m}} for the transition between the clock and Rydberg states of AEL realizable as recently demonstrated in experiments, and concerning that fast gates are easier to attain a higher fidelity in practice, the protocols can enable high-fidelity entanglement in nuclear spins of AEL atoms.

We also present two protocols to entangle electrons and nuclear spins in AEL. First, we show a super Bell state (SBS) entangled between the electrons and nuclei of two atoms. SBS is like a large “Bell” state entangled between three “small” Bell states and one product state. For three atoms, we show a protocol to realize a highly entangled state that includes W and GHZ states simultaneously. These two multi-atom entangling protocols have durations 7.69\pi /{\mathrm{\Omega }}_{\mathrm{m}} and 11.4\pi /{\mathrm{\Omega }}_{\mathrm{m}}, respectively, and can be used for preparing Bell, hyperentangled, and nuclear-spin GHZ states based on measurement. SBS can also enable dense coding where three classical bits of information can be securely shared when only one physical particle is sent in a public channel.

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