Single-site Rydberg addressing in 3D atomic arrays for quantum computing with neutral atoms

Neutral atom arrays have become a promising platform for large-scale quantum computing Jaksch et al. (2000); Saffman et al. (2010, 2011); Saffman (2016); Weiss and Saffman (2017); Adams et al. (2019). The attraction lies in that one can not only prepare large-scale qubit arrays and initialize the state for the qubits with high fidelity, but can also prepare high-fidelity single-qubit gate and easily read out the quantum information stored in the qubits Aljunid et al. (2009); Leung et al. (2014); Nogrette et al. (2014); Xia et al. (2015); Zeiher et al. (2015); Ebert et al. (2015); Wang et al. (2016); Barredo et al. (2016). The fidelity of two-qubit entangling gates is growing steadily recently Wilk et al. (2010); Isenhower et al. (2010); Zhang et al. (2010); Maller et al. (2015); Zeng et al. (2017); Picken et al. (2019); Levine et al. (2018); Graham et al. (2019); Levine et al. (2019), from the initial value of $73\%$ Isenhower et al. (2010) to $97\%$ Levine et al. (2019), which points to the likeliness that high-fidelity universal gate set can be experimentally realized in the near future for the required accuracy of measurement-free fault-tolerant quantum computing Crow et al. (2016). Nevertheless, there are still many unsolved issues toward this goal as reviewed in Ref. Saffman (2016).

One issue for large-scale quantum information processing in a 3-dimensional (3D) qubit array is how to selectively excite one qubit deep in the lattice between ground and Rydberg states. To date, there has been demonstration of two-qubit gate in a two-dimensional (2D) array of neutral atoms Maller et al. (2015); Graham et al. (2019), where the single-site addressing without affecting nontarget atoms is guaranteed by sending laser fields perpendicular to the 2D array, and thus is not applicable in a 3D lattice. There have been several experiments on single-qubit gates in 3D Wang et al. (2015, 2016) or 2D Xia et al. (2015) atomic arrays by using a hybrid irradiation of laser and microwave fields. These methods depend on shifting the atomic transition frequency by sending laser fields upon target atoms, where the Stark shifts (divided by the Planck constant) should be comparable to the inverse of the gate durations of several hundred $\mu s$. To the best of our knowledge, however, there is no method of single-site Rydberg addressing in a 3D lattice where nontarget atoms in the light path are not influenced.

In this work, we present two methods for selectively exciting one qubit in any site of a 3D qubit array to Rydberg state. The first method relies on a well-known phenomenon: the action of two symmetrically detuned laser pulses is equivalent to that of a monochromatic field whose amplitude is sinusoidally modulated in time, as studied in Ref. Goreslavsky et al. (1980). This means that a full dipole transition can also occur by absorbing two fractional photons, i.e., by absorbing half of a photon with energy $E_{ge}+\hbar\Delta$, and half of another photon with energy $E_{ge}-\hbar\Delta$, where $\hbar$ and $\Delta$ are the reduced Planck constant and frequency detuning, respectively. This resonance can be called off-resonance-induced resonance (ORIR). By ORIR, one laser light is sent along one direction, while the other sent along another direction, and they together excite a target atom. The nontarget atom can accumulate phase shift or even experience state evolution in general, but we show that our ORIR-based theory can eliminate its effect when the state of the target atom is restored. Besides the ORIR-based method, we show a second method with a three-level ladder type system, where one target atom can be excited to Rydberg state while preserving the state of any nontarget atom. In the second method, the state of the target atom can pick a $\pi$ phase shift upon the completion of the ground-Rydberg-ground state transfer; such a $\pi$ phase is crucial in the controlled-phase gate based on Rydberg blockade. These methods make it possible to couple only one atom to the Rydberg state in 3D.

We further show that ORIR can lead to high fidelity in the Rydberg blockade gate. In quantum computing with neutral atoms and Rydberg interactions Jaksch et al. (2000); Saffman et al. (2010, 2011); Saffman (2016); Weiss and Saffman (2017); Adams et al. (2019), it has been an outstanding challenge to design a practical high-fidelity entangling gate Goerz et al. (2014); Theis et al. (2016); Shi (2017); Petrosyan et al. (2017); Shi (2018a); Shen et al. (2019); Shi (2019); Yu et al. (2019); Levine et al. (2019). A traditional method to achieve an entangling gate by Rydberg interactions is via the blockade mechanism Jaksch et al. (2000), in which there is a fundamental blockade error. We show that when the Rabi frequency $\Omega$ in the usual method is replaced by an ORIR-induced Rabi frequency $i\Omega\sin(\Delta t)$, the blockade error can be suppressed by more than two orders of magnitude. More important, this reduction of error by ORIR is robust against the variation of the blockade interaction, which compares favorably to other pulse-shaping-based methods for suppressing the blockade error. Thus, ORIR can effectively remove the blockade error, making it possible to realize a high-fidelity neutral-atom entangling gate with quasi-rectangular pulses that are easily attainable by pulse pickers.

The remainder of this work is organized as follows. In Sec. II, we give details about how ORIR appears. In Sec. III, we study two methods for single-site Rydberg addressing in a 3D optical lattice; a comparison between the two methods is given at the end of Sec. III. In Sec. IV, we show that ORIR can suppress a fundamental rotation error in the Rydberg blockade gate. In Sec. V, we discuss other applications of ORIR. Section VI gives a brief summary.

The reason for ORIR to occur lies in that the dipole coupling between an atom and two symmetrically detuned laser fields can be characterized by a sinusoidal Rabi frequency $2i\Omega\sin(\Delta t)$ or $2\Omega\cos(\Delta t)$ Goreslavsky et al. (1980). To put it in perspective we consider a pair of external coherent electromagnetic fields of equal strength but with opposite detunings $\pm\Delta$ applied for the dipole transition $|g\rangle\leftrightarrow|e\rangle$. In the dipole approximation, these two fields lead to a pair of counter-rotating Rabi frequencies $\Omega_{\pm}=\Omega e^{\pm i\Delta t}$. When only one of these two detuned fields is applied, the expectation value of the population in $|e\rangle$ has an upper bound $\Omega^{2}/(\Omega^{2}+\Delta^{2})$ if the initial state is $|g\rangle$ Shi (2017, 2018a), as shown in Fig. 1(a). But when the two oppositely detuned fields are applied simultaneously and because $\Omega_{+}+\Omega_{-}=2\Omega\cos(\Delta t)$, a full transition between $|g\rangle$ and $|e\rangle$ becomes attainable within a time of less than $|\pi/\Delta|$ as long as $|\Omega/\Delta|\geq\pi/2$. The population dynamics in the two-level system in response to the detuned driving is shown in Fig. 1 when $\Omega/\Delta=\pi/2$. Figures 1(a) and 1(b) show that the dynamics in the two-level system driven by the two oppositely detuned fields dramatically differs from that when only one detuned driving is present. This phenomenon results from the quantum interference between the two Rabi oscillations with opposite detunings. Remarkably, the speed of this off-resonance-induced resonant transition is comparable to its resonant counterpart: when $|\Omega/\Delta|=\pi/2$, a full transition between the two states is achieved within a time of $t_{\pi}\pi/2$ that is comparable to the duration, $t_{\pi}=\pi/(2\Omega)$, of a $\pi$ pulse required for a full transition in the resonant case with the Rabi frequency $2\Omega$. This type of multi-photon ORIR can occur in various electric or magnetic dipole transitions of an atom or molecule, either natural or artificial.

We give a detailed mathematical argument for the above discussion. For a two-level system in Fig. 1, the eigenenergies of $|g\rangle$ and $|e\rangle$ are zero and $E_{\text{ge}}$, respectively. The electric dipole transition between $|g\rangle$ and $|e\rangle$ is driven by two laser fields, one with central (angular) frequency $\omega+\Delta$, and the other with central frequency $\omega-\Delta$, where $\omega=E_{\text{ge}}/\hbar$. The Hamiltonian (divided by $\hbar$) of the matter-light coupling in dipole approximation is

	$\displaystyle\hat{H}(t)$	$\displaystyle=$	$\displaystyle\omega|e\rangle\langle e|+[\Omega_{1}(e^{it(\omega+\Delta)}+e^{-it(\omega+\Delta)})|e\rangle\langle g|$		(1)
			$\displaystyle+\Omega_{2}(e^{it(\omega-\Delta)}+e^{-it(\omega-\Delta)})|e\rangle\langle g|+\text{H.c.}]/2,$

where we assume that $\Omega_{1}$ and $\Omega_{2}$ are real, and “H.c.” denotes the Hermitian conjugate. In principle, there is a phase difference $\varphi_{0}=(\mathbf{k}_{1}-\mathbf{k}_{2})\cdot\mathbf{r}$ between $\Omega_{1}$ and $\Omega_{2}$, where the subscript $1(2)$ distinguishes the two transitions with opposite detunings. If the Rydberg state $|e\rangle$ is excited by couterpropagating fields along $\mathbf{z}$ via two-photon excitation through an intermediate state with a GHz-scale detuning $\delta_{1(2)}$, and if meanwhile the fields for $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ copropagate, $\varphi_{0}$ becomes $2(\Delta+\delta_{2}-\delta_{1})z/c$, where $z$ is the z-component coordinate of the atom, $c$ the speed of light. The difference between $\delta_{1}$ and $\delta_{2}$ is much larger than $\Omega_{1(2)}$ so that a common intermediate state is used for the two two-photon transitions; for instance, $\delta_{1}$ and $\delta_{2}$ can have different signs. Because $\Delta\sim\Omega_{1(2)}$ and the effective two-photon Rabi frequency $\Omega_{1(2)}$ is on the order of MHz, $\varphi_{0}\approx 2(\delta_{2}-\delta_{1})z/c$. For atoms cooled to temperatures around $1$ K or below, the change of $z$ within several $\mu$s is on the order of $\mu$m, which means that $\varphi_{0}$ is almost constant; furthermore, $|\delta_{2}-\delta_{1}|/c\ll k_{1},k_{2}$, and thus we can assume that this phase has been compensated by adjustment of the overall phases carried by the laser beams. This is technically possible since the phase fluctuation of a laser field can be made negligible Saffman and Walker (2005); Wineland et al. (1998). But for single-site Rydberg addressing with our methods (shown in Sec. III), the fields for $\Omega_{1}$ and those for $\Omega_{2}$ propagate along different directions, e.g., one along $\mathbf{z}$ and the other along $\mathbf{x}$, then $\varphi_{0}\approx\mathcal{K}(z-x)$, where $\mathcal{K}$ is the difference of the wavevectors of the upper and lower transitions. For configurations with $\mathcal{K}$ of several $10^{6}m^{-1}$ Isenhower et al. (2010), the fluctuation of the qubit locations lead to large errors in $\varphi_{0}$, and it is necessary to cool atoms to very low temperatures to establish ORIR.

Using the operator $\hat{\mathcal{R}}=\omega|e\rangle\langle e|$ for a rotating-frame transform, $e^{i\hat{\mathcal{R}}t}\hat{H}e^{-i\hat{\mathcal{R}}t}-\hat{\mathcal{R}}$, the Hamiltonian becomes,

	$\displaystyle\hat{H}(t)$	$\displaystyle=$	$\displaystyle\Omega_{1}(e^{it(2\omega+\Delta)}+e^{-it\Delta})|e\rangle\langle g|/2$
			$\displaystyle+\Omega_{2}(e^{it(2\omega-\Delta)}+e^{it\Delta})|e\rangle\langle g|/2+\text{H.c.}.$

When $\Omega_{1}=\Omega_{2}=\Omega$, the above equation simplifies to,

$\displaystyle\hat{H}(t)$

$\displaystyle=$

$\displaystyle\Omega(e^{2it\omega}+1)\cos(t\Delta)|e\rangle\langle g|+\text{H.c.}.$

We further assume $\Delta\sim|\Omega|\ll\omega$ so that the Bloch-Siegert shift is negligible Cohen-Tannoudji et al. (1998). In this case, $e^{\pm 2it\omega}$ is rapidly oscillating and can be discarded according to the rotating wave approximation. This leads to the following Hamiltonian,

$\displaystyle\hat{H}(t)$

$\displaystyle=$

$\displaystyle\Omega\cos(t\Delta)(|e\rangle\langle g|+|g\rangle\langle e|).$

Starting from the initial state, $|\psi(0)\rangle=|g\rangle$, the system wavefunction $|\psi(t)\rangle=C_{g}(t)|g\rangle+C_{e}(t)|e\rangle$ evolves according to

	$\displaystyle C_{g}(t)$	$\displaystyle=$	$\displaystyle\cos\left[\frac{\Omega}{\Delta}\sin(t\Delta)\right],$
	$\displaystyle C_{e}(t)$	$\displaystyle=$	$\displaystyle-i\sin\left[\frac{\Omega}{\Delta}\sin(t\Delta)\right].$		(2)

When $\frac{|\Omega|}{\Delta}\geq\frac{\pi}{2}$ is satisfied, the transition probability from the ground state to the excited state can reach 1. A critical condition for a full transition is

$\displaystyle\frac{|\Omega|}{\Delta}=\frac{\pi}{2},~{}~{}t=\frac{\pi}{2\Delta}.$

Remarkably, the time for the transition from the ground state to the excited state is $\frac{\pi}{2\Delta}$, which is only $\pi/2$ times longer than the time $\frac{\pi}{2|\Omega|}$ of a $\pi$ pulse in a resonant transition with a Rabi frequency $2\Omega$.

ORIR is a useful method in quantum control with atomic ions Sørensen and Mølmer (1999). Below, we show its applicability in neutral atoms.

ORIR can be used for Rydberg addressing of a single atom in a 3D optical lattice. In general, the difficulty of Rydberg addressing in a 3D lattice lies in two aspects. First, sending lasers to a target atom in a dense 3D lattice can influence other atoms along the light path, and it is unlikely to leave the state of a nontarget atoms intact after fully Rydberg exciting and deexciting a target atom. Second, the atoms along the light path can exhibit Rydberg blockade, bringing the problem a many-body complexity. One may image that for a two-photon excitation of $s$- or $d$-orbital Rydberg atoms, the two laser beams can be sent along different directions; because both laser fields are largely detuned, only the target atom at the intersection of the two beams are Rydberg excited. However, phase twist can occur for the nontarget atom even if it is pumped only by an off-resonant field [see, for example, the dashed curve in Fig. 3(b) of Ref. Shi (2019)]. So, there can be a phase shift to the atomic state of a nontarget qubit illuminated by the laser fields addressing the lower transition in our problem. In fact, even if the detuning of the laser field addressing the lower transition is 10 times larger than its Rabi frequency $\Omega_{\text{g}}$, the phase shift to the ground state can reach $\pi/2$ if the atom is irradiated by a time of $20\pi/\Omega_{\text{g}}$. These issues can be tackled by using ORIR-based optical spin echo in a very small lattice, and a microwave spin echo assisted by ORIR in a relatively large one. Here, the optical spin echo is for restoring the state of nontarget atoms if no Rydberg interaction exists between the nontarget atoms, while the microwave spin echo is for removing many-body effect if there is Rydberg blockade in the nontarget atoms.

We take a system shown in Fig. 2 for illustration. For a small array, the laser spot is small enough so that only an atom at the beam axis can be irradiated, and the laser intensity is of similar magnitude at the target atom and any nontarget atom along the light path. So, the Rabi frequencies are of similar magnitude for all the irradiated atoms in the problem. In Fig. 2, a cubic lattice with $3\times 5\times 3$ sites is shown, the lattice constant is $\mathcal{L}$, and the relevant transition is between a hyperfine state of $F=2$ and a high-lying $s$-orbital Rydberg state of rubidium-87. The quantization axis is along $[101]$ which can be specified by a magnetic field. All laser fields are $\pi$-polarized along the quantization axis. We want to excite the central atom (denoted by the red ball), which is located at the origin of the Cartesian coordinate (the $\mathbf{x}-\mathbf{y}-\mathbf{z}$ arrows are only for clarifying the directions). In a rotating-frame as in Sec. II, a two-photon transition $|1\rangle\leftrightarrow|r\rangle$ with Rabi frequency $\Omega_{\perp}=\Omega e^{i(t\Delta_{\perp}+\varphi_{\perp})}$ is created by sending focused Gaussian beams along $\mathbf{l}_{\perp}=[\overline{1}21]$, with the foci at the center of the target site. Here, $\varphi_{\perp}$ is determined by the phases of the laser oscillators and the distance that the light travels from the laser sources to the target atom; for brevity, we write $\varphi_{\perp}=\mathbf{k}_{\perp}\cdot\mathbf{r}$, where $\mathbf{k}_{\perp}$ is the wavevector of the fields. Furthermore, $\Delta_{\perp}$ is the overall detuning of the two-photon transition between the ground state $|1\rangle$ and the Rydberg state $|r\rangle$, which should be much smaller than the detuning $\delta$ for the transition $|1\rangle\rightarrow|e\rangle$. In our method, $\Delta$ is several MHz, thus $\delta$ should be at least several hundred MHz. Meanwhile, another set of lasers are sent along $\mathbf{l}_{\shortparallel}=[12\overline{1}]$ which also focus at the same target atom. In method I, the lasers along $\mathbf{l}_{\perp}$ and $\mathbf{l}_{\shortparallel}$ are of similar wavelengths, but with some difference so that the two two-photon transitions are built via very different one-photon detunings at the intermediate state. In the same rotating frame, the lasers along $\mathbf{l}_{\shortparallel}$ drive the transition $|1\rangle\leftrightarrow|r\rangle$ by a Rabi frequency $\Omega_{\shortparallel}=\Omega e^{i(t\Delta_{\shortparallel}+\varphi_{\shortparallel})}$.

Our scheme requires the condition of $e^{i\varphi_{\perp}},~{}e^{i\varphi_{\shortparallel}}=\pm 1$. Because both $\varphi_{\perp}$ and $\varphi_{\shortparallel}$ are determined by the laser sources and the length of the light path, it is possible to tune the laser phases to the condition of $e^{i\varphi_{\perp}},~{}e^{i\varphi_{\shortparallel}}=\pm 1$ if the position fluctuation of qubits is negligible. For optically cooled qubits before each experimental cycle, there is fluctuation of the qubit positions, leading to fluctuation in $\varphi_{\perp}$ and $\varphi_{\shortparallel}$. For the configuration in Fig. 2(b), we have $|\mathbf{k}_{\perp}|=|\mathbf{k}_{\shortparallel}|\approx 5\times 10^{6}m^{-1}$ when the upper and lower fields propagate oppositely. This means that the fluctuation of the qubit position along the light path should be much smaller than $1~{}\mu$m to validate the method. In the experiment of Ref. Graham et al. (2019), the transverse (longitudinal) position fluctuation of the qubit is $0.27~{}(1.47)~{}\mu$m, which will lead to large fluctuation of $\varphi_{\perp}$ and $\varphi_{\shortparallel}$ if similar traps are used here. Thus it is necessary to use sufficiently deep traps and efficient cooling. This is why the methods shown below can not be implemented by sending one laser field along $\mathbf{l}_{\perp}$ for a one-photon transition $|1\rangle\rightarrow|e\rangle$, another field along $\mathbf{l}_{\shortparallel}$ for the other one-photon transition $|e\rangle\rightarrow|r\rangle$, where $|e\rangle$ is the intermediate state shown in Fig. 2(b). This is because the wavevector for these two one-photon transitions are much larger, which makes the condition of $e^{i\varphi_{\perp}}=e^{i\varphi_{\shortparallel}}=\pm 1$ even more challenging to be satisfied.

An alternative solution to the above issue is to use naturally existing transitions with negligible wavevector. For instance, the Rydberg excitation via two counterpropagating fields for $6^{1}S_{1}\rightarrow 6^{1}P_{1}\rightarrow n^{1}S_{0}$ (or $n^{1}D_{2}$) of ytterbium suffers from a negligible Doppler dephasing because $k\lesssim 10^{5}m^{-1}$ (see, for example, Fig.1 of Ref. Lehec et al. (2018)). In this latter case, it is necessary to store quantum information in the nuclear spin states of, e.g., ¹⁷¹Yb (or ¹⁷³Yb) for the purpose of quantum computing with Rydberg interaction. For Rydberg excitation of one of the two nuclear spin qubit states, one can first put the other nuclear spin qubit state to the metastable excited state ${}^{3}P_{0}$ to avoid leakage. However, it is beyond the scope of this work to give detail about this latter scheme of Rydberg gate based on ¹⁷¹Yb. We assume that the fluctuation of $\varphi_{\perp}$ and $\varphi_{\shortparallel}$ has been suppressed.

Below, Secs. III.1 and III.2 show two optical spin-echo methods for the excitation of target atoms, termed as method I and method II. The above discussion is applicable to method I. In method II, the lasers along $\mathbf{l}_{\perp}$ is for the transition $|1\rangle\leftrightarrow|r\rangle$, but those along $\mathbf{l}_{\shortparallel}$ is for the transition between $|r\rangle$ and another Rydberg state $|R\rangle$ via a low-lying intermediate state. Because $|r\rangle$ is near $|R\rangle$, the phase $\varphi_{\shortparallel}$ can be easily set in method II. The discussion about the phase above is applicable for both $\varphi_{\shortparallel}$ and $\varphi_{\perp}$ in method I and $\varphi_{\perp}$ in method II.

It is useful to briefly show why two methods are introduced. For brevity, we use “one pulse” when the laser fields along $\mathbf{l}_{\perp}$ and $\mathbf{l}_{\shortparallel}$ irradiate the system simultaneously for a certain duration. In method I, one pulse is used for the Rydberg excitation, and the nontarget atoms can have some residual population in the Rydberg state when the target atom is excited to the Rydberg state. After the second pulse, both the target and nontarget atoms return to the ground state. No phase shift occurs for the target atom because of the spin echo. In method II, two pulses are used for Rydberg excitation. The two pulses form an optical spin echo sequence for the nontarget atoms so that they have no population in the Rydberg state when the target atom is in the Rydberg state. Similarly, two similar pulses can pump the target atom back to its ground state. Because the target atom experiences no spin echo, it can accumulate a $\pi$ phase shift upon its state restoration. When there is Rydberg interaction between nontarget atoms, a microwave spin echo is used to reverse the sign of the Rydberg interaction, shown in Sec. III.3. In Sec. III.4, we study the issue of divergence of the laser beam. In Sec. III.5, we study the application of the methods in the Rydberg blockade gate. We give a detailed comparison between the two methods in Sec. III.6.