⇦Chinese Physics Letters, 2020, Vol. 37, No. 4, Article code 044209Express Letter High-Fidelity Manipulation of the Quantized Motion of a Single Atom via Stern–Gerlach Splitting * Kun-Peng Wang (王坤鹏)1,2,3, Jun Zhuang (庄军)1,2,3, Xiao-Dong He (何晓东)1,2**, Rui-Jun Guo (郭瑞军)1,2,3, Cheng Sheng (盛诚)1,2, Peng Xu (许鹏)1,2, Min Liu (刘敏)1,2, Jin Wang (王谨)1,2, Ming-Sheng Zhan (詹明生)1,2** Affiliations 1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071 2Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071 3University of Chinese Academy of Sciences, Beijing 100049 Received 18 January 2020, online 06 March 2020 *Supported by the National Key Research and Development Program of China (Grant Nos. 2017YFA0304501, 2016YFA0302800 and 2016YFA0302002), the Key Research Program of Frontier Science of the Chinese Academy of Sciences (CAS) (Grant No. ZDBS-LY-SLH012), the National Natural Science Foundation of China (Grant No. 11774389), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB21010100), and the Youth Innovation Promotion Association CAS (Grant No. 2019325).
**Corresponding authors. Email: hexd@wipm.ac.cn; mszhan@wipm.ac.cn Citation Text: Wang K P, Zhuang J, He X D, Guo R J and Sheng C et al 2020 Chin. Phys. Lett. 37 044209    Abstract We demonstrate high-fidelity manipulation of the quantized motion of a single $^{87}$Rb atom in an optical tweezer via microwave couplings induced by Stern–Gerlach splitting. The Stern–Gerlach splitting is mediated by polarization gradient of a strongly focused tweezer beam that functions as fictitious magnetic field gradient. The spatial splitting removes the orthogonality of the atomic spatial wavefunctions, thus enables the microwave couplings between the motional states. We obtain coherent Rabi oscillations for up to third-order sideband transitions, in which a high fidelity of larger than $0.99$ is obtained for the spin-flip transition on the first order sideband after subtraction of the state preparation and detection error. The Stern–Gerlach splitting is measured at a precision of better than $0.05$ nm. This work paves the way for quantum engineering of motional states of single atoms, and may have wide applications in few body physics and ultracold chemistry. DOI:10.1088/0256-307X/37/4/044209 PACS:42.50.Dv, 32.80.Qk, 42.50.Ct © 2020 Chinese Physics Society Article Text With recent booming developments of atom sorting techniques,[1–8] single neutral atoms[9,10] or molecules[11,12] in optical tweezers offer a promising platform for quantum simulations and computations and ultracold chemistry. For the aforementioned applications, full quantum state control over the single atom is highly demanded including internal hyperfine state and the quantized motion.[13,14] Especially controlling the quantized motion is the first essential step to coherently associating a single molecule from exactly two atoms and manipulating few-atom systems for cold chemistry. The conventional techniques for manipulating atomic motional states is to drive Doppler sensitive Raman laser couplings at a large detuning of resonant optical transitions, which could lead to decoherence caused by photon scattering and fluctuations of optical phases and intensities.[15–20] An alternative route is based on direct microwave couplings between motional states which can bypass the technical limitations encountered in Raman couplings. The key idea of the microwave coupling is to engineer Zeeman state-dependent displacements in the atomic wavefunctions by using Stern–Gerlach effect in a real gradient magnetic field[15,21] as demonstrated with trapped ions.[15–20] For optically trapped neutral atoms, vector light shifts can be conveniently utilized to generate the Stern–Gerlach effect and engineer the microwave driven motional state transitions.[21,22] In a constant homogeneous magnetic field the motional wavefunctions for all the hyperfine Zeeman states would be essentially the same. Thus the different trap states are orthogonal and have no wavefunction overlaps to induce transitions. The state-dependent displacement removes the orthogonality between motional states and induces non-zero Franck–Condon factors, thus leads to motional sidebands appearing in addition to the carrier transitions even with a homogeneous external magnetic field. Previous demonstrations of this scheme have been implemented in state-dependent optical lattices[21,23–25] as well as nanofiber-based traps.[26,27] However, various technical fluctuations in lattice potentials make it challenge to control the motional states with high fidelity. To date, the benchmarking of the fidelity of motional state control for neutral atoms has not yet been reported. In this work, we report a high fidelity of larger than $0.99$ for the motional state control of single neutral atoms. We demonstrate long-lived coherent Rabi oscillations between the motional states of a single trapped $^{87}$Rb atom with microwave couplings in an optical tweezer. To turn on the couplings between motional states, we utilize the Stern–Gerlach splitting by changing the trap polarization and the external magnetic field to desire directions and driving microwave transitions between two hyperfine states. Long-lived coherent Rabi oscillations between the motional ground state and the excited states are obtained. The fidelity of motional state control is measured to be $0.996(1)$ with a state preparation and measurement (SPAM) error of $0.05(4)$. The reported fidelity agrees with theoretical simulations. The atomic wavefunction displacements are then precisely measured at a precision of better than $0.05$ nm.
cpl-37-4-044209-fig1.png
Fig. 1. Experimental scheme. (a) Single $^{87}$Rb atoms are trapped in an optical tweezer (contours) that has polarization gradient (density plots) around the focus of the beam. The corresponding maximum values are set to $1$. A linearly polarization along the $x$ direction leads to wavefunction displacements along this axis. The ground state wavefunction has a typical width of about $50$ nm depending on the trap oscillating frequency. (b) The displaced harmonic oscillator model (not to scale). For $^{87}$Rb atoms, the $|2,-2\rangle$ and $|1,-1\rangle$ states have a relative displacement $d$ along the $x$ axis.
Figure 1 depicts the experimental scheme. The details of the experimental setup and ground state cooling have been described in our previous work.[28] Single $^{87}$Rb atoms are loaded from a magneto-optical trap (MOT) with an $852$-nm trapping beam which passes through a high numerical aperture (${\rm NA} = 0.6$) microscope and focused to a waist of about $0.75\,µ$m. When a single atom is detected, it is further cooled to about $15\,µ$K with the standard optical molasses method. Then the trapped $^{87}$Rb atoms are initialized into the hyperfine state of $|F,m_{\rm F}\rangle \equiv |2,-2\rangle$ with optical pumping. Then the three-dimensional Raman sideband cooling[22,29] is performed to prepare single $^{87}$Rb atoms into the motional ground state in the trap and a three-dimensional ground state probability of $0.91(5)$ is obtained.[28] An optical tweezer is typically formed by a strongly confined Gaussian beam through a high numeric aperture microscope with diffraction-limited performance, thus a longitudinal polarization component emerges around the focus of the beam leading to the polarization gradient effect.[22,26] The polarization gradient functions as a fictitious magnetic field along a specific axis which is perpendicular to both the polarization vector and the wave vector of the beam. The maximum gradient in the trap center is approximated by $2.6 {\rm NA} \sin({\rm NA})/\lambda \approx 1.03/\,µ$m, where $\lambda$ is the wavelength of the trap beam.[22] Such a polarization gradient can be expressed as a fictitious magnetic field gradient of $2.1$ G$/\mu$m in a $1.6$-mK trap which causes vector light shifts and changes the effective potential depth. Due to the Stern–Gerlach effect, the fictitious magnetic field gradient displaces the position of the potential well for Zeeman states of $m_{\rm F} \neq 0$ along the direction of the input polarization of the trap beam. The atomic wavefunction for an atom in a specific $m_{\rm F}$ state has a displacement $d$ relative to the focus of the tweezers. The displacement $d$ depends on the hyperfine state $|F,m_{\rm F}\rangle$ via[26,30] $$ d=\frac{1}{4\pi} \frac{\alpha_{\rm v}}{\alpha_{\rm s}} \frac{m_{\rm F}}{F} \lambda ,~~ \tag {1} $$ where $\alpha_{\rm s}$ and $\alpha_{\rm v}$ are respectively the scalar and vector polarizabilities depending on the atomic energy levels and the specific wavelength of the tweezer. For the $^{87}$Rb atom in an $852$-nm trap, the ratio $\alpha_{\rm v} / \alpha_{\rm s}$ is $0.083$ for the $F=1$ ground state and $-0.166$ for $F=2$, thus the relative displacement between the wavefunctions in the $|2,-2\rangle$ and $|1,-1\rangle$ state is estimated to be $17$ nm. This relative displacement $d$ enables the coupling between motional states when driving a microwave transition between the two states with a Lamb–Dicke (LD) parameter of $\eta = d \sqrt{m \omega_x / (2 \hbar)}$,[21] where $m$ is the atomic mass and $\omega_x$ is the trap oscillating frequency. We calculate the wavefunction overlaps between the ground state and excited states, as shown in Fig. 2, by numerically solving the Schrödinger equation for a Gaussian trap. We note that the wavefunction displacement leads to a decrease in the carrier transition strength. The strength of the first (second) order sideband transition has a maximum at a displacement of $\sqrt{2\hbar/m \omega_x}$ ($2\sqrt{\hbar/m \omega_x}$), which corresponds to $38$ nm ($54$ nm) in a $1.6$-mK trap.
cpl-37-4-044209-fig2.png
Fig. 2. The calculated wavefunction overlap as a function of wavefunction displacement. The trap depth is set at $1.6$ mK and the beam waist is $0.75\,µ$m. The overlaps between two ground states ($| n=0 \rangle$ and $| n'=0 \rangle$, where the index prime denotes displaced states) $\langle n'=0 | n=0 \rangle$ are shown as filled circles, and the overlaps between the first excited state ($|n'=1 \rangle$) and the ground state $\langle n'=1 | n=0 \rangle$ are shown as squares and the overlaps $\langle n'=2 | n=0 \rangle$ ($\langle n'=3 | n=0 \rangle$) are shown with triangles (open circles). Positive position part of the ground state wavefunction is plotted as the solid line to guide the eyes (rescaled to keep the height of about $1$ at the origin).
We firstly demonstrate the control of the Stern–Gerlach splitting by changing the polarization of the trapping beam. To observe both red and blue motional sideband transitions, we do not apply the ground state cooling here. After molasses cooling and optical pumping, the atomic state is initialized into $|2,-2\rangle$. Then the trap polarization is set along $x$ direction controlled by a liquid-crystal variable wave plate and the quantization field is set along $y$ direction, so that the motional state coupling is turned on. Subsequently, we record the carrier and the sideband transitions by applying rectangular shape pulses to drive the spin transition $|2,-2\rangle$ to $|1,-1\rangle$. The resulting spectra are shown in Fig. 3(a). The coupling is strong so that second order sideband transitions can be observed clearly. In this spectrum, the peak at zero detuning is resonant with the carrier transition and the data are fitted with a Gaussian multi-peak function. When both the trap polarization and the quantization field are set along the $y$ direction, the trap polarization gradient effect and the coupling between motional states is suppressed. As the squares indicated in Fig. 3(a), we do not observe clear sideband transitions. For this spectrum, we fit the data with a standard Rabi sinc-function of ${\it\Omega}^2/({\it\Omega}^2+\Delta^2) \sin^2(\sqrt{{\it\Omega}^2+\Delta^2}t)$.
cpl-37-4-044209-fig3.png
Fig. 3. (a) The microwave spectra of thermal $^{87}$Rb atoms in a linearly polarized trap where the polarization vector is set along the $y$ direction (squares) and $x$ direction (filled circles) respectively. When the atoms are trapped in the $x$-direction polarized trap, both the red ($|n\rangle \rightarrow |n'-1\rangle$) and blue ($|n\rangle \rightarrow |n'+1\rangle$) sideband transitions can be observed. Even second order transitions can be observed obviously. (b) The microwave carrier and sideband transitions of single $^{87}$Rb atoms in the quantum ground state of an optical tweezer. For the carrier transition, the microwave (MW) pulse duration is $0.03$ ms, and for the $\Delta n=\{1,2,3\}$ sideband transitions, the corresponding pulse durations are $\{0.07,0.15,0.3\}$ ms respectively. The inset shows the coherent Rabi oscillations on the first (squares) and second (filled circles) order sideband transitions.
Next we demonstrate the coherent manipulation of the quantized motion of a single $^{87}$Rb atom. After preparing a motional ground-state single atoms in the state dependent potential, we drive the microwave transition with a rectangular pulse shape to obtain the carrier and sideband spectra, as shown in Fig. 3(b). Higher order transitions $|n\rangle \rightarrow |n'+2\rangle$ and $|n\rangle \rightarrow |n'+3\rangle$ can also be observed clearly with pulse durations as long as $0.15$ ms and $0.3$ ms, respectively. Compared with the case of thermal atoms, red sideband transitions that decrease the motional quanta are suppressed nearly perfectly due to the high fidelity of ground-state preparation. Then we observe the coherent Rabi oscillations on the sideband transitions shown in Fig. 3(b), where the carrier oscillation is omitted for simplicity. The corresponding Rabi frequencies are ${\it\Omega}_c=2\pi \times 15.7(6)$ kHz and ${\it\Omega}_{sb1}=2\pi \times 7.64(1)$ kHz for the carrier and the first order sideband transitions respectively. The resulting ratio ${\it\Omega}_{sb1}/{\it\Omega}_c = 0.490(2)$ amounts to the spatial Lamb–Dicke parameter. The second order sideband transition has a Rabi frequency of $2\pi \times 2.87(2)$ kHz. We can also drive the Rabi oscillations on the third order $|n\rangle \rightarrow |n'+3\rangle$ sideband transitions.[31]
cpl-37-4-044209-fig4.png
Fig. 4. (a) The excitation probabilities $P$ as a function of number of $\pi$ pulses $l$ for the first order sideband transitions $|n\rangle \rightarrow |n'+1\rangle$. The solid line is the model fitting and the dashed area is the $95\%$ confidence band. The open circles are the Monte Carlo simulations. (b) The dependence of operation error $\epsilon=1-F$ on the magnetic field noise with a constant $\sigma_{\it\Omega}$ of $0.012$ ${\it\Omega}$. The red points are the Monte Carlo simulations of the sideband $\pi$-pulse fidelity. Two blue squares are the experimental results of $\pi$-pulse fidelity obtained from the analytical fittings. The green circles are the average fidelities obtained from the simulation of randomized benchmarking under the same parameters as the $\pi$-operation.
Then we benchmark the fidelity for the motional state manipulation by performing a multi-pulse sequence measurement and comparing with Monte carlo simulations. Experimentally, we apply odd numbers of $\pi$-pulses on the first-order sideband transition and obtain the decay of population as the number of pulses increases, as shown in Fig. 4(a). We fit the experimental data with a formula of $P=\frac{1}{2}+\frac{1}{2}(1-d_{\rm if})(1-2\epsilon)^l$, where $d_{\rm if}$ is the depolarization probability associated with SPAM, while $\epsilon$ is the average error per pulse.[32,33] The fitted $\epsilon=0.004(1)$ and $d_{\rm if}=0.05(4)$ lead to a fidelity of $F=1-\epsilon=0.996(1)$ for the $\pi$-pulses. To understand the sources contributing to the above measured fidelity, we examine two main sources of error during the experiment: frequency detuning caused by magnetic field noise and fluctuations of Rabi frequency caused by microwave power. To this end, the fluctuations of the magnetic field $B$ and Rabi frequency ${\it\Omega}$ (proportional to the square root of microwave power) are modeled with Gaussian distributions of $f(x)=\exp (-(x-\mu_f)^2/(2 \sigma_f ^2))$ ($f = B$ or ${\it\Omega}$), where $\mu_f$ is the mean value and $\sigma_f$ is the standard deviation. We extract the fluctuations of magnetic field and microwave power from a set of measured Rabi oscillations by Monte Carlo simulations[31] and obtain $\sigma_{\it\Omega} = 0.012(1) {\it\Omega}$ and $\sigma_B=0.19(1)$ mG.[31] With these two parameters the calculated fidelity is $0.9973(3)$ and the corresponding simulation of multi-$\pi$ sequence measurement is also shown in Fig. 4(a). Thus the simulations agree well with the measurement result. We note that before upgrading the current supply that generates the constant magnetic field the measured $\pi$-pulse fidelity is only $0.981(3)$ (the corresponding $\sigma_B$ is fitted to be $0.57$ mG), which also agrees with the simulation. A multi-$\pi$ sequence measurement may experience the error cancellation when there is an offset in the pulse area which is typically caused by the long term drifts in the experiment. Thus we calibrate the experimental conditions to suppress the error cancellation.[31] Moreover, we calculate the average error of single-qubit Clifford gates by simulating the randomized benchmarking (RB) process under the same noise, as shown in Fig. 4(b). The resulting average fidelity of $0.995(1)$. The RB processes is typically applied to obtain an average fidelity for $24$ single qubit Clifford gates.[33] Due to the difference in sensitivity to magnetic field noise and pulse area errors for different gates, the average fidelity is typically lower than the $\pi$-transition fidelity. In the end, we describe the measurement of the Stern–Gerlach splittings or the wavefunction displacements by using wavefunction overlaps. The ratio of the wavefunction overlap for the sideband transition $|n=0\rangle\rightarrow|n'=1'\rangle$ to that of the carrier transition $|n=0\rangle\rightarrow|n'=0'\rangle$ is related to the spatial Lamb–Dicke parameter $\eta$, which is defined by[24] $$ \eta \equiv {\it\Omega}_{\langle 1'|0\rangle}/{\it\Omega}_{\langle 0'|0\rangle} = \frac{d}{\sqrt{2}} \sqrt{\frac{m \omega_x}{\hbar}},~~ \tag {2} $$ where the index prime denotes the final states, ${\it\Omega}_{\langle 0'|0\rangle}$ and ${\it\Omega}_{\langle 1'|0\rangle}$ are the Rabi frequencies of the carrier and sideband transitions. Experimentally, we measure the ratio ${\it\Omega}_{\langle 1'|0\rangle}/{\it\Omega}_{\langle 0'|0\rangle}$ for different trap depths (or the square root of the harmonic oscillation frequency $\sqrt{\omega_x}$), as shown in Fig. 5. We then extract the slope of $0.04005(6)/\sqrt{{\rm\,kHz}}$ by a linear fit. From Eq. (2), the resulting relative displacement between the $|2,-2\rangle$ and $|1,-1\rangle$ states is $d=19.32(3)$ nm, which is consistent with the previous estimation. We measure the displacements at different magnetic fields and obtain the average value of $19.30(4)$ nm as shown in Fig. 5. Because the absolute displacements are proportional to the magnetic moments of each state, the corresponding displacement is $12.88(3)$ nm for the $|2,-2\rangle$ state, and $-6.44(2)$ nm for the $|1,-1\rangle$ state. The single atom wavefunction has a typical width of about $50$ nm, thus our measurement uncertainty of the Stern–Gerlach splittings via wavefunction overlaps is well below this width. The Stern–Gerlach splittings for other atomic species can also be determined using this method. The splitting can be controlled by changing the direction of the magnetic field or the trap polarization and the maximum splitting can be further tuned by changing the laser wavelength of the tweezers.
cpl-37-4-044209-fig5.png
Fig. 5. Measurement of wavefunction displacements in the tweezer trap. The displacements are measured at several magnetic fields which are along the $y$ direction. The inset shows the measured $\eta$ as a function of the square root of the trap frequency $\omega_x$ at a magnetic field of $5.546$ G.
In conclusion, we have demonstrated the high-fidelity manipulation of the motional states of single atoms by utilizing the Stern–Gerlach splitting in an optical tweezer. The achieved fidelity is larger than $0.99$, which is limited by technical fluctuations and drifts. Furthermore, the Stern–Gerlach splitting is measured precisely by probing wavefunction overlaps between motional states. Together with the high fidelity single qubit gates on the internal states,[34] our work represents the full quantum state control of a single neutral atom qubit with high quality which will be beneficial to quantum computation with neutral atoms. When two atoms are prepared in a single trap, the collision dynamics are strongly dependent on their relative motions, thus the Stern–Gerlach splitting can be used to engineer the quantized motion of heteronuclear two-atom systems[35,28] and even few-body mixture systems. Therefore it may have wide applications in the investigations of few-body physics and ultracold chemistry.[36–38] Because the polarization gradient is inherent within strongly focused optical tweezers, the microwave manipulation can also be extended to single molecules as proposed recently.[39] References An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arraysSingle-atom trapping in holographic 2D arrays of microtraps with arbitrary geometriesIn situ single-atom array synthesis using dynamic holographic optical tweezersThree-dimensional rearrangement of single atoms using actively controlled optical microtrapsLow-Entropy States of Neutral Atoms in Polarization-Synthesized Optical LatticesSynthetic three-dimensional atomic structures assembled atom by atomSorting ultracold atoms in a three-dimensional optical lattice in a realization of Maxwell’s demonGray-Molasses Optical-Tweezer Loading: Controlling Collisions for Scaling Atom-Array AssemblyExperimental investigations of dipole–dipole interactions between a few Rydberg atomsQuantum information with Rydberg atomsA quantum dipolar spin liquidAn optical tweezer array of ultracold moleculesEntangling two transportable neutral atoms via local spin exchangeMeasurement-Based Entanglement of Noninteracting Bosonic AtomsTrapped-Ion Quantum Logic Gates Based on Oscillating Magnetic FieldsMicrowave quantum logic gates for trapped ionsMicrowave Control of Trapped-Ion Motion Assisted by a Running Optical LatticeHigh-Fidelity Quantum Logic Gates Using Trapped-Ion Hyperfine QubitsHigh-Fidelity Universal Gate Set for Be 9 + Ion QubitsTrapped-Ion Spin-Motion Coupling with Microwaves and a Near-Motional Oscillating Magnetic Field GradientMicrowave Control of Atomic Motion in Optical LatticesCoherence and Raman Sideband Cooling of a Single Atom in an Optical Tweezer3D Projection Sideband CoolingMicrowave control of atomic motional states in a spin-dependent optical latticeStern–Gerlach detection of neutral-atom qubits in a state-dependent optical latticeFictitious magnetic-field gradients in optical microtraps as an experimental tool for interrogating and manipulating cold atomsObservation of Ultrastrong Spin-Motion Coupling for Cold Atoms in Optical MicrotrapsPreparation of a heteronuclear two-atom system in the three-dimensional ground state in an optical tweezerCooling a Single Atom in an Optical Tweezer to Its Quantum Ground StateDynamical polarizability of atoms in arbitrary light fields: general theory and application to cesiumCharacterizing quantum gates via randomized benchmarkingRandomized benchmarking of quantum gatesHigh-Fidelity Single-Qubit Gates on Neutral Atoms in a Two-Dimensional Magic-Intensity Optical Dipole Trap ArrayMolecular Assembly of Ground-State Cooled Single AtomsFew-body physics with ultracold atomic and molecular systems in trapsUniversal few-body physics and cluster formationOne-dimensional mixtures of several ultracold atoms: a reviewSideband cooling of molecules in optical traps
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