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Coherent spin manipulation of individual atoms on a surface

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Science  25 Oct 2019:
Vol. 366, Issue 6464, pp. 509-512
DOI: 10.1126/science.aay6779

Coherent surface spin manipulation

Spin-based quantum information processing requires coherent spin manipulation. Yang et al. demonstrate coherent control of surface titanium and iron atom spins on a magnesium oxide surface with a magnetic scanning tunneling microscope tip. Arbitrary sequences of fast electrical pulses delivered to the top induced large electric fields. These fields drove metal-atom movement, which then modulated the tip-atom exchange interaction to create an oscillating effective magnetic field. Advanced spin-control protocols such as Ramsey fringes and Hahn spin echoes revealed quantum dynamics, such as coherent oscillations in a titanium atom dimer assembled on the surface with the tip.

Science, this issue p. 509

Abstract

Achieving time-domain control of quantum states with atomic-scale spatial resolution in nanostructures is a long-term goal in quantum nanoscience and spintronics. Here, we demonstrate coherent spin rotations of individual atoms on a surface at the nanosecond time scale, using an all-electric scheme in a scanning tunneling microscope (STM). By modulating the atomically confined magnetic interaction between the STM tip and surface atoms, we drive quantum Rabi oscillations between spin-up and spin-down states in as little as ~20 nanoseconds. Ramsey fringes and spin echo signals allow us to understand and improve quantum coherence. We further demonstrate coherent operations on engineered atomic dimers. The coherent control of spins arranged with atomic precision provides a solid-state platform for quantum-state engineering and simulation of many-body systems.

Detecting and controlling the coherent dynamics of artificial spin systems with single-spin resolution could provide fundamental insight into quantum magnetism (1, 2). A key technique for realizing coherent control is pulsed spin resonance, which traditionally uses short bursts of oscillating magnetic fields to induce transitions between selected quantum spin states (3). Coherent spin manipulation lies at the heart of spin-based quantum information processing and coherent spintronic devices (4). In the solid-state environment, coherent control of single spins has been performed on molecules in break junctions (5), quantum dots and dopants in semiconductors (6, 7), and nitrogen-vacancy centers in diamond (8, 9). It would be desirable to image and precisely modify the local environment and energy-level structures at the atomic scale.

Atomic-scale structures can be constructed and imaged with a scanning tunneling microscope (STM), which can probe spin-spin interactions (10, 11), spin dynamics (1214), and single-molecule motion (15, 16). Notably, artificial nanostructures built by STM have developed into a fruitful testing ground for exploring quantum magnetism (11, 12, 14, 1722). Recently, continuous-wave (CW) electron spin resonance (ESR) was combined with spin-polarized STM to detect spin states with highly improved energy resolution (2224).

The time evolution of a spin state can be accurately controlled by applying pulsed electromagnetic fields (3). In semiconductor heterostructures, coherent spin dynamics have been measured with STM by using femtosecond light pulses (13). Single-spin coherent control has not been achieved in STM because the spin manipulation time was long compared to the coherence time (22). Here, we demonstrate coherent spin manipulations at the single-atom scale, achieved by modulating the magnetic exchange interaction between the STM tip and the Ti atom on a surface using the time-varying electric field in the STM junction (Fig. 1A) (25).

Fig. 1 An STM circuit to manipulate spins of individual atoms.

(A) Schematic showing the low-temperature (T = 1.2 K) STM integrated with a radiofrequency (RF) generator and an arbitrary waveform generator (AWG), as well as an STM image (8.4 nm by 10 nm) of Ti and Fe atoms on a bilayer MgO on Ag(001) (setpoint: VDC = 60 mV, IDC = 10 pA). Exchange coupling (red wavy line) with the tip magnetic moment (white arrow) drives Ti spin resonance. The tip magnetoresistively senses the different Ti spin orientations, indicated by the red (|↓〉 state), blue (|↑〉 state), and green (superposition state) arrows. (B) Top: Orbital occupancy of the 3d1 configuration. Bottom: Zeeman energy of the Ti spin. (C) CW-ESR spectrum of a single Ti atom, fitted to an asymmetric Lorentzian (eq. S14) (VDC = 50 mV, IDC = 5 pA, VRF = 50 mV, Bext = 0.82 T). Inset: resonant frequency f0 as a function of IDC at a constant VDC = 50 mV. Fitting (red curve) to eq. S17 yields an angle θ = 73° between the tip’s magnetic moment and Bext.

The Ti atoms were separated from the Ag substrate with a bilayer MgO film and were most likely hydrogenated (supplementary materials and methods) (24). We refer to them simply as Ti. Each Ti atom has an electron spin S = 1/2 and was adsorbed either on top of a surface oxygen atom or at a bridge site between two oxygen atoms (24, 26). The two spin states |↓〉 and |↑〉, with electron spin quantum number ms = ±1/2, were separated by the Zeeman energy in the presence of an applied magnetic field (Fig. 1B). In addition to the direct-current (DC) voltage VDC, a radiofrequency (RF) voltage was applied with an amplitude VRF at the STM tunnel junction (22, 24).

The magnetic field experienced by a Ti spin (S) is the vector sum of the externally applied static field Bext and the effective tip field, which, in the presence of VRF with frequency f, has a static component Btip and an oscillatory component ΔBtipcos(2πft). The Hamiltonian is given by (25)H=γ[B+ΔBtipcos(2πft)]S(1)where γ is the gyromagnetic ratio and ℏ is Planck’s constant. B = Bext + Btip is the total static field. As long as ΔBtip is noncollinear with B, it can drive transitions between |↓〉 and |↑〉 states when hf matches the Zeeman splitting set by B (Fig. 1B, bottom). This change in magnetic states is visible as a change in the steady-state current in the CW-ESR spectrum (Fig. 1C).

To access the fast coherent spin dynamics of a Ti atom, we applied a series of RF pulses at the resonant frequency to induce Rabi oscillations between the two spin states (Figs. 1A and 2A). Between pulses, the Ti spin was reinitialized to the |↓〉 state by a spin-polarized current. The degree of polarization of the initial state was determined by the spin polarization of the tip (27). During the RF pulse, the Ti spin rotated coherently between the |↓〉 and |↑〉 states at the Rabi frequency Ω, so increasing the pulse width τ yielded an oscillatory current signal ΔI (Fig. 2, A and B). The exponential decay of ΔI gives the coherence time T2Rabi≈ 40 ns. When the RF frequency was detuned away from the Larmor frequency, Ω increased and the oscillation amplitude was reduced (fig. S10).

Fig. 2 Coherent control of a single Ti spin.

(A) Rabi oscillations of a Ti spin at different values of VRF, as indicated (VDC = 60 mV, IDC = 4 pA, Bext = 0.90 T). The RF frequency is on resonance at f0 = 22.57 GHz (fig. S6A). Solid curves are sinusoidal fits to the tunnel current signal ΔI. The data are offset vertically for better visibility. Each spectrum was measured for ~20 min. (B) Rabi oscillations represented in the color scale shown, over a fine grid of τ and VRF. Inset: Ω at different values of VRF fitted to a quadratic polynomial with zero offset. (C) Schematics showing the homodyne detection of the Rabi oscillations by measuring current Ihomo, the time-averaged product of GRF and VRF. GRF(t) is the instantaneous tunnel conductance when VRF (t) is on. Pulse width τ was controlled by the AWG. (D) Ω and 1/T2Rabi at different tip–atom distances set by different IDC at VDC = 50 mV (VRF = 80 mV, Bext = 0.82 T). Solid lines are linear fits. Lower panels show the Rabi oscillations with fitting at IDC = 3 pA (left) and 9 pA (right).

The Rabi oscillation of the Ti spin was read out magnetoresistively by using a homodyne detection technique (Fig. 2C and supplementary text section 4), a standard technique for coherent detection in physics and engineering (28, 29). Here the asymmetric CW-ESR line shape is the result of homodyne demodulation (26). When the moment of the magnetic tip was canted with respect to the quantization axis of the Ti spin, the Larmor precession of the Ti spin yielded an RF conductance GRF. The Rabi oscillations of the Ti spin modulated the amplitude of GRF and were then extracted by measuring the time-average current due to VRF (Fig. 2C). This homodyne detection yielded a better signal-to-noise ratio than measurements using DC pulses (supplementary text section 5), so homodyne detection results are reported below.

The spatial orientation of the tip magnetic moment was crucial for observing the Rabi oscillations (supplementary text section 6), because both the driving field and the amplitude of GRF increased with the angle θ between the tip magnetic moment and Bext (inset of Fig. 1C).

We gained additional control of Ω by adjusting the tip–atom distance dtip, in addition to varying VRF. Ω increased linearly with the setpoint current IDC at a constant VDC and VRF (Fig. 2D). Because IDC depended exponentially on dtip, this result demonstrated the exponential dependence of Ω on dtip, an important characteristic of exchange-field–driven ESR (25). We note that T2Rabi increased with larger dtip (Fig. 2D), because one of the dominant decoherence sources was the tunneling current (supplementary text section 7) (30). For any given tip, T2Rabi and Ω scaled inversely with varying tip height. Thus, for a given tip and VDC, the number of observable Rabi cycles was nearly independent of the tip height and could possibly be increased by using the magnetic dipolar coupling to the tip, or by placing a surface spin nearby to supply the magnetic field gradient (23).

The coherence time of Rabi oscillations T2Rabi is, in principle, sensitive to (i) scattering electrons from tip or substrate (31), (ii) tunneling current arising through VDC (30) and VRF, (iii) variations of the resonant frequency caused by the slow-varying tip magnetic field, and (iv) variations in Ω caused by changes of tip–atom distance (Fig. 2D) or variations in VRF. To understand and improve the spin coherence time, we applied multipulse spin manipulation sequences (Fig. 3).

Fig. 3 Ramsey fringes and spin echo of a single Ti spin.

(A) Pulse scheme (top) and Bloch sphere representation (bottom) of the spin evolution in the rotating frame for Ramsey fringes measurement. Red and blue arrows represent the Ti spin’s initial or final state. Purple dots indicate free evolution between pulses. Orange dots represent a rotation about the x axis. (B) Ramsey signals and Ramsey fringe frequency ΩRam (inset) at different detuning frequencies ff0. (C) Pulse scheme (top) and Bloch sphere representation (bottom) for Hahn-echo measurement. (D) Spin-echo signals at different values of total free evolution time τ. The exponential fit (red curve) gives a coherence time of T2 ~189 ns. Measured at VDC = 50 mV, IDC = 3 pA; VRF = 190 mV, f0 = 20.55 GHz, Bext = 0.82 T.

Figure 3B shows the Ramsey fringes measured on a single Ti atom by applying two π/2 pulses separated by a time delay τ, where π represents an RF pulse that flips the spin from |↓〉 to |↑〉. The RF frequency was intentionally detuned from the resonance so that during the free evolution τ, a phase was accumulated between the two spin components. The time delay τ thus determined whether the spin returned to the |↓〉 state or continued to evolve to the |↑〉 state, leading to interference fringes. We directly controlled the Ramsey fringe frequency by choosing the frequency detuning ff0 (Fig. 3B, inset).

The decay of the Ramsey signal with τ also gave a coherence time T2*, which was about 40 ns. T2* was not substantially longer than T2Rabi, indicating that decoherence was dominated by sources other than current induced by VRF or slow variations in Ω.

To improve the coherence time, we performed a Hahn-echo pulse sequence, consisting of a π pulse placed between two π/2 pulses: π/2−τ/2−π−τ/2−π/2, where τ is the duration of free precession (Fig. 3C). This sequence decoupled the Ti spin from any slowly changing magnetic fields (32). Here, the last π/2 pulse served as a homodyne detection pulse (7). The echo signal decayed exponentially with a time constant T2 ≈ 189 ± 23 ns (Fig. 3D), which is several times longer than T2Rabi or T2*. This result indicates that the Ramsey coherence time T2* was limited by slow field variations that can be canceled by the echo technique. The main remaining decoherence source is the tunneling current and scattering electrons from the tip and substrate, and thus we would expect lower tunnel current and thicker MgO (31) to further extend T2.

We further demonstrate coherent operations on coupled-spin states in designed atomic structures by assembling Ti atom pairs at chosen spacings (Fig. 4) (24). The spin Hamiltonian of a Ti spin dimer (S1 and S2) is H=HZee+Hint(2)which consists of the Zeeman energy HZee=γ(Bext+Btip)S1z+γBextS2z and the interaction energyHint=JS1S2+D(3S1zS2zS1S2) (supplementary text section 11) (24). The Ti spin under the tip is S1. Hint contains an exchange coupling with strength J and dipolar coupling with strength D and produced a four-level energy spectrum with an unequal energy spacing, which could be characterized by CW-ESR (24, 26).

Fig. 4 Coherent control of spin dimers.

(A) Left: Schematic showing an STM image (4 nm by 4 nm) of one oxygen-site and one bridge-site Ti atom (VDC = 60 mV, IDC = 30 pA), and the magnetic tip. Middle: Atom positions of Ti dimer on MgO. Gray circles represent oxygen atoms. Right: Energy level diagram with a ferromagnetic coupling of ~0.1 GHz. Dashed arrows indicate ESR transitions. (B) CW-ESR measured on the oxygen-site Ti atom (VDC = 50 mV, IDC = 3 pA, VRF = 10 mV). (C) Pulsed ESR measured on the oxygen-site Ti atom at f = 20.50 GHz and 20.58 GHz, corresponding to the blue and red peaks in (B) (VDC = 50 mV, IDC = 3 pA), showing the coherent transitions from |↓⇑〉 to |↑⇑〉 (left), and from |↓⇓〉 to |↑⇓〉 (right). (D) Dependence of Ω on VRF for transitions iii and iv. Ω is extracted from a sinusoidal fit in (C). (E to H) Same as in (A) to (D) except measured on two oxygen-site Ti atoms having a closer spacing, with an antiferromagnetic coupling of ~0.9 GHz. (G) The coherent transitions from |↓⇓〉 to |↓⇑〉–α|↑⇓〉 (left) and from |↓⇓〉 to |↑⇓〉+α|↓⇑〉 (right), where α ≈ 0.7. (H) Dependence of Ω on VRF for transitions I and III. VRF = 25 mV in (F). Measured at Bext = 0.82 T.

The spin dimer in Fig. 4A had an interaction energy of ~0.1 GHz, much smaller than the Zeeman energy difference between the two spins induced by the tip field (~0.35 GHz). The energy eigenstates thus remained approximated as Zeeman product states (|↓⇓〉, |↓⇑〉, |↑⇓〉, and |↑⇑〉) (24). We performed controlled rotation of the spin under the tip (target spin), conditionally on the state of the other spin (control spin). When a pulse was applied at frequency fiv (Fig. 4B), matching the energy spacing between the first and third energy levels (Fig. 4A, right panel), only the |↓⇓〉 ⇢ |↑⇓〉 transition was resonant with the RF field. Correspondingly, the RF field flipped the target spin if, and only if, the control spin was in state |⇓〉 (Fig. 4C, right panel). Similarly, a π pulse at frequency fiii flipped the target spin when the control spin was in state |⇑〉. This capability provides a possible realization of a CNOT gate (33).

The spin dimer in Fig. 4E had a larger interaction of ~0.9 GHz, which overcame the magnetic asymmetry created by the tip and caused two of the quantum eigenstates to strongly mix to form states |↓⇑〉−α|↑⇓〉 and |↑⇓〉+α|↓⇑〉 with α ≈ 0.7 (24). The other two eigenstates remained the Zeeman product states (Fig. 4E, right panel). We coherently excited the system from the ground state |↓⇓〉 to either of the mixed states. When the RF pulse was tuned to transition I (Fig. 4F), |↓⇓〉 evolved into |↓⇑〉−α|↑⇓〉 for a π pulse (Fig. 4G, left). In comparison, an RF pulse at the frequency of transition III coherently rotated |↓⇓〉 into |↑⇓〉+α|↓⇑〉 (Fig. 4G, right). The Rabi frequency Ω grew in proportion to VRF for these two transitions (Fig. 4H). For any given VRF, the ratio of Ω for these two transitions depended only on the state mixing α (supplementary text section 11). As we increased the tip field, the state mixing was reduced (24), and thus the ratio of Ω between I and III became smaller (fig. S13). This tunable tip magnetic field thus provided control of the energy-level structures and Rabi rates.

Combining pulsed ESR with STM manipulation paves the way for the coherent excitation and detection of many-body states of artificial spin structures, such as topological quantum states of atomic chains (34). The exchange-field–driven coherent manipulation should be applicable to other solid-state spin systems (fig. S14) (5, 7). The time-domain ESR spectroscopy could extend the sensing capabilities of a single-atom quantum sensor (8, 9, 32) and possibly also be used for quantum information applications using surface atoms (33, 35).

Supplementary Materials

science.sciencemag.org/content/366/6464/509/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S14

References (3646)

References and Notes

Acknowledgments: We thank B. Melior for expert technical assistance. Funding: We acknowledge financial support from the Office of Naval Research. S.-H.P., P.W., Y.B., T.C., T.E., and A.J.H. acknowledge support from the Institute for Basic Science (IBS-R027-D1). A.A. acknowledges support from the Engineering and Physical Sciences Research Council (EP/L011972/1 and EP/P000479/1) and the QuantERA European Project SUMO. Author contributions: K.Y. and C.P.L. designed the experiment. K.Y., W.P., S-H.P., P.W., Y.B., T.C., and T.E. performed STM measurements. K.Y. and C.P.L. performed the analysis and wrote the manuscript with help from all authors. All authors discussed the results and edited the manuscript. Competing interests: None declared. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the supplementary materials. Additional data related to this paper may be requested from the authors.
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