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Electrically driven nuclear spin resonance in single-molecule magnets

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Science  06 Jun 2014:
Vol. 344, Issue 6188, pp. 1135-1138
DOI: 10.1126/science.1249802

Electrical control of nuclear spin qubits

Quantum bits of information (qubits) that are based on spins of atomic nuclei are an attractive option for quantum information processing. It can sometimes be tricky to manipulate these qubits using magnetic fields directly. Thiele et al. developed a technique for electrically controlling a nuclear spin qubit in the single-molecule magnet TbPc2. When they hit the qubit with a microwave pulse, the microwave's electric field generated effective magnetic fields much larger than those available previously.

Science, this issue p. 1135

Abstract

Recent advances in addressing isolated nuclear spins have opened up a path toward using nuclear-spin–based quantum bits. Local magnetic fields are normally used to coherently manipulate the state of the nuclear spin; however, electrical manipulation would allow for fast switching and spatially confined spin control. Here, we propose and demonstrate coherent single nuclear spin manipulation using electric fields only. Because there is no direct coupling between the spin and the electric field, we make use of the hyperfine Stark effect as a magnetic field transducer at the atomic level. This quantum-mechanical process is present in all nuclear spin systems, such as phosphorus or bismuth atoms in silicon, and offers a general route toward the electrical control of nuclear-spin–based devices.

The realization of a functional quantum computer is currently one of the most ambitious technological goals. Among existing concepts (13), devices in which the quantum bits (qubits) are encoded by spins are very attractive, as they benefit from the steady progress in nanofabrication and allow for electrical readout of the qubit states (46). Nuclear-spin–based devices are better isolated from the environment than their electron spin counterparts (7), but their detection and manipulation remain challenging.

Operating nuclear spin qubits have been demonstrated with devices based on nitrogen vacancy centers (8), single-molecule magnets (911), and silicon (12). Yet, their integration remains limited by the on-chip microcoils (13) used to manipulate the spin. The parasitic crosstalk to neighboring spin qubits and the large currents necessary to perform quantum operations are serious limiting factors. Using electric fields instead of magnetic fields to manipulate the spin would alleviate this problem, as only small displacement currents are required; in addition, electric fields can be easily focused and shielded within a small volume. The coupling of the spin to the electric field is established by the hyperfine Stark effect, which transforms the electric field into a local magnetic field. Moreover, the static hyperfine Stark effect can be used to tune individual nuclear qubits in and out of resonance (14) and thus allows for the individual addressability of different nuclear spin qubits.

To perform our experiments, we used a three-terminal nuclear spin qubit transistor (9) (Fig. 1A). We studied the transistor, consisting of a TbPc2 single-molecule magnet coupled to source, drain, and gate electrodes, by performing electrical transport measurements inside a dilution refrigerator at 40 mK. We can associate the device with three coupled quantum systems (Fig. 1B):

Fig. 1 Nuclear spin qubit transistor and its detection scheme.

(A) Artist’s view of a nuclear spin qubit transistor based on a single TbPc2 molecular magnet. The molecule, consisting of a Tb3+ ion (pink) sandwiched between two Pc-ligands (white), is coupled to source, drain, and gate (not shown) electrodes. The four anisotropic nuclear spin states of the Tb3+ (colored circles) can be manipulated with an electric field pulse. (B) Three coupled subsystems of the transistor: (i) The four-level nuclear spin qubit is hyperfine (HF) coupled to (ii) an Ising-like electronic spin, which in turn is antiferromagnetically exchange (Ex) coupled to (iii) a readout quantum dot.

(i) A nuclear spin qubit emerging from the atomic core of the Tb3+ ion. It possesses a nuclear spin Embedded Image leading to four different qubit states: Embedded Image, Embedded Image, Embedded Image, and Embedded Image.

(ii) An electronic spin arising from the 4f electrons of the terbium. Its electronic configuration is [Xe]4f8 resulting in a total spin of S = 3 and a total orbital momentum of L = 3. A strong spin-orbit coupling yields an electronic spin with a total angular magnetic moment of J = 6. In addition, the ligand field, generated by the two Pc’s, leads to a well-isolated electron spin ground state doublet of Embedded Image and Embedded Image with a uniaxial anisotropy axis perpendicular to the Pc plane. The degeneracy of the doublet is lifted by the hyperfine coupling to the nuclear spin qubit and splits each electronic spin ground state into four different quantum states. At zero external field, the energy levels are intrinsically separated by Embedded Image GHz, Embedded Image GHz, and Embedded Image GHz, where the index 0 corresponds to the ground state, and indices 1, 2, and 3 to the first, second, and third excited states, respectively (Fig. 2A).

Fig. 2 Zeeman diagram and nuclear spin detection procedure.

(A) Zeeman diagram of the TbPc2 molecular magnet, showing the hyperfine split electronic spin ground state doublet Embedded Image and Embedded Image as a function of the external magnetic field Embedded Image parallel to the easy-axis of magnetization. The ligand-field–induced avoided level crossings (colored rectangles) allow for tunneling of the electron spin. (B) The jumps of the conductance g of the readout quantum dot during magnetic-field sweeps are nuclear spin dependent. (C) Histograms of the positions of about 75,000 conductance jumps, showing four nonoverlapping Gaussian-like distributions; each conductance jump can be assigned to a nuclear spin state.

(iii) A readout quantum dot created by the Pc ligands. Their delocalized π-electron system is tunnel-coupled to the source and drain terminals, creating a quantum dot in the vicinity of the electronic spin carried by the Tb3+ ion. Furthermore, an overlap of the delocalized π-electron system with the terbium’s 4f wave functions gives rise to an exchange coupling of Embedded Image T between the readout quantum dot and the electronic spin (15).

To perform the readout of the single–nuclear spin state, we exploit the different interactions between the three quantum systems.

First, the hyperfine interaction splits each electronic ground state doublet into four nuclear-spin–dependent levels (Fig. 2A). Electronic levels corresponding to the same nuclear spin state are mixed owing to the off-diagonal terms in the ligand-field Hamiltonian; in an external magnetic field, this results in avoided level crossings of Embedded Image (rectangles in Fig. 2A). Sweeping the magnetic field slowly enough over such an anticrossing gives rise to the quantum tunneling of magnetization (QTM) (16, 17), which reverses the electronic spin according to the Landau-Zener probability (18, 19). Because the magnetic field position of the QTM is nuclear spin dependent, we can use this process to measure the state of the nuclear spin qubit (911).

In the second stage, the electronic spin is mapped onto the readout dot’s conductance through use of the exchange interaction (15). It induces a slight modification of the readout quantum dot’s chemical potential depending on whether the electronic spin is Embedded Image or Embedded Image. Therefore, when sweeping the magnetic field at constant bias and gate voltages, the reversal of the electronic spin results in a conductance jump (Fig. 2B). The amplitude of the jump is typically about 3% and its position in the magnetic field is nuclear spin state dependent.

For statistical analysis, we swept the magnetic field back and forth 75,000 times while monitoring the conductance of the readout quantum dot. By plotting the magnetic field position of all the detected conductance jumps into a histogram, we obtained four nonoverlapping peaks (Fig. 2C), which enabled us to unambiguously assign a nuclear qubit state to each detected jump. The error induced by our nuclear spin readout procedure is mainly due to inelastic electronic spin reversals, which were misinterpreted as a QTM event, and is estimated to be less than 4% (20). For the device presented here, we found relaxation times T1 of ≈34 s for Embedded Image and T1’s of ≈17 s for Embedded Image (15).

We turn now to the electrical manipulation of a single nuclear spin. The hyperfine Stark effect describes the change of the hyperfine constant A in the Hamiltonian Embedded Image as a function of an external electric field (21, 22). Writing the Hamiltonian as Embedded Image demonstrates how the modification of A is converted into a change of the effective magnetic field Embedded Image at the center of the nucleus. Given a HF constant of A = 24.9 mK (23) and Embedded Image, we obtain an effective static field of 329 T. Thus, a periodic modulation of A by 1/1000 of its value is sufficient to generate local magnetic field oscillations of Embedded Image329 mT. Because the orientation of the quantization axis of the molecule with respect to the electric field is not well determined, the effective magnetic field will have components in the x and z direction. However, in terms of oscillating fields, only the component in the x direction can rotate the nuclear spin, whereas the z component induces additional decoherence. Moreover, even moderate electric field amplitudes of 1 mV/nm are sufficient to induce a controlled fine tuning of the HF constant, which is on the order of 1% (15).

For the experimental demonstration of the single–nuclear-spin manipulation, we focused on the nuclear qubit subspace of Embedded Image and Embedded Image, whose eigenstates are separated by Embedded Image GHz [the exact value is device dependent (23, 24)]. We initialize the nuclear spin qubit by sweeping the external magnetic field back and forth between ±60 mT at 80 mT/s (Fig. 3A) until a QTM transition is measured at –38 mT, which is the signature of the Embedded Image qubit state (Fig. 2C). We then apply a microwave (MW) pulse of duration τ and a local field amplitude on the order of ≈1 mV/nm while keeping the external magnetic field constant (Fig. 3A); the pulse modulates the hyperfine constant A at the MW frequency. Finally, we detect the resulting state by sweeping back the external magnetic field on a time scale faster than the measured relaxation times of both nuclear spin states. The entire sequence is rejected when the final state is not detected because of a missing QTM transition. Repeating this procedure resulted in coherent Rabi oscillations (Fig. 3, B and C). The visibility of the Rabi oscillations as a function of the applied MW frequency (Fig. 4A) has a maximum at the resonant frequency Embedded Image and decreases for increasing detuning Embedded Image. In addition, a clear dependence of the nuclear qubit resonance frequency on the gate voltage is observed in Fig. 4A. This effect can be attributed to the static HF Stark shift, owing to the additional electric field induced by the gate voltage, which shows our ability to tune the HF constant A between the electronic spin and the nuclear spin qubit. Only the z component of the effective magnetic field will modify the level splitting. Applying a static gate voltage offset of 16 mV shifts the resonant frequency of the nuclear spin qubit by Embedded Image MHz, corresponding to Embedded Image (Fig. 4B, inset).

Fig. 3 Rabi oscillations of a single nuclear spin qubit.

(A) Time-dependent external magnetic field Embedded Image and pulse sequence. The nuclear spin is first initialized in the lower Embedded Image state (init sequence). A subsequent MW pulse of frequency Embedded Image and duration τ induces an effective oscillating magnetic field resulting in coherent manipulation of the two lower states of the nuclear spin qubit. Finally, Embedded Image is swept back to probe the final state of the nuclear spin qubit. (B and C) Rabi oscillations between Embedded Image and Embedded Image states obtained by repeating the above sequence 100 times at each τ, for two different MW powers, (B) PMW = 1 mW and (C) PMW = 1.58 mW.

Fig. 4 Stark shift of the hyperfine coupling and Ramsey fringes.

(A) Rabi oscillations visibility measured at different MW frequencies for three different gate voltages Embedded Image. (B) Rabi frequencies Embedded Image corresponding to the visibility of (A). The continuous lines are fit to the experimental points following the theoretical expression of the Rabi frequency dependence (see text). (Inset) The relative change Embedded Image with respect to the applied Embedded Image. (C) Time-dependent external magnetic field Embedded Image and pulse sequence. Initialization and probe of the nuclear spin qubit are performed by using the identical protocol as in Fig. 3A. The MW sequence consists of two π/2 pulses, with an increasing interpulse delay τ. (D) Ramsey interference fringes obtained by repeating the procedure of (C) 100 times. Embedded Image = 2.205 V, corresponding to a Rabi frequency Embedded Image = 1.136 MHz and a resonant frequency Embedded Image = 2.449 MHz of the nuclear spin qubit. The measured coherence time Embedded Images.

To extract the Stark shift–induced effective ac magnetic driving field at the nuclei, used to coherently manipulate the nuclear spin qubit, we measured the Rabi frequency Embedded Image evolution as a function of the applied MW frequency for the three different gate voltages (Fig. 4B). The horizontal shift of the minimum is again induced by the static gate voltage, whereas the vertical evolution indicates an increasing effective ac field in the x direction, which is probably caused by the nonlinearity of the HF Stark effect. The solid lines are fits to Embedded Image, with Embedded Image being the nuclear g-factor [≈1.354 for Tb (25)], Embedded Image the nuclear magneton, and Bx the effective magnetic field in the x direction. The equation gives Bx = 62, 98, and 183 mT for Embedded Image 2.205, 2.215, and 2.221 V, respectively, up to two orders of magnitude higher than magnetic fields created by on-chip microcoils. The electric driving field is induced along the source-drain direction, and only the x component of the corresponding effective magnetic field is responsible for the nuclear spin rotation.

We turn now to the measurements of the Ramsey fringes to assess the dephasing time Embedded Image of the nuclear spin qubit, which is tantamount to the average duration over which the coherence of the quantum superposition is preserved. As shown by the pulse sequence presented in Fig. 4C, the nuclear spin qubit is first initialized in the Embedded Image state. Subsequently, two π/2 MW pulses are generated with an interpulse delay τ. Finally, the readout of the final state is probed with the same procedure as explained previously. Repeating this procedure results in the Ramsey fringes shown in Fig. 4D. The data follow an exponentially decaying cosine function revealing a coherence time Embedded Image. Detailed studies suggest that the major contribution to the decoherence was caused by charge noise, which induced magnetic field fluctuations of about 10 mT via the HF Stark effect. Therefore, we expect that more stable gate oxides would increase Embedded Image by one or two orders of magnitude.

Our results show the general feasibility of establishing an all-electrical control of a single nuclear spin through use of the hyperfine Stark effect and should be transferable to other spin qubit devices with a large hyperfine interaction.

Supplementary Materials

www.sciencemag.org/content/344/6188/1135/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S10

References (2635)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: This work was partially supported by MoQuaS FP7-ICT-2013-10, the Deutsche Forschungsgemeinschaft Programs no. SPP 1459 and TRR 88 3Met, ANR-12-JS10-007, ANR-13-BS10-0001 MolQuSpin, and European Research Council Advanced Grant MolNanoSpin no. 226558. The samples were manufactured at the NANOFAB facilities of the Neel Institute. We acknowledge E. Bonet, O. Buisson, Y. Deschanels, F. Evers, E. Eyraud, O. Gaier, M. Ganzhorn, C. Godfrin, C. Grupe, C. Hoarau, D. Lepoittevin, T. Meunier, N. Roch, C. Thirion, M. Urdampiletta, and R. Vincent.
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