Nonlinear effects in electrically driven spin resonance

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1 Nonlinear effects in electrically driven spin resonance András Pályi Eötvös University udapest, Hungary with Gábor Széchenyi (Eötvös) Rabi frequency (a.u.) amplitude of ac E-field (a.u.) G. Széchenyi and A. Pályi, Phys. Rev. (4)

2 Magnetic spin resonance single-electron QD experiment: Koppens et al., Nature 6 Energy scales: orbital level spacing: ω (ω ) Zeeman splitting: (g µ ) excitation frequency: ω excitation strength: ac (g µ ac ) ω ac sin ωt if ω = (resonance condition), then P (t) ΩRabi RWA time drive strength, ac

3 Part Electrically driven spin resonance in quantum dots

4 Electrically driven spin resonance (simple, `local ) Coherent Control of a Single Electron Spin with Electric Fields K. C. Nowack,* F. H. L. Koppens, Yu. V. Nazarov, L. M. K. Vandersypen* 3 NOVEMER 7 VOL 38 SCIENCE A I dot E Energy.3 µm ext DC+ + dot ee(t) Position

P (dm) spin blo of zero states s ediated between emiconductor nanowire Here sp ro field ( " " or L.EDSR P.

5 P (dm) spin blo of zero states s ediated between emiconductor nanowire Here sp ro field ( " " or L.EDSR P. Kouwenhoven Spin orbit qubit in a semiconductor nanowire ESR Kavli Institute of Nanoscience, Delft University of Technology, 6 GA Delft, The Netherlands. Department of Applied Physics, Eindhoven University of Technolog dient as *These authors contributed equally work. S. Nadj-Perge *, S. M. Frolov *,toe.this P. A. M. akkers & L. P. Kouwenhoven magnet a funis estiin the le 8 4 N AT U R E VO L / 3 D E C E M E R on pro- a b Motion of electrons can influence their spins through a fun-macmillan Publishers Limited. All rights reserved at the changes damental effect called spin orbit interaction. This interaction pro- a mation, b ctrons, vides a way to control spins electrically and thus lies at the D between foundation nt spin the 3 of spintronics. Even at the level of single electrons, om the spin orbit interaction has proven promising for coherent spin 3 ubit) in 4 states w rotations. Here we implement a spin orbit quantum bit (qubit) in 4 5 tion spin is ar Δr where the spin orbit interaction is an 5indium arsenide nanowire, S Express so strong that spin and motion can no longer be separated. In this In this and universal single-qubit c μm regime, we realize fast qubit rotationstime extract -qubit c superpo I (fa) 45 5 control using only electric fields; the qubits are hosted in singlesingleelectron quantum dots that are individually addressable. We.7 compon tuating e. We enhance coherence by dynamically decoupling the qubits from 38 the environment. Nanowires offer various advantages for quantum s from the righ computing:5 they can serve as one-dimensional templates!for scalable EDSR 5 (ns) antum burst qubit registers, and it is possible to vary the material even during evolves calable wire growth. Such flexibility can be used to design wires with supgduring due to d Rabi frequency vs. drive strength can mo pressed decoherence and to push semiconductor qubit e fidelities towards error correction levels. Furthermore, electrical dots6 can th supcentral be integrated with optical dots in p n junction nanowires. The parallel delities coherence times achieved here are sufficient for the conversion of ots can an electronic qubit into a photon, which can serve as a flying qubit from pa 7 for long-distance quantum communication. P = 3 dm s. The ciently Figure a shows a scanning electron microscope image of our nanosionindiof ), either a (X, X) or a doi:.38/nature968 (Y, Y) state, because antiparallel states decay quickly to a non-blocked singlet state,. y idling the qubits in the parameter range of spin blockade, they will be initialized in one of the two parallel states with equal probability. We note that spin orbit and hyperfine interactions also mediate a slower decay of parallel states into (, )7,9,. This reduces the read-out fidelity to 7 8% (Supplementary Information, section 5.). electric field applied to gate four oscillates both inducing EDSR. c, Spin blockade is lifted near f 5 gm/h. Here the microwave power is P 5 extracted from c at f 5 9 GHz. e, Magnified view doi:.38/nature968 at high values owing to the difference betwee frequency is swept in a fixed range around f 5 at resonance varies owing to non-monotonic m Electrically LETTERdriven spin resonance (simple, `local ), 3 4 3,4 D S 5 Δr μm Time R wire device. Two electrodes, source and drain, are used to apply a voltage bias of 6 mv across the InAs nanowire. Voltages applied to five closely spaced, narrow gates underneath the nanowire create a confinement potential for two electrons separated by a tunnelling barrier. The defined structure is known as a double quantum dot in the (, ) charge configuration7. Each of the two electrons represents a spin orbit qubit (Fig. b). In the presence of strong spin orbit coupling, neither spin nor orbital number is separately well defined. Instead, each qubit state is a spin orbit doublet, X and Y. Similar to pure spin states, a magnetic field,, controls the energy splitting, EZ 5 gm, between spin orbit states, where g is the Lande g-factor in a quantum dot and m is the ohr (mt)the crucial difference from a spin magneton. qubit is that in a spin orbit qubit e the orbital part of the spin orbit wavefunction is used for f (GHz) f (MHz) f (GHz) (mt) R I (pa) I (fa) Figure Device and ESR detection scheme. a, ScanningVelectron (mv) μw microscope (SEM) P image of adm devicef with the same gate pattern as used in = 34 (mt) the experiment. The Ti/Au gates are4 deposited on top of a GaAs/AlGaAs e d I (fa) heterostructure containing a two-dimensional electronf =gas 9 nm below the 9 GHz 4 surface. White arrows indicate current flow.8 through the two coupled dots g = 37 dm (dotted circles). ThePright side gate is fitted with a homemade bias-tee (riseg magnetic (mt) time 5field ps) to allow fast pulsing of the dot levels. b, SEM image of a device I (fa) d 6 L ESR sp The res measur satellite the RF inhomo 8. f (GHz) gr, are r nanobecause apply a plied to he reatecona nelling transdot in. mev b). In I (pa) g qubit 5 current I (pa) (pa) 6 microwave frequency (GHz).7 3

6 Model # (spin-orbit) ψ (z) W so θ nanowire H = p z m + mω z = ω E(t) =E ac cos(ωt) a a + x H = σ z z H E = e E ac z cos(ωt). H SO = ασ x p z = SO σ x i(a a) Questions: - mechanism of electrically driven spin resonance? - what is the character of spin dynamics? - spin-flip time vs. drive strength?

7 The mechanism (in a perturbation picture) ω SO E ω ω Golovach et al., PR 6

8 The mechanism (in a perturbation picture) ω SO E ω ω SO E Golovach et al., PR 6

9 The mechanism (in a perturbation picture) ω SO E ω ω SO E Ω Rabi SOE ω Golovach et al., PR 6

10 The mechanism (in a perturbation picture) ω SO E ω ω SO E Ω Rabi SOE ω Golovach et al., PR 6

11 The mechanism (in a perturbation picture) ω SO E ω ω SO E SO E Ω Rabi SOE ω Golovach et al., PR 6

12 The mechanism (in a perturbation picture) ω SO E ω ω SO E SO E Ω Rabi SOE ω Ω Rabi SOE (ω + ) E SO (ω ) SOE ω Golovach et al., PR 6

13 Part Electrically driven spin resonance due to a disordered transverse magnetic field G. Széchenyi and A. Pályi, Phys. Rev. (4) see also hyperfine-mediated EDSR: Laird et al., PRL 7, Semicond. Sci. Tech. 9 Rashba PR 8

14 The model ( disordered transverse field ) ψ (z) W Szechenyi and Palyi, PR 4 ξ i x z (z) Ingredients: E(t) =E ac cos(ωt) ξ i dis = ξ i ξ j dis = ξ δ ij - single electron trapped in D parabolic quantum dot - static -field σ z - ac E-field oscillation amplitude A - disordered transverse field (z)σ x, (z) =a i ξ i δ(z z i ),

15 Collective transverse field ψ (z) W ξ i x z (z) E(t) =E ac cos(ωt) collective transverse field: = ψ (z) ψ energy scale of collective transverse field: a dis ξ W energy-scale hierarchy: ω ω

16 Time-periodic collective transverse field A =.W weak drive A = W strong drive A = W position, z/w color: ψ (z,t) collective transverse field, (t)/ T T T T T T π/ω 4π/ω π/ω 4π/ω time, t time, t

17 Time-periodic collective transverse field A =.W Get typical Rabi freq: weak drive A = W strong drive A = W () Fourier transform position, z/w (discrete freq spectrum) () take Fourier component at color: ψ (z,t) ω (3) avg for realizations collective transverse field, (t)/ T T T T T T π/ω 4π/ω π/ω 4π/ω time, t time, t

18 Result #: Maximal Rabi frequency typical Rabi frequency, ΩRabi/ oscillation amplitude, A/W

19 Result #: Maximal Rabi frequency typical Rabi frequency, ΩRabi/ saturation due to anharmonicity: Seigo Tarucha s talk on Monday oscillation amplitude, A/W

20 Slower spin-flips at stronger drive: why? simplify: A >> W, zig-zag oscillation center of wave packet, z(t)/a t } } W π ω time Ω Rabi π/ω dt (t) cos ωt floor(a/w ) j= t (j t) cos ωj t } W A W/A } A/W [ (j )-s uncorrelated]

z N-photon subharmonic resonances x b y x y x E A Laird et al 8 Laird et al., SST 9 (GaAs) 9 8 7 I (pa) 36 5 (b) I (pa) Vlp (mv) = V V ac e e.

Spin resonance signal (measured in conductance) in the device of figure 5(a). The EDSR signal shows up as a decrease in φ (deg) 4 g = 6.

Nanotech (Nanotubes) e Microwave frequency (GHz) I (pa) (nv) 3 y z (T) conductance as expected at frequency corresponding to g =th:.

) An additional signal of opposite sign Figure Effect half of the bend. a, (dotted Current at constant appears4at exactly this frequency line).

) signals below mt is consistent with a greater 3 non-blockaded bias direction the current is isotropic. d, Sc axis as inoffig.

The contribution of x to whorizontal ¼ 8 andfeatures w ¼ 98.

5ghz result from resonances of the eff varies g = 6.8 n=3 + Scarlino, Kawakami,...,Vandersypen circuit.

V-shaped lines with g ¼ are electric dipole spin resonances (EDSR).

frequency is subtracted.) The resonance lines with transitions7,9. EDSR is me blockade lifting around = at[9].

21 z N-photon subharmonic resonances x b y x y x E A Laird et al 8 Laird et al., SST 9 (GaAs) I (pa) 36 5 (b) I (pa) Vlp (mv) = V V ac e e.8 m 8 7 I (pa) 5 mv bias = T week ending 6 JUNE 4 (c) g/ g/3 g/4 5 g/5 8 6 (5,)T(5,)S (5,)T (5,)T+ g = Figure 6. Spin resonance signal (measured in conductance) in the device of figure 5(a). The EDSR signal shows up as a decrease in φ (deg) 4 g = I (pa) 64 5 PHYSICAL REVIEW LETTERS f (GHz) tot (mt) (a) Detuning (mev) c PRL, 76 (4) eff g φ (deg) eff 4 Field (mt) y Fei Pei et al., Nat. Nanotech (Nanotubes) e Microwave frequency (GHz) I (pa) (nv) 3 y z (T) conductance as expected at frequency corresponding to g =th:.45danon & 5 exp: Stehlik et al. PRL (InAs nanowire), Rudner arxiv orating a V (mv) (mev) (mt) f (GHz) rp (marked with the dashed line.) An additional signal of opposite sign Figure Effect half of the bend. a, (dotted Current at constant appears4at exactly this frequency line). The larger magnetic field ¼ T applied in the x y plane as a function of detuning and field angle w n = n= splitting bothg.) signals below mt is consistent with a greater 3 non-blockaded bias direction the current is isotropic. d, Sc axis as inoffig. mag b, Cut along w at the detuning marked by the arrow in a. c, In the gthe = 7.8 total field difference between dots. The contribution of x to whorizontal ¼ 8 andfeatures w ¼ 98. across the bend, precession of effective spin states about different axes leads to partial lifting of valley at.5when and.5ghz result from resonances of the eff varies g = 6.8 n=3 + Scarlino, Kawakami,...,Vandersypen circuit. As in figure, field- and frequency-independent e,microwave Current as a function of microwave frequency and z. V-shaped lines with g ¼ are electric dipole spin resonances (EDSR). (To make the re backgrounds have been subtracted, including any signal due to spin n=4 unpublished slopeg/n arise from n-photon clearer, the mean current each frequency is subtracted.) The resonance lines with transitions7,9. EDSR is me blockade lifting around = at[9].... blockade transition in the many-electron regime of a second device with similar geometry. which would be expected if the field gradient were the primary 5 (mt) EDSR mechanism. n=8 I (pa) (d) and the8x (left axis), ng z (out of is shown in lower onfined (Si/SiGe) 5 S(T)v sviassa¼function ( vsofvvsand l V (þ) v An interesting consequence of the level structure in tofig. address svsl)n (n, normalization factor). (color online). (a) lp rp. Microwave driving with f ¼ 4 GHz and ev ac ¼.8 mev broadens the allow use lowest energy K! axis. (,) statelevelasdiagram a spectroscopic that featuresthis in thewe charge stabilitythe diagram. A white line indicatestk! the detuning (b) Energy with the ac excitation drawn to valley spin blockade appears also for the initially 7. Open issues and discussion

22 N-photon subharmonic resonance A =.W A = W A = W position, z/w color: ψ (z,t) collective transverse field, (t)/ T T T T T T π/ω 4π/ω π/ω 4π/ω time, t time, t

23 N-photon subharmonic resonance A =.W A = W A = W position, z/w color: ψ (z,t) Get typical Rabi freq: () Fourier transform (discrete freq spectrum) () take Fourier component at Nω (3) avg for realizations collective transverse field, (t)/ T T T T T T π/ω 4π/ω π/ω 4π/ω time, t time, t

24 Result #: Subharmonic resonances (experiments: Marcus, Kouwenhoven, Petta groups) typical Rabi frequency, ΩRabi/ N = N = N = oscillation amplitude, A/W power-law turn-on; maximal Rabi frequencies; efficient subharmonics

25 Part 3 Spin-valley qubit and disordered transverse field in carbon nanotubes Kuemmeth, Ilani et al., Nature 8 Flensberg and Marcus, PR Rudner and Rashba PR Laird et al, Nat. Nanotech review: Laird et al., arxiv 4

26 Disordered transverse field in carbon nanotube QD K K energy splitting: spin-orbit, ~ mev K K

27 Disordered transverse field in carbon nanotube QD K K energy splitting: spin-orbit, ~ mev K K spin-valley qubit

28 Disordered transverse field in carbon nanotube QD K K energy splitting: spin-orbit, ~ mev K K

29 Disordered transverse field in carbon nanotube QD K K K K energy splitting: spin-orbit, ~ mev K K K K

30 Disordered transverse field in carbon nanotube QD K K K K energy splitting: spin-orbit, ~ mev K K K K K K K K

31 Disordered transverse field in carbon nanotube QD K K K K energy splitting: spin-orbit, ~ mev K K K K impurity K K K K

32 Disordered transverse field in carbon nanotube QD K K K K energy splitting: spin-orbit, ~ mev K K K K K K impurity K K K K K K

33 Disordered transverse field in carbon nanotubes K K H imp V imp e i(k K )x imp SO K K ψl (z imp ) K K +h.c. valley resonance in nanotubes: Palyi and urkard, PRL

34 Disordered transverse field in carbon nanotubes K K H imp V imp e i(k K )x imp SO K K ψl (z imp ) K K +h.c. this is the random tranverse field ξ valley resonance in nanotubes: Palyi and urkard, PRL

35 Summary ) disordered transverse field model ) strong driving: A ~ W 3) maximal Rabi frequency 4) multi-photon spin control typical Rabi frequency, ΩRabi/ N = N = N = oscillation amplitude, A/W Thanks to: - OTKA (Hungarian NSF) PD Grant - EU FP7 Marie Curie Career Integration Grant `CarbonQubits - János olyai Scholarship of the Hungarian Academy of Sciences

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