Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

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56 Rashba spin-orbit coupling in quantum dots requires a suitably large change in α. In experiments with 2DEGs, changes in α of a factor of 2 are reported, and in a recent Letter by Koga et al., α was shown to vary in the range ≈ (0.3 − 1.5) × 10 −12 eVm (a factor of 5) in one InGaAs sample [25]. Grundler [27] has shown that the large back-gate voltages usually used to change α can be drastically reduced by placing the gates closer to the 2DEG. Thus, it is conceivable that changes in α of a factor between 2 and 5 could be produced with voltages small enough to be pulsed with rise times substantially shorter than a typical coherent oscillation period. In Fig. 3.3a we plot time-traces of the transition probability P(t p ) calculated for the first anticrossing as a function of magnetic field. We have used the values α 1 = 1.5×10 −12 and α 2 = 0.3×10 −12 eVm from the Koga experiment [25]. The amplitude of the oscillations P max for different ratios of α 2 /α 1 is presented in Fig. 3.3b, which shows a node at B = B 0 (δ = 0). This is because, for δ = 0, the eigenstates of JCM are 2 −1/2 (|n,↑〉 ± |n + 1,↓〉) for all α ≠ 0. Therefore, a finite detuning is required to obtain the maximum amplitude, which concurs with δ > k B T,Γ R to overcome broadening effects. Both the amplitude P max and frequency Ω show non-trivial dependencies on α 1 and α 2 as well as on the magnetic field. This latter behaviour stems from the parametric dependence on B of all three parameters in H JC . For our model parameters with α 2 /α 1 = 1/5 and with the detuning set such that the amplitude is maximised, we have P max ≈ 0.45 with a Rabi frequency of Ω = 2GHz, which corresponds to a period of about 3 ns. This is within accessible range of state-of-the-art experimental technique. Note that the period can be extended by using weaker confinement and SO coupling. For both the observation of coherent oscillations, and the operation as a qubit, it is essential that the lifetime of state ψ + α is long. This is the case for a pure electronic spin in a QD [37, 110], and we now show that the hybridisation of the spin with the orbitals, and the ensuing interaction phonons, does not affect this. We assume a piezo-electric coupling to acoustic phonons via the potential V ep = λ q e iq·r (b q + b † −q), with phonon operators b q and |λ q | 2 = P/2ρcqV , with coupling P, mass density ρ, speed of sound c, and volume V [111]. For n = 0, a Golden Rule calculation yields the rate Γ ep /ω 0 = √ mP 2l0 8π(ω s ) 2 sin ρl 0 ˜l 2 θ + sin 2 θ − ξ 5 I(ξ), (3.9) with ω s = c/l 0 , ξ = 2 −1/2 (˜l/l 0 )(∆/ω s ), and I(ξ) ≤ 8/15. Close to B 0 , ξ ≪ 1, and thus the rate is extremely small Γ ep ≈ 10 4 s −1 (Fig. 3.3c). Therefore, the robustness of spin qubits is not significantly weakened by the SO hybridisation. In general, residual relaxation affects our measurement scheme in two ways. During the oscillation (Fig. 3.2c), the system may relax to the eigenstate ψ − α 2 . This
0 0.5 1.5 3.1 Spin-orbit driven coherent oscillations in a few-electron quantum dot 57 2 0 1 2 tp in ns 8 6 4 2 0 (a) P (t p ) 0.4 0.2 0.0 0.8 1 1.2 1.4 B/B 0 B/B 0 0.5 1 1.5 (b) (c) 0 1 2 B/B 0 0.4 0.2 0.0 1e–5 1e–7 Pmax Γep/ω0 Figure 3.3: Characteristics of the Rabi oscillation. (a) Probability P(t p ) of finding electron in upper level after time t p following the non-adiabatic change α 1 = 1.5 → α 2 = 0.3 × 10 −12 eVm as function of magnetic field. (b) Amplitude of oscillation as function of B/B 0 for changing from α 1 = 1.5, 0.8, 0.6 to α 2 = 0.3 × 10 −12 eVm (top to bottom). (c) Phonon-induced relaxation rate for InAs parameters α = 1.5 × 10 −12 eVm, P = 3.0 × 10 −21 J 2 /m 2 , ρ = 5.7 × 10 3 Kg/m 3 , c = 3.8 × 10 3 m/s. Close to B 0 the rate is suppressed to Γ ep < 10 −7 ω 0 . damps the oscillation by a factor exp(−Γ 1 t p ) to the constant value I = eΓ R P max /2. Relaxation during the read-out phase (Fig. 3.2d) simply reduces the overall amplitude of the signal by a factor exp(−Γ 2 /Γ R ). Clearly then, to observe oscillations, we require Γ 1 < Ω and Γ 2 < Γ R . In summary, we have outlined a proposal for the observation of spin-orbit driven coherent oscillations in a single quantum dot. We have derived an approximate model, inspired by quantum optics, that shows the oscillating degree of freedom to represent a novel, composite spin-angular momentum qubit. This work was supported by the EU via TMR/RTN projects, and the German and Dutch Science Foundations DFG, NWO/FOM. We are grateful to T. Brandes, C.W.J. Beenakker and D. Grundler for discussions, and to B. Kramer for guidance and hospitality in Hamburg.
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Interaction and confinement in nano
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Abstract iii Abstract It is the pur
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Publications v Publications Some of
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2 Table of Contents 3 Rashba spin-o
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4 Introduction (phonons). Therefore
Page 12 and 13: 6 Introduction Figure 1: Scanning e
Page 15 and 16: Chapter 1 The Rashba effect Effects
Page 17 and 18: 1.2 The model 11 the position-depen
Page 19 and 20: 1.3 Rashba effect in a perpendicula
Page 21 and 22: Chapter 2 Rashba spin-orbit couplin
Page 23 and 24: 2.1 Rashba effect and magnetic fiel
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Page 33 and 34: 2.2 Spectral properties, various li
Page 39 and 40: 2.3 Electron transport in one-dimen
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Page 51 and 52: 2.4 Summary 45 For weak SO coupling
Page 53 and 54: 2.4 Summary 47 splitting can become
Page 55 and 56: Chapter 3 Rashba spin-orbit couplin
Page 57 and 58: 3.1 Spin-orbit driven coherent osci
Page 59 and 60: 3.1 Spin-orbit driven coherent osci
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Page 65 and 66: 3.2 Introduction to quantum dots an
Page 75 and 76: PSfrag replacements 3.2 Introductio
Page 77 and 78: 3.3 Effects of relaxation 71 page 5
Page 79 and 80: 3.3 Effects of relaxation 73 The re
Page 81 and 82: 3.3 Effects of relaxation 75 with q
Page 83 and 84: 3.3 Effects of relaxation 77 eh 14
Page 85 and 86: Chapter 4 Conclusion In this part o
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Page 89: Part II Phonon confinement in nanos
Page 92 and 93: 86 Introduction to electron-phonon
Page 94 and 95: 88 Introduction to electron-phonon
Page 96 and 97: 90 Coupled quantum dots in a phonon
Page 102 and 103: ) ) ) ) 3 4 + ) * - 96 Coupled quan
Page 106 and 107: 100 Coupled quantum dots in a phono
Page 108 and 109: 102 Coupled quantum dots in a phono
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106 Coupled quantum dots in a phono
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108 Coupled quantum dots in a phono
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110 Conclusion due to phonon van-Ho
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Appendix A Jaynes-Cummings model Th
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Appendix B Fock-Darwin representati
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Appendix C Evaluation of the phonon
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Appendix D Matrix elements of dot e
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121 with λ ± l (q ‖) = F n B pz
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Bibliography [1] G. Bergmann, Phys.
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Bibliography 125 [31] M. Pletyukhov
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Bibliography 127 [72] T. Schäpers,
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Bibliography 129 [109] U. Merkt, J.
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Bibliography 131 [144] B. Auld, Aco
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Bibliography 133 [182] A. A. Kisele
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Acknowledgements Finally, I would l
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