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

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70 Rashba spin-orbit coupling in quantum dots α 1 ↔ α 2 , all possible ratios are shown in the figure. For the ratios of ˜λ 2 /˜λ 1 = 2 and 5, as in chapter 3.1, we find amplitudes of the order of 12% and 45%, respectively. In a possible experimental setup, the magnetic field is used to change the detuning δ. Unlike the JCM in quantum optics, the parameters δ, ω and ˜λ are connected via the magnetic field. Thus, changing B effectively alters the coupling between the pseudo-spin states in the JCM. The non-linear dependence of δ/λ as function of B leads to non-trivial features in the oscillation pattern. In the following, we calculate how the characteristics of the oscillation depend on B. Whenever material parameters are needed we use InGaAs values [25], |g| = 4, m/m 0 = 0.05, α 1,2 = (0.3...1.5) × 10 −12 eVm. The confinement length is set to l 0 = 150nm, corresponding to an energy scale ω 0 ≈ 0.1meV. With these numbers we have l SO,2,1 = (500...2500)nm. Compared to bulk InAs the InGaAs parameter for the strength of the SO coupling is rather small. This is to make sure that the period of Rabi oscillations is in a technologically accessible range of > 100ps. Although, a large ratio of ˜λ 2 /˜λ 1 is needed to get a significant oscillation amplitude – a small value for ˜λ 2 is needed to have a low Rabi frequency. We now evaluate the oscillation in P(t p ) as a function of magnetic field. The frequency of coherent oscillations Eq. (3.34) depends non-trivially on B. Rewriting the frequency in units of the resonance Rabi frequency yields Ω(B) Ω 0 = 1 2 ˜λ 2 (B) ˜λ 2 (B 0 ) √ [ δ ] 2 (B) + 4, Ω0 = Ω(B 0 ). (3.46) ˜λ 2 This non-linear function leads to the result that away from resonance (B ≠ B 0 ), an increase of λ 2 by a factor 5 does not translate into a 5 times larger frequency. Only at resonance a linear λ 2 -dependence is recovered. In Fig. 3.3a the probability of finding an electron in the upper state after a pulse time t p as a function of the detuning is shown. The asymmetric course of the oscillation amplitude (Fig. 3.3b) is a consequence of the parametric dependence on the magnetic field. 3.2.5 The current We now establish a relation between oscillations in the occupation probability and the transport properties via sequential tunnelling by rate equation arguments. The time scales in the different steps of the operation scheme (as shown in Fig. 3.2 on
3.3 Effects of relaxation 71 page 55) fulfil the conditions 1. initialisation time (panel Fig. 3.2b) t i ∼ Γ −1 R > Γ−1 L , 2. pulse time (panel Fig. 3.2c) t p < Ω −1 , 3. read-out time (panel Fig. 3.2d) t o ∼ Γ −1 R . The time to initialise the dot in state Ψ − is limited by the time it takes to get rid of an electron which by chance occupies the state Ψ + . Requiring Γ L > Γ R leads to a preferential filling of the dot from the left lead. In addition, the pulse time needs to be smaller than the inverse Rabi frequency to observe oscillation behaviour. The read-out time is given by Γ −1 R . Thus, the period for one cycle is T cycle = 2Γ −1 R +t p and the transferred charge is 2eP(t p ). The factor 2e corresponds to the fact that if P(t p ) = 1 then two electrons are transferred per cycle on average; every electron which undergoes the oscillation is accompanied by an additional electron simply tunnelling through the dot via state Ψ + . Thus, the current can be estimated as I(t p ) = 2eP(t p) +t ≈ eΓ R P(t p ), for Γ R ≪ Ω. (3.47) p 2Γ −1 R By measuring the current as a function of the pulse length t p one can map the oscillation in the occupation probability onto a transport quantity. 3.3 Effects of relaxation Phase coherence is essential for the observation of probability oscillations which is outlined above. However, in every real physical system decoherence is always present due to coupling to the environment. Even if the system is prepared in the ground state the phase will be randomised after times longer than the dephasing time. Our scheme for the observation of coherent oscillations requires the systems to remain in an excited state for a rather long time. Thus, any relaxation from the excited state to the ground state will lead to dephasing and destroy the CO signal. The relevant mechanisms for relaxation and dephasing of spin states in QDs are still being investigated [37]. A lower bound for the relaxation time of 50µs has been measured at 20mK in a one-electron GaAs dot with an in-plane field of 7.5T [131]. This time is orders of magnitude larger than in a 2D system. Notably, the measurement was limited by the signal-to-noise ratio and thus, the real value of the relaxation time may even be larger, substantiating the proposal to consider the spin state of quantum dots as a possible physical realisation of a quantum computing architecture [122]. The theoretical understanding of spin-flip transitions in QDs is important in order that one can estimate spin coherence times which need to be sufficiently long
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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
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6 Introduction Figure 1: Scanning e
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Chapter 1 The Rashba effect Effects
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1.2 The model 11 the position-depen
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1.3 Rashba effect in a perpendicula
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Chapter 2 Rashba spin-orbit couplin
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2.1 Rashba effect and magnetic fiel
Page 25 and 26: 2.1 Rashba effect and magnetic fiel
Page 31 and 32: PSfrag replacements 2.1 Rashba effe
Page 33 and 34: 2.2 Spectral properties, various li
Page 39 and 40: 2.3 Electron transport in one-dimen
Page 41 and 42: PSfrag replacements 2.3 Electron tr
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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
Page 61 and 62: £££££££££££££¢¢¢¢
Page 63 and 64: 0 0.5 1.5 3.1 Spin-orbit driven coh
Page 65 and 66: 3.2 Introduction to quantum dots an
Page 75: PSfrag replacements 3.2 Introductio
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
Page 87 and 88: Tuneable quantum dots connected to
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 110 and 111: PSfrag replacements 1042 Coupled qu
Page 116 and 117: 110 Conclusion due to phonon van-Ho
Page 119 and 120: Appendix A Jaynes-Cummings model Th
Page 121 and 122: Appendix B Fock-Darwin representati
Page 123 and 124: Appendix C Evaluation of the phonon
Page 125 and 126: 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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