Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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90 J.M. Elzerman et al. suitable for ESR detection, as the dot levels are far above EF (so are not affected by photon-assisted tunnelling or heating). If B is perpendicular to the surface, Bac must run through the dot in a direction parallel to the surface, so we must place the wire above the dot rather than to its side. The wire could be located on top of an insulating dielectric layer that covers the gate electrodes. 6.6 Coherent Spin Interactions: √ swap Two electron spins S1 and S2 in neighbouring quantum dots are coupled to each other by the exchange interaction, which takes the form J(t)S1 · S2. If the double dot is filled with two identical spins, the interaction does not change their orientation. However, if the left electron spin starts out being |↑〉 and the right one |↓〉, then the states of the two spins will be swapped after a certain time. An interaction active for half this time performs the √ swap gate, which has been shown to be universal for quantum computation when combined with single qubit rotations [94]. In fact, the exchange interaction is even universal by itself when the state of each qubit is encoded in the state of three electron spins [17]. The strength J(t) of the exchange interaction depends on the overlap of the two electron wavefunctions, which varies exponentially with the voltage applied to the gate controlling the inter-dot tunnel barrier. By applying a (positive) voltage pulse with a certain amplitude and duration, we can temporarily turn on the exchange interaction, thereby performing a √ swap gate. We expect that J may correspond to a frequency of ∼10 GHz, so two-qubit gates could be performed in ∼100 ps. A much larger value would not be convenient experimentally, as we would have to control the exact amplitude and duration of the pulse very precisely. On the other hand, a very slow exchange operation would be more sensitive to decoherence resulting from fluctuations in the tunnel rate, due to charge noise. The value of J can in principle be determined in a transport measurement [33], or alternatively by using the QPC tunnel spectroscopy technique developed in Sect. 3. However, in practical situations J might be too small to be resolved. To explore the operation of the swap gate, we only need reliable initialization and read-out, without requiring ESR [2]. Imagine qubit 1 is prepared in a pure state |↑〉and qubit 2 is prepared in a statistical mixture of |↑〉and |↓〉. Measurement of qubit 1 should then always give |↑〉, while measurement of qubit 2 should give probabilistically |↑〉 or |↓〉. After application of the swap gate, in contrast, measurement of qubit 2 should always give |↑〉, while measurement of qubit 1 should give a probabilistic outcome. This and other spin-interaction experiments are probably easiest in a parallel magnetic field, where initialization to a statistical mixture is convenient. In addition, a large perpendicular field shrinks the electron wavefunctions, lowering the tunnel coupling and thus the exchange interaction between the two dots.
6.7 Unresolved Issues Semiconductor Few-Electron Quantum Dots as Spin Qubits 91 Several issues are not yet fully resolved, both experimentally and theoretically. One of these is the question of electron spin resonance in the reservoir. There are indications that the g-factor in the dot is different from that in the reservoir [95] (disregarding enhancement due to exchange interactions, which are not relevant for a “global” excitation such as ESR). However, if the two gfactors are equal, then any coherent operation of the spin on the dot will also influence the spin population outside the dot. This has not been taken into account in this section, but it could lead to complications for the proposed ESR experiments. Another question is related to the ∼10 6 nuclear spins in the quantum dot that couple to the electron spin via the hyperfine coupling. Through the Overhauser effect they produce an effective magnetic field, which can be very large (∼5 T) for a fully polarized nuclear spin ensemble. Statistical fluctuations in the Overhauser field could lead to changes in the phase of the electron spin. It is not yet clear what the influence will be on spin manipulation experiments. If it turns out to be a problem, we may have to polarize the nuclear system completely in order to suppress the fluctuations. A more practical consideration is the effect of charge switches in the heterostructure, which make any experiment more difficult. This is particularly true for two-spin interaction experiments, as charge noise can affect the interdot tunnel barrier and therefore the exchange interaction, resulting in decoherence. In collaboration with the group of Prof. Wegscheider in Regensburg, we have started to investigate the possible origin of charge switching, in an effort to produce more quiet heterostructures and devices. Finally, so far we have used at most two quantum dots, not paying much attention to the scalability of our spin qubit approach. For instance, the ESRfield generated by the big wire runing next to the double dot will also influence other spins in nearby dots. We may therfore have to develop techniques to locally control the g-factor felt by the electron spin in a dot, in order to shift particular dots in or out of resonance. 6.8 Conclusion and Outlook In summary, we have demonstrated that single electrons trapped in GaAs lateral quantum dots are promising candidates for implementing a spin qubit. We have realized the “hardware” for such a system: a device consisting of two coupled quantum dots that can be filled with one electron spin each, flanked by two quantum point contacts. Using these QPCs as charge detectors, we can determine all relevant parameters of the double dot. In addition, we have developed a technique to measure the spin orientation of an individual electron. Now we can proceed to combine all these ingredients with the ability to generate strong microwave magnetic fields close to the dot, and gate voltage
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Lecture Notes in Physics Editorial
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W. Dieter Heiss (Ed.) Quantum Dots:
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Preface Nanoscale physics, nowadays
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Contents A Guide for the Reader ...
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A Guide for the Reader Quantum dots
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The Renormalization Group Approach
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RG for Interacting Fermions 5 where
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where � is a shorthand: � ≡ 3
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RG for Interacting Fermions 9 These
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RG for Interacting Fermions 11 is i
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RG for Interacting Fermions 13 the
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RG for Interacting Fermions 15 this
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RG for Interacting Fermions 17 equa
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� p � dθp = g 2π RG for Inter
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RG for Interacting Fermions 21 the
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References RG for Interacting Fermi
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Semiconductor Few-Electron Quantum
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1.2 Implementations Semiconductor F
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GaAs n-AlGaAs AlGaAs GaAs Semicondu
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142 C.W.J. Beenakker largely indepe
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144 C.W.J. Beenakker The effective
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146 C.W.J. Beenakker Fig. 6. Ensemb
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148 C.W.J. Beenakker 1.4 1.2 1.0 0.
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150 C.W.J. Beenakker 〈ρ(E)〉 =
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152 C.W.J. Beenakker P 0.5 0.4 0.3
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154 C.W.J. Beenakker P(x) 0.4 0.3 0
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156 C.W.J. Beenakker tunnel/ Egap 4
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158 C.W.J. Beenakker its momentum).
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160 C.W.J. Beenakker E00 = π� =
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162 C.W.J. Beenakker 〈ρ(E)〉 δ
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164 C.W.J. Beenakker As described i
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166 C.W.J. Beenakker 0.30 � Egap
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168 C.W.J. Beenakker The τE-depend
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170 C.W.J. Beenakker that a fully m
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172 C.W.J. Beenakker (τ−,τ+), w
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174 C.W.J. Beenakker 61. P. G. Silv
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