ANÁLISIS DE LOS RESULTADOS DE LA MUESTRA FINAL

In this section, some very recent ideas from the article of Falciet al. [110] will be presented. The total model Hamiltonian Htotal looks now a bit different than before

Htotal =Hqb− 1 2 ξ(t) ˆσz, (9.9) where ˆHqb=−1₂Ω~~σ. Ω = p ε2

as+ ∆2 is the splitting in the qubit and θ (see below) will be the angle between the z axis and ~Ω. ξ(t) describes a classical stochastic process.

For weak coupling, one can find the expression for the dephasing rate as [41, 135]

Γ2 = 1

4S(Ω) sin

2_θ+ 1

2S(0) cos

2_{θ, (9.10)}

Ifθ =π/2, the pure dephasing or adiabatic part of Γ2 vanishes. By this, part of the effect of slow noise is eliminated. This is called the optimal working point. Such a system has been realized experimentally [37] for a superconducting phase qubit (“Quantronium”).

In Ref. [110], the behavior of aboutNBF = 2000 bistable fluctuators has been simulated with a stochastic Schr¨odinger equation. ξ(t) is generated as a sum of NBF RTN processes. This represents experimental data showing decaying coherent oscillations. Then, different techniques are used to determine the envelope function of these decaying oscillations.

For slow noise, ξ(t) can be treated in the adiabatic approximation. Observables are then given by path integrals over a weight P[ξ(t)]. Within the static-path approximation

ξ(t) =ξ0, which accounts for the lack of control on the environment preparation, one can obtain the following expression for the phase Φ(t)

−iΦ(t) =−1 2 (cosθ σξt)2 1 +isin2θ σ2 ξt/Ω − 1 2 ln µ 1 +isin2θ σ 2 ξt Ω ¶ , (9.11)

where σξ = ¯v2NBF/4 is the variance of ξ0. This approximation [Eq. (9.11)] is valid close to θ = 0 and θ = π/2. The resulting suppression factor exp(₌Φ) gives a exp¡

−1 2σ

2 ξt2

¢ behavior forθ = 0 and a power law h1 +¡

σ2 ξt/Ω

¢2i−1/4

behavior for θ=π/2.

If slow and fast fluctuators are mixed, one can study the interplay between both by a two-stage elimination [110]. One decomposesξ(t)→ξ(t)+ξf(t), whereξ(t) represents slow fluctuations that can be treated with the adiabatic approximation. ξf(t) stands for the fast fluctuations, whose influence can be determined within weak coupling theory, which is the next step. Then, the static-path approximation for θ = π/2 leads to the decay of coherences exp · −1₄ Sf(Ω)t− 1 2 ln ¯ ¯ ¯ ¯ 1 + µ iΩ +Sf(0)− 1 2Sf(Ω) ¶_σ2 ξt Ω2 ¯ ¯ ¯ ¯ ¸ , (9.12)

where Sf(ω) refers to the set of fast bistable fluctuators (BFs), whereas σ2ξ refers to the set of slow BFs. This result is very generic and can e.g. be applied, when slow impurity noise is combined with fast electromagnetic noise. It is quite remarkable that Eq. (9.12) describes both exponential decay (coming from the fast BFs), which could also be derived by a weak coupling approach like Bloch-Redfield theory [115], and non-exponential decay (coming from the slow BFs), which cannot be obtained by weak coupling methods.

Ref. [110] and the ideas in it are important to understand and analyze experiments, whether and how they depend on 1/f noise. The critical aspect is that one needs enough coherent oscillations (or a large quality factor) of the qubit in order to determine the envelope function for the decay. But once this behavior is characterized, one could think about schemes to compensate this RTN or 1/f noise. Ideally, the oscillations should be measured in a Ramsey-fringe experiment [111]. Unfortunately, the current implementations of charge qubits in double quantum dots did not show enough coherent oscillations to perform such an analysis along the lines of Ref. [110]. This is also the reason, why this chapter is in the part “Perspectives” of this thesis. The only experiment until now that could be analyzed with these methods was the already mentioned measurement on the Quantronium circuit, done by Vion et al. [37].

Charge qubits in other

semiconducting nanostructures

The main challenge for improving the intrinsic quantum coherence of charge states in semiconductors is to further reduce the impact of phonons. Within the present design, this can be accomplished by phonon cavities [123].

Besides the well-established way of defining double quantum dots in a 2DEG later- ally next to each other, alternative designs have become feasible in recent years. The first, very recent approach is a mixture of vertical and lateral quantum dot design in a GaAs/AlGaAs/InGaAs heterostructure [186]. Another recently proposed way to a double quantum dot structure is using confined electrons in a carbon nanotube [187]. Semicon- ducting nanowires [188] also show the characteristic Coulomb blockade behavior and might be a promising design as well. In the following, we will shortly present these three alter- natives to the lateral design of a double quantum dot.

10.1

Phonon cavities for lateral quantum dots

From theoretical considerations [122, 124], one expects a smaller dephasing rate due to the electron-phonon interaction in double quantum dot charge qubit, if the double dot could be realized in a phonon cavity. Unfortunately, a working underetched double quantum dot charge qubit has not been demonstrated yet. For single quantum dots, however, the first transport experiment [123] of a quantum dot in a phonon cavity looks very promising. A phonon blockade has been found in the sequential transport through the quantum dot. Figure 10.1 (a) shows a picture of the sample and Figure 10.1 (b) provides level diagrams to illustrate the phonon blockade phenomenon. Further improvements in the fabrication of similar structures should lead to a working laterally coupled double quantum dot charge qubit in a phonon cavity. Further details on the fabrication of these devices can be found in Refs. [189, 190].

Figure 10.1: From Ref. [123]: (a) suspended quantum dot cavity and in-plane gate formed in the 130 nm thin GaAs/AlGaAs membrane. The inset shows the blocked differential con- ductance in the linear transport regime. (b) Level diagrams for single electron tunneling including phonon blockade (upper row: tunneling processes into the dot; lower row: tun- neling processes out of the dot): (i) electrons can tunnel sequentially through the dot, if the local dot level µ(N + 1) is aligned between the reservoirs. (ii) Tunneling into the phonon cavity results in the excitation of a cavity phonon with energy ~_{Ωph, leading to a level} mismatch ε0 and thus to phonon blockade. (iii) Single electron tunneling is reestablished by a resonant higher electronic state µ∗_{(N + 1) which is enabled to coherently reabsorb} the phonon and to hereby replace the ground state.