5. Discusión de resultados
5.2. Estrategias de políticas de desarrollo rural
As already mentioned in the introduction, we now present another approach to describe the double dot potential as a symmetric double-well potential, namely a quartic potential [131] [Eq. (5.1)] as depicted in Figure 5.2. In this case, the qubit is only described by the coupling between the two dots. In the localized basis of the charge qubit (electron on the left or right dot), the dot Hamiltonian would read
Hd= ~∆
2 σˆx . (5.15)
We, on the other hand, start from the microscopic single-particle Hamiltonian
Hd = p2
Therefore, we use a microscopic derivation of the inter-dot coupling ∆ by determining the charge eigenstates of this HamiltonianHd. This should work as long as the dots are well- defined,i.e. the Bohr radius of the single dotsaB =
q ~
mω0 should be much smaller than the distance [131] r from Eq. (5.1). We will use the characteristic distance a=aB/
√
2 in the following. ~ω0is the energy scale for this parabolic confinement potential andm= 0.067me is the effective mass of an electron in GaAs. Since the Bloch-Redfield approach only works in the eigenbasis of the considered system, we calculate the two lowest eigenfunctions of (5.16). For this, we consider p and x in Hd as quantum mechanical operators acting on the eigenfunctions of single harmonic oscillators (details see Appendix C). The oscillator wavefunctions are chosen as a complete basis and the exact wavefunctions for the charge eigenstates will be superpositions of these. The resulting eigenfunctions only depend on the confinement frequencyω0, the half distance between the dot centers r and the number N of the excited levels of the harmonic oscillators that we allow to be populated. One then finds the generic forms
ψgs(x, y, z) = N X n=1 a2nΦ2n(x)Φ0(y)δ(z) (5.17) ψes(x, y, z) = N X n=1 a2n−1Φ2n−1(x)Φ0(y)δ(z) , (5.18)
whereψgs denotes the symmetric ground state of the artificial molecule andψes the asym- metric first excited state of the coupled system. Φn(x) is the nth eigenfunction of a one- dimensional harmonic oscillator in x-direction. a2n and a2n+1 are the coefficients for the harmonic oscillator states |2ni and |2n+ 1i. In y-direction only the lowest eigenfunction plays a role, because in this direction, we only have a usual harmonic oscillator. Since we consider electrons in a 2DEG in a GaAs/AlGaAs heterostructure, we assume that there is no significant contribution of electrons that are not in the x-y-plane. More details on the charge eigenstates and the potential can be found in Appendix C.
The electron-phonon interaction, taking only the piezoelectric contribution into ac- count, here looks as follows [141, 142]
He−ph = X ~ q,α s ~ 2ρV ω~q,α eA~q,αei~q~r ³ b†~q,α+b−~q,α ´ , (5.19)
where ρ is the crystal mass density andA~q,α is an effective piezoelectric modulus
A~q,α =ξiξkβikjej~q,α. (5.20)
Hereξ~=~q/q is the phonon wave vector,~eis the phonon unit polarization vector and βikj is the piezotensor. These tensor elements are only nonzero for this kind of crystals, if all three indices are different, βxyz =βxzy =...=h14 = 1.2·109 eV/m (for GaAs) [141].
Figure 5.2: (Color online) Sketch of the considered system: two coupled quantum dots realizing a position charge qubit. 2r is the distance of the dot centers in one qubit. One can see the localized states and the molecular states.
In order to calculate the Golden Rule rates like Eq. (5.4), we need to determine the matrix elements ofHe−ph in the eigenbasis of the system, i.e. in terms of ψ+ and ψ−. But since these functions consist of Hermite polynoms, we have to calculate the matrix elements between two eigenfunctions of the harmonic oscillator and then add all contributions in the functions ψ+ and ψ− according to their weight. To do this, we determine only the matrix elements h`|eiκxˆx|ni, with |`i and |ni being two Eigenfunctions of the harmonic oscilla- tor. κx = aqx is a dimensionless prefactor describing the spatial quantities involved here, namely thex-component of the wavevector of the phononsqx and the distancea=aB/
√ 2 representing the confinement potential. The exponential function in this matrix element is the only point, where the position of the electron enters the HamiltonianHe−ph, Eq. (5.19). We also evaluate the three spatial directions separately, i.e. we only treat one-dimensional problems that are combined again later on. The above mentioned matrix element describes only the x-direction, which is also the most interesting one due to the form of the poten- tial V(x, y), Eq. (5.1). To use the relations for the harmonic oscillator, we furthermore substitute the spatial x by the harmonic oscillator operator ˆx that acts on the states |ni and |li. The matrix element for this exponential function (see above) in x-direction can be determined by identifying eiκxˆx with the displacement operator ˆD(α) = exp(αˆa†−α∗aˆ) for the harmonic oscillators with α = iκx. ˆa† and ˆa operate on the eigenfunctions of the harmonic oscillators. For the displacement operator, one can use the following relation [143] for `≥n h`|Dˆ(α)|ni= r n! `!α `−ne−|α|22L(`−n) n (|α|2) , (5.21)
where L(n`−n) is an associated Laguerre polynom.
Combining all these expressions for the contributions to the Eigenfunctionsψ+and ψ−, one can also find spectral densities J`mnk(ω) similar to Eq. (5.10). Due to the non-trivial contributions of all considered harmonic oscillator states (we usually use N = 40) and the anisotropy factorsA~q,α, the J(ω) is in general different for all combinations of ψ+ and ψ−.
The real parts of the Golden Rule rates for this model then read <³Γ(+)`mnk´ = J`mnk(ωnk) 2~ µ coth µ~ ωnk 2kBT ¶ −1 ¶ (5.22) <³Γ(`mnk−) ´ = J`mnk(ω`m) 2~ µ coth µ~ ω`m 2kBT ¶ + 1 ¶ . (5.23)
Also in this case, we apply the secular approximation and get the same definition for the relaxation rate [Eq. (5.13)] and the dephasing rate [Eq. (5.14)]. The secular approximation is actually exact here to the selection rules for the J`mnk(ω). In this approach, one again finds no significant pure dephasing contribution to dephasing, because the most important parts of the spectral functions J`mnk(ω) are again super-Ohmic.