Simulations of electric-dipole spin resonance for spin-orbit-coupled quantum dots in Overhauser field: fractional resonances and selection rules

Simulations of electric-dipole spin resonance for spin-orbit-coupled quantum dots in Overhauser field: fractional resonances and selection rules

E.N. Osika, B. Szafran and M.P. Nowak AGH University of Science and Technology,
Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Kraków, Poland
July 12, 2019
Abstract

We consider spin rotations in single- and two-electron quantum dots that are driven by external AC electric field with two mechanisms that couple the electron spatial motion and the spin degree of freedom: the spin-orbit interaction and a random fluctuation of the Overhauser field due to nuclear spin bath. We perform a systematic numerical simulation of the driven system using a finite difference approach with an exact account taken for the electron-electron correlation. The simulation demonstrates that the electron oscillation in fluctuating nuclear field is translated into an effective magnetic field during the electron wave packet motion. The effective magnetic field drives the spin transitions according to the electric-dipole spin resonance mechanism. We find distinct signatures of selection rules for direct and higher-order transitions in terms of the spin-orbital symmetries of the wave functions. The selection rules are violated by the random fluctuation of the Overhauser field.

I Introduction

Control and coherent manipulation of electron spin in quantum dotshanson () are extensively studied in the context of information storage and processing.lossdivi () The most basic operation on the single electron spin is its rotation. One of applicable procedures for spin rotation is the electron spin resonance (ESR). The ESR appears in constant magnetic field which splits the spin-up and spin-down energy levels. In presence of an additional AC magnetic field with frequency tuned to the Zeeman energy splitting, the electron undergoes the Rabi oscillation between the opposite spin states.ESR () It has been proposed for quantum wells EDSRQW () and quantum dots EDSR () that the oscillating external magnetic field can be replaced by oscillating electric field according to electric-dipole spin resonance (EDSR) mechanism.EDSRo () In presence of the spin-orbit (SO) coupling the electron motion induced by the electric field is translated into an effective magnetic field that depends on the electron momentum. The effective SO magnetic field induces Rabi oscillations and the corresponding spin rotations in a similar manner as the external field. The EDSR was observed in experiments that probe lifting of the Pauli blockade of the current across double quantum dots.EDSRm (); Sch (); Nadj () It has also been demonstrated that the role of SO coupling that translates the electron motion into appearance of an effective magnetic field can be played by a gradient of external magnetic field. Space () Successful experiments, in which EDSR is achieved due to the spatial fluctuation of the GaAs nuclear spin field created by the hyperfine interaction (HF) were also performed.EDSRnuc ()

The purpose of the present paper is the simulation of the EDSR that results from the random fluctuation of the HF field and determination of its signatures as compared to the process governed by SO interaction. We consider single and two-electron systems confined in quantum dots and transitions that are induced by oscillating electric field. We perform numerical simulations for systems with strong lateral confinement using a finite-difference approach that takes an exact account for the electron-electron correlation. We discuss the characteristic features of the resonant transitions due to the HF field and/or SO interaction. In particular the results indicate existence of selection rules for transitions governed by SO interaction. We discuss the symmetries behind the selection rules also for fractional resonances that have been observed in experiments with both SO coupling Sch (); Nadj () and HF field EDSRnuc () and were recently explainednowak () as second-order processes similar to multi-photon transitions in quantum optics.optics () We demonstrate that the selection rules are violated by the Overhauser field fluctuations. This finding is a relevant information for dynamical nuclear polarizationot (); johnson2005 (); inni () which reduces the randomness of the Overhauser field for which restoration of the selection rules should be observed.

Ii Theory

The EDSR requires application of an external magnetic field to induce the Zeeman splitting. At non-zero the direct electron-nuclear spin flip-flopsjohnson2005 () become suppressed.hanson () Then, the field of nuclear spins separates from the electron wave function. The characteristic time for the fluctuation of the nuclear spin field due to the dipole-dipole and HF interactions is of the order of 10-100 s which is much longer hanson () than the time scales for the evolution of the electron spin. For the above reasons we will treat the HF field as a static Overhauser magnetic field.

ii.1 Hamiltonians

We consider a quasi one-dimensional quantum dot with a strong lateral confinement – see Fig. 1 with external magnetic field oriented along the axis of the wire . In this configuration the orbital effects of the external magnetic field can be neglected, and the single-electron Hamiltonian takes the form,

(1)

where is the confinement potential, is the potential of the AC electric field which will drive the spin transitions, , describes the interaction of the electron spin with the Overhauser field

(2)

and introduces the linear Rashba spin-orbit interaction due to the electric field oriented perpendicular to the length of the quantum dot

(3)

We apply GaAs material parameters with the effective electron mass , and the Lande factor . We assume kV/cm which for the bulk SO coupling constant nm (after Ref. silva, ) gives meV nm. For this value of , SO coupling produces comparable effects to the Overhauser field fluctuation (see below).

Figure 1: Schematics of the considered system: quantum dot of length nm assumed in form of an infinite quantum well, with strong electron confinement near the axis of the dot , with external magnetic field vector oriented axially, external electric field perpendicular to the axis , and AC electric field along the dot .

We assume that the confinement potential is separable into a lateral and longitudinal components: . We consider a quantum dot with a strongly an elongated geometry (see Fig. 1). For the length of the dot which largely exceeds the lateral extent of the wave function all the low-energy phenomena appear in the ground-state of the lateral quantization.pt () We assume that the quantum dot is defined in the direction by an infinite quantum well of length 158.2 nm: (see Fig. 1) and that the lateral confinement potential is strong enough to induce localization around the axis of the cylinder within a region of radius 10 nm. We assume that the lateral wave function can be described by a Gaussian for which an analytical form of the effective electron-electron interaction potential is known.pt () Namely, we take the single-electron lateral wave function of form

(4)

where nm is adopted for the localization parameter. Upon integration over the lateral degrees of freedompt (); nowak () the single-electron Hamiltonian takes a 1D form,

For the electron pair with a double quantum dot is considered in Sec. III. B we take , where is barrier applied centrally to the system of height 10 meV and width 13.56 nm.

For the electron-pair we consider the Hamiltonian including the electron-electron interaction

(6)

with the dielectric constant of . After integration of the Hamiltonian with the lateral wave function (4) one obtains

(7)

where the last term is the effective 1D interaction derived in Ref. [pt, ] as .

Numerical calculations are performed with a finite difference approach with discretized versions of the differential operators and a finite mesh constant . For a single electron we look for eigenstates and time evolution of the spinor components . For the two-electron wave function we solve the equations for the bi-spinor in form

(8)

The fermion symmetry with respect to the electron interchange implies that and . The spin unpolarized parts have no definite symmetry with respect to the exchange of the electron spatial coordinates separately, instead one has . The evolution of the system in time is given by the Schrödinger equation , which is solved with the explicit Askar-Cakmak schemeacs ()

ii.2 Effective magnetic field due to nuclear spins

In GaAs each nucleus of the crystal lattice carry an uncompensated spin that interacts with the electron spin via the Fermi contact hyperfine interaction,hanson () , where is the position of k-th nucleus, and are the nuclear and electron spin operators, and is the coupling constant which is proportional to the magnetic moment of nucleus and the probability density of finding the electron therein.

In nonzero when the entanglement between the electron wave function and the nuclear spins can be neglected, the electron interacts with the ensemble of spins through the Overhauser effective magnetic field , with . For fully polarized nuclei the maximal value of the Overhauser field is 5T. For the purpose of the numerical simulation we need the distribution of the Overhauser field along the quantum dot. The field is generated in the following manner. We consider all the nuclei present within the volume of the quantum dot. With each nucleus at position (eight nuclei per cubic unit cell of lattice constant nm) we attribute a local vector of an effective magnetic field of length 5T with orientation taken at random with the uniform distribution. For the purpose of the present study we do not distinguish between isotopes of Ga and As. The electron lateral wave function averages the magnetic field due to the nuclei.kh (); merk () We calculate the magnetic field in the cell of the finite difference mesh, between and points

(9)

The simulated field due to nuclear spins is plotted in Fig. 2. For nm, the field amplitude is of the order of 20 mT. The actual effective field perceived by the electron spin will be further reduced to a few mili TeslamT () by the spread of the wave function along the quantum dot (see below).

Figure 2: Components of the Overhauser field as integrated with the lateral electron wave function on a finite difference mesh along the quantum dot (see Section II.2).

Iii Results and Discussion

Figure 3: (a) Energy spectrum of the single electron in a single quantum dot of length 158.2 nm (see Fig. 1) as a function of the external magnetic field for a static potential . The enlarged fragments of weak magnetic fields are given in (b) and (c). In (b) no HF field is assumed. In (c) we assume the HF field of Fig. 2. In (b) and (c) the results plotted with solid (dashed) lines were obtained with (without) SO coupling.

iii.1 Single-electron electric-dipole spin resonance

Figure 3(a) shows the single-electron energy spectrum for a quantum dot of length 158.2 nm as a function of the magnetic field. Figures 3(b) and (c) present zoom at the ground-state energy level that is split by the Zeeman interaction. The spin-orbit coupling [Fig. 3(b)] lowers the energy levels and preserves the double (Kramers) degeneracy of the ground-state. When the Overhauser field is introduced [Fig. 3(c)], the energy levels are no longer degenerate at 0T. For the adopted parameters () the energy effects due to both spin-orbit coupling and hyperfine field are comparable and of the order of eV.

Figure 4: Results for a single electron in a single quantum dot of length 158.2 nm (as in Fig. 3) with (static potential). Spin-up (a,c) and spin-down (b,d) wave functions components of the ground state (a,b) and the first excited state (c,d) as obtained with SO coupling (no HF field) for the external field of T. The wave functions are given in atomic units.

Figure 5: Results for a single electron in a single quantum dot of length 158.2 nm (as in Fig. 3) but with the amplitude of AC field kV/cm. EDSR induced by SO coupling (no HF field) for resonant AC frequency for the transition from the ground-state to the first-excited state and T oriented along the direction. The black (red) lines show the projections of wave functions on the two lowest-energy single electron states and the dashed line gives the mean value of the spin component that is referred to the right axis.

iii.1.1 EDSR due to SO coupling

Let us now consider the spin rotations due to spin-orbit coupling only (no hyperfine field). Figure 4 shows the real part of the wave functions of the two lowest energy levels for T. The ground-state [Fig. 4(a,b)] is nearly spin polarized. Its spin-down component [Fig. 4(b)] is of the odd spatial parity and corresponds to the first-excited energy level of the quantum dot without SO coupling. The SO coupling Hamiltonian commutes with the s-parity operator , where is the parity operator with respect to point inversion through the center of the dot. Therefore, the components of the spin-orbitals possess a definite but opposite spatial parities. The SO coupling appears through coupling of spatial energy levels of opposite parities.

Now, we apply a harmonic AC field of amplitude kV/cm – for which the potential drop along the dot is meV. We consider the ground-state as the initial condition and apply the resonant frequency for the transition to the first excited energy level with inverted spin, , which corresponds to AC oscillation period of about 0.6 ns. We see in Fig. 5 that after 60 ns (nearly 100 periods of the external electric field) the electron spin gets inverted and the electron occupies the first excited state.

Figure 6: Results for a single electron in a single quantum dot of length 158.2 nm with the amplitude kV/cm of the AC field (as in Fig. 5). Maximal probability of the spin inversion as a function of the driving frequency for 500 ns of the simulation. Separate plots correspond to: SO interaction without HF field (a), HF field without SO interaction (b), both HF and SO interaction present (c). Insets show the enlarged fragments for second and third order transitions.

Figure 6(a) shows the maximal probability of the spin inversion obtained during 500 ns of the simulation as a function of the frequency of the AC field. We observe a pronounced resonance near 7.5 eV, that corresponds to the simulation of Fig. 5. We also notice a sign of fractional resonance near 2.5 eV. Note, that the half-resonancenowak () due to second-order transition that could be expected near 3.8 eV is missing. The reason for the missing half-resonance is the symmetry of the wave function for the initial and final states. The oscillator strength of the direct (first-order) transition from initial state to the final state which results from the harmonic perturbation of form , is determined by the Fermi golden rule with the matrix element . For the wave functions of Fig. 4 the components of the sum are non-zero due to the spin-orbit coupling that introduces non-zero overlap between the components of wave functions for the same spin orientation. Moreover, since the initial and final states have opposite parities for each of the components, and is the odd parity function, the integrands are even functions with respect to point inversion through the center of the dot. In general, the first-order transition are allowed between states of opposite s-parities.

According to the time-dependent perturbation theory,shankar () the second-order transitions are allowed if one the products is non-zero, where is any of the intermediate eigenstates of the unperturbed Hamiltonian. Now, since all the eigenstates of the unperturbed Hamiltonian possess a definite s-parity, and in our case and states are of the opposite s-parity, none of these products can be non-zero. The second-order transitions are allowed only between the states of the same s-parity (see below for EDSR in the two-electron system). The third order transitions that involve two intermediate states are again allowed between states of opposite s-parity, hence the 1/3 peak observed in Fig. 6(a).

Figure 7: Results for a single electron in a single quantum dot of length 158.2 nm with the amplitude kV/cm of the AC field (as in Fig. 5). (a,b) Electron densities at opposite phases of the driving AC electric field. (c) The red line shows the mean position of the electron. The three other lines near the bottom of the plot indicate the average value of the Overhauser field as perceived by the spin of the electron as a function of time

iii.1.2 EDSR due to the hyperfine field

We now consider the hyperfine field only and apply the harmonic perturbation to the potential (same resonant frequency and amplitude as in Fig. 5). Charge density at opposite phases of the AC field is given in Fig. 7(a,b). Figure 7(c) shows that the center of the electron packet oscillates with an amplitude of about 10 nm. As the packet oscillates, the electron spin perceives a different effective magnetic field that is averaged by the moving charge density – see Fig. 7(c) for the components of the effective field. As a consequence, the electron spin is inverted after about 200 periods of the oscillating field.

The probability of the spin inversion as a function of are plotted in Fig. 6(b). In presence of the nuclear field, the s-parity of eigenstates is no longer defined, and we obtain the direct, second and third order transitions to the excited state. A reduced width of the direct transition can be noticed with respect to the SO case.

Figure 8: Results for a single electron in a single quantum dot of length 158.2 nm with the static potential , as in Fig. 3. Solid lines show the wave function components for the ground-state (a,b) and the first excited state as obtained with both SO coupling and HF field present for the external field of T. In the minority spin components (b,c) the results obtained without HF are displayed for comparison. In the majority spin components (a,d) the lines with and without HF coincide.

iii.1.3 EDSR: combined nuclear field and SO coupling

Figure 8 shows the wave functions with both SO coupling and the nuclear field (solid lines). For comparison, the results without nuclear field are also plotted (dashed lines). The nuclear field shifts the zero of the minority spin components, which are no longer of a strict odd spatial parity. The impact of the hyperfine field on the majority wave function is not visible at this scale. In spite of the fact, that the SO coupling seems to dominate, the transition probability as function of the frequency of Fig. 6(c) resembles rather the case of the nuclear field, due to the presence of the half-resonance. Nevertheless, the width of the direct transition is similar to the one observed for pure SO coupling.

iii.2 Two-electron results

The experiments EDSRm (); EDSRnuc () probe the consequences of EDSR in lifting the Pauli blockade of the current in double quantum dots. The blockade occurs when electrons in both the dots acquire the same spin. When the spin of one of electrons is inverted the current restarts to flow. For the rest of this paper we consider a double quantum dot [see the inset to Fig. 9(a)]. We increased the amplitude of the AC field twice to 0.1 kV/cm – to maintain the same potential drop within a single dot as compared to the results of the previous section. Figure 9 shows the energy spectrum for the electron pair. The Coulomb blockade experiments on lifting the Pauli blockade are performed for the spin-polarized triplet as the initial state.

Figure 9(b) compares the spectrum without SO and without the hyperfine field (dashed lines), to the case of pure SO coupling. The avoided-crossing opened by SO interaction between the and energy levels is distinctly enhanced by the HF field [see Fig. 9(c)], which also lifts the degeneracy of the triplet states at T.

Figure 9: (a) Energy spectrum for two electrons in the double quantum dot (within the large quantum dot of Fig. 1 of length 158.2 nm we put a central barrier of width 13.56 nm and height 10 meV – see the inset for the applied potential) as a function of the external magnetic field. The static potential is assumed (). Arrows indicate the transitions that appear in EDSR. The enlarged fragments of weak magnetic fields are given in (b) and (c). In (b) no HF field is assumed. It is accounted for in (c). In (b) and (c) the results plotted with solid (dashed) lines were obtained with (without) SO coupling.

Figure 10: Results for two electrons in the double quantum dot considered in Fig. 9 for the amplitude of AC field kV/cm. Results for SO coupling present but without the Overhauser field. (a) Minimal spin obtained during 100 ns of the simulation as function of the AC frequency for the as the initial state and T. (b) Maximal probabilities of finding the electron in , and states. The probabilities are less than 1 due to the finite simulation time.

Figure 11: Results for two electrons in the double quantum dot considered in Fig. 9 for the static potential . Components of the two-electron eigenfunctions as functions of both electron coordinates over the entire length of the double dot for (first row of plots), , and at T. The color scales give the real part of the wave function in atomic units. Spin-orbit coupling is present, and Overhauser field is absent. The length of and axes corresponds to the length of the double dot nm.

Figure 12: Results for two electrons in the double quantum dot considered in Fig. 9. Black, green and blue lines show the projections of the wave functions on the triplets states for resonant frequency tuned to transition with SO but without HF field. The purple line shows the average value of the component of the spin.

Figure 13: Results for two electrons in the double quantum dot [as in Fig. 9] for Overhauser field present but without the SO coupling. (a) Minimal spin obtained during 100 ns of the simulation as function of the AC frequency for the as the initial state and T. (b) Maximal probabilities of finding the electron in , and states.

Figure 14: Same as Fig. 13 only for Overhauser field without SO coupling. Minimal spin obtained during 100 ns of the simulation for the amplitude of kV/cm (a) and kV/cm (b) and various values of the magnetic field. The right scale corresponds to the black curve obtained for external magnetic field of T. The blue solid lines are shifted by 1 down on the spin scale with subsequent values of the magnetic field. The dashed lines indicate the energies of the transitions from as calculated from the spectra without perturbation.

Figure 15: Same as Figs. 13 only for both Overhauser field and SO coupling present. (a) Minimal spin obtained during 100 ns of the simulation as function of the AC frequency for the as the initial state and T. (b) Maximal probabilities of finding the electron in , and states.

iii.2.1 Two-electron EDSR: pure SO coupling

We have performed time dependent simulations starting from the ground state – the triplet at T. The minimal spin that was obtained during 100 ns of the simulation is displayed in Fig. 10(a). Fig. 10(b) shows the maximal value of the square of projection of the wave function on the Hamiltonian eigenstates. We can see that the direct transition , that is most relevant for the lifting of the spin Pauli blockade,EDSRm () is missing in this plot. The reason for this is again the symmetry of the wave functions, Fig. 11 shows the real parts of the components of , , and wave functions. Each of the four components possesses a definite parity with respect to simultaneous inversion of both electron coordinates. In the basis of , , , , the two-electron s-parity operator has the form

(10)

As it can be noticed in Fig. 11, the only state of positive s-parity is . and (the last state was not included in Fig. 11), have negative s-parity. Hence, the matrix element for the direct transition from to both and final states vanishes. On the other hand the second-order transition to state with half resonance is observed near eV. No half resonance is observed for the transition to , in consistence with the discussion of selection rules given above for the single electron.

Figure 10(b) shows that an admixture of state is observed in the final state for the energy equal half of the energy splitting between and . Note, that in this case the energy spacing between and is exactly the same as between and . Fig. 12 shows the occupation probability as a function of time for the frequency of the external field tuned to resonance. We can see that first a finite probability of occupation appears, but it never approaches 1. Instead the probability occupancy of increases. Hence in this case we observe a two-step process, first pumping the electron from to state and next from to .

iii.2.2 Two-electron EDSR with nuclear field

Figure 13 shows the transitions for the HF field without the SO coupling. As compared to the previous case we notice: a distinct transition to the state (s-parities are no longer definite), and reduced probability of finding in the final state, due to the energy difference of the transitions between the triplets. Also, a half resonance for transitions to can be observed. The peaks for half resonances are much smaller than in the SO case. Note, that also the direct transitions occur significantly slower (cf. the height of the peak).

In Fig. 14 we show the scans obtained as a function of both AC frequency and the magnetic field for AC amplitude of the electric field kV/cm that was considered above (a) and for a stronger amplitude of kV /cm (b). In Fig. 14(a) at this scale we can see only the direct transitions to and , which exactly agree in energy with spectral separation of the eigenstates from the ground state (dashed lines). In Fig. 14(b) we notice also fractional transitions to the singlets. However, there is a detectable redshift of resonant frequencies for singlets. This shift is due to appearance of the double occupancy of the dot that is induced by stronger slope of the confinement potential. The double occupancy lowers the energy of the singlets with respect to the triplets, for which the double occupancy is forbidden by the Pauli exclusion.szafranexchange ()

The scan of the transitions for both SO and hyperfine field present is given in Fig. 15. As compared to the pure HF case, the height of peak is reduced with respect to . In contrast to the pure SO coupling case the half-transitions occur for both and .

Iv Summary and Conclusions

In summary, we have performed a systematic numerical simulation for the EDSR mechanisms in single electron (electron spin flip) and two-electron (transitions from the spin-up polarized triplet to excited states) driven by AC electric field and mediated by both spin-orbit coupling and the abrupt fluctuations of the static Overhauser field. We have demonstrated that the latter is translated into a smooth effective magnetic field felt by the electron spin due to the wave packet oscillation. The simulation indicates the presence of both integer and fractional resonances that correspond to first and higher-order transitions. The results for transitions in presence of the pure spin-orbit coupling bear distinct signatures for the selection rules which result from the spin-orbital symmetry of wave functions. In presence of the Overhauser field the selection rules no longer hold. In experiments the randomness of the Overhauser field can be reduced or eliminated by the dynamical nuclear polarization ot (); inni (); johnson2005 () or in the strong external magnetic field of a few Tesla. As the hyperfine field becomes ordered, the selection rules should be restored and the observed spectrum for the spin transitions should start to resemble the case of pure SO coupling.

Acknowledgements.
This work was supported by the funds of Ministry of Science and Higher Education (MNiSW) for 2012–2013 under Project No. IP2011038671, and by PL-Grid In- frastructure. M.P.N. gratefully acknowledges the sup- port from the Foundation for Polish Science (FNP) un- der START and MPD programme co-financed by the EU European Regional Development Fund.

References

  • (1) R. Hanson, L.P. Kouwenhoven, J.R. Petta, S. Tarucha, and L.M.K. Vandersypen, Rev. Mod. Phys. 79, 1217 (2007)
  • (2) D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
  • (3) J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, B. Witkamp, L. M. K. Vandersypen and L. P. Kouwenhoven, Nature (London) 430, 431 (2004); J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 30, 2180 (2005).
  • (4) E. I. Rashba and A. L. Efros, Phys. Rev. Lett. 91, 126405 (2003).
  • (5) V.N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B 74, 165319 (2006).
  • (6) G. Dresselhaus, Phys. Rev. 100, 580 (1955); Y. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).
  • (7) F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, Nature (London) 442, 766 (2006); K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarov, and L. M. K. Vandersypen, Science 318, 1430 (2007); S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Nature (London) 468, 1084 (2010); M. Pioro-Ladriere, T. Obata, Y. Tokura, Y.-S. Shin, T. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Nat. Phys. 4, 776 (2008); R. Brunner, Y.-S. Shin, T. Obata, M. Pioro-Ladriere, T. Kubo, K. Yoshida, T. Taniyama, Y. Tokura, and S. Tarucha, Phys. Rev. Lett. 107, 146801 (2011).
  • (8) M. D. Schroer, K. D. Petersson, M. Jung, and J. R. Petta, Phys. Rev. Lett. 107, 176811 (2011).
  • (9) S. Nadj-Perge, V. S. Pribiag, J. W. G. van den Berg, K. Zuo, S. R. Plissard, E. P. A.M. Bakkers, S.M. Frolov, and L. P. Kouwenhoven, Phys. Rev. Lett. 108, 166801 (2012).
  • (10) Y. Tokura, W.G. van der Wiel, T. Obata, and S. Tarucha, Phys. Rev. Lett. 96, 047202 (2006).
  • (11) E. A. Laird, C. Barthel, E. I. Rashba, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 99, 246601 (2007); Semicond. Sci. Technol. 24, 064004 (2009).
  • (12) M.P. Nowak, B. Szafran, F.M. Peeters, Phys. Rev. B 86, 125428 (2012).
  • (13) J. H. Shirley, Phys. Rev. 138, B979 (1965).
  • (14) K. Ono and S. Tarucha, Phys. Rev. Lett. 92, 256803 (2004).
  • (15) M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 99, 246602 (2007); H. O. H. Churchill, A. J. Bestwick, J.W. Harlow, F. Kuemmeth, D. Marcos, C. H. Stwertka, S. K. Watson, and C. M. Marcus, Nat. Phys. 5, 321 (2009); T. Kobayashi, K. Hitachi, S. Sasaki, and K. Muraki, Phys. Rev. Lett. 107, 216802 (2011); J. Baugh, Y. Kitamura, K. Ono, and S. Tarucha, Phys. Rev. Lett. 99, 096804 (2007); A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq, Phys. Rev. Lett. 99, 036801 (2007); S. M. Frolov, J. Danon, S. Nadj-Perge, K. Zuo, J.W.W. van Tilburg, V. S. Pribiag, J.W. G. van den Berg, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Phys. Rev. Lett. 109, 236805 (2012).
  • (16) A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Nature (London) 435, 925 (2005); F.H.L. Koppens, J.A. Folk, J.M. Elzerman, R. Hanson, L.H.W. van Beveren, I.T Vink, H.P. Tranitz, W. Wegscheider, L.P. Kouwenhoven, and L.M.K. Vandersypen, Science 309, 1346 (2005).
  • (17) E.A. de Andrada e Silva, G.C. La Rocca, and F. Bassani, Phys. Rev. B 55, 16293 (1997).
  • (18) S. Bednarek, B. Szafran, T. Chwiej, and J. Adamowski, Phys. Rev. B 68, 045328 (2003).
  • (19) A. Askar and A. C. Cakmak, J. Chem. Phys., 68, 2794 (1978)
  • (20) R. Shankar, Principles of Quantum Mechanics, Springer, 1994.
  • (21) A.V. Khaetskii, D. Loss and L. Glazman, Phys. Rev. Lett. 88, 18602 (2002).
  • (22) I.A. Merkulov, A.L. Efros, and M. Rosen, Phys. Rev. B 65, 205309 (2002).
  • (23) P.F. Braun, X. Marie, L. Lombez, B. Urbaszek, T. Amand, P. Renucci, V.K. Kalevich, K.V. Kavokin, O. Krebs, P. Voisin, and Y. Masumoto, Phys. Rev. Lett. 94, 116601 (2005); M.V.G. Dutt, J. Cheng, B. Li, X. Xu, X. Li, P.R. Berman, D.G. Steel, A.S. Bracker, D. Gammon, S.E. Economou, Phys. Rev. Lett. 94, 227403 (2005).
  • (24) B. Szafran, F. M. Peeters, and S. Bednarek, Phys. Rev. B 70, 205318 (2004).
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