Pumped double quantum dot with spin-orbit coupling
© Khomitsky and Sherman; licensee Springer. 2011
Received: 13 August 2010
Accepted: 11 March 2011
Published: 11 March 2011
We study driven by an external electric field quantum orbital and spin dynamics of electron in a one-dimensional double quantum dot with spin-orbit coupling. Two types of external perturbation are considered: a periodic field at the Zeeman frequency and a single half-period pulse. Spin-orbit coupling leads to a nontrivial evolution in the spin and orbital channels and to a strongly spin- dependent probability density distribution. Both the interdot tunneling and the driven motion contribute into the spin evolution. These results can be important for the design of the spin manipulation schemes in semiconductor nanostructures.
PACS numbers: 73.63.Kv,72.25.Dc,72.25.Pn
Quantum dots, being one of the most intensively studied examples of natural and artificial nanostructures, attract attention due to the richness in the properties they demonstrate in the static and dynamic regimes . A possible realization of qubits for quantum information processing can be done by using spins of electrons in semiconductor quantum dots . Spin- orbit coupling makes the dynamics even in the basic systems such as the single-electron quantum dots extremely rich both in the orbital and spin channels. If the frequency of the electric field driving the orbital motion matches the Zeeman resonance for electron spin in a magnetic field, the spin-orbit coupling causes a spin flip. This effect was proposed in refs. [3, 4] to manipulate the spin states by electric means. The efficiency of this process is much greater than that of the conventional application of a periodic resonant magnetic field. The ability to cause coherently the spin flip in GaAs quantum dots was demonstrated in ref.  where the gate-produced electric field induced the spin Rabi oscillations. In ref.  periodic electric field caused the spin dynamics by inducing electron oscillations in a coordinate-dependent magnetic field. In addition, these results confirmed that the spin dephasing in GaAs quantum dots, arising due to the spin-orbit coupling [7, 8] is not sufficiently severe to prohibit a coherent spin manipulation.
The spin dynamics experiments [5, 6] necessarily use at least a double quantum dot to detect the driven spin state relative to the spin of the reference electron. Multiple quantum dots realizations become nowadays the subject of extensive investigation . In double quantum dots an interesting charge dynamics occurs and requires theoretical understanding. In this article we address full driven by an external electric field spin and charge quantum dynamics in a one-dimensional double quantum dot [10–13]. Despite the simplicity, these systems show a rich physics. In the wide quantum dots, where the tunneling is suppressed, and the motion is classical, the interdot transfer occurs only due to the over-the-barrier motion, and a chaos-like behavior is usually expected. The irregular driven behavior in the spin and charge dynamics in these systems was studied in ref. . In the quantum double quantum dots, the tunneling between single quantum dots is crucial and the spin-orbit coupling makes the interdot tunneling spin-dependent [15–17]. In quantum systems a finite set of energy eigenstates allows only for a strongly irregular rather than a real chaotic behavior. These orbital and spin dynamical irregularities are important for the understanding of the quantum processes in multiple quantum dots.
In this article we consider various regimes for a one-dimensional double quantum dot with spin-orbit coupling driven by an external electric field and analyze the probability and spin density dynamics in these systems.
Hamiltonian, time evolution, and observables
We use the highly numerically accurate approach to describe the dynamics with the sum of Hamiltonians in Equations (2)-(4). As the first step we diagonalize exactly the time-independent H 0 + H so in the truncated spinor basis ψ n (x) |σ〉 of the eigenstates of the quartic potential in magnetic field without spin-orbit coupling with corresponding eigenvalues E nσ . As a result, we obtain the basis set |ψ n 〉 where bold n incorporates the spin index. For the presentation, it is convenient to introduce the four-states subset: |ψ 1 〉 = ψ 1(x)|↑〉, |ψ 2 〉 = ψ 1(x)|↓〉, |ψ 3 〉 = ψ 2(x)|↑〉, |ψ 4 〉 = ψ 2(x)|↓〉, and to note that the spin-dependent bold index may not correspond to the state energy due to the Zeeman term in the Hamiltonian. The wavefunction ψ 1(x) (ψ 2(x)) is even (odd) with respect to the inversion of x. In the case of weak tunneling, assumed here, these functions can be presented in the form: , where ψ L(x) and ψ R(x) are localized in the left and in the right dot, respectively.
and the spin-dependent velocity in Equation (5).
where ω R(t) is the probability to find electron and is the analog of expectation value of the spin component.
Calculations and results
As the electron wavefunction at t = 0 we take linear combinations of two out of four low-energy states. The initial state in the form is localized in the left quantum dot, corresponding to the parameters .
Where T z (B z ) = 2πħ/Δ z is the Zeeman period. The second type is a half-period pulse, same as in Equation (13), but acting at the time interval 0 < t < T z (B z )/2 only. The spectral width of the pulse covers both the spin and the tunneling splitting of the ground state, thus, driving the spin and orbital dynamics simultaneously. Since Tz(Bz)ω ≫ 1, that is the corresponding frequencies are much less than those for the transitions between the orbital levels corresponding to a single dot, the higher-energy states follow the perturbation adiabatically. The field strength ℰ 0 is characterized by parameter f such that |e|ℰ 0 ≡ f × U 0/2d. Here we concentrate on the regime of a relatively weak coupling (f =≫ 1)
where the shape of the quartic potential remains almost intact in time, and the interdot tunneling is still crucially important. For the magnetic field we consider two different regimes Δ z = ΔE g /2 and Δ z = 2ΔE g to illustrate the role of the Zeeman field for the entire dynamics.
We consider a nanostructure with nm and U 0 = 10 meV. The four lowest spin-degenerate energy levels are E 1 = 3.938 meV, E 2 = 4.030 meV, E3 = 9.782 meV, E 4 = 11.590 meV counted from the bottom of a single quantum dot with the tunneling splitting ΔE g = E 2 - E 1 = 0.092meV, and the corresponding timescale 2π ħ/ΔE g = 45ps. The spin-orbit coupling is described by parameters α = 1.0 · 10-9 eVcm and β = 0.3 · 10-9 eVcm. The field parameter f = 0.125, corresponding to ℰ 0 = 177 V/cm. We use the truncated basis of 20 states with the energies up to 42 meV.
We have studied the full driven quantum spin and charge dynamics of single electron confined in one-dimensional double quantum dot with spin-orbit coupling. Equations of motion have been solved in a finite basis set numerically exactly for a pulsed field and by the Floquet technique for the periodic fields. We explored here the regime of relatively weak coupling to the external field, where a nontrivial dynamics already occurs. Our results are important for the understanding of the effects of spin-orbit coupling for nanostructures as we have demonstrated a possibility to achieve a controllable spin flip at various time scales and in various regimes by the electrical means only.
D.V.K. is supported by the RNP Program of Ministry of Education and Science RF (Grants No. 188.8.131.5286, 184.108.40.20678, 220.127.116.11/4297, 2.1.1/2833), by the RFBR (Grant No. 09-02-1241-a), by the USCRDF (Grant No. BP4M01), by "Researchers and Teachers of Russia" FZP Program NK-589P, and by the President of RF Grant No. MK-1652.2009.2. E.Y.S. is supported by the University of Basque Country UPV/EHU grant GIU07/40, Basque Country Government grant IT-472-10, and MCI of Spain grant FIS2009-12773-C02-01. The authors are grateful to L.V. Gulyaev for assistance.
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