Flying electron spin control gates

The control of "flying” (or moving) spin qubits is an important functionality for the manipulation and exchange of quantum information between remote locations on a chip. Typically, gates based on electric or magnetic fields provide the necessary perturbation for their control either globally or at well-defined locations. Here, we demonstrate the dynamic control of moving electron spins via contactless gates that move together with the spins. The concept is realized using electron spins trapped and transported by moving potential dots defined by a surface acoustic wave (SAW). The SAW strain at the electron trapping site, which is set by the SAW amplitude, acts as a contactless, tunable gate that controls the precession frequency of the flying spins via the spin-orbit interaction. We show that the degree of precession control in moving dots exceeds previously reported results for unconstrained transport by an order of magnitude and is well accounted for by a theoretical model for the strain contribution to the spin-orbit interaction. This flying spin gate permits the realization of an acoustically driven optical polarization modulator based on electron spin transport, a key element for on-chip spin information processing with a photonic interface.


SM1. SPIN POLARIZATION MEASUREMENTS
The spectroscopic photoluminescence (PL) studies of the spin transport were carried out in a helium ow cryostat (10-20 K) with optical access and radio-frequency wiring for SAW excitation. The spins were optically excited using a circularly polarized laser beam from a tunable Ti-sapphire laser (wavelengths λ L between 760 and 808 nm) focused onto a ∼ 2 µm wide spot on the sample surface by a 20x objective [L in Fig. 2(c) of the main text].
The same objective collects the PL emitted from along the SAW path and directs it to a triple-grating spectrometer operating in the subtractive mode and equipped with a cooled CCD detector. Spatially resolved PL maps are recorded by imaging the acoustic transport path onto the input slit of the spectrometer. An optical arrangement consisting of a quarter wave plate and a beam polarization displacer placed in front of the spectrometer slit shifts the PL images with right (I R ) and left (I L ) hand circular polarizations along the slit direction. In this way, we simultaneously record PL images of the spatial distribution of the two polarizations on the CCD.
The spin polarization ρ s during acoustic transport can be determined by exciting spins with a right hand circularly polarized laser beam and detecting the PL intensities I R and I L according to: Since hole-spin relaxation in QWs and QWRs is typically much faster than for electrons [1], we assume that ρ s reects only the electron spin dynamics.
In order to correct for artifacts arising from dichroism of the optical components of the setup, the polarization experiments were carried out by recording PL maps excited with a right-hand and then a left-hand circularly polarized laser beam. We will denote the corresponding * santos@pdi-berlin.de   [3,4], the latter also decreases of the electron-hole exchange interaction and its associated spin relaxation ratio [5].

SM4. SAW-INDUCED SO FIELDS
In this section, we briey summarize the SO mechanisms related to the strain and piezoelectric elds induced by a SAW. In the absence of a SAW, the conduction band spin splitting for electrons moving along the ⟨110⟩-directions of an intrinsic GaAs QW on a (001) substrate is given by the Dresselhaus term: where γ is a material constant and w z = d QW +2d 0 is the total extension of the electronic wave function, assumed to be equal to the QW width (d QW ) plus the penetration depth in the barrier layers (d 0 ). For an electron moving with the SAW velocity v SAW , the wave vector k is obtained from the electron momentum according to: where m * is the conduction band (CB) eective mass.
A uniaxial strain leads to a linear term in the CB splitting given by: cyclic permutations. Theσ i denote the Pauli matrices.
The apostrophe ( ′ ) indicates that the strain components are relative to the conventional axes x ′ , y ′ , and z ′ . This Hamiltonian gives rise to a strain-induced spin precession frequencies given by [9] ℏΩ In order to address the SAW strain eld, it is convenient to use a reference frame with the x-axis along the SAW propagation direction, the y-axis on the sample surface and perpendicular to x, and the z-axis perpendicular to the surface. A SAW propagating along the [110]or [110]-directions of the (001) surface induces three nonvanishing engineering strain components u xx , u zz , and u xz . Furthermore, we use u ij to denote the engineering strain components and, thus, to distinguish them from the physical strain ones ε i,j (ε ij = u ij for i = k and ε ij = u ij /2 for i ̸ = j).
In a piezoelectric material, the strain eld from the SAW generates a piezoelectric polarization eld D given where the superscript T denotes the vector transposition operation. When the SAW elds are transformed to the Cartesian reference frame, the strain and electrical polarization elds become: Note that the strain induces a SO component perpendicular to the SAW propagation direction.
A SAW in a piezoelectric material also induces a transverse piezoelectric eld F z associated with D, which gives rise to a Rashba spin-orbit contribution. As a consequence, SAWs along a ⟨110⟩ induce three contributions for the CB spin splitting, all oriented in-plane and perpendicular to the SAW propagation direction, given by: The upper and lower signs apply for SAWs along the [110] and [110] directions, respectively. We will show in Sec. SM5 A that the C ′ TABLE SM1. Tight-binding parameters used in the calculation of spin-orbit conduction band splitting. The notation used corresponds to the one of Ref. 10. In order to account for the valence band discontinuity we subtracted 0.007 eV from the diagonal parameters (i.e., Es, Ep, and Es * ) of the Al0.15Ga0.85As barrier layers.
Parameter GaAs [11] (eV) Al0.15Ga0.85As (eV)  this can be carried out using ab-initio approaches or empirical approaches using a large number of bands (e.g., a 14 band k · p method). Here, we determine the splittings using the empirical tight-binding (TB) method [13] following the approach described in Refs. 14 and 15. This atomistic approach enables the determination of the band structure of QWs with widths up to a few tens of nanometres using moderate computational eorts. Although empirical in the sense that it uses parameters tted to the bulk band structure, it can be regarded as microscopic when compared with other approaches such as the k · p eective mass calculations.
The TB calculations were carried out using a basis consisting of sp 3 s * orbitals [16] including spin-orbit coupling [17,18]. This orbital basis consists of 10 orbitals per atom and includes only nearest-neighbour interactions. It has been shown to reproduce very well the highest valence and the lowest conduction bands of most bulk semiconductors [16]. The tight-binding parameters employed for the GaAs QW and the Al 0.15 Ga 0.85 As barrier layers are summarized in Table SM1. A. Tight-binding calculation of the spin-orbit strain coecients As discussed in Sec. SM4, the SAW strain induces SO contributions described by two material constants C 3 and C ′ 3 . Reported values for these contributions span a wide range: C 3 = 0.47 eVnm [19], C 3 = 0.52 eVnm [6], and C 3 = 0.81 eVnm [9]. C ′ 3 has not been measured (nor calculated), and should be negligible in comparison with C 3 .In this section, we use the TB approach to estimate the magnitude of these contributions.
We determined the strain coecients C 3 and C ′  Fig. SM3(b). We nd that C ′ 3 is essentially zero.
In order to calculate C 3 , we took u zz = 0, u xz = 0 and performed calculations for dierent u xx [ Fig. SM3(a)]. The TB calculations yield |C 3 | = 1.65 eVnm. This C 3 value agrees very well with the experimental one obtained from the QWR data in the main text. It is, however, twice as large as the one experimentally determined by Beck (0.81 eVnm), [9] and also larger then the one estimated from D'yakonov's calculations. [7,20] Here, k x is given by Eq. SM5 and w z = d QW + 2d B As mentioned in the main text, F z is proportional to u xx . In order to determine the proportionality con- Note that the second term on the right-hand-side (corresponding to Bychkov-Rashba) turns positive due to the negative F z .
The ts of the experimental data for QWRs in Fig. 4 where, as mentioned in the main paper, Ω D , Ω R , and Ω S are the magnitudes of the precession frequency for a single SAW beam along the x-direction. As in the previous case, the amplitude of the transverse piezoelectric eld can be related to the strain amplitude u xx by a factor r SF = 21 × 10 3 kV/cm. Due to the quadratic dependence of Ω ′ SO on Ω D in Eq. (SM17), it is not possible to determine from the DQD data in Fig. 4(b) of the main paper the relative signs of the Dresselhaus and SAW-related SO elds. The red line in Fig. SM5 shows, however, that the SAW power dependence of Ω ′ SO is reasonably well accounted for using a value of C 3 = −2.6 eV nm that is comprable to the QWR results. Table SM3 where Ω BIA and Ω SIA are the components of the SOeld due to the bulk (BIA) and structural (SIA) inversion asymmetry, respectively, and r s is a factor weighting the amplitude of the two components (see further details below). For small magnitudes of the electron momentum ℏk, these components can be expressed in the k = (k x , k y ) = k(cos θ, sin θ) basis, where θ is the propagation angle with respect to the x axis, as: where v p = 2k B T m * e and T is the temperature. The diusion along y proceeds until the spins are backreected at the edges of the channel.
within the channel boundaries, the electron spins precess around the spin orbit eld with an angular precession frequency Ω SO [cf. Eq. (SM18)] determined by the momentum ℏk until they are scattered with a scattering time τ . The scattering time τ is determined from the mobility µ according to: τ = m * e µ/q e , where q e is the electron charge. The motion along y is randomized at each scattering event.
Due to the linear dependence of the precession frequencies on k, the spin precession angle for propagation along a xed direction only depends on the distance between the initial and nal points of the trajectory. If (cos θ, sin θ)∆ℓ is the displacement vector between two successive scattering events, than the corresponding spin precession angle under a SO eld Ω SIA will be δϕ s = 2π∆ℓ/L SIA is indeed expected from Eqs. (SM18) for r s > 0, when the magnitude |Ω SO | is constant. Since the time ∆t for propagation over a short distance ∆r = (∆x, ∆y), given by ∆t = ∆r/v = m * e ∆r/(ℏk), is inversely proportional to k, then the precession angle around y can be expressed as δϕ = Ω SOy ∆t = Ω SO cos θ∆t = (ℏ/m * e )Ω SO ∆x. As a consequence, the precession period along x does not depend on the spin propagation angle θ.
The dependence of ℓ s on w y displayed in Fig. SM8(b) shows two regions with dierent behaviors. For wide channels, ℓ s saturates at a value which reduces with increasing temperature. For narrow channels, in contrasts, ℓ s is essentially independent of temperature and increases according to ℓ s ∝ w −2 . Such a dependence agrees with previous simulation results from Refs. 3 and 23 and is attributed to the increased role of motional narrowing for w y < L (2D) SO .
The results of this section thus show that while the spin precession frequency during acoustic transport is essentially independent of the channel width, the spin decoherence lengths increase signicantly in narrow channels (w y << L SO ).