Parametric spin excitations in lateral quantum dots
PParametric spin excitations in lateral quantum dots.
Jamie D. Walls ∗ Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 (Dated: November 10, 2018)In this work, the spin dynamics of a single electron under parametric modulation of a lateral quantum dot’selectrostatic potential in the presence of spin-orbit coupling is investigated. Numerical and theoretical calcu-lations demonstrate that, by squeezing and/or moving the electron’s wave function, spin rotations with Rabifrequencies on the order of tens of megahertz can be achieved with experimentally accessible parameters inboth parabolic and square lateral quantum dots. Applications of parametric excitations for determining spin-orbit coupling parameters and for increasing the spin polarization in the electronic ground are demonstrated.
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I. INTRODUCTION
Single spin operations are an important component for possible realizations of quantum dot-based quantum computers . Inprinciple, single spin manipulations can be performed using electron spin resonance (ESR) techniques with the application ofoscillating magnetic fields. A recent experiment utilizing ESR methods was used to generate spin rotations in a single electronlateral quantum dot at low static magnetic field strengths, where the required frequencies were on the order of hundreds of mega-hertz. However, extending ESR methods to microwave frequencies at low temperatures while minimizing the accompanying ACelectric field in the semiconductor quantum dots is experimentally challenging.Instead of considering the AC electric fields as something to be avoided, AC electric fields can be used to control the electronspin in a quantum dot. Modulations of the electrostatic potential can readily couple to the electronic degree of freedom in asemiconductor quantum dot and can be performed on the nanosecond timescale , which is roughly the timescale associatedwith the Zeeman energy in static fields on the order of a Tesla for GaAs quantum dots. However, in order to translate electroniccontrol into spin control, there must exist an additional coupling between the spin and electronic degrees of freedom.The two main methods of coupling the spin and electronic degrees of freedom are either through the Zeeman interaction orthrough spin-orbit coupling. Consider first the Zeeman interaction, which is given by (cid:98) H Z = (cid:126)B ( (cid:126)r ) · (cid:16)(cid:98) g ( (cid:126)r ) µ B (cid:126)S (cid:17) , where (cid:126)B ( (cid:126)r ) isthe applied magnetic field, (cid:98) g ( (cid:126)r ) is the effective g-tensor, µ B is the Bohr magneton, and (cid:98) S is the electron’s spin vector. Previousexperimental work has demonstrated that electric fields can be used to move the electron around in the presence of a spatiallydependent g-tensor, (cid:98) g ( (cid:126)r ) , in order to generate an effective time-dependent Zeeman interaction, (cid:98) H Z ( t ) = ¯ h(cid:126)ω ( t ) · (cid:126)S , which wasused to perform spin rotations . Alternatively, (cid:98) H Z ( t ) can also be generated by using an electric field to move an electronaround in the presence of a spatially inhomogeneous magnetic field .Besides manipulating the effective Zeeman interaction, modulating the electrostatic potential can be used to manipulate theelectron’s spin through the spin-orbit interaction. Previous theoretical studies have proposed using electric fields to generate spinrotations via an electric dipole spin resonance (EDSR); theoretical calculations indicate that EDSR methods can be used to rotatean electron’s spin on the order of tens of nanoseconds for electrons in quantum wells and quantum dots . For parabolicquantum dots, application of an electric field is equivalent to modulating the center of the quantum dot which, in the presence ofspin-orbit coupling, is able to induce spin transitions when the modulation frequency is equal to the Zeeman frequency. Besidesutilizing EDSR for spin excitation, electric fields can also be used to modulate the Rashba spin-orbit coupling constant in orderto perform combined spin and orbital excitations in quantum dots .In this work, the spin dynamics under the combined action of spin-orbit coupling and parametric modulations of the electro-static confining potential (such as squeezing and moving the electronic wave function) in both parabolic and square quantumdots is examined. Theoretical and numerical calculations utilizing Floquet theory and effective Hamiltonian theory are used todemonstrate that parametric modulations of the electrostatic potential can generate single spin rotations on the timescale of tensof nanoseconds using experimentally accessible parameters. In order to maximize the amplitude of the Rabi oscillations, themodulation frequency must be chosen with precision on the order of the Rabi frequency (order of 1-10 MHz). Such parametricexcitations therefore can provide better energy resolution of the quantum dot’s energy levels over transport measurements, wherethe measured energy levels are often poorly resolved due to thermal effects. Besides performing single spin rotations, we haveshown that measurement of the observed Rabi oscillations and the required modulation frequencies can be used to determinethe spin-orbit coupling parameters in the quantum dot. Finally, we have demonstrated that squeezing and expanding the elec-tronic wave function in a parabolic quantum dot can induce both orbital and spin excitations, which can be used to increase thespin polarization of the lowest electronic state. It should be noted that relaxation is neglected throughout most of the paper sincethe calculated Rabi frequencies (on the order of 1-10 MHz) are at least one order of magnitude greater than the experimentallyobserved /T values found in GaAs quantum dots . a r X i v : . [ c ond - m a t . o t h e r] M a y II. PARAMETRIC MODULATIONS AND FLOQUET THEORY
Before studying the case of parametric excitations in lateral quantum dots, the general formalism of using parametric mod-ulations to generate excitations in quantum systems is presented. Consider a Hamiltonian which is a function of the parameter ω : (cid:98) H ( ω ) = (cid:88) k E k ( ω ) | k ( ω ) (cid:105)(cid:104) k ( ω ) |(cid:104) k ( ω ) | k ( ω ) (cid:105) + (cid:88) j Using the formalism presented in Section II, we are now ready to begin studying parametric excitation in a single-electron,lateral quantum dot, which is taken to lie in the XY plane with the electron’s wave function strongly confined along the (cid:98) z direction. In the following, only an in-plane magnetic field, (cid:126)B = B cos( θ ) (cid:98) x + B sin( θ ) (cid:98) y , will be considered, which allows oneto neglect orbital effects associated with an out of plane magnetic field. The Hamiltonian for a single-electron lateral quantumdot defined by the electrostatic potential (cid:98) V ( (cid:98) X, (cid:98) Y ) , in the presence of spin-orbit coupling (for the moment, only Rashba andlinear Dresselhaus are considered), is given by: (cid:98) H = (cid:98) H + (cid:98) H SO (16) (cid:98) H = (cid:98) P X m ∗ + (cid:98) P Y m ∗ + (cid:98) V ( (cid:98) X, (cid:98) Y ) − ¯ hω Z (cid:98) σ Z (17) (cid:98) H SO = (cid:98) P X ¯ h ( ζ ( − θ ) (cid:98) σ X + ζ ( θ ) (cid:98) σ Z ) − (cid:98) P Y ¯ h ( ζ ( − θ ) (cid:98) σ X + ζ ( θ ) (cid:98) σ Z ) (18)where ¯ hω Z = | gµ B (cid:126)B | , ζ ( θ ) = α cos( θ ) − β sin( θ ) , ζ ( θ ) = α sin( θ ) − β cos( θ ) [where α and β are the Rashba and linearDresselhaus coupling constants respectively], and the vector potential, (cid:126)A = (cid:98) Z sin( θ ) (cid:98) x − (cid:98) Z cos( θ ) (cid:98) y , was chosen. Due to thestrong confinement along the (cid:98) z -direction, all terms linear in (cid:98) Z have been truncated/removed from (cid:98) H in Eqs. (16)-(18), with termsquadratic in (cid:98) Z being incorporated into the the confining potential along the (cid:98) z -direction. The electrostatic potential, (cid:98) V ( (cid:98) X, (cid:98) Y ) ,mostly results from voltages applied to surface gates above the 2DEG, which confines the electron within the quantum dot;changing the voltages of the surface gates can change (cid:98) V ( (cid:98) X, (cid:98) Y ) . The spin quantization axis has been taken to be along thedirection of the in-plane magnetic field. The eigenstates of (cid:98) H are denoted by | n, ±(cid:105) , where (cid:98) H | n, ±(cid:105) = ( E n ∓ ¯ hω Z / | n, ±(cid:105) .In the presence of spin-orbit coupling, the various | n, ±(cid:105) states are mixed; however, if the confinement strength is much largerthan the spin-orbit coupling strength, i.e., |(cid:104) n, ±| (cid:98) H SO | m, ±(cid:105)| (cid:28) | ∆ nm | and |(cid:104) n, ∓| (cid:98) H SO | m, ±(cid:105)| (cid:28) | ∆ nm ± ω Z | for all n and m , (cid:98) H SO is suppressed and can be treated as a perturbation to (cid:98) H .In this section, a lateral quantum dot defined by a parabolic electrostatic potential will be examined. Such parabolic potentialshave enjoyed tremendous success in describing transport and spectral properties in lateral quantum dots . The electrostaticpotential for a parabolic quantum dot can be written as: (cid:98) V ( (cid:98) X, (cid:98) Y ) = m ∗ ω X (cid:16) (cid:98) X − x c (cid:48) (cid:17) + m ∗ ω Y (cid:16) (cid:98) Y − y c (cid:48) (cid:17) − eF X (cid:98) X − eF Y (cid:98) Y = m ∗ ω X (cid:18) (cid:98) X − eF X m ∗ ω X − x c (cid:48) (cid:19) + m ∗ ω Y (cid:18) (cid:98) Y − eF Y m ∗ ω Y − y c (cid:48) (cid:19) − e F X m ∗ ω X − e F Y m ∗ ω Y (19)where (cid:126)r c (cid:48) = x c (cid:48) (cid:98) x + y c (cid:48) (cid:98) y is the center of the parabolic well, ω X and ω Y are the oscillator frequencies of the parabolicdot, and F X and F Y are static electric fields which do not alter the energy levels of the quantum dot but do shift the ef-fective center of the quantum dot to (cid:126)r c = [ x c (cid:48) + eF X / ( m ∗ ω X )] (cid:98) x + [ y c (cid:48) + eF Y / ( m ∗ ω Y )] (cid:98) y . For convenience, (cid:98) V ( (cid:98) X, (cid:98) Y ) FIG. 1: Parametric modulations of a parabolic quantum dot’s confining potential by either modulating (A) the confining frequency ( ω X and/or ω Y ) or (B) the center of the dot, (cid:126)r c = x c b x + y c b y . In the presence of spin-orbit coupling, if either δω X ( Y ) ( t ) or δ(cid:126)r c ( t ) ≡ δx c ( t ) b x + δy c ( t ) b y is modulated at roughly the energy difference between | , , + (cid:105) and | , , −(cid:105) , effective spin rotations can be performed as depicted in Fig. 1. in Eq. (19) was taken to be separable in the (cid:98) X and (cid:98) Y degrees of freedom; that is, the principal axes of (cid:98) V ( (cid:98) X, (cid:98) Y ) werechosen for convenience to be the same as those used in Eq. (16) (if this is not the case, inclusion of terms like λ (cid:98) X (cid:98) Y can be readily incorporated into the following theory). The eigenstates of (cid:98) H for parabolic confinement, | n, m, ±(cid:105) , satisfy (cid:98) H | n, m, ±(cid:105) = [¯ hω X ( n + 1 / 2) + ¯ hω Y ( m + 1 / ∓ ¯ hω Z / | n, m, ±(cid:105) and are centered about (cid:126)r c . Under conditions of strongconfinement (where ¯ hω X , ¯ hω Y (cid:29) | (cid:98) H SO | ), the states | , , + (cid:105) and | , , −(cid:105) are approximately the two lowest energy eigenstates.Transitions between these two states correspond to spin rotations within the ground electronic state.As discussed in Section II, efficient transitions between the states | , , + (cid:105) and | , , −(cid:105) in the quantum dot can occur wheneverthe Hamiltonian is parametrically modulated at roughly the energy difference between the two states. There are two naturalmodulation parameters to choose from in parabolic quantum dot potentials as shown in Fig. 1: either modulation of the oscillatorfrequencies [ ω X ( Y ) → ω X ( Y ) + δω X ( Y ) in Fig. 1(A)], or modulation of the center of the quantum dot [ (cid:126)r c → (cid:126)r c + δ(cid:126)r c in Fig.1(B)] (note that modulation of F X and/or F Y is equivalent to modulating (cid:126)r c ). Such parametric modulations can in principlebe performed by modulating the surface gate voltages which define (cid:98) V ( (cid:98) X, (cid:98) Y ) . Modulations of surface gate voltages have beenexperimentally performed on the nanosecond timescale . A. Parametric Modulations of the Confining Strength in a Parabolic Quantum Dot The modulation the the oscillator strengths of the quantum dot, ω X ( Y ) → ω X ( Y ) ( t ) = ω X ( Y ) + δω X ( Y ) ( t ) , corresponds tosqueezing and expanding the electron’s wave function in a time-dependent manner, as illustrated in Fig. 1(A). Using Eq. (4), theeffective time-dependent Hamiltonian during modulation of ω X and ω Y at a frequency ω r is given by: (cid:101) H ( t )¯ h = ω X ( t )( a † X a X + 1 / 2) + ω Y ( t )( a † Y a Y + 1 / − ω Z (cid:98) σ Z + i (cid:115) m ∗ ω X ( t )2¯ h (cid:16) a † X − a X (cid:17) ( ζ ( − θ ) (cid:98) σ X + ζ ( θ ) (cid:98) σ Z ) + i ω X ( t ) ∂ω X ( t ) ∂t (cid:16) ( a † X ) − a X (cid:17) − i (cid:115) m ∗ ω Y ( t )2¯ h (cid:16) a † Y − a Y (cid:17) ( ζ ( − θ ) (cid:98) σ X + ζ ( − θ ) (cid:98) σ Z ) + i ω Y ( t ) ∂ω Y ( t ) ∂t (cid:16) ( a † Y ) − a Y (cid:17) + i ∂ω X ( t ) ∂t eF X ω X ( t ) (cid:115) hm ∗ ω X ( t ) (cid:16) a † X − a X (cid:17) + i ∂ω Y ( t ) ∂t eF Y ω Y ( t ) (cid:115) hm ∗ ω Y ( t ) (cid:16) a † Y − a Y (cid:17) (20) ≡ h (cid:32) (cid:101) H ( ω X , ω Y ) + (cid:88) m (cid:98) V m exp( i mω r t ) (cid:33) (21)where a † X ( Y ) and a X ( Y ) are the creation and annihilation operators associated with the harmonic potential, and ω X ( Y ) ( t ) = ω X ( Y ) + δω X ( Y ) sin( ω r t + φ X ( Y ) ) is the time-dependent oscillator frequency. Note that (cid:101) H ( ω X , ω Y ) contains the time-independent contributions to the energy from the harmonic potential and the Zeeman and spin-orbit interactions. In the presenceof a static electric field (i.e., F X ( Y ) (cid:54) = 0 ), modulations of ω X and ω Y also results in a modulation of (cid:126)r c , which leads to the termslinear in a X ( Y ) and a † X ( Y ) in Eq. (20).Efficient transitions between the Floquet states | (cid:105) ≡ | , , + , F (cid:105) and | (cid:105) ≡ | , , − , F (cid:105) can be performed if the modulationfrequency, ω r , equals the energy difference between states | (cid:105) and | (cid:105) , i.e., ω r ≈ ω Z . Using Eq. (4), the effective Hamiltonian, (cid:98) H EFF , in the presence of parametric modulation of the oscillator strengths at ω r ≈ ω Z is given [up to δω X ( Y ) ]: (cid:98) H EFF = ¯ h (cid:0) ω r − ω Z − ∆ Z − δ Z (cid:1) (cid:98) σ Z + δ + (cid:98) σ + δ − (cid:98) σ − (22)where (cid:98) σ Z = | (cid:105)(cid:104) | − | (cid:105)(cid:104) | , (cid:98) σ = | (cid:105)(cid:104) | , (cid:98) σ − = | (cid:105)(cid:104) | , and ∆ Z = m ∗ ω Z ¯ h (cid:32) ω X ( ζ ( − θ )) ω Z − ω X + ω Y ( ζ ( − θ )) ω Z − ω Y (cid:33) δ Z = m ∗ ω Z δω X ( ζ ( − θ )) h ω X (cid:18) ω X + 4 ω X ω Z − ω X ω Z + 27 ω X ω Z + 4 ω Z ( ω X − ω Z )(4 ω X − ω Z ) ( ω X − ω Z ) (cid:19) + m ∗ ω Z δω Y ( ζ ( − θ )) h ω Y (cid:18) ω Y + 4 ω Y ω Z − ω Y ω Z + 27 ω Y ω Z + 4 ω Z ( ω Y − ω Z )(4 ω Y − ω Z ) ( ω Y − ω Z ) (cid:19) + ω Z ¯ h (cid:32) ( δω X eF X ) ( ω X + 3 ω Z )( ζ ( − θ )) ω X ( ω Z − ω X ) + ( δω Y eF Y ) ( ω Y + 3 ω Z )( ζ ( − θ )) ω Y ( ω Z − ω Y ) (cid:33) + 2 eF X eF Y δω X δω Y ¯ h cos( φ X − φ Y ) ω Z ζ ( − θ ) ζ ( θ ) ω Z ( ω X + ω Y ) + ω X ω Y − ω Z ω X ω Y ( ω X − ω Z ) ( ω Y − ω Z ) δ ± = − ω Z ¯ h (cid:18) ζ ( − θ ) δω X eF X exp( ± i φ X ) ω X ( ω X − ω Z ) − ζ ( − θ ) δω Y eF Y exp( ± i φ Y ) ω Y ( ω Y − ω Z ) (cid:19) ± i m ∗ ω Z ¯ h (cid:18) exp( ± i φ X ) ζ ( − θ ) ζ ( θ ) δω X ω X − ω Z + exp( ± i φ Y ) ζ ( θ ) ζ ( − θ ) δω Y ω Y − ω Z (cid:19) (23)where ∆ Z is the higher-order contribution of the spin-orbit coupling to the energy difference (in Eq. (23), ∆ Z is only writtento second-order in the spin-orbit coupling), and δ Z is a Bloch-Siegert shift which arises from the noncommutivity of (cid:101) H ( t ) atdifferent times. In the presence of a static electric field, the coupling between the states | (cid:105) and | (cid:105) , δ ± in Eq. (23), is first-orderin the spin-orbit coupling parameters, whereas it is second-order in the spin-orbit coupling when F X = F Y = 0 eV/m.Figure 2(A) presents the exact numerical simulation of the transition amplitude, |(cid:104) , , + | U ( t, | , , −(cid:105)| , for a parabolicquantum dot under harmonic modulation of ω X for F X (cid:54) = 0 . In the simulation, the eigenstates of (cid:101) H ( ω X , ω Y ) in Eq. (20) werenumerically found by diagonalizing (cid:101) H ( ω X , ω Y ) using a basis of four hundred | n, m, ±(cid:105) states. Next, (cid:101) H ( t ) in Eq. (21) wasexpanded in the n lowest energy eigenstates of (cid:101) H ( ω X , ω Y ) , and U ( t ) = T exp( − i / ¯ h (cid:82) t (cid:101) H ( t (cid:48) ) d t (cid:48) ) was then found numerically( n = 30 was found to give converged results for the simulations). The modulation frequency, ω r , was given by the numericallycalculated energy difference between the two states, ∆ E , plus the Bloch- Siegert shift given in Eq. (23), i.e., ω r = ∆ E / ¯ h + δ Z ≈ ω Z + ∆ Z + δ Z . This method was used throughout the paper when numerically calculating the exact transition amplitudes.The following parameters were used in the simulation shown in Fig. 2(A): ¯ hω Y = ¯ hω X = 1 meV, ¯ hω Z = 0 . meV(corresponding to an in-plane magnetic field of around five Tesla in GaAs), α = 4 × − eV-m, m ∗ = 0 . m (where m isthe free electron mass), δω X = ω X / , F X = 10 eV/m, and θ = 0 . With these parameters, the effective Rabi frequency for“on-resonance” modulation [ ( ω r − ω Z ) / (2 π ) = − . MHz] observed in Fig. 2(A) was 9.86 MHz, which is consistent withthe theoretical value given by Eq. (23), | δ ± | / (2 π ) = 9 . MHz. The fact that the Rabi frequency is directly proportional to thespin-orbit coupling arises because modulation of ω X results in a modulation of the effective center of the quantum dot (cid:126)r c by δ(cid:126)r c when F X (cid:54) = 0 . For the parameters used in Fig. 2(A), | δ(cid:126)r c | = 2 . nm.For the case when F X = F Y = 0 eV/m, (cid:126)r c remains fixed, and the electron wave function is only squeezed/expandedduring the modulation [Fig. 1(A)]. Figure 2(B) presents the exact numerical simulation of the transition amplitude, |(cid:104) , , + | U ( t, | , , −(cid:105)| , when F X = F Y = 0 eV/m. The following parameters were used: ¯ hω Y = 1 meV, ¯ hω X = 0 . meV, ¯ hω Z = 0 . meV, α = 8 × − eV-m, m ∗ = 0 . m (where m is the free electron mass), δω X = ω X / , and θ = π/ . Withthese parameters, the effective Rabi frequency observed in Fig. 2(B) was 355 kHz, which is consistent with the calculated valuegiven by Eq. (23), | δ ± | / (2 π ) = 354 kHz, for “on-resonance” irradiation, ( ω r − ω Z ) / (2 π ) = − . MHz. The observed Rabifrequency was over an order of magnitude smaller than the case of nonzero F X [Fig. 2(A)]. As stated earlier, the effective Rabifrequency is smaller since it is second-order in the spin-orbit coupling and contains terms like ( α − β ) / θ ) ± αβ cos(2 θ ) .Note that for θ = nπ/ , both α and β must be nonzero for δ ± to be nonzero. Furthermore, for the case when ω X = ω Y and foruniform modulation (i.e, δω Y = δω X ), the Rabi frequency in Eq. (23) is exactly proportional to αβ cos(2 θ ) . Finally, it shouldbe noted that ω r − ω Z is on the order of | δ ± | [Fig. 2(A)] or much greater than | δ ± | [Fig. 2(B)], so that ω r must be tuned withprecision given by | δ ± | in order to maximize the amplitude of the Rabi oscillations.Although Eq. (4) was used to generate (cid:98) H EFF in Eqs. (22)-(23), the dynamics under modulations of ω X and ω Y can becalculated using the following time-dependent Hamiltonian: (cid:98) H ( t ) = (cid:98) P X m ∗ + (cid:98) P Y m ∗ + (cid:98) H SO − ¯ hω Z (cid:98) σ Z + m ∗ (cid:16) ω X ( t ) (cid:98) X + ω Y ( t ) (cid:98) Y (cid:17) − eF X (cid:98) X − eF Y (cid:98) Y (24)where (cid:98) P X ( Y ) = (cid:112) m ∗ ¯ hω X ( Y ) / i ( a † X ( Y ) − a X ( Y ) ) , (cid:98) X ( (cid:98) Y ) = (cid:112) ¯ h/ (2 m ∗ ω X ( Y ) ) (cid:16) a X ( Y ) + a † X ( Y ) (cid:17) , and ω X ( Y ) ( t ) = ω X ( Y ) + δω X ( Y ) ( t ) . (cid:98) H ( t ) in Eq. (24) can be used to construct an effective Hamiltonian in the | , , + , F (cid:105) and | , , − , F (cid:105) subspace,which is also given by Eqs. (22)-(23). B. Parametric Modulations of a Parabolic Quantum Dot’s center In addition to modulating the confinement frequency, the center of the parabolic quantum dot can also be modulated, i.e, x c → x c + δx c ( t ) and y c → y c + δy c ( t ) . Unlike the case of modulating ω X and ω Y , displacements of the harmonic potential donot alter the energy spacings of the quantum dot; however, the simultaneous eigenstates of the oscillator do change under suchmodulation. Using Eq. (4), the Hamiltonian under time-dependent displacements of the parabolic potential’s center is given by: (cid:101) H ( t )¯ h = ω X ( a † X a X + 1 / 2) + ω Y ( a † Y a Y + 1 / − ω Z (cid:98) σ Z + i (cid:114) m ∗ ω X h (cid:16) a † X − a X (cid:17) ( ζ ( − θ ) (cid:98) σ X + ζ ( θ ) (cid:98) σ Z ) − i (cid:114) m ∗ ω X h ∂x c ( t ) ∂t (cid:16) a † X − a X (cid:17) − i (cid:114) m ∗ ω Y h (cid:16) a † Y − a Y (cid:17) ( ζ ( θ ) (cid:98) σ Z + ζ ( − θ ) (cid:98) σ X ) − i (cid:114) m ∗ ω Y h ∂y c ( t ) ∂t (cid:16) ( a † Y ) − a Y (cid:17) (25)For small displacements, x c → x c + δx c sin( ω r t + φ X ) and y c → y c + δy c sin( ω r t + φ Y ) , the effective Hamiltonian in the | , , + , F (cid:105) and | , , − , F (cid:105) subspace can be written as (for ω r ≈ ω Z ): (cid:98) H EFF ¯ h = 12 (cid:0) ω r − ω Z + δ cZ + ∆ Z (cid:1) (cid:98) σ Z + δ c + (cid:98) σ + δ c − (cid:98) σ − (26) FIG. 2: Numerical simulation of the transition amplitudes, |(cid:104) , , + | T exp( − i R t d t (cid:48) e H ( t (cid:48) )) | , , −(cid:105)| , caused by modulating a quantum dot’soscillator frequency, ω X ( t ) = ω X + δω X sin( ω r t ) , (A) with and (B) without a static electric field. In the presence of a static electric field,the effective Rabi frequency is first-order in the spin-orbit coupling [Eq. (23)], which lead to a large Rabi frequency as shown in Fig. 2(A).The following parameters were used in the simulation: ¯ hω Y = ¯ hω X = 1 meV, α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , δω X = ω X , F X = 10 eV/m, and φ X = 0 , which gave a Rabi frequency of 9.86 MHz compared with the calculated value of | δ ± | / (2 π ) ≈ . MHz by Eq. (23). In the absence of a static electric field, the effective Rabi frequency is second-order in the spin-orbitcoupling, which lead to a smaller Rabi frequency as shown in Fig. 2(B). The following parameters were used: ¯ hω Y = 1 meV, ¯ hω X = 0 . meV, ¯ hω Z = 0 . meV, α = 8 × − eV-m, δω X = ω X / , β = 0 eV-m, θ = π/ , F X = 0 eV/m, and φ X = 0 , which gave a Rabifrequency of 355 kHz compared with the calculated value of | δ ± | / (2 π ) = 354 kHz given by Eq. (23). where δ cZ = ( m ∗ ) ω Z h (cid:32) ( δx c ) ω X ( ω X + 3 ω Z )( ζ ( − θ )) ( ω Z − ω X ) + ( δy c ) ω Y ( ω Y + 3 ω Z )( ζ ( − θ )) ( ω Z − ω Y ) (cid:33) + ( m ∗ ω X ω Y ) h cos( φ cX − φ cY ) δy c δx c ω Z ζ ( − θ ) ζ ( θ ) ω Z ( ω X + ω Y ) + ω X ω Y − ω Z ( ω X − ω Z ) ( ω Y − ω Z ) (27) δ c ± = − m ∗ ω Z h (cid:18) exp( ± i φ cX ) δx c ω X ζ ( − θ ) ω Z − ω X − exp( ± i φ cY ) δy c ω Y ζ ( − θ ) ω Z − ω Y (cid:19) (28) FIG. 3: Numerical simulation of the transition amplitudes, |(cid:104) , , + | T exp( − i R t d t (cid:48) e H ( t (cid:48) )) | , , −(cid:105)| , caused by modulating the center of thequantum dot’s parabolic potential. In this case, the effective Rabi frequency is first-order in the spin-orbit coupling [Eq. (28)], leading to alarge Rabi frequency. The following parameters were used in the simulation: ¯ hω Y = ¯ hω X = 1 meV, α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , δx c = 11 . nm [equivalent to using an electric field of E X = 10 eV/m], and φ X = 0 , which gave a Rabi frequencyof 48.7 MHz compared with calculated value of | δ EF ± | / (2 π ) ≈ . MHz given by Eq. (28). The effective coupling between spin states, δ c ± , is first-order in the spin-orbit coupling, which leads to large Rabi frequencies.Figure 3 presents an exact numerical simulation of the transition amplitude, |(cid:104) , , + | T exp( − i (cid:82) t d t (cid:48) (cid:101) H ( t (cid:48) ) / ¯ h ) | , , −(cid:105)| , in thepresence of modulating the center of the parabolic well. The following parameters were used in the simulation: ¯ hω X = ¯ hω Y = 1 meV, α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , F X = F Y = 0 eV/m, and δx c = 11 . nm. Suchparameters gave an effective Rabi frequency of 48.7 MHz in Fig. 3 compared with the calculated value of | δ c + | / (2 π ) = 48 . MHz given by Eq. (28) for “on-resonance” modulation (i.e., ( ω r − ω Z ) / (2 π ) = − . MHz). As mentioned in the introduction,parametric modulation of a parabolic quantum dot’s center is equivalent to applying a time-dependent electric field, E ( t ) = E X ( t ) (cid:98) x + E Y ( t ) (cid:98) y , with the correspondence that δx c ( t ) = eE X ( t ) / ( m ∗ ω X ) and δy c ( t ) = eE Y ( t ) / ( m ∗ ω Y ) . The simulation inFig. 3 corresponds to E X = 10 eV/m. Previous theoretical work has suggested using such EDSR methods to perform efficientspin rotations in quantum wells and quantum dots . C. Determining the spin-orbit coupling constants, α and β In the above examples of parametric oscillations, both the effective Rabi frequency, δ ± , and the effective offset, ∆ Z , dependupon the spin-orbit coupling constants of the quantum dot ( α and β ), the oscillator frequencies ( ω X and ω Y ), and the directionof the magnetic field, θ . This dependence can potentially be used to determine both the spin-orbit coupling constants α and β .For instance, consider the effects of modulating the center of the parabolic dot, i.e., performing an EDSR experiment. Takingthe ratio of the “measured” Rabi frequency [Eq. (28)] for an experiment with δx c = λ mod and δy c = 0 to the “measured” Rabi0frequency for an experiment with δx c = 0 and δy c = λ mod gives ZZ ( θ ) = δ ± ( δx c = λ mod , δy c = 0) , θδ ± ( δx c = 0 , δy c = λ mod , θ )= − ζ ( − θ ) ζ ( − θ ) (29)For an in-plane magnetic field along the (cid:98) y axis (i.e., θ = π/ ), the relative ratio of the spin-orbit coupling strengths can be foundsince in this case ZZ ( π/ 2) = β/α . Furthermore, ∆ Z , which can be determined by tuning ω r to the frequency which maximizesthe amplitude of the Rabi oscillations and then approximating ∆ Z ≈ ω r − ω Z , can then be used to determine the absolute valueof α and β . Although the calculations used in this section were for a harmonic potential, more realistic (cid:98) V ( (cid:98) X, (cid:98) Y ) using the fullelectrostatic potential generated by the metallic surface gates could be used to more accurately characterize the spin dynamicsin terms of α and β under parametric modulation of (cid:98) V ( (cid:98) X, (cid:98) Y ) . IV. THE EFFECTS OF THE CUBIC DRESSELHAUS SPIN-ORBIT INTERACTION Recent work has indicated that the cubic Dresselhaus term can become the dominant spin-orbit interaction in highly confined(i.e., large ω X and ω Y ) quantum dots . For the coordinate system chosen in this work, the cubic Dresselhaus term is given by: (cid:98) H cub D = γ ¯ h (cid:16) (cid:98) P Y (cid:98) P X (cid:98) P Y (cos( θ ) (cid:98) σ Z − sin( θ ) (cid:98) σ X ) − (cid:98) P X (cid:98) P Y (cid:98) P X (sin( θ ) (cid:98) σ Z + cos( θ ) (cid:98) σ X ) (cid:17) (30)The linear Dresselhaus coupling, β in Eq. (18), is related to the cubic Dresselhaus coupling constant, γ in Eq. (30), by β = γ (cid:104) (cid:98) P Z (cid:105) / ¯ h [where for a quantum well of width l w , (cid:104) (cid:98) P Z (cid:105) / ¯ h ≈ ( π/l w ) ]. For a quantum well with l w = 30 nm, andfor ¯ hω X ( Y ) ≈ meV, the relative strength of the cubic and linear Dresselhaus terms is given roughly by (cid:104) (cid:98) P X ( Y ) (cid:105) / (cid:104) (cid:98) P Z (cid:105) ≈ m ∗ ω X ( Y ) l w / (2 π ¯ h ) = 0 . . The larger ω X and ω Y are for the quantum dot, the more the cubic Dresselhaus coupling con-tributes to the total spin-orbit interaction.Under parametric modulation of the center of the quantum dot (i.e., an EDSR experiment), the effect of the cubic Dresselhauscoupling can be found by simply adding (cid:98) H cub D to (cid:101) H ( t ) in Eq. (25). The effective Hamiltonian in the | (cid:105) = | , , + , F (cid:105) and | (cid:105) = | , , − , F (cid:105) subspace in this case is given by: (cid:98) H cub ¯ h = 12 (cid:0) ω r − ω Z − ∆ Z, cub (cid:1) (cid:98) σ Z + δ c + , cub (cid:98) σ + δ c − , cub (cid:98) σ − (31)where ∆ Z, cub = m ∗ ω Z ¯ h (cid:32) ω X (cid:0) ζ cub ( − θ ) (cid:1) ω Z − ω X + ω Y (cid:0) ζ cub ( − θ ) (cid:1) ω Z − ω Y (cid:33) − m ∗ ω Z h (cid:18) m ∗ γ ¯ h (cid:19) ω X ω Y (cid:18) ω Y sin ( θ )(2 ω Y + ω X ) − ω Z + ω X cos ( θ )(2 ω X + ω Y ) − ω Z (cid:19) δ c ± , cub = − m ∗ ω Z h (cid:18) exp( ± i φ cX ) δx c ω X ζ cub ( − θ ) ω Z − ω X − exp( ± i φ cY ) δy c ω Y ζ cub ( − θ ) ω Z − ω Y (cid:19) (32)with ζ cub ( − θ ) = α cos( θ ) + [ β − γm ∗ ω Y / (2¯ h )] sin( θ ) and ζ cub ( − θ ) = − ( α sin( θ ) + [ β − γm ∗ ω X / (2¯ h )] cos( θ )) . Since γ and β have the same sign, the cubic Dresselhaus interaction lessens the contribution of the linear Dresselhaus interaction tothe effective Rabi frequency under parametric modulation of the electrostatic potential. Similar results are also obtained inthe case of modulating ω X and ω Y ; in this case, the effective Rabi frequency is again proportional to the spin-orbit couplingparameters only in the presence of a nonzero static electric field, F X ( Y ) (cid:54) = 0 [otherwise, it is second-order in the spin-orbitcoupling parameters].Under parametric modulation of both the quantum dot’s center and oscillator frequency, Rabi oscillations, which depend onlyupon γ , can occur; measuring the Rabi frequency would therefore provide an independent measurement of γ . Consider thecase when the quantum dot’s center is modulated about the (cid:98) y direction at a frequency of ω r , y c ( t ) = δy c sin( ω r t + φ Y ) ,while at the same time the quantum dot is being periodically squeezed about the (cid:98) x direction at a frequency of ω r , ω X ( t ) = ω X + δω X sin( ω r t + φ X ) . In this case, efficient spin rotations can be performed in the relevant bimodal Floquet subspace, | (cid:105) = | , , + , F , F (cid:105) and | (cid:105) = | , , − , F , F (cid:105) when ω r + ω r ≈ ω Z [here F and F denote the Floquet states relativeto the oscillator frequencies ω r and ω r respectively]. The effective Hamiltonian in this subspace can be written as: (cid:98) H cub ¯ h = 12 (cid:0) ω r + ω r − ω Z − ∆ Z, cub − δ Z, cub (cid:1) (cid:98) σ Z + δ , cub (cid:98) σ + δ − , cub (cid:98) σ − (33)1 FIG. 4: Numerical simulation of the transition amplitudes, |(cid:104) , , + | T exp( − i R t d t (cid:48) e H ( t (cid:48) )) | , , −(cid:105)| , caused by the simultaneous modulationof the quantum dot’s oscillator frequency [ ω X ( t ) = ω X + δω X sin( ω r t ) ] and center [ y c ( t ) = δy c sin( ω r t ) ], (A) with and (B) without theRashba and linear Dresselhaus spin-orbit interaction. Efficient spin transitions, which depend upon the cubic Dresselhaus coupling to first-order, can be performed when ω r + ω r ≈ ω Z [Eq. (35)]. In both simulations, the following parameters were used: ¯ hω Y = 0 . meV, ¯ hω X = 3 meV, , ¯ hω Z = 0 . meV, θ = 0 , δω X = ω X , ω r = 2 ω r , δy c = 56 nm (equivalent to E Y = 10 eV/m), γ = 27 × − eV-m ,and φ X = φ Y = 0 . In (A), α = 4 × − eV-m and l w = 30 nm (giving β = 2 . × − eV-m), which gave a sizable Bloch-Siegert shiftof δ Z, cub / (2 π ) = 3 . MHz. In Fig. 4(A), the observed Rabi frequency was 736.6 kHz, which is within of the calculated theoretical valuegiven by Eq. (35), | δ ± , cub | / (2 π ) = 721 . kHz. In (B), the linear Dresselhaus and Rashba spin-orbit coupling constants were artificially set tozero, resulting in δ Z, cub / (2 π ) ≈ kHz; the observed Rabi frequency was 724.5 kHz, which is similar to that observed in Fig. 4(A) and iscloser to the calculated value given by Eq. (35). This demonstrates that the observed Rabi oscillations are mainly due to the cubic Dresselhauscoupling [differences between (A) and (B) are mainly due to higher-order coupling terms involving α and β ]. where δ Z, cub ≈ (cid:18) m ∗ δy c ¯ h (cid:19) ω r ω Y ω Z (cid:0) ζ cub ( − θ ) (cid:1) ( ω r − ω Y ) ( ω Z − ω r ) (34) δ ± , cub = ± i γ cos( θ )2¯ h exp( ± i ( φ X + φ Y )) δω X δy c ω r ( m ∗ ω X ω Y ) ( ω r − ω Y )( ω r − ω X ) (35)The effect of the linear Dresselhaus and Rashba spin-orbit coupling arises mainly in the Bloch-Siegert shift, δ Z, cub , and in ∆ Z, cub ;higher-order terms contributions to δ ± , cub arising from both the linear Dresselhaus and Rashba spin-orbit coupling can be madenegligible by using smaller δω X and δy c .Figure 4 shows the numerical simulation of |(cid:104) , , + | T exp( − i ¯ h (cid:82) t (cid:101) H ( t (cid:48) ) d t (cid:48) ) | , , −(cid:105)| under both periodic modulation of thequantum dot’s center about the (cid:98) y direction and modulation of the oscillator frequency about the (cid:98) x direction in (A) the presence of2and in (B) the absence of the Rashba and linear Dresselhaus spin-orbit interactions. In Fig. 4(A), the following parameters wereused: ¯ hω X = 3 meV, ¯ hω Y = 0 . meV, ¯ hω Z = 0 . meV, δω X = ω X / , ω r = 2 ω r , γ = 27 × − eV-m , α = 4 × − eV-m, l w = 30 nm (giving β ≈ . × − eV-m), and δy c = 56 nm (giving E Y ≈ eV/m). With the above parameters, δ Z, cub / (2 π ) ≈ . MHz. The observed Rabi frequency in Fig. 4(A) was 736.6 kHz, which is within of the calculated valuegiven by Eq. (35), | δ ± , cub | / (2 π ) = 721 . kHz. Since δ Z, cub > | δ ± , cub | , ω r and ω r must be tuned to include δ Z, cub in orderto maximize the amplitude of the Rabi oscillations [in Fig. 4(A), ( ω r + ω r − ω Z ) / (2 π ) = − . MHz+ δ Z, cub / (2 π ) ]. Notealso that in the numerical simulation of Fig. 4(A), the transition amplitude does not go exactly to 1 (maximum value of 0.9861),which is most likely due to higher-order corrections to the Bloch-Siegert shift, δ Z, cub .Better agreement between the Rabi frequency calculated using Eq. (35) to numerical simulation is obtained in Fig. 4(B)where both α and β were artificially set to zero, removing the dominant higher-order corrections to both δ Z, cub and δ ± , cub . InFig. 4(B), the observed Rabi frequency was 724.5 kHz, which is closer to the calculated value given by Eq. (35), | δ ± , cub | / (2 π ) =721 . kHz. It should be noted that both simulations shown in Fig. 4 give roughly the same Rabi frequency, which, to a goodapproximation, is given by Eq. (35). Thus measuring the Rabi frequency under such parametric modulations should enable γ tobe directly determined. It should be noted that for a quantum dot with a non-harmonic potential and in the presence of an out ofplane magnetic field, it has previously been found that the EDSR technique also generates spin rotations which, to first order,depend upon γ and the cyclotron frequency. V. PARAMETRIC MODULATIONS IN SQUARE QUANTUM DOTS Besides harmonic potentials, another model electrostatic potential for electrons in lateral quantum dots is that of a square-box(hard wall) potential defined by V ( x, y ) = 0 for ≤ x ≤ L X and ≤ y ≤ L Y and V ( x, y ) = ∞ everywhere else. Althoughthe hard wall potential is not very realistic in two-dimensional systems, previous studies have utilized such models in analyzingtransport in quantum dots , and such potentials are the quintessential model for studies of chaos in two-dimensional systems,such as in the stadium billiard . For the square-box potential, the eigenstates for (cid:98) H are simply the two-dimensional particle ina box states, (cid:104) (cid:126)r | n, m, ±(cid:105) = (cid:113) L X (cid:113) L Y sin (cid:16) πnxL X (cid:17) sin (cid:16) πmyL Y (cid:17) |±(cid:105) . In order to induce transitions between the various | n, m, ±(cid:105) ,small, periodic modulations in both the length and the width of the box, L X and L Y , can be made. Using Eq. (4), the effectivetime-dependent Hamiltonian during modulation of the box’s length and width, L X → L X ( t ) and L Y → L Y ( t ) , can be writtenas: (cid:101) H ( t )¯ h = ¯ hπ m ∗ L X ( t ) (cid:98) H X + ¯ hπ m ∗ L Y ( t ) (cid:98) H Y − ω Z (cid:98) σ Z + L X ¯ h L X ( t ) (cid:98) P X ( ζ ( − θ ) σ X + ζ ( θ ) (cid:98) σ Z ) − L Y ¯ hL Y ( t ) (cid:98) P Y ( ζ ( θ ) σ Z + ζ ( − θ ) (cid:98) σ X )+ ∂L X ( t ) ∂t hL X ( t ) (cid:16) (cid:98) P X (cid:98) X + (cid:98) X (cid:98) P X (cid:17) + ∂L Y ( t ) ∂t hL Y ( t ) (cid:16) (cid:98) P Y (cid:98) Y + (cid:98) Y (cid:98) P Y (cid:17) (36)where (cid:98) H X = (cid:88) n,m, ± n | n, m, ±(cid:105)(cid:104) n, m, ±| (cid:98) H Y = (cid:88) n,m, ± m | n, m, ±(cid:105)(cid:104) n, m, ±| (37)The form of (cid:101) H ( t ) arises naturally from Eq. (4); previous work on parametric deformations of hard wall potentials have alsoarrived at a similar form for the effective Hamiltonian . Note, however, that it is necessary to separate the transformation (cid:99) W ( t ) from the dynamics and only include its effect at the end of the calculation .By performing small modulations of the length and width, L X → L X + δL X ( t ) and L Y → L Y + δL Y ( t ) , spin transitionscan be performed due to the spin-orbit interaction. The effective Hamiltonian in the | (cid:105) ≡ | , , + , F (cid:105) and | (cid:105) ≡ | , , − , F (cid:105) Floquet subspace is given to first-order in δL X ( Y ) ( t ) and for ω r ≈ ω Z as: (cid:98) H EFF ¯ h = (cid:0) ω r − ω Z + ∆ Z (cid:1) (cid:98) σ Z + δ + (cid:98) σ + δ − (cid:98) σ − (38)3 FIG. 5: Numerical simulation of the transition amplitudes, |(cid:104) , , + | T exp( − i R t d t (cid:48) e H ( t (cid:48) )) | , , −(cid:105)| , in a square quantum dot caused byeither (A) modulating the size of the quantum dot or (B) by applying a time-dependent electric field. In both cases, the Rabi frequency is first-order in the spin-orbit coupling strength. The parameters used in both simulations were L X = L Y = 70 nm [giving ¯ h π / (2 m ∗ L X ( Y ) ) =1 . meV], α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , ( ω r − ω Z ) / (2 π ) = − . kHz, and φ X = 0 . In (A), L X ( t ) = L X + δL X sin( ω r t ) with δL X = 5 nm, which gave an effective Rabi frequency of 10.6 MHz, which is in excellent agreement withthe calculated value given by Eq. (40), | δ ± | / (2 π ) = 10 . MHz. In (B), the applied electric field was (cid:126)E ( t ) = E X sin( ω r t ) b x with E X = 10 eV/m, which gave an effective Rabi frequency of 3.94 MHz, which is in excellent agreement with the calculated value given by Eq. (42), | δ EF ± | / (2 π ) = 3 . MHz. where ∆ Z = 32 m ∗ π ¯ h ∞ (cid:88) n =2 n (1 + ( − n )( n − (cid:32) ( ζ ( − θ )) η X ( n − − ( η X ) + ( ζ ( − θ )) η Y ( n − − ( η Y ) (cid:33) (39) δ ± = 4¯ h ∞ (cid:88) n =2 n (1 + ( − n )1 − n ζ ( − θ ) η X δL X exp( ± i φ X ) L X (cid:16) ( n − − η X (cid:17) − ζ ( − θ ) η Y δL Y exp( ± i φ Y ) L Y (cid:16) ( n − − η Y (cid:17) (40)where η X ( Y ) = 2 m ∗ ω Z L X ( Y ) / (¯ hπ ) . By choosing ω r = ω Z − ∆ Z , efficient spin transitions can be generated in square quan-tum dots [Fig. 5(A)] since the Rabi frequency, | δ ± | / (2 π ) , is directly proportional to the spin-orbit coupling strength, in contrast4to a parabolic quantum dot undergoing modulations of confining frequency and in the absence of a static electric field [Eq. (23)].Figure 5(A) presents an exact numerical simulation of the transition amplitude, |(cid:104) , , + | T exp( − i (cid:82) t d t (cid:48) (cid:101) H ( t (cid:48) ) / ¯ h ) | , , −(cid:105)| , inthe presence of modulating the effective size of a square quantum dot. The following parameters were used in the simulation: L X = L Y = 70 nm [giving ¯ h π / (2 m ∗ L X ( Y ) ) ≈ . meV and η X ( Y ) = 0 . ], δL X = 5 nm, α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , and ( ω r − ω Z ) / (2 π ) = − . kHz. Such parameters gave an effective Rabi frequency of 10.6MHz, which is in excellent agreement with the calculated value given by Eq. (40), | δ + | / (2 π ) = 10 . MHz.Alternatively, an electric field, (cid:126)E ( t ) = E X sin( ω r t + φ X ) (cid:98) x + E Y sin( ω r t + φ Y ) (cid:98) y , applied to the quantum dot can also inducesingle spin rotations by using EDSR effects. Incorporating the interaction with the electric field, (cid:98) V ( t ) = − e (cid:126)E ( t ) · (cid:126)r , an effectiveHamiltonian in the | (cid:105) and | (cid:105) subspace [when ω r ≈ ω Z ] can be written as: (cid:98) H EFF ¯ h = (cid:0) ω r − ω Z + ∆ Z (cid:1) (cid:98) σ Z + δ EF + (cid:98) σ + δ EF − (cid:98) σ − (41)where δ EF ± = 32 m ∗ π ¯ h ∞ (cid:88) n =2 n (1 + ( − n )( n − (cid:32) L X E X ζ ( − θ ) η X exp( ± i φ X )( n − − η X − L Y E Y ζ ( − θ ) η Y exp( ± i φ Y )( n − − η Y (cid:33) (42)Like the case of modulating the size of the quantum dot, the effective Rabi frequency, | δ EF ± | / (2 π ) , is again propor-tional to the spin-orbit coupling strength. Figure 5(B) presents an exact numerical simulation of the transition amplitude, |(cid:104) , , + | T exp( − i (cid:82) t d t (cid:48) (cid:101) H ( t (cid:48) ) / ¯ h ) | , , −(cid:105)| , in the presence of an electric field, (cid:126)E ( t ) = E X sin( ω r t + φ X ) (cid:98) x . The follow-ing parameters were used in the simulation: L X = L Y = 70 nm [giving ¯ h π / (2 m ∗ L X ( Y ) ) ≈ . meV and η X ( Y ) = 0 . ], E X = 10 eV/m, α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , and ( ω r − ω Z ) / (2 π ) = − . kHz. Suchparameters gave an effective Rabi frequency of 3.94 MHz, which is in excellent agreement with the calculated value given byEq. (42), | δ EF + | / (2 π ) = 3 . MHz. VI. PARAMETRIC ORBITAL AND SPIN EXCITATIONS FOR INCREASING SPIN POLARIZATION In addition to single spin manipulations in parabolic quantum dots, combined transitions between both the spin and the orbitaldegrees of freedom can be generated under parametric modulation of (cid:98) V ( (cid:98) X, (cid:98) Y ) . In a parabolic quantum dot, application of anelectric field [ (cid:101) H ( t ) in Eq. (25)] and/or modulation of ω X and ω Y [ (cid:101) H ( t ) in Eq. (20)] can directly induce both orbital and spinexcitations. Such transitions are an essential component for proposals to increase spin polarization in a quantum dot’s groundelectronic state . Consider first the case of parametrically modulating a parabolic quantum dot’s oscillator frequency, ω X , inthe absence of any static electric fields [ (cid:101) H ( t ) in in Eq. (20) with F X = F Y = 0 eV/m]. In the following,we will neglect thecubic Dresselhaus interaction [ (cid:98) H cub D in Eq. (30)] and will take ω Y > ω X in order that the parametric modulation can be chosento connect states involving changes in only one of the orbital degrees of freedom, such as | , (cid:105) going to | , (cid:105) . If we wish togenerate transitions between the states | , , + (cid:105) and | , , −(cid:105) , ω X must be modulated at frequency of ω r ≈ ω X − ω Z . Labelingthe relevant states as | (cid:105) = | , , − , F (cid:105) and | (cid:105) = | , , + , F (cid:105) , the effective Hamiltonian in the {| (cid:105) , | (cid:105)} Floquet subspaceunder parametric modulation of ω X (at ω r ≈ ω X − ω Z ) is given by: (cid:98) H EFF ¯ h = 12 (cid:0) ω Z − ω X + ω r + ∆ Z + δ Z (cid:1) (cid:98) σ Z + δ (cid:98) σ + δ − (cid:98) σ − (43) ∆ Z = m ∗ ω Z ¯ h (cid:32) ω X ( ζ ( − θ )) ω Z − ω X + ω Y ( ζ ( − θ )) ω Z − ω Y (cid:33) δ Z ≈ δω X ω X (cid:18) ( ω X − ω Z ) ω X − ( ω X − ω Z ) (cid:19) δ ± = (cid:114) mω X h ω Z δω X ζ ( − θ ) ω X − ω Z exp( ± i φ X ) (44)By choosing ω r = ω X − ω Z − ∆ Z − δ Z , efficient spin and orbital transitions between the states | (cid:105) and | (cid:105) can be generated[Fig. 6(A)] since the Rabi frequency, | δ ± | , is directly proportional to the spin-orbit coupling. Note that there can be a significantBloch-Siegert shift, δ Z , since the transition involves different electronic orbitals.5Figure 6(A) presents an exact numerical simulation of the transition amplitudes, |(cid:104) , , + | T exp( − i (cid:82) t d t (cid:48) (cid:101) H ( t (cid:48) ) / ¯ h ) | , , −(cid:105)| (black curve) and |(cid:104) , , + | T exp( − i (cid:82) t d t (cid:48) (cid:101) H ( t (cid:48) ) / ¯ h ) | , , + (cid:105)| (red curve), in the presence of the modulation ω X ( t ) = ω X + δω X sin( ω r t ) . The following parameters were used in the simulation: ¯ hω Y = 1 meV, ¯ hω X = 0 . meV, α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , F X = F Y = 0 eV/m, and δω X = ω X / . Such parameters gave an effective Rabifrequency 48.1 MHz, which is in excellent agreement with the calculated value given by Eq. (44), | δ | / (2 π ) = 48 . MHz,under “on-resonant” modulation [ ( ω r − ( ω X − ω Z )) / (2 π ) = 32 . MHz − δ Z / (2 π ) = 17 . MHz]. Note that the state | , , + (cid:105) essentially does not evolve into any of the other states [the red line in Fig. 6(A)] thus confirming that the electron must be spindown in order to be excited from the state | , (cid:105) . As mentioned earlier, such spin selective transitions can be used to increase thespin polarization of a quantum dot , which will be discussed later in this section.Since the relevant transitions involve both orbital and spin excitation, the required modulation frequency is quite large, ω r ≈ ω X − ω Z ≈ . GHz, which can be experimentally challenging. However, higher-order processes which utilize smallervalues of ω r can be found in order to induce efficient spin and orbital transitions. The simplest higher-order process to inducetransitions between the states | , , + (cid:105) and | , , −(cid:105) occurs when ω r ≈ ( ω X − ω Z ) / . In this case, the Floquet states | , , + , F (cid:105) and | , , − , F (cid:105) are degenerate. The effective Hamiltonian in this subspace is given by [for ω r ≈ ( ω X − ω Z ) / ]: (cid:98) H ω ¯ h = 12 (cid:0) ω Z − ω X + 2 ω r + ∆ Z + δ ω,Z (cid:1) (cid:98) σ Z + δ ω, + (cid:98) σ + δ ω, − (cid:98) σ − (45)where δ ω,Z ≈ δω X ω X (cid:18) ( ω X − ω Z ) ω X − ( ω X − ω Z ) (cid:19) δ ω, ± = ∓ i (cid:115) m ∗ ω X ¯ h (cid:0) ω X + 2 ω X ω Z + ω Z (cid:1) δω X ζ ( − θ )( ω X − ω Z ) ( ω X + ω Z )(3 ω X + ω Z ) exp ( ± i φ X ) (46)In this case, the effective Rabi frequency is again directly proportional to the spin-orbit coupling strength and is also proportionalto the square of the modulation strength, ( δω X ) .Figure 6(B) presents an exact numerical simulation of the transition amplitudes, |(cid:104) , , + | T exp( − i (cid:82) t d t (cid:48) (cid:101) H ( t (cid:48) ) / ¯ h ) | , , −(cid:105)| (black curve) and |(cid:104) , , + | T exp( − i (cid:82) t d t (cid:48) (cid:101) H ( t (cid:48) ) / ¯ h ) | , , + (cid:105)| (red curve), in the presence of modulating the oscillator fre-quency of the parabolic well at ω r ≈ ( ω X − ω Z ) / . The following parameters were used in the simulation: ¯ hω Y = 1 meV, ¯ hω X = 0 . meV, α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , δω X = ω X / , and ( ω r − / ω X − ω Z )) / (2 π ) = − δ ω,Z / (4 π ) + 16 . MHz = 14.42 MHz. The above parameters gave an effective Rabifrequency of 8.77 MHz, which is within of the calculated value given by Eq. (46), | δ | / (2 π ) ≈ . MHz. Note also thatthe state | , , + (cid:105) doesn’t evolve during the parametric modulation of ω X [red curve in Fig. 6(B)].In addition to modulating the confinement frequency, the spin-orbit coupling constants, α and β , can also be modulated inorder to produce combined spin and orbital excitations which are first-order in α and β . By modulating an electric field generatedby a surface gate above the quantum dot, the Rashba spin-orbit coupling parameter within the quantum dot can be controlled ;additionally, for quantum dots formed using semiconductor heterostructures, surface gates could also be used to change (cid:104) (cid:98) P Z (cid:105) thereby changing the linear Dresselhaus coupling constant, β = γ (cid:104) (cid:98) P Z (cid:105) . Such modulations of the spin-orbit interaction havebeen previously suggested as a way to perform both spin and orbital excitations in the absence of a magnetic field . Finally,it should be noted that application of an electric field can only couple the states | , , + (cid:105) and | , , −(cid:105) to second-order in thespin-orbit coupling for a parabolic quantum dot; therefore, the EDSR technique cannot generate such transitions as efficiently asmodulating ω X when the spin-orbit coupling is weak. A. Increasing spin polarization in the lowest electronic oribtal Throughout this work, relaxation has been neglected when calculating the various spin transitions generated under parametricmodulation of the quantum dot’s electrostatic potential. In the previous sections where only effective spin rotations within thelowest electron state were considered, neglect of both T and T relaxation seemed justified since the calculated Rabi frequencies( − MHz) were one to two orders of magnitude larger than the measured /T values in quantum dots (we assumethat /T is on the same order as /T ). However, relaxation of excited electronic states in quantum dots occurs on thenanosecond time scale , which is much faster than the calculated Rabi frequencies (order of 10 MHz) associated with combinedspin and electronic excitation shown in Fig. 6; therefore, relaxation effects must be considered when examining such transitions.The ability to coherently couple the states | , , −(cid:105) and | , , + (cid:105) , coupled with relaxation, can be used to help spin polarize theelectronic ground state of a quantum dot as depicted in Fig. 7(A). From Fig. 6, parametric modulation of the oscillator frequencycan connect the state | , , −(cid:105) to the state | , , + (cid:105) while leaving the state | , , + (cid:105) relatively uncoupled from any other state of6 FIG. 6: (Color online) Numerical simulation of the transition amplitudes, |(cid:104) , , −| T exp( − i R t d t (cid:48) e H ( t (cid:48) )) | , , + (cid:105)| (black curve) and |(cid:104) , , + | T exp( − i R t d t (cid:48) e H ( t (cid:48) )) | , , + (cid:105)| (red curve) caused by modulating the oscillator strength defining the parabolic quantum dot, when(A) ω r ≈ ω X − ω Z and (B) ω r ≈ ( ω X − ω Z ) / . The following parameters where used in the simulations: ¯ hω Y = 1 meV, ¯ hω X = 0 . meV, α = 4 × − eV-m, β = 0 eV-m, ¯ hω Z = 0 . meV, θ = 0 , δω X = ω X , and φ X = 0 . In (A), modulation of ω X at a frequencyof ω r / (2 π ) ≈ ( ω X − ω Z ) / (2 π ) = 36 . GHz can generate effective transitions between the states | , , −(cid:105) and | , , + (cid:105) , which gave aRabi frequency of 48.1 MHz, which is in excellent agreement with the calculated value given by Eq. (44), | δ | / (2 π ) = 48 . MHz. In (B),modulation of ω X at a frequency of ω r / (2 π ) ≈ ( ω X − ω Z ) / (4 π ) = 18 . GHz resulted in a Rabi frequency of 8.77 MHz, which is within of the calculated value given in Eq. (46), | δ ω, ± | / (2 π ) ≈ . MHz. Although the resulting Rabi frequency is smaller in Fig. 6(B) thanin Fig. 6(A), the required ω r is a factor of two smaller, which makes such modulations more experimentally feasible. the quantum dot. An electron in the excited state | , , + (cid:105) can quickly relax to the state | , , + (cid:105) due to the direct coupling ofthe electron to piezo-phonons , whereas the relaxation of the state | , , + (cid:105) to the state | , , −(cid:105) , which requires an effectivespin-phonon coupling, occurs at a much smaller rate. This differential relaxation can be used to generate spin polarization in thelowest electronic state in a manner similar to optical pumping and Overhauser techniques.In the following work, the dynamics under parametric modulation of the confining potential has been restricted to thefollowing subspace for simplicity: {| (cid:105) ≡ | , , + (cid:105) , | (cid:105) ≡ | , , −(cid:105) , | (cid:105) ≡ | , , + (cid:105)} , and the calculations were per-formed using different values of δ ± taken from the Fig. 6. The density matrix in this subspace can be written as ρ ( t ) = (cid:80) i =1 p ii ( t ) | i (cid:105)(cid:104) i | + p ( t ) | (cid:105)(cid:104) | + p ( t ) | (cid:105)(cid:104) | (where only coherence between the states | (cid:105) and | (cid:105) have been considered).Defining W ij to be the transition rate from state i to state j , and Γ to be the decoherence rate for the coherence between states7 | (cid:105) and | (cid:105) , the various coefficients in ρ ( t ) can be found by solving:dd t p p p p p = − ( W + W ) W W W − ( W + W ) W i δ − i δ − W W − ( W + W ) − i δ i δ − i δ − − i δ − − Γ − i δ i δ − Γ p p p p p (47)where the parametric modulation of the quantum dot is assumed not to affect the values of W ij and Γ in Eq. (47). In theabsence of any term connecting the states | (cid:105) and | (cid:105) , i.e., δ ± = 0 , the various transition rates, W ij , must satisfy the followingconditions in order to ensure that the equilibrium density matrix, ρ eq = (cid:80) i p eq ii | i (cid:105)(cid:104) i | , is a solution to Eq. (47): W ij = p eq jj p eq ii W ji (48)where p eq jj is the equilibrium population for state j , with p eq > p eq > p eq and (cid:80) i p eq ii = 1 .In the presence of nonzero δ ± , the effective spin polarization of the ground electronic state, P Z = p − p p + p , can increase fromits equilibrium value, P eq Z = p eq − p eq p eq + p eq . Such an increase occurs since W involves only orbital relaxation, which is in generalmuch faster than the transition rate W , which requires both orbital relaxation and a spin flip. Applying parametric modulationsto the quantum dot can therefore transfer population from state | (cid:105) to state | (cid:105) , which subsequently relaxes to state | (cid:105) , therebyincreasing the population difference between states | (cid:105) and | (cid:105) . This is shown in Fig. 7(A).The steady-state spin polarization under such parametric oscillations can be found by setting the left-hand side of Eq. (47) tozero and solving for p and p , which gives: P steady-state Z = W (cid:16) W + W + 2 | δ ± | Γ (cid:17) + ( W − W − W ) (cid:16) W + W + 2 | δ ± | Γ (cid:17) W (cid:16) W − W + 2 | δ ± | Γ (cid:17) + ( W + W + W ) (cid:16) W + W + 2 | δ ± | Γ (cid:17) (49)The transition rate, W , is mainly dominated by the coupling of the electron to piezo-phonons. From Eq. (4) of Ref. , thisrate is given as W ≈ . × s − (for ¯ hω X = 0 . meV and ¯ hω Y = 1 meV); using Eq. (7) of Ref. as a rough estimatefor W , one obtains W ≈ × − s − , which satisfies W (cid:29) W . Although no theory has been attempted in this workto calculate Γ , as long as ≥ W , the diagonal elements, p ii , will be nonnegative (as seen from numerically integratingEq. (47)). Although the value of Γ does not drastically affect P steady-state Z , it does affect the time scale in which P steady-state Z is reached. In the following we simply take, for illustrative purposes, Γ = 10 W . Finally, since the effective T for GaAsquantum dots is on the order of milliseconds to microseconds , a conservative value of T = 100 µ s was chosen, whichprovided the values of W and W from the condition that /T = W + W = W (1 + exp(¯ hω Z / ( k B T ))) [where thepopulations were assumed to be given by the Boltzmann distribution ]. For T = 2 K and ¯ hω Z = 0 . meV, W ≈ . × s − . Using the parameters from Fig. 6(A) ( | δ ± | / (2 π ) = 48 . MHz), P Z ( t ) was found by integrating Eq. (47) and is shown(solid curve) in Fig. 7(B), with P stead-state Z ≈ . ≈ . P eq Z , which is reached within . µ s. No oscillations are present in P Z ( t ) since Γ (cid:29) | δ ± | .For the case of modulating ω X at a frequency of ω r ≈ ( ω X − ω Z ) / [with | δ ω, ± | / (2 π ) ≈ . MHz as shown in Fig. 6(B)],it took roughly twenty times as long ( ≈ µ s) to reach P steady-state Z ≈ . , which is shown by the dashed curve in Fig. 7(B).It should be noted that P steady-state Z is approximately equal to the initial equilibrium population difference between the states | (cid:105) and | (cid:105) , p eq − p eq p eq + p eq ≈ . . Thus turning on the coupling between | (cid:105) and | (cid:105) allows the initial “orbital” polarization between thestates | (cid:105) and | (cid:105) to be transferred into spin polarization between the states | (cid:105) and | (cid:105) . This process is similar to the Overhausereffect, where, by coupling the electron and nuclear spins, the initial electron spin polarization can be converted into nuclearpolarization . Therefore, in order to increase P steady-state Z , experiments should be performed at low temperatures, large magneticfields, and with increased lateral confinement, i.e., large ω X . However, increasing ω X requires modulating the quantum dot ata higher ω r (which may be unfeasible experimentally) in addition to the fact that | δ ± | decreases with increasing ω X , therebyincreasing the time it takes to reach P steady-state Z .Finally, it should be noted that the above calculations assumed that the quantum dot was isolated and absolutely closed fromthe leads, i.e., no electrons could enter or exit from the dot. However, if electrons are able to tunnel in/out of the quantum dot,modulating the electrostatic potential could be used to selectively “kick” out spin down electrons. “Re-zeroing” the state of thequantum dot to spin up could be used as an initialization step for a possible quantum computation . Such a process wouldrequire the tunneling rate out of the dot for an electron in state | (cid:105) to be much faster than W . However, the evolution under8 FIG. 7: Method for increasing the spin polarization of the lowest electronic state, | , (cid:105) . In (A), parametric modulations are performed inorder to coherently connect the states | (cid:105) ≡ | , , −(cid:105) to the state | (cid:105) ≡ | , , + (cid:105) (which is denoted by δ ± ), thereby transferring populationfrom state | (cid:105) to state | (cid:105) . The population in state | (cid:105) predominately relaxes to the state | (cid:105) ≡ | , , + (cid:105) due to the electron directly couplingto piezo-phonons, thereby increasing the spin polarization of the ground electronic state, P Z = p − p p + p . (B) Integration of Eq. (47) gave P Z ( t ) for parametric modulations of ω X at the frequency ω r = ω X − ω Z (solid curve) and at ω r = ( ω X − ω Z ) / (dashed curve). Alongwith the parameters used in Fig. 6, the following parameters were used in solving Eq. (47): W = 3 . × s − , W = 5 × − s − , W = 3 . × s − , T = 2 K, Γ = 10 W , and | δ ± | / (2 π ) ≈ . MHz (solid curve) and | δ ± | / (2 π ) ≈ . MHz (dashed curve).In both cases, P steady-state Z ≈ . [Eq. (49)], although for larger | δ ± | , P steady-state Z was reached on a faster time scale [ . µ s (solid curve)compared to µ s (dashed curve)]. parametric modulations of the confining potential would have to be reconsidered, since in the case of a non-isolated quantumdot, there would be a finite probability during the parametric modulation that an electron in the state | (cid:105) would get excited andkicked out of the dot if the tunneling rate out of the dot is very large. VII. DISCUSSION AND CONCLUSIONS In this work, a general formalism combining Floquet theory with effective Hamiltonian theory was used to study the spin dy-namics under parametric modulations of a lateral quantum dot’s electrostatic potential in the presence of spin-orbit coupling. Inparabolic quantum dots, modulating the center of the quantum dot, i.e., performing an EDSR experiment, was found to generatelarger Rabi frequencies than parametrically modulating the confining frequency (in the absence of static electric fields), sincethe latter is second-order in spin-orbit coupling. However, in the presence of a static electric field, both methods gave similarRabi frequencies for a parabolic quantum dot. For square dots, both EDSR and modulating the width/length of the quantum dotgenerate Rabi frequencies which are first-order in spin-orbit coupling. The modulation frequency must be chosen with precisionon the order of the Rabi frequency (on the order of tens of MHz) in order to maximize the amplitude of the Rabi oscillations,thereby providing better spectroscopic precision of the quantum dot’s energy levels relative to transport measurements, wherethermal effects decrease the spectral resolution.Inclusion of the cubic Dresselhaus spin-orbit coupling didn’t dramatically alter the results obtained for parametric modula-tion in a parabolic quantum dot. However, the cubic Dresselhaus interaction tends to decrease the contribution of the linearDresselhaus interaction to the Rabi frequency. Furthermore, measurement of the Rabi frequency under, for example, an EDSR9experiment using different orientations of the in-plane magnetic field could be used to determine the ratio of the Rashba spin-orbit coupling constant to the difference of the linear Dresselhaus spin-orbit coupling constant with the product of the cubicDresselhaus coupling constant and the oscillator frequency ( ω X or ω Y ). For example, using Eq. (29) and including the cubicDresselhaus coupling from Eq. (32) gives: ZZ (cid:16) π (cid:17) = β − m ∗ ω Y γ h α (50)In order to separate γ from β in Eq. (50), experiments could be repeated for different values of ω Y , which would leave the linearDresselhaus’s contribution to ZZ ( π/ unchanged but would alter the cubic Dresselhaus’s contribution. Alternatively, it wasshown that applying a time-dependent electric field along the (cid:98) y direction coupled with parametric modulation of the confinementfrequency along the (cid:98) x direction could be used to generate Rabi frequencies which were proportional to γ and were independentof α and β to first-order. Measurement of the resulting Rabi frequency would provide another independent measure of the cubicDresselhaus spin-orbit coupling constant.Parametric modulations of the confining potential were also shown to generate combined spin and orbital excitations in aparabolic quantum dot with Rabi frequencies on the order of tens of MHz. However, since electronic relaxation times are on theorder of nanoseconds, the effects of orbital relaxation had to be taken into account. A combination of coherent spin and orbitalexcitation with orbital relaxation was shown to be able to increase the spin polarization of the ground electronic state [Fig. 7]by transferring the initial “electronic polarization” between the ground and excited electronic state into spin polarization of theground state. For the parameters chosen in this work, this corresponded to a three-fold increase in the spin polarization of theground electronic state. Larger spin polarizations could be achieved by either using a more confined quantum dot (larger ω X ),larger magnetic fields, or by going to lower temperatures.Finally, the spin control developed in this work used only parametric modulations of idealized electrostatic potentials(parabolic and square-box), which should only be considered as a model for spin control in lateral quantum dots. For suchmethods to be used in actual experimental quantum dots, more realistic electrostatic potentials for a quantum dot, i.e., non-parabolic and non-square-box potentials, are required. Furthermore, additional work is needed in order to better characterize theactual time-dependent electrostatic potentials generated by modulating the surface gate voltages of the quantum dot along withdesigning optimal configurations of surface gates in order to generate a desired transition. The results presented in this workcould furthermore be extended to many-electron quantum dots, where the effects of electron-electron coupling on perform-ing spin excitations should be examined. Recent theoretical work has demonstrated that electronically controlled, magneticdipole-like couplings between spins in different quantum dots can also be generated. 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B , 12639 (2000). C.W.J. Beenakker, Phys. Rev. B , 1646 (1991). M. Trif, V.N. Golovach, and D. Loss, Phys. Rev. B , 085307 (2007). APPENDIX A: THE EFFECTIVE HAMILTONIAN If for a given Hamiltonian (written for convenience in Floquet space), (cid:98) H F = (cid:98) H F + (cid:98) V F , there exists a subspace of interest, Q , which is weakly coupled by (cid:98) V F to the rest of the Floquet space, U , then the dynamics within Q can be separated from U byconstructing an effective Hamiltonian from (cid:98) H F , (cid:98) H EFF F , which is block-diagonal in the Q and U subspaces . Defining (cid:98) P Q tobe the projection operator onto the subspace Q , (cid:98) P Q = (cid:80) | α,m F (cid:105)∈ Q | α, m F (cid:105)(cid:104) α, m F | , and (cid:98) P U to be the complementary projectionoperator onto the subspace U , (cid:98) P U = (cid:98) F − (cid:98) P Q , (cid:98) H EFF F can be determined by constructing a unitary transformation, exp( (cid:98) S F ) , suchthat exp (cid:16) (cid:98) S F (cid:17) (cid:98) H F exp (cid:16) − (cid:98) S F (cid:17) = (cid:98) H EFF F , where in order to ensure that (cid:98) H EFF F is block diagonal in both the Q and U subspaces, (cid:98) H EFF F must satisfy the following: (cid:98) P Q (cid:98) H EFF F (cid:98) P U = (cid:98) P U (cid:98) H EFF F (cid:98) P Q = (cid:98) (A1)Using Eq. (A1), a perturbation expansion for (cid:98) S F in powers of (cid:98) V F , (cid:98) S F = (cid:80) ∞ m =1 (cid:98) S ( m ) F , can be constructed. Separating (cid:98) V F intoits diagonal and off-diagonal components, (cid:98) V F = (cid:98) V SF + (cid:98) V DF , with (cid:98) V SF = (cid:98) P Q (cid:98) V F (cid:98) P U + (cid:98) P U (cid:98) V F (cid:98) P Q and (cid:98) V DF = (cid:98) P U (cid:98) V F (cid:98) P U + (cid:98) P Q (cid:98) V F (cid:98) P Q , (cid:98) S F can be written up to m = 3 as: (cid:104) α, m F | (cid:98) S (1) F | β, n F (cid:105) = (cid:104) α, m F | (cid:98) V SF | β, n F (cid:105) E α − E β + ¯ h ( m − n ) ω r (cid:104) α, m F | (cid:98) S (2) F | β, n F (cid:105) = (cid:104) α, m F | [ (cid:98) S (1) F , (cid:98) V DF ] | β, n F (cid:105) E α − E β + ¯ h ( m − n ) ω r (cid:104) α, m F | (cid:98) S (3) F | β, n F (cid:105) = (cid:104) α, m F | [ (cid:98) S (2) F , (cid:98) V DF ] | β, n F (cid:105) E α − E β + ¯ h ( m − n ) ω r + (cid:104) α, m F | [ (cid:98) S (1) F , [ (cid:98) S (1) F , V SF ]] | β, n F (cid:105) E α − E β + ¯ h ( m − n ) ω r ) (A2)1Using the above values of (cid:98) S F in Eq. (A2), the effective Hamiltonian in the subspace Q , (cid:98) P Q (cid:98) H EFF F (cid:98) P Q = (cid:80) ∞ m =0 (cid:98) H EFF ( m ) F , is given(up to order ( (cid:98) V F ) ) as: (cid:104) α , n F | (cid:98) H EFF (0) F | α , m F (cid:105) = (cid:104) α , n F | (cid:98) H F | α , m F (cid:105)(cid:104) α , n F | (cid:98) H EFF (1) F | α , m F (cid:105) = (cid:104) α , n F | (cid:98) V DF | α , m F (cid:105)(cid:104) α , n F | (cid:98) H EFF (2) F | α , m F (cid:105) = 12 (cid:104) α , n F | [ (cid:98) S (1) F , (cid:98) V SF ] | α , m F (cid:105)(cid:104) α , n F | (cid:98) H EFF (3) F | α , m F (cid:105) = 12 (cid:104) α , n F | [ (cid:98) S (2) F , (cid:98) V SF ] | α , m F (cid:105)(cid:104) α , n F | (cid:98) H EFF (4) F | α , m F (cid:105) = 12 (cid:104) α , n F | [ (cid:98) S (3) F , (cid:98) V SF ] | α , m F (cid:105) − (cid:104) α , n F | [ (cid:98) S (1) F , [ (cid:98) S (1) F , [ (cid:98) S (1) F , (cid:98) V SF ]]] | α , m F (cid:105)(cid:105)