Electric-field control and noise protection of the flopping-mode spin qubit
Monica Benito, Xanthe Croot, Christoph Adelsberger, Stefan Putz, Xiao Mi, Jason R. Petta, Guido Burkard
EElectric-field control and noise protection of the flopping-mode spin qubit
M. Benito, X. Croot, C. Adelsberger, S. Putz, ∗ X. Mi, † J. R. Petta, and Guido Burkard Department of Physics, University of Konstanz, D-78457 Konstanz, Germany Department of Physics, Princeton University, Princeton, New Jersey 08544, USA (Dated: August 27, 2019)We propose and analyze a novel “flopping-mode” mechanism for electric dipole spin resonancebased on the delocalization of a single electron across a double quantum dot confinement potential.Delocalization of the charge maximizes the electronic dipole moment compared to the conventionalsingle dot spin resonance configuration. We present a theoretical investigation of the flopping-mode spin qubit properties through the crossover from the double to the single dot configurationby calculating effective spin Rabi frequencies and single-qubit gate fidelities. The flopping-moderegime optimizes the artificial spin-orbit effect generated by an external micromagnet and draws onthe existence of an externally controllable sweet spot, where the coupling of the qubit to charge noiseis highly suppressed. We further analyze the sweet spot behavior in the presence of a longitudinalmagnetic field gradient, which gives rise to a second order sweet spot with reduced sensitivity tocharge fluctuations.
I. INTRODUCTION
Control of individual electron spins is one of the cor-nerstones of spin-based quantum technology. Althoughstandard single-electron spin resonance has been demon-strated [1], there is a strong incentive to avoid the useof local oscillating magnetic fields since these are tech-nically demanding to generate at the nanoscale, hinderindividual addressability, and limit the Rabi frequencydue to sample heating issues. Electric dipole spin reso-nance (EDSR) techniques offer a more robust method toelectrically control the electron spin state. Traditionally,successful implementations have used spin-orbit coupling[2], hyperfine interaction [3] and g-factor modulation [4].The transition from GaAs to Si-based spin qubits hasled to dramatic advances in the field of spin-based quan-tum computing. Site-selective single-qubit control [5–7],two-qubit operations with high fidelity [8–13], electronshuttling [14], and strong coupling to microwave pho-tons [15, 16] have been demonstrated. Recent demon-strations of strong spin-photon coupling have used dou-ble quantum dot (DQD) structures where the charge ofone electron is delocalized between both dots (“flopping-mode”; Fig. 1(a)), thus enhancing the coupling strengthto the cavity electric field beyond the decoherence rate[17–19] and enabling the transfer of information betweenelectron-spin qubits and microwave photons [15, 16, 20].This suggests that the manipulation of electron spinswith classical electric fields will also be efficient in theflopping-mode configuration.The scalability of spin qubit processors hinges upon theuse of resources that permit fast control without a sig-nificant degradation in coherence times. The same prop-erties that make silicon based QDs extremely attractivefor quantum information processing make it challengingto use its intrinsic properties for electrical spin manipu-lation. Not only is the hyperfine interaction to nuclearspins largely reduced, but the intrinsic spin-orbit cou- (a) (b) ~B ≠ ≠ ≠ ≠
50 0 50 100 E ( µ e V ) Á ( µ eV) ≠ ≠ ≠ ≠
50 0 50 100 ‡ + · + ‡ + · + ‡ + · + ‡ ≠ · + E E E E zx B z ~E ac ( t ) Figure 1. (a) Schematic illustration of the flopping-modeEDSR mechanism, where the spin of an electron (shown asgreen circles) delocalized between two QDs is driven via anelectric field (purple line) in a magnetic field gradient (rep-resented with red arrows). (b) Energy levels E ,..., of theHamiltonian (17) as a function of the interdot detuning ε , cal-culated with t c = 20 µ eV, E z = 24 µ eV, gµ B b x = 15 µ eV, and gµ B b z = 4 µ eV. The asymmetry with respect to ε is due to thelongitudinal magnetic field gradient. Around zero detuning, | ε | (cid:28) t c , the electron delocalizes across the DQD, yieldinga larger electric dipole moment p compared to the single dotregime. The arrows represent the electrically addressable spin(solid line), charge (dashed line) and spin-charge (dotted line)transitions. pling for electrons in Si is very weak [21]. Recently, thisweak effect combined with the rich valley physics in Sihas been harnessed to achieve EDSR for single-electronspin qubits [22, 23] and singlet-triplet qubits [24, 25]. Amore flexible solution applicable to any semiconductor isthe mixing of orbital motion and spin via an externallyimposed magnetic field gradient [7, 26, 27]. Beyond thiseffective spin-orbit effect, the control over the magneticfield profile allows for selective addressing of spins placedin neighboring dots, since the resonance frequency variesspatially [6, 26, 28–32]. Here we investigate the effectof the micromagnet stray field on the coherence of the a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug flopping-mode spin qubit.In this work we envision the generation of single-electron spin rotations via a flopping-mode approach,which benefits from the electron delocalization betweentwo gate-defined tunnel coupled QDs [33], and track itsperformance as the electron is spatially localized in a sin-gle quantum dot (SQD). The electron tunneling in such adouble dot potential has a large electric dipole moment,which is partially transferred to the spin via the magneticfield gradient induced by the stray field of a micromag-net placed over the DQD, see Fig. 1(a). Moreover, dueto the spatial separation between the two QDs, obtaininga sizable magnetic field inhomogeneity, with the result-ing large effective spin orbit coupling, becomes relativelyeasy. A driving field on one of the gate electrodes thatshapes the QD modulates the potential and allows fullelectrical spin control via EDSR.The paper is organized as follows: In Sec. II we in-troduce the flopping-mode spin qubit and derive theRabi frequency and the relevant relaxation and dephas-ing rates under the effect of a transverse magnetic fieldgradient for the case of zero energy level detuning. InSec. III we take into account the effect of a general de-tuning and analyze the electrical control of the flopping-mode spin qubit as a function of externally controllableparameters. In Sec. IV we investigate the behavior of theflopping-mode spin qubit in the presence of a longitudinalmagnetic field gradient and how this affects the workingpoints with maximal single-qubit average gate fidelity. InSec. V we summarize our results and conclude. II. FLOPPING-MODE SPIN QUBIT
An electron trapped in a symmetric DQD, with zeroenergy level detuning (cid:15) = 0 between the left (L) andright (R) QDs will form bonding and antibonding chargestates, which are separated by an energy 2 t c , where t c isthe interdot tunnel coupling. The transition dipole mo-ment between the bonding and antibonding states, |∓i =( | R i ∓ | L i ) / √
2, is proportional to the electronic charge e and the distance between the two QDs d [15, 18, 19, 34],therefore an electric field with amplitude E ac at the po-sition of the DQD can drive transitions with Rabi fre-quency Ω c = edE ac / (cid:126) . Spins can be addressed via elec-tric fields by splitting the spin states via a homogeneousmagnetic field, B z , and inducing an inhomogeneous mag-netic field perpendicular to the spin quantization axis,i.e., transverse ( ± b x in the left/right QD). We model thespin and charge dynamics with the Hamiltonian H ε =00 = t c ˜ τ z + E z σ z − gµ B b x σ x ˜ τ x , (1)where ˜ τ α and ˜ σ α ( α = x, y, z ) are the Pauli matrices inthe charge ( |±i ) and spin subspace, respectively, E z is the Zeeman energy E z = gµ B B z , g is the electronic g-factor and µ B the Bohr magneton. The magnetic fieldgradient acts as an artificial spin-orbit interaction andhybridizes bonding and antibonding states with oppo-site spin direction via the two spin-orbit mixing angles φ ± = arctan [ gµ B b x / (2 t c ± E z )] ( φ ± ∈ [0 , π ]). As a con-sequence of this mixing, the electric dipole moment oper-ator acquires off-diagonal matrix elements in the eigenba-sis of Eq. (1) which involve spin-flip transitions [35, 36].In particular, given the four eigenenergies E ,..., , with2 E = − E = p (2 t c ± E z ) + ( gµ B b x ) , if τ de-notes the two-level-system with energy splitting E τ = E − E and σ the one with splitting E σ = E − E (seeFig. 1(b)), the electric dipole moment operator reads p = ed (cid:2) − cos ¯ φτ x + sin ¯ φσ x τ z (cid:3) , (2)where ¯ φ = ( φ + + φ − ) /
2, and τ ( σ ) α ( α = x, y, z ) arethe Pauli matrices in the corresponding τ ( σ ) subspace.This implies that the electric field can drive transitionsbetween the ground state and the first and second ex-cited states with Rabi frequency Ω σ = Ω c sin ¯ φ and Ω τ =Ω c cos ¯ φ , respectively; see the center part of Fig. 1(b),where we have defined 2 τ ± = τ x ± iτ y and 2 σ ± = σ x ± iσ y .For 2 t c < E z (2 t c > E z ), we define the spin qubit as s = τ ( s = σ ), i.e., as the ground state and the second(first) excited state, with Rabi frequency Ω s = Ω τ (Ω s =Ω σ ). If the transverse magnetic field is small, gµ B b x (cid:28)| t c − E z | , the expansion to first order yieldsΩ s = 2 t c gµ B b x Ω c / | t c − E z | + O ( b x ) (3)for both 2 t c < E z and 2 t c > E z . For a very small (or verylarge) tunnel splitting, 2 t c , the qubit is an almost purespin qubit and it is hardly addressable electrically, whilein the region 2 t c ≈ E z the spin-electric field coupling ismaximal [35] but the spin qubit coherence suffers to someextent from charge noise (see below).The spin or charge character of the qubit will be re-flected in the decoherence time. The spin-charge mix-ing mechanism also couples the spin to the phononsin the host material, therefore the relaxation rates viaphonon emission are γ ,σ = γ ,c sin ( ¯ φ ) and γ ,τ = γ ,c cos ( ¯ φ ) [37], respectively, where we have introduced γ ,c as the relaxation rate from the antibonding to thebonding state evaluated at the qubit energy. Since thespin qubit energy is essentially given by the Zeeman split-ting E z (weakly corrected by the spin-charge mixing), wecan safely assume a constant value for γ ,c , neglectingboth oscillations of the form cos ( qd ) (q is the phononquasimomentum) and polynomial dependences on thetransition frequency [38–42]. The expansion to the lowestorder in b x yields γ ,s = γ ,c (cid:2) t c gµ B b x / (4 t c − E z ) (cid:3) + O ( b x ) , (4)where we can evaluate γ ,c at the Zeeman splitting en-ergy E z . In the symmetric configuration ε = 0, puredephasing is strongly suppressed since the qubit is in asweet spot protected to some extent from charge fluctu-ations [43, 44]. Although the qubit energy splitting isfirst-order insensitive to electrical fluctuations in detun-ing ε , we account here for pure dephasing due to second-order coupling to charge fluctuations, which induces aGaussian decay of coherences ( ∝ e − ( γ (2) φ,σ ( τ ) t ) ) with rates γ (2) φ,σ = ( γ φ /E σ ) sin ¯ φ and γ (2) φ,τ = ( γ φ /E τ ) cos ¯ φ , where γ φ is the magnitude of the low-frequency detuning chargefluctuations (see Appendix A). The expansion to thelowest order in b x yields γ (2) φ,s = γ φ E z (cid:2) t c gµ B b x / (4 t c − E z ) (cid:3) + O ( b x ) . (5)Note that far from the resonant point 2 t c ≈ E z , otherdecoherence sources related to the spin, such as the hy-perfine interaction with nuclear spins, would start dom-inating the dephasing. The dephasing corresponding toquasistatic magnetic noise [45, 46] with magnitude γ M isalso quadratic, and the corresponding rates are γ M,σ = γ M (cos φ + +cos φ − ) / γ M,τ = γ M (cos φ + − cos φ − ) / b x , thespin qubit magnetic noise dephasing rate is γ M,s = γ M (cid:20) − ( gµ B b x ) (4 t c + E z )2(4 t c − E z ) (cid:21) + O ( b x ) . (6)In this architecture the electric field can induce spinrotations with Rabi frequency Ω s . We focus on the short-est single-qubit spin rotation ( X π gate), performed in thegate time t g = π/ Ω s . Using a master equation with qubitrelaxation and a noise term, we calculate the average gatefidelity (see Appendix C) and average this result over aGaussian distribution for the noise with standard devi-ation given by the total magnitude of the low-frequencynoise, Var( δ ) = 2 (cid:16) γ (2) φ,s + γ M,s (cid:17) . The optimal tunnelcoupling value to achieve the best single-qubit averagegate fidelity depends on the relation between the charge-induced dephasing and the magnetic noise (see Sec. III).Note that if the DQD is coupled to a microwave res-onator the spin qubit couples also to the confined elec-tric field and the Purcell effect opens another relaxationchannel via photon emission. Single-spin control wasdemonstrated in Ref. [15] in a detuned DQD configu-ration, where the spin-charge mixing, and therefore thecoupling of the spin to the electric field is much weaker.In the following we analyze the crossover from a symmet-ric (DQD) to a far detuned (SQD) configuration.
III. CROSSOVER FROM DQD TO SQD
In this section we calculate the spin Rabi frequencyand the single-qubit average gate fidelity for a generaldetuning ε and study the crossover from the molecular 00 . . − − −
15 0 15 30 45 Ω s / Ω c . . t c ( µ e V ) ε ( µ eV)48121620 − − −
15 0 15 30 45 0 . . . . . . − ¯ F (b) Ω = E z Figure 2. (a) Ratio between the spin Rabi frequency Ω s and the charge Rabi frequency Ω c as a function of detun-ing ε for t c = 15 µ eV. The spin Rabi frequency is max-imized for ε = 0. (b) Single-qubit average gate infidelityas a function of ε and t c . As expected, ¯ F is symmetricabout ε = 0, with the highest values achieved at ε = 0 andslightly away from the line with maximal spin-charge mixing,Ω = E z (black dashed line). The other parameters are cho-sen to be E z = 24 µ eV, gµ B b x = 2 µ eV, Ω c / π = 500 MHz, γ ,c / π = 18 MHz, γ φ / π = 600 MHz, and γ M / π = 2 MHz. or DQD regime ( ε = 0) to the SQD regime with theelectron strongly localized in the left or right QD ( | ε | (cid:29) t c ). An electron trapped in a detuned DQD, with energydetuning ε between the left and the right QDs, formscharge states separated by an energy Ω = p ε + 4 t c .The detuning reduces the off-diagonal matrix elements ofthe transition dipole moment operator in the eigenbasisresulting in a Rabi frequency Ω c = Ω c cos θ , where wehave introduced the orbital angle θ = arctan ( ε/ t c ), andincorporates diagonal matrix elements. With a magneticfield profile as explained above, the model Hamiltonianreads [35] H = Ω2 ˜ τ z + E z σ z − gµ B b x ˜ σ x θ ˜ τ x − sin θ ˜ τ z ) . (7)The eigenenergies, labelled as E ,..., read 2 E = − E = q (Ω ± b ) + ( gµ B b x cos θ ) , with b = p E z + ( gµ B b x sin θ ) , and all the off-diagonal matrixelements of the electric dipole moment operator in theeigenbasis are non-zero. Therefore all the transitions canbe addressed electrically, as shown in Fig. 1(b) via coloredarrows. The Rabi frequencies for the transitions involv-ing the lower energy states are (see Appendix A) Ω σ =Ω c cos Φ sin ¯ φ , and Ω τ = Ω c cos Φ cos ¯ φ , where the angleΦ = arctan ( b x sin θ/B z ) describes an orbital-dependentspin rotation, φ ± = arctan [ gµ B b x cos θ/ (Ω ± b )] ( φ ± ∈ [0 , π ]) generalize the spin-orbit mixing angles, and ¯ φ =( φ + + φ − ) / s = τ ( s = σ ) for Ω < E z (Ω > E z ), i.e., as theground state and the second (first) excited state. Asexpected, the spin qubit Rabi frequency is reduced as ε increases. The expansion of Ω s for small b x ( gµ B b x (cid:28)| Ω − E z | ) yieldsΩ s = 2 t c gµ B b x Ω c / | Ω − E z | + O ( b x ) , (8)generalizing Eq. (3) to ε = 0. In Fig. 2(a), we plot theratio Ω s / Ω c as a function of ε for tunnel coupling t c =15 µ eV and fixed magnetic field profile, E z = 24 µ eV and gµ B b x = 2 µ eV. As expected, for a given amplitude ofthe applied electric field the Rabi frequency is larger atzero detuning, which implies that at ε ≈ b x reads γ ,s = γ ,c (2 t c / Ω) (cid:2) t c gµ B b x / (Ω − E z ) (cid:3) + O ( b x ) . (9)In this detuned situation, the second excited state canalso decay to the first excited state via phonon emission,which opens another spin relaxation channel for the case E z > Ω (see Appendix A). However, the correspondingdecay rate is lower than γ ,c due to the smaller energygap between these two states and it can be neglectedfor the relevant parameters. Moreover the low-frequencycharge fluctuations (with magnitude γ φ ) induce pure de-phasing with rates proportional to the first derivative ofthe transition frequencies with respect to ε , γ (1) φ,τ ( σ ) = γ φ cos θ { tan θ (cos φ + ± cos φ − )+ sin Φ (sin φ + ∓ sin φ − ) } (10)(see Appendix A), which yields γ (1) φ,s = γ φ | ε | E z (cid:2) t c gµ B b x / (Ω − E z ) (cid:3) + O ( b x ) . (11)The second order contribution to spin dephasing is pro-portional to the second derivatives of the transition fre-quencies, as calculated from second order perturbationtheory [47–49]. The full expression for this spin contri-bution is given in Appendix A. Including terms to lowestorder in b x , we find γ (2) φ,s = γ φ E z (cid:20) t c gµ B b x (Ω − E z ) (cid:21) (cid:20) − ε Ω − E z (cid:21) + O ( b x ) . (12)Finally, the dephasing rates corresponding to quasistaticmagnetic noise are given in Appendix B and accountingfor terms to lowest order in b x , we find γ M,s = γ M p (cid:15) + 4 t c Ω (cid:20) − ( gµ B b x ε ) E z Ω − ( gµ B b x ) t c (Ω + E z )(2 t c + (cid:15) )(Ω − E z ) (cid:21) + O ( b x ) . (13) In Fig. 2(b), we show the single-qubit average gate fi-delity as a function of ε and t c , calculated by averagingthe X π average gate fidelity in the presence of Gaus-sian distributed noise with standard deviation given bythe total magnitude of the low-frequency noise, Var( δ ) =2 (cid:16) γ (1) φ,s + γ (2) φ,s + γ M,s (cid:17) . First, we can observe the op-timal values of t c mentioned in Sec. II and a reductionin the fidelity when Ω = E z (indicated by the dashedline) due to large spin-charge mixing. Moreover, we cansee the detrimental effect of working slightly away fromthe sweet spot ( ε = 0). The qubit not only suffers froma lower Rabi frequency but the first order charge noisecontribution dominates, abruptly decreasing the averagegate fidelity.As an estimate of the number of Rabi oscillations thatcan be observed with high visibility in a EDSR experi-ment we can use the quality factor Q , defined as the ratioof spin Rabi frequency and decay rates Q = 2Ω s γ ,s / q γ (1) φ,s + γ (2) φ,s + γ M,s . (14)This expression should be viewed as an approximate in-terpolation between the limiting cases where relaxationrate γ ,s or the low-frequency noise are dominating [50].Increasing the detuning localizes the electron more ina single QD and the flopping-mode EDSR mechanismdescribed above may compete with other EDSR mech-anisms that take place in a SQD, via excited orbital orvalley states [23, 27, 51–56]. Also in a DQD structure, ifthe intervalley interdot tunnel coupling [57–59] is strongcompared to the valley splittings [59], the effective spinRabi frequency will be modified. In this work we fo-cus on the micromagnet-induced flopping-mode EDSRmechanism, which dominates if the excited orbital andvalley energy splittings are large enough. For a discus-sion of the interplay between micromagnet-induced SQDand flopping-mode EDSR mechanisms we refer the readerto Appendix D.In more realistic setups, where the micromagnet strayfield is not perfectly aligned with the DQD, there can bemagnetic field gradients in the z direction (longitudinal)and a finite average field in the x direction (transverse).Given the importance of the protection against chargefluctuations, we investigate the sweet spot behavior usinga more general model in the following section. IV. FLOPPING-MODE CHARGE NOISESWEET SPOTS
In this section, we examine the optimal working pointsfor flopping-mode spin qubit EDSR operation. For themodel used in Sec. III, the zero detuning point consti-tutes a first order sweet spot with respect to fluctuationsin the detuning, since the qubit energy is insensitive to ε variations to first order. In this case, it is importantto account for the second order contribution to qubit de-phasing which, as mentioned above, is related to the sec-ond derivative of the qubit energy with respect to thedetuning. The micromagnet could be designed to inducea longitudinal magnetic field gradient between the leftand the right QDs with the aim of obtaining a differentspin resonance frequency depending on the electron posi-tion. Fabrication misalignments can also give rise to bothlongitudinal gradients and overall transverse magneticfields [16, 50, 60], i.e., the magnetic field components inthe right and left QD positions may be B ( L,R ) z = B z ± b z and B ( L,R ) x = B x ± b x , where B z (cid:29) B x , b x , b z . Viaa rotation of the spin quantization axis, given by thesmall angle ζ = arctan ( B x /B z ), it is always possibleto rewrite the latter as B ( L,R ) z = p B z + B x ± b z and B ( L,R ) x = ± b x , with b z = b z cos ζ + b x sin ζ , (15) b x = b x cos ζ − b z sin ζ , (16)therefore a model containing a homogeneous field andtwo gradients is sufficient. In the following we work ina rotated coordinate system and rename the variables as p B z + B x → B z , b x → b x and b z → b z . This allows usto use the model Hamiltonian in Eq. (7), with a homoge-neous field B z and a transverse inhomogeneous compo-nent b x , and add a term accounting for the longitudinalgradient ( ± b z in the left/right QD), H = H − gµ B b z ˜ σ z θ ˜ τ x − sin θ ˜ τ z ) . (17)Note that the relative values of b x and b z can becontrolled via the direction of the external magneticfield [60].For simplicity we analyze first this model in the limit ofsmall inhomogeneous fields, gµ B b x,z (cid:28) | Ω − E z | . Whilethe transverse gradient corrects the spin qubit energysplitting E s (from the value E s = E z for b x,z = 0) tosecond order, the longitudinal gradient has an effect tofirst order, leading to E s ’ E z − E z − ε E z (Ω − E z ) ( gµ B b x ) − ε Ω gµ B b z . (18)From this simplified expression, we can explore the exis-tence of first order sweet spots. Unless b z = 0, the spinqubit does not have a first order sweet spot at zero de-tuning. For an arbitrary value of t c , if b z < b x /B z thespin qubit should be operated at a first order sweet spotslightly shifted from zero detuning (see below). For alarger longitudinal gradient, b z > b x /B z , there are twofirst order sweet spots for a given value of tunnel split-ting below the Zeeman energy, i.e., 2 t c < E z . For largertunnel splitting, 2 E z > t c > E z , there are also two first 23.823.8423.880 10 20 30 402323.52424.50 10 20 30 40 ε ( µ eV) . µ eV0 . µ eV0 . µ eV E s ( µ e V ) ε ( µ eV) . µ eV1 . µ eV Figure 3. Spin qubit energy splitting E s as a function of thedetuning ε , for various values of the longitudinal gradient field( gµ B b z ), as indicated, increasing from top to bottom. Theinterdot tunnel splitting amounts to (a) 2 t c = 18 µ eV and (b)2 t c = 30 µ eV, while the homogeneous field Zeeman energy is E z = 24 µ eV and the transverse inhomogeneous componentis gµ B b x = 2 µ eV. The thin shaded areas indicate first ordersweet spots for the corresponding color line and the wide blueshaded area in (b) indicates the region around the secondorder sweet spot for gµ B b z = 0 . µ eV. The discontinuity in(a) occurs at Ω = E z due to a level crossing of the upperqubit state. order sweet spots if b x B z < b z < b z = 3 √ t c E z (4 t c − E z ) / b x B z (19)and none otherwise.In Fig. 3, the exact spin qubit energy splitting E s , cal-culated from the eigenenergies of the Hamiltonian (17), isshown as a function of the DQD detuning ε for differentvalues of b z . For negative values of b z the sweet spotswill occur at negative values of ε . The panels (a) and(b) represent a generic case with tunnel splitting belowand above the Zeeman energy, respectively. The black(solid) lines are for b z = 0 and the red (dashed) lines cor-respond to b z < b x /B z , showing therefore one first ordersweet spot in both panels (a) and (b). In Fig. 3(a), since2 t c < E z , we expect two first order sweet spots for largeenough values of longitudinal gradient, which can be seenin the green (dash-dotted) line. In Fig. 3(b), we analyzea case with 2 E z > t c > E z . The green (dash-dotted)line corresponds to the intermediate region of two firstorder sweet spots, b x /B z < b z < b z . Finally, the blue(dotted) line is obtained for b z ∼ b z . At this point, E s becomes very flat, which would protect the qubit even tohigher order from fluctuations in the detuning.To confirm this, we show in Fig. 4 the second deriva-tive of the spin qubit energy splitting with respect todetuning. In panel (a) b z < b x /B z , while in panel (b) b z > b x /B z . The superimposed black dashed line in-dicates the position of the first order sweet spots. InFig. 4(a), the value of the second derivative along theexpected first order sweet spot (black dashed line) does81216 − −
17 0 17 34 − −
17 0 17 34 t c ( µ e V ) ε ( µ eV)81216 − −
17 0 17 34 − − ∂ ε E s ( m e V − ) (a) ∂ ε E s = 0 b z < b x /B z ε ( µ eV) − −
17 0 17 34(b) ∂ ε E s = 0 b z > b x /B z Figure 4. Second derivative ∂ ε E s of the spin qubit energysplitting with respect to the detuning ε as a function of t c and ε for (a) gµ B b z = 0 . µ eV and (b) gµ B b z = 0 . µ eV. Theblack dashed lines indicate the first order sweet spot positionsand the circle in panel (b) indicates the position of the secondorder sweet spot. The homogeneous field Zeeman energy is E z = 24 µ eV and the transverse inhomogeneous component is gµ B b x = 2 µ eV. not change significantly. Increasing the value of b z cangive rise to a situation as shown in Fig. 4(b), where theline indicating the position of the first order sweet spot(black dashed line) crosses the line of zero second deriva-tive, allowing for a second order sweet spot and a qubitprotected against charge noise up to second order.The longitudinal magnetic field gradient may also in-fluence the electric dipole moment operator and there-fore the Rabi frequencies of the different transitions. InAppendix E we treat the transverse component b x per-turbatively and calculate the correction of the spin Rabifrequency due to the longitudinal magnetic field gradient,Ω s ’ Ω c t c gµ B b x | Ω − E z | (cid:20) εb z Ω B z (cid:21) , (20)i.e., b z (cid:28) B z incorporates a small correction. This meansthat b z does not have a noticeable effect on the spin Rabifrequency and the phonon induced spin dephasing rate,but it strongly affects the pure spin dephasing rate dueto charge fluctuations via a drastic modification of thequbit energy detuning dependence, as shown in Figs. 3and 4.To examine the overall performance of the qubit indifferent regimes, we show in Fig. 5 the single-qubit av-erage gate fidelity as a function of ε and t c . The chargenoise induced spin dephasing rate has been calculatednumerically from the derivatives of the spin qubit en-ergy splitting E s with respect to detuning ε . The effectof the small longitudinal gradient on the spin Rabi fre-quency, the phonon induced spin relaxation rate and themagnetic noise induced rate is very small, therefore wehave neglected it here. Since we have assumed that thepure dephasing rate induced by charge noise fluctuationsis the dominant source of decoherence, the condition for 81216 − −
17 0 17 34 − −
17 0 17 34 t c ( µ e V ) ε ( µ eV)81216 − −
17 0 17 34 0 . . . . . . − ¯ F (a) ∂ ε E s = 0 b z < b x /B z ε ( µ eV) − −
17 0 17 34(b) ∂ ε E s = 0 b z > b x /B z Figure 5. Single-qubit average gate infidelity 1 − ¯ F as afunction of detuning ε and interdot tunnel coupling t c for(a) gµ B b z = 0 . µ eV and (b) gµ B b z = 0 . µ eV. The homo-geneous field Zeeman energy is E z = 24 µ eV and the trans-verse inhomogeneous component is gµ B b x = 2 µ eV. The otherparameters are chosen to be Ω c / π = 500 MHz, γ ,c / π =18 MHz, γ φ / π = 600 MHz, and γ M / π = 2 MHz. The blackdashed lines indicate the first order sweet spot positions. Inpanel (b) the squares mark the position of the first order sweetspots for t c = 13 µ eV and the circle indicates the position ofthe second order sweet spot. the best quality qubit coincides with the position of thefirst order sweet spots, which, as opposed to the casewith b z = 0 shown in Fig. 2, does not occur at ε = 0.Although for a fixed tunnel coupling t c the two first or-der sweet spots exhibit high single-qubit average gatefidelity, their properties are very different. For example,for t c = 13 µ eV the spin Rabi frequency at the sweetspot at (cid:15) = 3 . µ eV is four times larger than at the oneat (cid:15) = 18 . µ eV (these two first-order sweet spots are in-dicated by squares in Fig. 5(b)), but the phonon-inducedrelaxation rate and the charge noise dephasing rates arealso 16 and 9 times higher, respectively. The first ordersweet spot situated at larger detuning could thereforeserve as idle point, while the one at lower detuning isused as operating point. Finally, as shown in Fig. 5(b),an even larger average gate fidelity can be achieved byoperating close to the second order sweet spot. Notethat the best fidelity does not correspond exactly to thesecond order sweet spot, since phonon relaxation and nu-clear spin induced dephasing are also present. V. CONCLUSIONS
The flopping-mode configuration is shown to be usefulnot only for achieving a strong coupling between cavityphotons and single spins [15, 16, 20], but also for coher-ent electrical spin manipulation. We have analyzed thevariation of the performance of the flopping-mode EDSRmethod from the symmetric ( ε = 0) DQD to the highlybiased ( | ε | (cid:29) t c ) SQD regime. Importantly, the appliedpower of the electric field necessary to obtain a givenRabi frequency will be reduced by orders of magnitude byworking in the DQD regime. This efficient single spin ma-nipulation implemented in silicon QDs would constitutea fundamental step towards a fully electrically control-lable quantum processing architecture for spin qubits, aplatform which already benefits from mature silicon pro-cessing technology.Given the presence of environmental charge noise intypical QD devices, it is important to know the positionof the exact first order sweet spot, which can be shifteda few µ eVs away from zero detuning in the presence ofa longitudinal magnetic field gradient. Interestingly, itis also possible to find two first order sweet spots forthe same value of tunnel coupling, with different Rabifrequency and decoherence rate, which could be poten-tially exploited for different steps of qubit manipulation.Finally, we predict the existence of second order sweetspots, where the qubit is insensitive to electrical fluctua-tions up to second order. Acknowledgments.—
This work has been supported bythe Army Research Office grant W911NF-15-1-0149 andthe DFG through SFB767. We would also like to ac-knowledge B. D’Anjou and M. Russ for helpful discus-sions.
Appendix A: Electric dipole moment and dephasing
In this Appendix we calculate the Rabi frequencies forthe different transitions in the flopping-mode spin qubit,the phonon-induced spin relaxation rates and the puredephasing rates due to low-frequency electrical fluctua-tions in the DQD detuning. In Eq. (2) we have expressedthe electric dipole moment operator in the eigenbasis ofEq. (1), which is the model Hamiltonian for ε = 0 and b z = 0. For detuned QDs ( ε = 0), we can write theelectric dipole moment in the eigenbasis of the Hamilto-nian in Eq. (7) and find that the electric field couples toall possible electronic transitions, as shown in Fig. 1(b),since the electric dipole moment operator has the form p = ed cos θ ( T + Z / T = − cos Φ cos ¯ φτ x + cos Φ sin ¯ φσ x τ z (A1)+ (sin Φ cos φ − + tan θ sin φ − ) ( σ + τ − + h.c. ) − (sin Φ cos φ + − tan θ sin φ + ) ( σ + τ + + h.c. ) , and the diagonal component Z = { tan θ (cos φ + + cos φ − )+ sin Φ (sin φ + − sin φ − ) } τ z + { tan θ (cos φ + − cos φ − )+ sin Φ (sin φ + + sin φ − ) } σ z . (A2)The first terms in the off-diagonal component determinethe Rabi frequencies Ω τ ( σ ) and the direct phonon relax- ation rates γ ,τ ( σ ) given in Sec. III. The term in the sec-ond line of Eq. (A1) corresponds to transitions betweenthe first and second excited states, and it opens a newchannel for spin relaxation in the case E z > Ω. We haveneglected this channel here because the correspondingphonon emission rate is suppressed by the small energygap between these two states for the relevant parameterregimes.The electrical fluctuations also couple to the systemvia the electric dipole moment. If the amplitude δ ε andfrequency of these fluctuations is small, we can calculatethe spin qubit dephasing rate by treating them withintime-independent perturbation theory [47–49], obtainingthe dephasing Hamiltonian H δ ε = X η = τ,σ (cid:18) ∂E η ∂ε δ ε + 12 ∂ E η ∂ε δ ε (cid:19) η z , (A3)where the first order contribution relates directly to thediagonal components in Eq. (A2), since ∂E τ ( σ ) ∂ε = cos θ { tan θ (cos φ + ± cos φ − )+ sin Φ (sin φ + ∓ sin φ − ) } . (A4)and all the terms of the off-diagonal component Eq. (A1)contribute to second order [49]. More precisely, the sec-ond derivatives read ∂ E τ ( σ ) ∂ε = cos θ (cid:26) cos Φ cos ¯ φE τ ( σ ) + (sin Φ cos φ + − tan θ sin φ + ) E τ + E σ ) ± (sin Φ cos φ − + tan θ sin φ − ) E τ − E σ ) ) . (A5)Assuming Gaussian distributed low frequency noiseleads to a Gaussian decay of coherence ∝ e − Γ φ t with thetotal pure spin dephasing rate related to the variance ofthe noise functionΓ φ = (cid:20) Var (cid:18) ∂E s ∂ε δ ε + 12 ∂ E s ∂ε δ ε (cid:19) / (cid:21) / = h γ (1) φ,s + γ (2) φ,s i / , (A6)where γ (1) φ,s = γ φ ∂ ε E s ∂ε , γ (2) φ,s = γ φ ∂ ε E s ∂ε , and γ φ = σ ε / √ σ ε is the standard deviation of the fluctuations δ ε . Appendix B: Quasistatic magnetic noise
In this Appendix we calculate the dephasing rate ofthe flopping-mode spin qubit due to hyperfine interac-tion with the nuclear spins. For this we use the qua-sistatic approximation [45], which assumes that the fluc-tuations in the Overhauser field occur in a time scalemuch longer than the system dynamics. Then we treatthe noise Hamiltonian term˜ V = ξ L ( t )˜ σ z (1 + ˜ τ z ) / ξ R ( t )˜ σ z (1 − ˜ τ z ) / , (B1)with two random variables for the noise in the left andright QDs, to first order in time-independent perturba-tion theory. First we transform Eq. (B1) into the eigen-basis of Eq. (7), obtaining the diagonal component Z = ξ + cos Φ4 { (cos φ + − cos φ − ) τ z + (cos φ + + cos φ − ) σ z } + ξ − cos Φ sin θ σ z τ z , (B2)where ξ ± = ξ L ( t ) ± ξ R ( t ).If we assume now Gaussian distributions with zeromean value and σ M = Var ( ξ R ( t )) = Var ( ξ L ( t )), thecoherences decay as ∝ e − ( γ M,σ ( τ ) t ) , with the dephasingrates due to nuclear spins γ M,σ ( τ ) = γ M cos Φ2 q (cos φ + ± cos φ − ) + 4 sin θ , (B3)where γ M = σ M , whose expansion to lowest order in b x yields Eq. (13). Appendix C: Single-qubit average gate fidelity
We determine the quality of the quantum gate, rep-resented by the operator E , via the average fidelity ¯ F = h ψ |E [ | ψ i i ] | ψ i , which compares the targeted pure state | ψ i and the obtained mixed state density matrix E [ | ψ i i ], av-eraged over all possible pure input states | ψ i i . In thiscase the real quantum gate is determined by the simpletwo-level system master equation˙ ρ = − i (cid:20) δ σ z , ρ (cid:21) + γ ,s σ − ρσ + − { σ + σ − , ρ } ] (C1)for the qubit density matrix ρ , where δ is the noise mag-nitude.We now calculate the entanglement fidelity F e for thegate applied to only one qubit of a two-qubit state pre-pared in a maximally entangled state, since this relatesto the average fidelity as ¯ F = (2 F e + 1) / F ( δ ) = 13 (cid:8) e − t g γ ,s (C2)+ e − t g γ ,s h cosh ( t g γ ,s ) − cosh (cid:16) t g q γ ,s − δ (cid:17)io . Finally, since we consider only low-frequency noise, themeasurable and interesting quantity is the average of thisfidelity over the randomly distributed noise variable δ . Appendix D: Low power EDSR
In this Appendix, we analyze the power necessary todrive Rabi oscillations at a given frequency by taking intoaccount both SQD and flopping-mode EDSR induced bythe micromagnet. Following Refs. [26, 27], we can com-plete Eq. (8) by including the SQD contribution to theRabi frequency,Ω s ≈ edE ac (cid:126) gµ B b x (cid:18) t c Ω | Ω − E z | + (cid:126) m ∗ e d E (cid:19) , (D1)where E orb is the orbital energy, E orb ≈ − m ∗ e is the electron effective mass. Since the drive power isproportional to the square of the electric field, P ∝ E ac ,the power necessary to drive the spin qubit at a givenRabi frequency follows [50] P ∝ Ω s (cid:20) ed (cid:126) gµ B b x (cid:18) t c Ω | Ω − E z | + (cid:126) m ∗ e d E (cid:19)(cid:21) − . (D2) Appendix E: Effect of b z on the spin Rabi frequency In this Appendix we investigate the effect of a lon-gitudinal magnetic field gradient on the flopping-modeRabi frequencies. Since b z is the difference in longitudi-nal magnetic field between the left and the right QDs, itcan be seen as a detuning parameter (similar to ε ) thatdepends on the spin, therefore its effect can be includedin the form of a spin-dependent orbital basis transforma-tion, | + , σ i = cos ( θ σ / | + , σ i − sin ( θ σ / |− , σ i , |− , σ i = sin ( θ σ / | + , σ i + cos ( θ σ / |− , σ i , (E1)with orbital angles θ ↑ ( ↓ ) = arctan [( ε ± gµ B b z ) / t c ] andorbital energies Ω ↑ ( ↓ ) = p ( ε ± gµ B b z ) + 4 t c , instead ofthe θ and Ω used in Sec. III. With this, we can treat b x perturbatively and find the spin Rabi frequencyΩ s ’ t c gµ B b x Ω c cos ¯ θ E z / [ E z − (Ω ↑ − Ω ↓ ) / ↑ + Ω ↓ ) / − E z , (E2)that generalizes the result in Eq. (8). Here, ¯ θ = ( θ ↑ + θ ↓ ) /
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