Single-spin manipulation in a double quantum dot in the field of a micromagnet
Stefano Chesi, Ying-Dan Wang, Jun Yoneda, Tomohiro Otsuka, Seigo Tarucha, Daniel Loss
SSingle-spin manipulation in a double quantum dot in the field of a micromagnet
Stefano Chesi,
1, 2, ∗ Ying-Dan Wang,
3, 2
Jun Yoneda,
2, 4
Tomohiro Otsuka,
2, 4
Seigo Tarucha,
2, 4, 5, 6 and Daniel Loss
2, 7 Beijing Computational Science Research Center, Beijing 100084, China Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan Quantum-Phase Electronics Center, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan Institute for Nano Quantum Information Electronics,University of Tokyo, Meguro, Tokyo 153-8505, Japan Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland (Dated: September 28, 2018)The manipulation of single spins in double quantum dots by making use of the exchange interactionand a highly inhomogeneous magnetic field was discussed in [W. A. Coish and D. Loss, Phys. Rev.B , 161302 (2007)]. However, such large inhomogeneity is difficult to achieve through the slantingfield of a micromagnet in current designs of lateral double dots. Therefore, we examine an analogousspin manipulation scheme directly applicable to realistic GaAs double dot setups. We estimatethat typical gate times, realized at the singlet-triplet anticrossing induced by the inhomogeneousmicromagnet field, can be a few nanoseconds. We discuss the optimization of initialization, read-out, and single-spin gates through suitable choices of detuning pulses and an improved geometry.We also examine the effect of nuclear dephasing and charge noise. The latter induces fluctuationsof both detuning and tunneling amplitude. Our results suggest that this scheme is a promisingapproach for the realization of fast single-spin operations. PACS numbers: 75.75.-c, 71.10.Ca, 75.70.Tj, 71.23.-k
I. INTRODUCTION
Single electron spins confined in quantum dots can con-stitute building blocks to realize quantum informationprocessing. The challenges of realizing accurate spin ma-nipulation and the need to achieve easier integration intoscalable architectures have stimulated a detailed study ofa wide variety of setups and decoherence mechanisms.
In particular, a general strategy to implement a singlequbit relies on relatively complex states of several elec-trons in multiple quantum dots, instead of the spin-1/2of single electrons. In this approach, it becomes easier torealize single-qubit gates through electric manipulation,at the expense of more cumbersome schemes for the two-qubit gates. A well-studied example is the singlet-triplet(ST) qubit, based on the spin states | ↑↓(cid:105) , | ↓↑(cid:105) of adouble dot, for which universal control and a long life-time exceeding 200 µ s were demonstrated. Protocolsfor the CNOT gate were proposed in Refs. 7 and 8 andrecently an entangling operation of a pair of ST qubitswas realized. For the more direct approach of relying on spin-1/2qubits, the two-qubit operations can be realized on a few-hundred ps time scale but to achieve selective spin ma-nipulation of individual dots has proved to be more chal-lenging. Electric-dipole spin resonance (EDSR) based onspin-orbit interactions was demonstrated with an oper-ation time ∼
100 ns in GaAs lateral dots and ∼
10 nsin InAs nanowire dots. Another promising route relieson the slanting field of a micromagnet, which has al-lowed coherent rotations with ∼
100 ns period and was integrated with the two-qubit exchange gate. Recently,thanks to a better electrical coupling and design of themicromagnet, (cid:38)
100 MHz high-fidelity Rabi oscillationswere achieved. However, strong motivations still existto explore alternative single-spin manipulation schemes,which could achieve a better performance.An early idea based on inhomogeneous magnetic fieldsmakes use of the spin states | ↑↑(cid:105) , | ↑↓(cid:105) of a double quan-tum dot and can be considered as a compromise be-tween the two strategies outlined above. In fact, the firstspin simply acts as an ancillary spin to realize the univer-sal control of the ‘target’ spin through exchange pulses.The two-qubit gates can be realized as usual through theexchange interaction between target spins, with directtunneling or long-range coupling elements. The spinmanipulation is achieved with pulsed electric control in-stead of oscillating fields, and ∼ We consider here a similar approach to control the | ↑↑(cid:105) , | ↑↓(cid:105) states through detuning pulses in a differentparameter regime. We specialize to GaAs lateral dots inthe slanting filed of a micromagnet but, unfortunately,we find that the conditions of Ref. 18 (with negligiblehybridization to | ↓↑(cid:105) ) are not satisfied in current setups.Therefore, we have explored an alternative limit which in-troduces strong hybridization with the | ↓↑(cid:105) spin configu-ration and still allows one to achieve ∼ a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec ing charge dephasing could be suppressed by a relativelylarge interdot tunneling amplitude. Such large tunnelingalso helps to achieve faster rotation times.Interestingly, the so-called S − T + qubit is basedon a quite related spin manipulation scheme. Landau-Zener interferometry through the S − T + anticrossing,induced by a gradient of the Overhauser field, hasbeen demonstrated. Recent experiments in silicondouble dots have explored the same type of Landau-Zener dynamics, but with the anticrossing induced bya micromagnet. As micromagnets can realize a deter-ministic slanting field with larger gradient than typicalnuclear fields, these advances suggest that single-spin ma-nipulation of the type discussed here offers interestingprospects for a scalable architecture.Finally, we note that the use of an additional quantumdot for single-spin manipulation should not be seen as asignificant overhead, since the auxiliary dot can be usedfor efficient readout and single-spin initialization (see,e.g., Sec. IV B 1).The paper is organized as follows: In Sec. II we intro-duce the model and define the logical states. In Sec. IIIwe discuss the spin manipulation scheme through an ef-fective two-level system, which clarifies the analytic de-pendence on system parameters. In Sec. IV we collectour numerical simulations. In particular, Sec. IV A ison the micromagnet slanting field in a recent experimen-tal geometry, and a simple variation more suitable toour purposes; Sec. IV B is on the unitary dynamics forthe double dot initialization, readout, and spin manipu-lation; Sec. IV C is on the effect of the nuclear and chargenoise, which within our approximations can be describedwith simple analytic expressions. Section V contains ourfinal remarks. Some technical details are presented in theAppendices.
II. MODEL AND EIGENSTATES
We describe the double dot with the same effectiveHamiltonian of Ref. 18. The parameters entering thismodel can be derived from a more microscopic descrip-tion following Ref. 26 (see also Ref. 27). H is given by: H = H C + H T + H Z . (1)where H C takes into account the electrostaticenergies: H C = − (cid:88) l,σ V l n lσ + U c (cid:88) l n l ↑ n l ↓ + U (cid:48) c n n . (2) l = 1 , σ = ↑ , ↓ the two spin di-rections. The operator n lσ = d † lσ d lσ describes the occu-pation with spin σ of the l -th dot lowest orbital state, and n l = n l ↑ + n l ↓ gives the total occupation of the l -th dot.The charge configurations are indicated as ( n , n ) andwe restrict ourselves to a region in the stability diagramwhere only (1,1) and (0,2) are of interest. The first term in Eq. (2) is the local electrostatic potential; the secondterm is the on-site repulsion, which is zero for the (1 , ,
2) configuration. Asa result, the (1 ,
1) configuration has electrostatic energy E (1 , = − V − V + U (cid:48) c and the (0 ,
2) configuration has E (0 , = − V + U c . As usual, we introduce the detuningparameter ε = E (1 , − E (0 , = V − V + U (cid:48) c − U c , whichcan be controlled through V l . If V + V is held constantand we set E (1 , = 0, then E (0 , = − ε . H T is the tunneling Hamiltonian between the two dotswhich is assumed to be spin-conserving: H T = t (cid:88) σ (cid:16) d † σ d σ + d † σ d σ (cid:17) , (3)while H Z is the Zeeman energy: H Z = − | g | µ B (cid:88) l =1 , S l · b l , (4)with S l = (cid:80) ρ,ρ (cid:48) σ ρ,ρ (cid:48) d † lρ d lρ (cid:48) the spin operator for the l -th dot ( σ are Pauli matrices) and b l the two localmagnetic fields. b (cid:54) = b , due to the presence of theslanting field of a micromagnet (see Sec. IV A). In Eq. (4)we use a negative g -factor, appropriate for electrons inGaAs where g = − . H terms arising from the spin or-bit interaction because it is rather weak in GaAs. In fact,our analysis will yield an energy scale ∆ E for spin manip-ulation of the order of a few µ eV (e.g., ∆ E (cid:39) µ eV withparameters as in Fig. 10). On the other hand, the rele-vant matrix elements of the spin-orbit interaction [mod-ifying Eq. (9)] were discussed in detail in Ref. 27. Theyare (cid:46)
100 neV, thus we expect their effect to be small.Nuclear fluctuations are characterized by a similar energyscale and indeed we will find in Sec. IV C 1 that theireffect can be made much smaller than the perturbationdue to the micromagnet. A. Local spin basis
To discuss the properties of H , it is useful to introducethe local spin basis which diagonalizes H C + H Z . Witha suitable choice of coordinate axes in the spin space,we can always assume b along z , while b lies in the x − z plane. The angle θ between the two magnetic fieldssatisfies: cos θ = b · b b b . (5)We then define the following operators, where the tildesign indicates the rotated spin quantization axes: (cid:18) ˜ d ˜ d − (cid:19) = (cid:18) cos( θ/
2) sin( θ/ − sin( θ/
2) cos( θ/ (cid:19) (cid:18) d ↑ d ↓ (cid:19) , (6) E n e r gy ( μ e V ) E n e r gy ( μ e V ) -10-14-18-20Detuning ɛ ( μ eV)-100 -50 0 50 100 Detuning ɛ ( μ eV)0 5 10 15 20 ɛ B FIG. 1. Spectrum of the double quantum dot as function ofdetuning ε . The solid lines are computed from the full Hamil-tonian Eq. (9). The dashed circle in the left panel highlightsthe anticrossing region around ε B , shown in more detail inthe right panel. The dashed curves are obtained from theeffective two-level system of Eq. (13) and describe well theanticrossing region. We used t = 5 µ eV and b , obtainedfrom the geometry of Fig. 3(b) with B = 0 . ϕ = 0. while ˜ d = d ↑ , ˜ d − = d ↓ . It is then natural to dis-cuss the Hamiltonian in the subspace generated by thefollowing basis (with µ, ν = ± ): | ˜ ψ µ,ν (cid:105) = ˜ d † µ ˜ d † ν | (cid:105) , (7) | ˜ S (0 , (cid:105) = ˜ d † ˜ d † − | (cid:105) . (8)where | (cid:105) is the (0 ,
0) charging state. The matrix repre-sentation of H is immediately obtained: H = | g | µ B b t sin θ −| g | µ B ∆ b t cos θ | g | µ B ∆ b − t cos θ −| g | µ B b t sin θ t sin θ t cos θ − t cos θ t sin θ − ε , (9)where b = ( b + b ) /
2, ∆ b = b − b . H can be diagonal-ized easily and an example of the numerical eigenvaluesas function of the detuning ε is shown in Fig. 1 for suit-able parameters. B. Logical states
The regime of large negative detuning ε (cid:28) − t , is es-pecially simple since tunneling becomes a small pertur-bation. More precisely, when | gµ B ∆ b | (cid:29) t / | ε | , (10)the effect of H Z dominates over tunneling and Eqs. (7)and (8) become a good approximation of the eigenstates.Similarly to Ref. 18, by working at detuning ε A deep intothis region, we identify the logical states |±(cid:105) (which wedefine as eigenstates of the logical σ z operator) with thetwo lowest eigenstates of the double dot. We have: |±(cid:105) (cid:39) | ψ + , ± (cid:105) , (11) where we supposed for definiteness b > b . In the ideallimit ε A → −∞ , |±(cid:105) are products of the spin states onthe two dots and the logical states coincide with the stateof the second spin. An operation in the logical subspaceamounts to a single-spin rotation where the electron inthe first dot acts as a frozen ancillary spin. As the de-tuning cannot be made arbitrary large, small correctionsexist to the factorized form in Eq. (11). These are dis-cussed in Appendix A. III. SPIN MANIPULATION
In this section our main result is discussed, i.e., wedescribe how to realize universal operations in the logicalsubspace and obtain their typical timescales. For a moredetailed analysis, including decoherence mechanisms, werefer to the following Sec. IV.For universal control of |±(cid:105) , two rotations with inde-pendent axes are necessary. The Zeeman splitting at ε A provides a natural way to implement z -rotations. By achange of detuning away from ε A (such that the systemevolves adiabatically in the subspace of the lowest twoeigenstates, but the energy gap between them is modi-fied) a controllable phase shift with respect to the evo-lution at ε A can be generated. On the other hand, the ε B anticrossing point is of special interest, as it allowsone to implement rotations about an axis independentof ˆ z . This region, around detuning ε B , is highlighted inFig. 1. To characterize the relevant splitting ∆ E at ε B ,which determines the rotation timescale, we develop asimple analytical treatment of the eigenstates based onfirst-order perturbation theory, which is appropriate forthe parameter regime of current experiments. A. Effective Hamiltonian
Our perturbation scheme is based on the fact thatin most realizations ∆ b (cid:28) b . In fact, setups involvinga micromagnet to generate an inhomogeneous slantingfield also apply a uniform magnetic field B largerthan the saturation field of the micromagnet ( B (cid:38) . ). On the other hand, a field gradient of order ∼ µ m allows one to achieve differences of at mostfew hundreds mT for typical quantum dot separations.In practice, the applied B is a few Tesla and the val-ues of ∆ b realized so far are in the range of 10 − which justifies considering ∆ b (cid:28) b (cid:39) B . Forthe same reason, the transverse component (with respectto B ) of the difference in local fields typically satisfies∆ b ⊥ (cid:28) b (cid:39) B , where:∆ b ⊥ = | ( b − b ) × B /B | . (12)Thus, θ (cid:39) ∆ b ⊥ /b (cid:28)
1. With the spin coordinates ofSec. II A we have in this regime that B is approximatelyalong ˆ z and ∆ b ⊥ (cid:39) b ,x . τ W ɛ A τ R τ G τ R ɛ B ɛ I = = τ I τ FIG. 2. Detuning pulses to initialize the system from ε I intothe logical subspace at ε A (first blue pulse), to execute single-spin rotations using the anticrossing at ε B (second red pulse),and to read-out the logical states (third green pulse). As discussed in more detail in Appendix B, the unper-turbed Hamiltonian H is simply obtained from Eq. (9)by setting to zero the diagonal elements proportionalto ∆ b , as well as the off-diagonal terms t sin( θ/ H can be easily diagonalized in terms of | ˜ T , ± (cid:105) , | ˜ S ± (cid:105) eigen-states, see Appendix B. Around ε B we rewrite H in the | ˜ T + (cid:105) , | ˜ S − (cid:105) subspace, which gives an effective two-levelsystem described by: H eff = −| g | µ B b − (cid:113) ∆+ ε t sin( θ/ − (cid:113) ∆+ ε t sin( θ/ − ε/ − ∆ / , (13)with ∆ = (cid:112) ε + 8 t cos ( θ/ . (14)The detuning at anticrossing is easily obtained: ε B = | g | µ B b − t | g | µ B b cos ( θ/ . (15)The eigenstates at ε B are |±(cid:105) B = (cid:16) | ˜ T + (cid:105) ± | ˜ S − (cid:105) (cid:17) / √ E = 2 t sin( θ/ (cid:114) (cid:16) t cos( θ/ gµ B b (cid:17) . (16)As discussed below, ∆ E is the main parameter whichdeterimins the spin manipulation time. B. Spin rotations
We now consider the detuning pulse for spin rotationsillustrated in Fig. 2 (in red). The first step is a change indetuning from ε A to ε B , with ramp-time τ R . Ideally, wewould like the evolution to be adiabatic in the lower twoenergy branches, except in the vicinity of ε B . Here, dueto the small energy scale ∆ E , a diabatic transformationcan be realized. As a consequence, after a time τ R : U R |±(cid:105) (cid:39) e ± iφ/ | + (cid:105) B ± |−(cid:105) B √ , (17) where φ is a phase which depends on the detailed formof the ramp. The U R |±(cid:105) states evolve for a time τ W under H eff and it is easy to show that the total effect ofthe pulse, U † R e − iH eff τ W / (cid:126) U R , is a rotation about ˆ x cos φ +ˆ y sin φ by an angle ∆ Eτ W / (cid:126) . In particular, a π rotationis realized when: τ W = τ π = (cid:126) π ∆ E , (18)which gives τ π ∼
10 ns with b = 1 T, t = 5 µ eV, and | gµ B ∆ b | = 1 µ eV (we also assumed ∆ b ⊥ ∼ ∆ b giving θ (cid:39) ∆ b ⊥ /b (cid:39) . E has a signif-icant dependence on t . From Eq. (16) we have:∆ E (cid:39) (cid:26) t sin( θ/
2) for t (cid:28) | g | µ B b √ | g | µ B b tan( θ/
2) for t (cid:29) | g | µ B b , (19)which shows that the splitting increases linearly at small t , until it saturates when the tunneling and Zeeman split-ting become comparable. The saturation value of ∆ E gives a lower bound on the operation time. By takinginto account the approximate value of θ (cid:39) ∆ b ⊥ /b : τ min π (cid:39) √ (cid:126) π | g | µ B ∆ b ⊥ . (20)Therefore, the limiting factor for the fastest operationtime is given by ∆ b ⊥ in this approximation. Using | gµ B ∆ b ⊥ | ∼ µ eV gives τ min π ∼ b ⊥ , giving τ min π (cid:46) t there is typically a clear separation of energy scales∆ E (cid:28) | g | µ B b (cid:46) t around ε B , since ∆ E ∼ | g | µ B ∆ b ⊥ .Therefore, the pure Hamiltonian dynamics can realizethe operation of Eq. (17) and the associated π -rotationaccurately. A quantitative characterization of the fidelityfor this rotation gate will be discussed later in Sec. IV Bthrough numerical simulation. IV. NUMERICAL STUDIES
We now apply the spin manipulation scheme discussedabove to a specific setup, which we study by numericalmeans. In particular, we consider the micromagnet anddouble dot geometries shown in Fig. 3. The setup ofFig. 3(a) is very close to the one of Ref. 17, speciallydesigned to optimize the performance of ESR rotationsusing the micromagnet stray field.
We consider sim-ple variations of the geometry and time-dependence ofthe detuning pulses to represent the typical performanceof the spin manipulation scheme. Substantial improve-ment could be realized by carrying out a more systematicoptimization. (a) (b) O xy z FIG. 3. Geometries used in the simulations. The micro-magnet extends vertically from z = − . µ m to z = 0, andis symmetric with respect to the x = 3 µ m plane. Panel(a) indicates relevant dimensions of the micromagnet (in µ m). The coordinates of the two quantum dots are (a) x = (3 . , . , − . µ m, x = (3 . , . , − . µ m, and (b) x = (3 , . , − . µ m, x = (3 . , . , − . µ m. We also in-dicate the angle ϕ defining the directions of B and m , seeEq. (22). A. Slanting field of the micromagnet
We consider here magnetostatic simulations of thestray field b m ( x ) produced by the micromagnet of Fig. 3,from which the local fields b , are extracted as follows: b , = B + b m ( x , ) , (21)where x , are the centers of the two quantum dots, seeFig. 3. The values of b , allow us to estimate throughEq. (18) the typical timescale for spin manipulation (seefurther below). We will also use the obtained values of b , for our simulations of the spin dynamics in the fol-lowing subsections.To obtain b m ( x ), we assume uniform magnetizationappropriate for Cobalt, µ | m | = 1 . µ the vac-uum permeability). The magnetization direction is deter-mined by the external field B and we simply assume that m is parallel to B . These approximations are justified ifthe magnetic field is sufficiently strong, such that themicromagnet is fully magnetized and shape anisotropyeffects can be neglected. For simplicity, we restrict our-selves to a magnetic field in the two-dimensional plane ofthe lateral quantum dots, where ϕ is the angle with the x -axes (with coordinates as in Fig. 3), thus: m = | m | (ˆ x cos ϕ + ˆ y sin ϕ ) . (22)As the original design of Fig. 3(a) was intended for ESRmanipulation based on a resonant electric drive displac-ing the dots along x , the field derivatives in this direc-tion are rather large (we obtain ∂b m ,z /∂x (cid:39) . / nmat the position of dot 1). To take better advantage ofthe micromagnet geometry, we consider a simple varia-tion in which the two quantum dots are aligned in the x -direction instead of along y , see Fig. 3(b). In this con-figuration faster spin rotations can be implemented withthe alternative scheme discussed here.In particular, the angle θ of Eq. (5) is an importantparameter since the anticrossing gap is ∆ E ∼ tθ at small Π Π Π Π (cid:106) Θ Π Π Π Π(cid:45) (cid:106) (cid:68) b (cid:72) m T (cid:76) FIG. 4. Difference in direction θ , see Eq. (5), and magnitudes,∆ b = b − b , between the local magnetic fields at the twoquantum dot locations, as functions of the direction of B .Solid (dashed) curves are for the setup of Fig. 3(a) (Fig. 3(b)).Each family of curves (solid/dashed in the left/right panel)are for B = 0 . , , . , B = 0 . , , . , tunneling, see Eq. (19). The value of θ is plotted in theleft panel of Fig. 4 as a function of the magnetizationdirection and with several values of the external mag-netic field B . As expected, the largest values of θ areobtained at the smaller external field B = 0 . ϕ , with anoptimal value around ϕ (cid:39) π/ ϕ (cid:39) θ obtained from setup (b) are significantly largerthan those from setup (a).The advantage of setup (b) in realizing faster rotationscan also be seen by computing τ π with θ, b obtained inthe simulation (we use b (cid:39) B = 0 . ϕ = 5 π/ τ π (cid:39) t = 10 µ eV and τ min π (cid:39) . t (cid:29) µ eV [see Eqs. (18) and (20),respectively]. Therefore, the original design (a) alreadyyields relatively fast timescales for single-spin manipula-tion. By considering setup (b) with ϕ = 0 and t = 10 µ eVwe obtain an improved operation time of τ π (cid:39) b which, as discussed in Appendix A,helps to disentangle the two spins at large negative de-tuning, thus to realize more faithful single-spin rotations.The values of ∆ b are plotted in the right panel of Fig. 4for the two geometries. Similarly to θ , ∆ b has a strongvariation with ϕ . The dependence on B is much less pro-nounced as it is mainly determined by the fixed differencein the longitudinal components (i.e., parallel to B ) of themicromagnet slanting fields at the two dot locations. B. Unitary dynamics
We now consider the unitary dynamics under the de-tuning pulses illustrated in Fig. 2. The effect of the anti-crossing at ε B on initialization and readout is discussed. Τ I (cid:72) ns (cid:76) P (cid:45) (cid:72) Τ I (cid:76) FIG. 5. Plot of the initialization fidelity P − ( τ I ) = |(cid:104)−| ψ ( τ I ) (cid:105)| for a linear ramp in detuning ε ( τ ) starting at ε I = − ε A = 200 µ eV (with | ψ (0) (cid:105) the ground state) andending at ε ( τ I ) = ε A . We used the geometry of Fig. 3(b)with ϕ = 0. The thin solid curves are for t = 10 µ eVwhere the lower green (upper orange) curve is for B = 0 . B = 2 T). The thick dashed red curve is the approximate for-mula (cid:104) − exp (cid:16) − πt τ I (cid:126) ε A (cid:17)(cid:105) exp (cid:16) − π ∆ E τ I (cid:126) ε A (cid:17) , obtained from theLandau-Zener probabilities, for t = 10 µ eV and B = 0 . t = 20 µ eV and B = 1 T. We also confirm that fast single spin manipulation withgate time ∼
1. Initialization
We first discuss an initialization procedure into the |−(cid:105) logical state by a detuning sweep starting from a largepositive ε I (first pulse in Fig. 2). This method is usu-ally more efficient than the initialization at ε A into | + (cid:105) (which is the ground state), based on relaxation: due tothe larger gap at positive detuning, the | S (0 , (cid:105) groundstate can be prepared faster and with higher fidelity. Ananalogous procedure, with a detuning pulse from ε A tolarge positive values, allows one to read-out the |±(cid:105) statesvia charge sensing.Starting from the ground state at ε I , it is straight-forward to evaluate numerically the time-evolution of | ψ ( τ ) (cid:105) . The probabilities of the logical states at time τ I are given by: P ± ( τ I ) = |(cid:104)±| ψ ( τ I ) (cid:105)| . (23)The fidelity P − ( τ I ) is plotted in Fig. 5 as function ofthe initialization time τ I (we have used a linear rampin detuning as illustrated in Fig. 2). Due to the an-ticrossing induced by tunneling around ε = 0, a suf-ficiently long τ I is necessary to guarantee adiabaticityin the two lowest energy branches. This time scale isgiven by τ I (cid:38) (cid:126) | ε A | /t , as seen by the comparison to theLandau-Zener probability in Fig. 5. However, a decrease of fidelity is obtained at large τ I which can be attributedto the presence of the anticrossing point at ε B . In fact,the requirement of a fully diabatic evolution at the ε B anticrossing is violated at small sweeping rate (large τ I ).To improve the maximum fidelity, it is necessary to have t (cid:29) ∆ E , and a possible strategy shown in Fig. 5 is toincrease the external magnetic field B , since this leads toa suppression of ∆ E .If, on the other hand, we want to improve the initial-ization fidelity by retaining the same value of the ∆ E (which determines the π -rotation time, as discussed inSec. III B), an alternative strategy is to increase simulta-neously t and B , as exemplified in Fig. 5. It is seen that,by doubling both B and t , ∆ E and the long- τ I decayof the two curves are left essentially unchanged. On theother hand, the curve with larger t shows a marked im-provement at shorter τ I and allows one to achieve a fasterinitialization with a higher maximum fidelity. Furtherimprovement could be achieved by using an optimizedpulse shape instead of a simple linear ramp. For example,if the two anticrossing regions | ε | (cid:46) t and | ε − ε B | (cid:46) ∆ E are well separated, a pulse with a different rate dε/dτ ineach region could be helpful to improve the fidelity.Finally, it is worth noting that, as far as the unitaryevolution is concerned, the fact that P − ( τ I ) < S ,z is not required. Rather, a longer τ I al-lows to achieve a higher initialization fidelity into thelogical subspace but the initialization axis (and readoutaxis as well, by considering the inverse detuning pulse)will be tilted with respect to the logical ˆ z . The tiltangle can be made larger with a longer τ I , before de-phasing mechanisms become relevant. Initialization ina superposition of eigenstates can be addressed experi-mentally with a pulse which returns to large positive de-tuning, to readout the S (0 ,
2) probability through chargesensing.
The resulting quantum oscillation are il-lustrated in Fig. 6.
2. Single-spin rotations As z -rotations can be easily implemented through theZeeman splitting (see the discussion at the beginning ofSec. III), we focus here exclusively on the spin manipula-tion realized using the anticrossing at ε B . The detuningpulse, with total gate time τ G , is illustrated in Fig. 2 andwe show numerical results with the system initialized inthe |−(cid:105) state. This is sufficient to illustrate the gate per-formance if the total fidelity F ( τ G ) = P + ( τ G ) + P − ( τ G )is close to 1, thus a nearly unitary operation is real-ized within the logical subspace [ P ± ( τ G ) are defined asin Eq. (23)].As expected from the discussion in Sec. III B, the de-tuning pulse should realize a rotation about an axis per-pendicular to ˆ z , thus allowing one to implement a π -rotation equivalent to a NOT-gate. The two upper plotsin Fig. 7 show the behavior of P + ( τ G ) for two represen- τ I (ns) P S ( τ I ) , P _ ( τ I ) τ W (ns) P S ( τ I + τ W ) FIG. 6. Main panel: The solid red line is the return prob-ability P S (2 τ I ) to the ε ( τ = 0) = 200 µ eV ground stateafter two linear detuning ramps of duration τ I : first to ε ( τ I ) = − µ eV and than back to ε (2 τ I ) = 200 µ eV.We used t = 10 µ eV, and the geometry of Fig. 3(b) with ϕ = 0, B = 1 T. The growing oscillations at large τ I re-flect the larger amplitude of the | + (cid:105) state at the initializationtime τ I . The dashed blue line shows P − ( τ I ), reproduced fromFig. 5. The inset shows the return probability by introduc-ing a waiting time τ W between the two linear ramps. Thethin blue, thick black, and dotted red curves correspond re-spectively to τ I = 1 , , τ I = 2 ns. Τ w (cid:72) ns (cid:76) P (cid:43) (cid:72) Τ G (cid:76) , F (cid:72) Τ G (cid:76) Τ w (cid:72) ns (cid:76) P (cid:43) (cid:72) Τ G (cid:76) , F (cid:72) Τ G (cid:76) Τ w (cid:72) ns (cid:76) F H (cid:72) Τ G (cid:76) , F (cid:72) Τ G (cid:76) Τ w (cid:72) ns (cid:76) F H (cid:72) Τ G (cid:76) , F (cid:72) Τ G (cid:76) FIG. 7. Upper panels: the blue lower curves show P + ( τ G )as function of τ W (i.e., the waiting time at ε B such that τ G = τ W + 2 τ R , see Fig. 2). The left panel is for the setup inFig. 3(a) with ϕ = 5 π/ ϕ = 0. Lower panels: the blue lower curves show F H ( τ G )as defined in Eq. (24) for the same parameters as the upperpanels (as above, left and right panels are for the setups ofFig. 3(a) and (b), respectively). In all panels the upper redcurves are F ( τ G ), t = 25 µ eV, B = 0 . τ R = 0 . |−(cid:105) at ε A = − µ eV. Fidelity τ G ( n s ) F i d e lit y τ R (ns) FIG. 8. Main panel: Maximum value of P + ( τ G ) obtained withan optimum value τ ∗ R , the initial state |−(cid:105) ( ε A = − µ eV),and a waiting time τ W = τ π at ε B , see Eq. (18). The plotillustrates the relation between the π -rotation fidelity P + ( τ G )and the corresponding total gate time τ G = τ π + 2 τ ∗ R . Thethree curves are a guide for the eye through the numericaldata (dots), computed for B = 0 . , , . , . . . T from left toright. The dashed curve refers to the geometry of Fig. 3(a)with ϕ = 5 π/ t = 10 µ eV. The other two curves are forFig. 3(b) with ϕ = 0 and t = 10 µ eV (solid) or t = 20 µ eV(dot-dashed). The inset shows two examples on how to deter-mine the optimum value τ ∗ R from the maximum in P + ( τ G ) asfunction of τ R . The slanting field is the same as in Fig. 3(b), t = 10 µ eV, and the thick (thin) curve is for B = 0 . tative cases. As expected, P + ( τ G ) displays oscillationsin τ G with period 2 τ π , which are significantly faster forthe setup (b) of Fig. 3 than for setup (a). We obtain P + ( τ G ) (cid:39) π -rotation, we find a non-monotonic de-pendence on τ R similar to the optimization of the initial-ization fidelity with respect to τ I . An example is shownin the inset of Fig. 8: the maximum in fidelity occurs at τ ∗ R , which is determined by the competition of the tworelevant anticrossings (at ε = 0 and ε B ) in requiring anadiabatic/diabatic evolution within the lower two energybranches. Similarly as before, an increase in the externalfield B leads to higher values of the fidelity due to thesuppression of ∆ E , thus to a better energy scale sepa-ration t (cid:29) ∆ E . However, a larger B also degrades thegate time τ G = τ π + 2 τ ∗ R due to the longer τ π .To clarify the typical interplay between relevant pa-rameters, we show in Fig. 8 the relation between the op-timum fidelity of a π -rotation and the corresponding gatetime τ G . As the external field B is increased, a betterfidelity approaching 1 is obtained at the expense of alonger τ G . In the geometry of Fig. 3(b) the same fidelityof setup (a) can be achieved with a shorter gate time,as seen by a comparison between the solid and dashedcurves of Fig. 8. The gate time can be further improvedif the tunneling energy is made larger, as seen by a com-parison of the solid and dot-dashed curves of Fig. 8.The reduced fidelity of the π -rotation in the favor-able regime of larger values of ∆ E (and shorter gatetimes) does not prevent in general to achieve effectivespin-manipulation since we obtain F ( τ G ) (cid:39) P + ( τ G ) (see the topright panel of Fig. 7) can be simply attributed to a ro-tation axis which is not perpendicular to ˆ z , due to theimperfect realization of the diabatic evolution Eq. (17).In particular, when P + ( τ ) = 1 / π/ z to the xy -plane (equivalent to an Hadamard gate) isrealized with high accuracy. We can characterize the fi-delity of this π/ | φ (cid:105) = (cid:0) | + (cid:105) + e − iφ |−(cid:105) (cid:1) / √ φ an arbitrary phase).Choosing φ to maximize the overlap with | ψ ( τ ) (cid:105) , we ob-tain for the Hadamard gate: F H ( τ G ) ≡ max φ |(cid:104) φ | ψ ( τ G ) (cid:105)| = 12 (cid:32)(cid:88) ± (cid:112) P ± ( τ G ) (cid:33) . (24)This quantity is plotted in the two lower panels of Fig. 7and is simply related to P + of the upper panels by the ap-proximate relation F H (cid:39) / (cid:112) P + (1 − P + ) (using F (cid:39) F H ( τ G ) approaches one when P + ( τ G ) = 1 / F ( τ G ). The NOT-gate can be alternatively realized by making use of a com-position of z -rotations and two of these π/ C. Decoherence mechanisms
We estimate here the decoherence timescales and theexpected analytic form of decay induced by the hyperfineinteraction and charge noise. From the resulting param-eter dependence, we suggest under what conditions thesedecoherence effects can be made small.
1. Hyperfine interaction
As the π -rotations can be realized on a rather shorttime scale (cid:28)
10 ns, see Figs. 7 and 8, it becomes justi-fied to approximate the nuclear environment with staticrandom fields, which modify the values of b , . Also a re-cently discussed nuclear dephasing mechanism, inducedby the inhomogeneous magnetic field, becomes only rel-evant at much longer times. We consider nuclear fieldswhich are uncorrelated between the two dots and havea Gaussian probability distribution with zero mean andvariance σ for each component, as discussed in previousworks. We have computed the result in Fig. 9 for aparticular set of parameters, where it can be seen thatthe effect on the fidelity at the first maximum is small.In fact, fluctuations induced by the nuclei are of order σ ∼ b (cid:39)
100 mTthrough the slanting field of the micromagnet, the effectof the nuclei on the anticrossing can be very small. Τ W (cid:72) ns (cid:76) P (cid:43) (cid:72) Τ G (cid:76) FIG. 9. Plot of the fidelity P + ( τ G ) = |(cid:104) + | ψ ( τ G ) (cid:105)| for a rota-tion starting from | ψ (0) (cid:105) = |−(cid:105) ( ε A = − µ eV) as a functionof the waiting time τ W at ε B . We used τ R = 2 ns, t = 10 µ eV,and the geometry of Fig. 3(a) with B = 1 T and ϕ = 5 π/ σ = 2 mT (average over 400 runs). The thick black dashedline is Eq. (25) with τ N given by Eq. (28). To obtain a quantitative expression, we assume thatthe evolution between ε A,B is realized as in Eq. (17).Since the random change in ∆ E is small, it is justified toconsider only the linear correction from the nuclear field.In this approximation ∆ E has a Gaussian distribution,which yields the following expression for P + ( τ G ): P + (2 τ R + τ ) (cid:39) (cid:104) − e − ( τ/τ N ) cos (∆ Eτ / (cid:126) ) (cid:105) . (25)The overline indicates the average over nuclear configu-rations and τ N = 2 (cid:126) / (∆ E − ∆ E ) = 2 (cid:126) /σ E . Toestimate ∆ E , we can simply use the unperturbed valuesof θ, b in Eq. (16). We also obtained to lowest order in σ : σ θ = (cid:18) b + 1 b (cid:19) σ , σ b = σ , (26)while the covariance is zero to the same order of approx-imation. From these results, σ E can be obtained fromEq. (16) as usual: σ E = (cid:18) ∂ ∆ E∂θ (cid:19) σ θ + (cid:18) ∂ ∆ E∂b (cid:19) σ b , (27)which is easily evaluated and yields results in good agree-ment with the numerical evaluation.A relevant regime which is more transparent to discussis when ∆ E (cid:39) tθ , see Eq. (19). In this case the fluctua-tions in ∆ E are directly related to the fluctuations in theangle θ . By using σ θ (cid:39) σ /b , we obtain the followingdecay time scale: τ N (cid:39) (cid:126) bσt , (28)which, using θ (cid:39) ∆ b ⊥ /b , can be compared to the char-acteristic gate time from Eq. (18): τ π (cid:39) π (cid:126) b ∆ b ⊥ t . (29)In Fig. 9 we have used Eq. (28), together with Eq. (25),and obtained a satisfactory description of the decayingoscillations. An interesting feature of Eq. (28) is thatfor this problem the relevant nuclear dephasing timescale τ N is proportional to b (cid:39) B . This is easily understoodsince ∆ E (cid:39) tθ and the typical change in the angle θ due to the nuclear field is δθ N ∼ σ/b , thus the fluctua-tions in θ (and in ∆ E ) are suppressed by a larger valueof b . Similarly, a larger magnetic field will increase theoscillation period, which is also proportional to b . There-fore, the ‘quality factor’ remains constant since the ratio τ N /τ π = ∆ b ⊥ / ( πσ ) is independent of b . In particular, theratio of timescales is governed by ∆ b ⊥ /σ , where ∆ b ⊥ isthe difference in local fields transverse to B , see Eq. (12).As discussed, this ratio can be made large since ∆ b ⊥ canbe of order ∼
100 mT (cid:29) σ ∼
2. Charge noise
As recent experiments have demonstrated theimportant role played by low-frequency chargefluctuations, we consider the effect of chargenoise on the single-spin rotations. We introduce arandom shift δε in detuning, assuming a Gaussian dis-tribution with variance σ ε . This type of noise displacesthe operating points from the desired values. Especially,the π -rotations are now realized at a detuning ε B + δε which does not coincide with the anticrossing point. Acertain degree of protection against dephasing arisesfrom the fact that ε B is a stationary point for the energygap and, to lowest order, the change in the gap energy∆ E is quadratic in δε . The relevant scale for σ ε is setby ∆ E . Figure 10 shows a strong suppression of thevisibility when σ ε (cid:38) ∆ E , while the coherent oscillationsare significantly more robust when σ ε (cid:46) ∆ E .A more precise description of this effect can be ob-tained by noticing, from the effective model in Eq. (13),that the noise in ε induces a perturbation along the ef-fective z -direction. The variance of this perturbation isobtained from: σ (cid:48) ε = 12 (cid:32) ∂ ∆ ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ε B (cid:33) σ ε = σ ε (cid:16) t cos( θ/ | g | µ B b (cid:17) . (30)On the other hand, fluctuations in the off-diagonal ele-ment of Eq. (13) can be readily calculated to linear order Τ W (cid:72) ns (cid:76) P (cid:43) (cid:72) Τ G (cid:76) Τ W (cid:72) ns (cid:76) P (cid:43) (cid:72) Τ G (cid:76) FIG. 10. Plot of the fidelity P + ( τ G ) = |(cid:104) + | ψ ( τ G ) (cid:105)| for arotation of |−(cid:105) ( ε A = − µ eV), as function of the waitingtime τ W at ε B . For both panels we assumed the slanting fieldof Fig. 3(b) with ϕ = 0. The solid curves are averages of 400runs using σ ε = 1 µ eV for the thin blue curves and σ ε = 3 µ eVfor the thick red curves. In the upper panel τ R = 2 ns, B = 1T, and t = 10 µ eV, which give ∆ E = 1 . µ eV. In the lowerpanel τ R = 0 . B = 0 . t = 20 µ eV, which give∆ E = 2 . µ eV. The dashed curves refer to the asymptoticexpression Eq. (32). The σ ε = 1 µ eV curve of the secondpanel has small decay and Eq. (32) is not plotted, as it isapplicable only for τ W (cid:29) (cid:126) ∆ E/σ (cid:48) ε (cid:39)
35 ns. in δε , which gives: σ (cid:48)(cid:48) ε = (cid:32) ∂∂ε (cid:114) ∆ + ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε B t sin( θ/ (cid:33) σ ε = 2 tan( θ/ (cid:20) (cid:16) | g | µ B bt cos( θ/ (cid:17) (cid:21) / σ (cid:48) ε < tan( θ/ √ σ (cid:48) ε . (31)We see that typically σ (cid:48)(cid:48) ε /σ (cid:48) ε (cid:28)
1, due to the small angle θ (an additional small factor appears for t (cid:28) | g | µ B b ).If we neglect the effect of gate noise in the off-diagonalelement, the problem is formally equivalent to the theoryof Ref. 35 describing the decay of Rabi oscillations due tothe transverse fluctuations of the Overhauser field. This0correspondence yields the following asymptotic result: P + ( τ ) (cid:39) √ π ∆ E σ (cid:48) ε exp (cid:18) ∆ E σ (cid:48) ε (cid:19) erfc (cid:18) ∆ E √ σ (cid:48) ε (cid:19) − (cid:115) ∆ E (cid:126) σ (cid:48) ε τ cos (cid:16) ∆ Eτ / (cid:126) + π (cid:17) , (32)with erfc( x ) the complementary error function. This ex-pression is characterized by a power-law decay and auniversal π/ t and smaller B , as in the second panel ofFig. 10, several assumptions in deriving the simple formof Eq. (32) become less accurate: θ is larger (leading to anincrease of σ (cid:48)(cid:48) ε /σ (cid:48) ε ), higher-order corrections to the H eff ofEq. (13) become more relevant, and even without chargenoise the amplitude of the P + ( τ G ) oscillations is signif-icantly smaller than one. Nevertheless, the power-lawdecay is still in qualitative agreement with the numericalresults. The asymptotic formula Eq. (32) is valid for: τ (cid:29) ∆ E (cid:126) σ (cid:48) ε , (33)when ∆ E/σ (cid:48) ε >
1. Equation (33) also provides the rele-vant time scale for a significant reduction in visibility dueto the τ − / prefactor, see the second line of Eq. (32),thus gives a quality factor τ /τ π ∼ (∆ E/σ (cid:48) ε ) . The valueof σ (cid:48) ε could be effectively reduced if t / | gµ B B | (cid:29) t . In other words, our scheme for rota-tions at ε B is based on the | ˜ S − (cid:105) state of Eq. (B6). While | ˜ S − (cid:105) necessarily introduces a superposition of differentspin states, it is still possible to suppress in Eq. (B6) theamplitude of | ˜ S (0 , (cid:105) through a large value of t , whichmakes this state closer to a pure (1,1) charging configu-ration, thus less sensitive to fluctuations in ε .Besides random shifts of ε , static fluctuations on thetunneling barrier can also be simply treated by introduc-ing a Gaussian variation of t , with variance σ t . Similar asdetuning noise, we obtain that the noise fluctuations inthe effective Hamiltonian Eq. (13) are prevalently alongthe effective z -direction. This leads to the same expres-sion Eq. (32) with σ (cid:48) ε replaced by the corresponding σ (cid:48) t ,obtained as follows: σ (cid:48) t = 12 ∂ ∆ ∂t (cid:12)(cid:12)(cid:12)(cid:12) ε B σ t = 2 σ t cos( θ/ | g | µ B b t cos( θ/ + t cos( θ/ | g | µ B b . (34)As seen in Fig. 11, the agreement with the numericsis good. As a result, although the parameter depen-dence is different, it might be difficult to distinguish thetwo possible effects of charge noise (tunnel and detun-ing fluctuations) from the asymptotic form of the coher-ent oscillations. For example, the decoherence of Fig. 10could also be interpreted as due to noise in t such that Τ W (cid:72) ns (cid:76) P (cid:43) (cid:72) Τ G (cid:76) FIG. 11. Plot of the fidelity P + ( τ G ) = |(cid:104) + | ψ ( τ G ) (cid:105)| with thesame parameters of the upper panel of Fig. 10, but includingfluctuations in the tunnel amplitude t instead of ε . The thickred (thin blue) solid curve is for σ t /t = 5% ( σ t /t = 1%). Thedashed curve is the asymptotic formula for the σ t /t = 5%curve, calculated using Eq. (34). σ (cid:48) t = σ (cid:48) ε . By combining Eqs. (30) and (34) we get thatthe σ ε = 1 , µ eV curves of the upper panel of Fig. (10)would correspond to σ t /t (cid:39) σ t /t (cid:39) V. DISCUSSION AND CONCLUSIONS
We have characterized a spin manipulation scheme in-volving the two lowest energy states of a double quan-tum dot in the slanting field of a micromagnet. Workingat sufficiently large negative detuning, this scheme ef-fectively realizes single-spin rotations in one of the twoquantum dots, with the other dot serving as an auxil-iary spin. The general principle of operation is similar toRef. 18 but the physical picture is different: the auxiliaryspin of Ref. 18 is “pinned” by a large local field b (cid:29) b and its role is to induce through the exchange interactionan effective local field (parallel to b ) on the second spin;instead, in our case we have b (cid:39) b (cid:39) B and at the ε B anticrossing the two spins become strongly entangled.In our parameter regime, fast spin manipulation ( ∼ b ,x (cid:28) b ,z (cid:28) b ,z , (35)a condition which in practice can turn to be too restric-tive. In fact, the timescale of x -rotations given in Ref. 18is (cid:126) / | g µ B b ,x | (as in Sec. II A we choose b along z and b ,y = 0). Since Eq. (35) implies b ,z (cid:39) b ,z − b ,z , itis difficult to realize b ,z much larger than (cid:39)
100 mT.Therefore, if Eq. (35) is strictly enforced in GaAs lat-eral quantum dots, b ,x could become comparable to the σ ∼ we have found that an alternative method1is to simply maximize b ,x (say, b ,x ∼
100 mT) whilesatisfying: b ,x (cid:28) b ,z (cid:39) b ,z , (36)which can be always realized with a sufficiently strongexternal field ( b i,z (cid:39) B ). In this case, the limiting opera-tion time ∼ (cid:126) / | g µ B b ,x | is approached when t (cid:39) | g | µ B B ,see Eq. (20). Thus in our case a relatively large tunnel-ing amplitude is favorable. A large tunneling amplitudeis also useful to suppress the effect of fluctuations in de-tuning, when t / | gµ B B | (cid:29)
1. Effectively, this schemetakes advantage of the large energy scales set by t and | g | µ B B , to achieve an improved fidelity and operationtime. An obvious limitation where this strategy breaksdown is given by the orbital energy scale of the quantumdots, but this is ∼ σ/b ,x ratio, which is typically small in the optimalregime. Thus, this approach could realize high fidelitysingle-spin gates on a timescale of ∼ ε B anticrossing,induced in that case by the nuclear fields. Landau-Zener interferometry yields an alternative approach forthe manipulation of |±(cid:105) . The gate time would be de-termined by the same timescale (cid:126) / ∆ E discussed so far,and the use of a micromagnet slanting field should al-low for significant improvements over nuclear fields. Arelevant process discussed in those works is the phonon-mediated relaxation at ε B , which yields a slow timescaleΓ − ∼ µ s when ∆ E ∼ k B T , by fitting a phenomeno-logical model. However, spin relaxation processes medi-ated by phonons can have a strong dependence on the gap∆ E and this estimate might not be appropriate in ourcase. A more detailed microscopic theory wouldbe necessary to assess this effect. We have also neglectedspin-orbit coupling terms, which have an effect on the an-ticrossing with a complicated dependence on the doubledot parameters. As their energy scale is comparable tothe nuclear field fluctuations, we expect a small influenceon our discussion of ∆ E and spin manipulation.In concluding, we stress again that the qubit is encodedhere into the single-spin states of one of the dots, even ifa double dot is used for spin manipulation. Thus, two-qubit gates could be implemented by simply controllingthe exchange interaction between two target spins, which is a potential advantage with respect to other typesof encoding using multiple quantum dots. ACKNOWLEDGMENTS
We thank W. A. Coish, S. N. Coppersmith, M. Del-becq, and P. Stano for helpful discussions. We acknowl-edge support from the IARPA project Multi-Qubit Co-herent Operations through Copenhagen University. S.C.and Y.D.W. acknowledge support from the 1000 Youth Fellowship Program of China. J.Y., T.O., and S.T. ac-knowledge support from ImPACT Program of Council forScience Technology and Innovation (Cabinet Office, Gov-ernment of Japan), Grants-in-Aid for Scientic Research S(No. 26220710), and FIRST. T.O. acknowledges supportfrom the Japan Prize Foundation, JSPS, RIKEN incen-tive project, and the Yazaki Memorial Foundation. D.L.acknowledges support from the Swiss NSF and NCCRQSIT.
Appendix A: Corrections to the factorized form ofthe logical states
In this Appendix, we discuss the leading correctionsto Eq. (11). For |−(cid:105) the probability of admixture with | ˜ S (0 , (cid:105) , introducing undesired entanglement betweenthe two spins, is of order t /ε A ∼ . t = 5 µ eV and ε A = − µ eV, and can besystematically reduced by increasing | ε A | (while to keep arelatively large value of t is desirable). | + (cid:105) (cid:39) | ψ + , + (cid:105) hasa much larger purity since typically sin( θ/ (cid:28)
1. Forthe |−(cid:105) logical state, we estimate that the probability ofmixing with | ψ − , + (cid:105) is of order t / ( ε A gµ B ∆ b ) . Since arealistic magnetic field gradient gives | gµ B ∆ b | ∼ µ eV,the factor t / ( gµ B ∆ b ) contributes to enhance the ad-mixture fraction with respect to t /ε A . The previousparameters give a probability ∼ .
6% in this case. De-spite the fact that the admixture with | ψ − , + (cid:105) can also besystematically reduced with | ε A | , this represents a moreserious limitation to realize |−(cid:105) (cid:39) | ψ + , − (cid:105) with high ac-curacy. Appendix B: Derivation of the effective Hamiltonianat the anticrossing point
We present here the derivation of H eff in Eq. (13). Byusing the local spin basis, the Zeeman Hamiltonian hasa simple diagonal form and can be separated as follows: H Z = −| g | µ B (cid:88) l =1 , b l ˜ S l,z = −| g | µ B b (cid:88) l ˜ S l,z − | g | µ B ∆ b S ,z − ˜ S ,z ) , (B1)where ˜ S l,z = ( ˜ d † l + ˜ d l + − ˜ d † l − ˜ d l − ) is the spin operator forthe l -th dot along the local field direction. In the secondline we have separated the homogeneous part, propor-tional to b , from the smaller contribution proportional to∆ b . Following this partition, we write H Z = H Z + δH Z .Similarly we define δH T from the tunneling Hamilto-nian. Applying the same spin rotation, we write: H T = t cos( θ/ (cid:88) µ = ± (cid:16) ˜ d † µ ˜ d µ + ˜ d † µ ˜ d µ (cid:17) − t sin( θ/ (cid:88) µ = ± µ (cid:16) ˜ d † µ ˜ d µ + ˜ d † µ ˜ d µ (cid:17) , (B2)2where ¯ µ = − µ . The physical meaning of this expression israther obvious: electrons maintain the original spin direc-tion upon tunneling but, due to the different quantizationdirections on l = 1 ,
2, the spin appears to have rotatedwhen expressed through the local spinor basis. There-fore, introducing the local quantization axes generatesin the second line of Eq. (B2) a spin-flip tunneling termanalogous to the one induced by spin-orbit interaction. For θ (cid:28) δH T and write H T = H T + δH T .The unperturbed Hamiltonian H ≡ H C + H T + H Z is formally equivalent to the familiar problem of a doubledot with uniform magnetic field of strength b and a mod-ified spin-preserving tunneling amplitude t cos( θ/ | ˜ T ± (cid:105) = | ˜ ψ ± , ± (cid:105) , (B3) | ˜ T (cid:105) = 1 √ (cid:16) | ˜ ψ + , − (cid:105) + | ˜ ψ − , + (cid:105) (cid:17) , (B4)with unperturbed eigenvalues ∓| g | µ B b and 0, respec- tively. The (1,1) “singlet” is: | ˜ S (1 , (cid:105) = 1 √ (cid:16) | ˜ ψ + , − (cid:105) − | ˜ ψ − , + (cid:105) (cid:17) . (B5)Notice that these states differ form the standard sin-glet/triplets since the spin quantization axes are dif-ferent on the two sites l = 1 ,
2. On the other hand, | ˜ S (0 , (cid:105) is the usual singlet state since it involves twoelectrons on the same dot. Diagonalization of H in the | ˜ S (1 , (cid:105) , | ˜ S (0 , (cid:105) subspace yields the eigenstates: | ˜ S ± (cid:105) = (cid:114) ∆ ± ε | ˜ S (1 , (cid:105) ± (cid:114) ∆ ∓ ε | ˜ S (0 , (cid:105) , (B6)with energy ( − ε ± ∆) /
2, where ∆ is given in Eq. (14)of the main text. For a large region of detunings, theseunperturbed eigenstates are a good approximation of theexact eigenstates. However, the effect of δH Z , δH T be-comes important at detunings ε A,B , which are particu-larly relevant for our single-spin manipulation scheme.Around ε B , it is appropriate to restrict ourselves to the | ˜ T + (cid:105) , | ˜ S − (cid:105) subspace, which gives Eq. (13) of the maintext. In particular, the off-diagonal terms in H eff are dueto δH T . ∗ [email protected] D. Loss and D. P. DiVincenzo, Phys. Rev. A , 120(1998)). R. A. ˙Zak, B. R¨othlisberger, S. Chesi, and D. Loss, Riv.Nuovo Cimento , 345 (2010). C. Kloeffel and D. Loss, Annu. Rev. Condens. Matter Phys. , 51 (2013). J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,and A. C. Gossard, Science , 2180 (2005). S. Foletti, H. Bluhm, D. Mahalu, V. Umansky, and A. Ya-coby, Nat. Phys. , 903 (2009). H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu,V. Umansky, and A. Yacoby, Nat. Phys. , 109 (2011). J. M. Taylor, H.-A. Engel, W. Dur, A. Yacoby, C. M. Mar-cus, P. Zoller, and M. D. Lukin, Nat. Phys. , 177 (2005). J. Klinovaja, D. Stepanenko, B. I. Halperin, and D. Loss,Phys. Rev. B , 085423 (2012). M. D. Shulman, O. E. Dial, S. P. Harvey, H. Bluhm,V. Umansky, and A. Yacoby, Science , 202 (2012). V. N. Golovach, M. Borhani, and D. Loss, Phys. Rev. B , 165319 (2006). K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, andL. M. K. Vandersypen, Science , 1430 (2007). S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, andL. P. Kouwenhoven, Nature , 1084 (2010). Y. Tokura, W. G. van der Wiel, T. Obata, and S. Tarucha,Phys. Rev. Lett. , 047202 (2006). M. Pioro-Ladri`ere, T. Obata, Y. Tokura, Y.-S. Shin,T. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Na-ture Phys. , 776 (2008). T. Obata, M. Pioro-Ladri`ere, Y. Tokura, Y.-S. Shin, T. Kubo, K. Yoshida, T. Taniyama, and S. Tarucha, Phys.Rev. B , 085317 (2010). R. Brunner, Y.-S. Shin, T. Obata, M. Pioro-Ladri`ere,T. Kubo, K. Yoshida, T. Taniyama, Y. Tokura, andS. Tarucha, Phys. Rev. Lett. , 146801 (2011). J. Yoneda, T. Otsuka, T. Nakajima, T. Takakura,T. Obata, M. Pioro-Ladri`ere, H. Lu, C. Palmstrøm, A. C.Gossard, and S. Tarucha, arXiv:1411.6738 (2014). W. A. Coish and D. Loss, Phys. Rev. B , 161302 (2007). L. Trifunovic, O. Dial, M. Trif, J. R. Wootton, R. Abebe,A. Yacoby, and D. Loss, Phys. Rev. X , 011006 (2012). L. Trifunovic, F. L. Pedrocchi, and D. Loss, Phys. Rev. X , 041023 (2013). H. Ribeiro, J. R. Petta, and G. Burkard, Phys. Rev. B , 115445 (2010). R. Petta, H. Lu, and A. C. Gossard, Science , 669(2010). H. Ribeiro, G. Burkard, J. R. Petta, H. Lu, and A. C.Gossard, Phys. Rev. Lett. , 086804 (2013). H. Ribeiro, J. R. Petta, and G. Burkard, Phys. Rev. B , 235318 (2013). X. Wu, D. R. Ward, J. R. Prance, D. Kim, J. K. Gamble,R. Mohr, Z. Shi, D. E. Savage, M. G. Lagally, M. Friesen,S. N. Coppersmith, and M. A. Eriksson, Proc. Natl. Acad.Sci. USA , 11938 (2014). G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev.B , 2070 (1999). D. Stepanenko, M. Rudner, B. I. Halperin, and D. Loss,Phys. Rev. B , 075416 (2012). W. G. van der Wiel, S. De Franceschi, J. M. Elzerman,T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev.Mod. Phys. , 1 (2002). T. Obata, M. Pioro-Ladri`ere, Y. Tokura, and S. Tarucha, New J. Phys. , 123013 (2012). RADIA Technical Reference Manual ESRF, Grenoble,France. F. Beaudoin and W. A. Coish, Phys. Rev. B , 085320(2013). W. A. Coish and D. Loss, Phys. Rev. B , 195340 (2004). O. E. Dial, M. D. Shulman, S. P. Harvey, H. Bluhm,V. Umansky, and A. Yacoby, Phys. Rev. Lett. , 146804(2013). V. Kornich, C. Kloeffel, and D. Loss, Phys. Rev. B ,085410 (2014). F. H. L. Koppens, D. Klauser, W. A. Coish, K. C. Nowack, L. P. Kouwenhoven, D. Loss, and L. M. K. Vandersypen,Phys. Rev. Lett. , 106803 (2007). A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B ,125316 (2001). V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev.Lett. , 016601 (2004). P. Stano and J. Fabian, Phys. Rev. Lett. , 186602 (2006). S. Amasha, K. MacLean, I. P. Radu, D. M. Zumb¨uhl, M. A.Kastner, M. P. Hanson, and A. C. Gossard, Phys. Rev.Lett. , 046803 (2008).40