Transfer of a quantum state from a photonic qubit to a gate-defined quantum dot
Benjamin Joecker, Pascal Cerfontaine, Federica Haupt, Lars R. Schreiber, Beata E. Kardynał, Hendrik Bluhm
TTransfer of a quantum state from a photonic qubit to a gate-defined quantum dot
Benjamin Joecker,
1, 2
Pascal Cerfontaine, Federica Haupt, Lars R. Schreiber, Beata E. Kardyna(cid:32)l, and Hendrik Bluhm ∗ JARA-FIT Institute Quantum Information, Forschungszentrum J¨ulichGmbH and RWTH Aachen University, 52074 Aachen, Germany Centre for Quantum Computation and Communication Technology,School of Electrical Engineering & Telecommunications, UNSW, Sydney, NSW, 2052, Australia Peter Gr¨unberg Institute, Forschungszentrum J¨ulich GmbH, 52425 J¨ulich, Germany
Interconnecting well-functioning, scalable stationary qubits and photonic qubits could substan-tially advance quantum communication applications and serve to link future quantum processors.Here, we present two protocols for transferring the state of a photonic qubit to a single-spin andto a two-spin qubit hosted in gate-defined quantum dots (GDQD). Both protocols are based onusing a localized exciton as intermediary between the photonic and the spin qubit. We use effectiveHamiltonian models to describe the hybrid systems formed by the the exciton and the GDQDs andapply simple but realistic noise models to analyze the viability of the proposed protocols. Usingrealistic parameters, we find that the protocols can be completed with a success probability rangingbetween 85-97 %.
I. INTODUCTION
Semiconductor quantum-dot devices have demon-strated considerable potential for quantum informationapplications. A prominent example are gate-definedquantum dots (GDQD), i.e. quantum dots realized insemiconductor heterostructures in which individual elec-trons are confined by an electrostatic trapping potential.Spin qubits based on GDQD in GaAs/Al x Ga x − As het-erostructures have demonstrated all key requirements forquantum information processing, such as qubit initializa-tion, readout, coherent control with high fidelity and two-qubit gates.
Moreover, thanks to their simi-larity to the transistors used in modern computer chips,these top-down fabricated quantum dots have goodprospects for realizing large scale quantum processingnodes. However, unlike self-assembled quantum dots,where excellent optical control and information transferhas been demonstrated,
GDQDs pose a number ofchallenges when it comes to couple them coherently withlight. The problems come from the lack of exciton con-finement: while the electron states are confined, the holestates are not. Since in the creation of an exciton the spinof the photo-excited electron is always entangled with theone of the hole, discarding the hole-spin inevitably leadsto decoherence of the electron spin. This limits consider-ably the possibility of optically controlling and manipu-lating spins in GDQDs, and it hinders their applicabilityin quantum communications.Despite these difficulties, first steps towards the goalof coherently coupling photons and electron spins inGDQDs have already been made, by trapping and detect-ing photo-generated carriers in GDQDs, and by prov-ing transfer of angular momentum between photons andelectrons. Much of this effort is motivated by the factthat robust spin-photon entanglement is a key require-ment for quantum repeaters for long-distance quantumcommunications, as well as for distributed quantumcomputing, where different computing nodes based on GDQD are connected by optical channels. One strat-egy to avoid entanglement between the spins of the elec-tron and the hole is to use g-factor engineering to ob-tain a much smaller g-factor for the electrons than forholes.
Here we propose a different strategy, which relies ona localized exciton in an optically active quantum dot(OAQD) as interface between a photonic qubit and a spinqubit in a GDQD. The OAQD could be a self-assembledquantum dot (SAQD) – as also proposed by Engel andcoworkers – an impurity, or a bound exciton localizedwith local electric-gates by exploiting the quantum Starkeffect. Using effective Hamiltonian models to describethe hybrid system formed by a bound exciton tunnel cou-pled to a GDQD, we analyse the feasibility of two differ-ent information transfer protocols. First, we consider thecase where the quantum state of the photon is mappedonto the state of a single-spin qubit, and then the casewhere the mapping is to a singlet-triplet qubit in a doubleGDQD. In both cases, the first step of the transfer pro-cess is the photo-excitation of an exciton in the OAQDin the Voigt configuration, i.e. in the presence of a strongin-plane magnetic field and normal incident light beam.We focus in particular on the effects that can hinder thecoherent transfer of the photo-excited electron from theOAQD to the GDQD, assuming a unitary mapping be-tween the photon state and the exciton state. Through-out the paper we use a InAs SAQD as a concrete exampleof OAQD (see Fig. 1). The described protocols can how-ever be straightforwardly extended to other OAQD withappropriate tunnel coupling to the GDQD. We estimatethe performance of the protocols using a realistic set ofparameters for InAs SAQDs. According to these esti-mates, the proposed protocols could be completed witha success probability of approximately 85% for the caseof the singlet-triplet qubit, and up to 97% probability forthe single-spin qubit.The paper is organized as follows. In Section II, wediscuss in detail the protocol for transferring information a r X i v : . [ qu a n t - ph ] D ec GD DD t c εε DD t DD H ex t c ε H ex b cOAQDGDQDa |◦↑⇑⟩ |↑◦⟩|↑⇑⟩↑⇑ ↑ ↑⇑ OAQD
FIG. 1. (color online). a) Schematic of a possible heterostruc-ture realising a hybrid device with a gate-defined double quan-tum dot tunnel coupled to a self-assembled quantum dot. A2DEG emerges in the conduction band minimum at the lowerAl . Ga . As/GaAs inverted interface. Metallic top gatescan be used to deplete the 2DEG and to create gate-defined(lateral) quantum dots. Addition of InAs during the growthof the GaAs layer leads to the formation of SAQDs. b) Modelof a single-level GDQD tunnel coupled to an optically activequantum dot, as discussed in Sec. II. Here t c represents thetunnel coupling between the GDQD and the OAQD, ε theenergy detuning between the electronic level in the GDQDand in the exciton, and H ex the excitonic exchange interac-tion. c) Model of a double dot tunnel coupled to an OAQD,as discussed in Sec. III. In addition to the quantities definedbefore, ε DD and t DD represent the detuning and the tunnelcoupling in the double dot, respectively. to a single-spin qubit, including the possible error sources(section IIB). Section III is dedicated to the protocolfor transferring information to a singlet-triplet qubit en-coded in a double dot. Details on how we deal with thedifferent noise sources and on the model Hamiltonian em-ployed in Sec. III are given in Appendix A and B, respec-tively. II. INFORMATION TRANSFER TO A SINGLESPIN-QUBITA. Transfer protocol
Transferring the information encoded in the polariza-tion of one photon to the spin of one electron in a GDQDusing an OAQD as intermediary requires two steps: (i)the creation of a bound exciton in the OAQD by absorp-tion of the incident photon; (ii) the adiabatic transferof the photo-excited electron into the GDQD. Here, wedo not model explicitly the absorption process. Rather,we assume that the process is coherent and the photo-generated exciton in the OAQD reflects the state of theabsorbed photon, and discuss under which conditions thephoto-excited electron can be coherently transferred tothe GDQD.In OAQDs embedded in GaAs, the light- and theheavy-hole bands are split in energy by several tens ofmeV due to strain or confinement. In the follow-ing we will use the notation |↓⇑(cid:105) z , |↑⇓(cid:105) z , . . . to indi-cate the electron-hole states, where the regular arrowrepresents the projection of the electron spin along thegrowth-direction z ( S (e) z = ± / (cid:126) ), and double arrowsthe projection of the heavy-hole spin ( S (h) z = ± / (cid:126) ).In this system, electron and hole states with antiparal-lel spins, {|↓⇑(cid:105) z , |↑⇓(cid:105) z } , have angular momentum ± (cid:126) ,and are optically addressable with circularly polarizedlight. Hence, they are referred to as bright states. Stateswith parallel spin, {|↑⇑(cid:105) z , |↓⇓(cid:105) z } , are optically inactiveand are referred to as dark states. The Hamiltonian ofthe electron-hole exchange interaction takes the block-diagonal form H ,z = 12 ∆ ∆ ∆ − ∆ ∆ − ∆ , (1)with respect to the basis {|↓⇑(cid:105) z , |↑⇓(cid:105) z , |↑⇑(cid:105) z , |↓⇓(cid:105) z } .Here, ∆ is the energy splitting between the dark andthe bright states originating from the electron-hole ex-change interaction. The off-diagonal terms ∆ and ∆ are responsible for the energy splitting of the bright anddark excitons, respectively.The excitation of the bright-states by photo-absorptioninduces entanglement between the spins of the electronand that of the hole as follows: α | σ + (cid:105) + β | σ − (cid:105) → α |↓⇑(cid:105) z + β |↑⇓(cid:105) z , where α and β are complex numbersand | σ + (cid:105) and | σ + (cid:105) represent left and right circularly po-larized photons, respectively. This poses a fundamentalproblem if we want to map the state of the photon ontothe spin of the electron only, and discard the hole. Toavoid this problem, it is necessary to eliminate the entan-glement between the spins of the electron and of the hole.One way to achieve this is via g-factor engineering. However, one difficulty with this approach is that the re-sulting small Zeeman splitting between the electron spin-states makes the system very susceptible to nuclear spinfluctuations. Furthermore, this approach is strictly lim-ited to (Al,Ga)As-based systems and cannot be extendedto other material system (e.g. II/VI) . Also, it can-not be simply extended to two-electron spin qubits withsinglet-triplet encoding, which have the advantage of fullelectrical control. Here, we investigate a different approach, which isbased on applying a strong in-plane magnetic field thatmixes bright and and dark states making all of them optically accessible , as described below. We assumethe in-plane magnetic field to be along the x -direction.Taking this as as the spin-quantization axis, the excitonHamiltonian takes the form H ex = H ,x + H Ze ,x , (2)where H ,x = 14 − ∆ − ∆ − + ∆ − ∆ − + ∆ − ∆ − ∆ − ∆ + ∆ − − ∆ + ∆ − − ∆ + ∆ ∆ + ∆ , (3)represents the electron-hole exchange interaction with respect to the basis {|↓⇑(cid:105) x , |↑⇓(cid:105) x , |↑⇑(cid:105) x , |↓⇓(cid:105) x } , and H Ze ,x = µ B B − g e − g h g e + g h g e − g h
00 0 0 − g e + g h , (4)accounts for the Zeeman splitting induced by the in-planemagnetic field (cid:126)B = B ˆ e x . Here, g e and g h are the g-factorsfor the electron and for the hole, respectively (with thesign convention that g e is negative and g h is positive).In the limit of large magnetic field ( | g e ± g h | µ B B (cid:29) ∆ , ∆ , ∆ ), the eigenstates of H ex almost coincide withthe basis kets {|↓⇑(cid:105) x , |↑⇓(cid:105) x , |↑⇑(cid:105) x , |↓⇓(cid:105) x } . We will there-fore denote them by their dominant basis-state contribu-tion, e.g. | (cid:102) ↑⇑(cid:105) x = √ − δ |↑⇑(cid:105) x + √ δ |↓⇓(cid:105) x , where δ issmall for large B . All eigenstates have a contributionfrom the bright-states, i.e. they are all optically active.In general, the bright-state contribution (BC) of a state | Ψ (cid:105) can be quantified as follow:BC(Ψ) = |(cid:104) Ψ | ↑⇓(cid:105) z | + |(cid:104) Ψ | ↓⇑(cid:105) z | . (5)The BC is a factor determining how fast a photon canbe absorbed (or reemitted). BC substantially smallerthan one are not fundamentally problematic, as theycan be compensated by longer photon wave-packets.Eigenstates with parallel-spins (cid:16) | (cid:102) ↑⇑(cid:105) x , | (cid:102) ↓⇓(cid:105) x (cid:17) can be ad-dressed only with horizontally polarized photons, whileeigenstates with antiparallel-spins (cid:16) | (cid:102) ↓⇑(cid:105) x , | (cid:102) ↑⇓(cid:105) x (cid:17) require vertically polarized photons. The idea is now to use thepair {| (cid:102) ↑⇑(cid:105) x , | (cid:102) ↓⇑(cid:105) x } to map a photon state as follow: α | ω , H (cid:105) + β | ω , V (cid:105) → α | (cid:102) ↑⇑(cid:105) x + β | (cid:102) ↓⇑(cid:105) x (6)(or, alternatively, α | ω (cid:48) , V (cid:105) + β | ω (cid:48) , H (cid:105) → α | (cid:102) ↑⇓(cid:105) x + β | (cid:102) ↓⇓(cid:105) x ), where α and β are complex numbers and | ω, H(V) (cid:105) represents a photon state with energy ω andhorizontal (vertical) polarization. In this kind of map-ping the whole information on the state of the photon isentirely mapped on the spin of the electron alone, sincethe excitonic states | (cid:102) ↑⇑(cid:105) x and | (cid:102) ↓⇑(cid:105) x have the same spinprojection for the hole.The next step of the protocol – and the main subjectof our analysis – is the coherent transfer of the photo-excited electron from the OAQD to a GDQD. If theOAQD and the GDQD are tunnel coupled, the coherenttransfer between the two can be achieved by adiabaticallyincreasing the detuning ε between the electronic levels inthe two system (see Fig. 1b). Ideally, the whole transferprotocol will then work as follows: α | ω , H (cid:105) + β | ω , V (cid:105) photo − excitation −→ α |◦ (cid:102) ↑⇑(cid:105) x + β |◦ (cid:102) ↓⇑(cid:105) x adiabatic transfer −→ α |↑ ◦ ⇑(cid:105) x + β |↓ ◦ ⇑(cid:105) x , (7)where now |◦ (cid:102) ↑⇑(cid:105) x represents the state where the GDQDis empty and there is an exciton with the parallel spins in the OAQD (see schematic in Fig.1b), while |↑ ◦ ⇑(cid:105) x rep-resents the state where the electron has been transferredinto the GDQD, leaving a hole alone in the OAQD (andsimilarly for the other states).We model the GDQD as a single electronic level anduse the basis {|◦ ↓⇑(cid:105) x , |◦ ↑⇓(cid:105) x , |◦ ↑⇑(cid:105) x , |◦ ↓⇓(cid:105) x , |↓ ◦ ⇑(cid:105) x , |↑ ◦ ⇓(cid:105) x , |↑ ◦ ⇑(cid:105) x , |↓ ◦ ⇓(cid:105) x } , (8)to represent the states of the coupled exciton-GDQD sys-tem. With respect to this basis the Hamiltonian of thecoupled exciton-GDQD system reads H = (cid:18) H ,x + H Ze ,x + ε t c t c ˜ H Ze ,x − ε (cid:19) , (9)where t c is a spin-conserving tunneling matrix element,and ε is the gate-dependent energy detuning betweenthe OAQD and the GDQD, see Fig. 1b. We call thestates where the electron and the hole are both on theOAQD excitonic states , and those where the electron hasbeen transferred to the GDQD separated states . Theexcitonic exchange-interaction, H ,x , has non-vanishingmatrix elements only between excitonic states. ˜ H Ze ,x has the same structure as in Eq.(4), but it can differnumerically from H Ze ,x because of a different Zeemansplitting in the GDQD in the OAQD (e.g., because of adifferent g -factor, ˜ g e , in the GDQD). Spin-orbit effects,which in principle can lead to spin-flip processes duringthe adiabatic transfer, are not included in Eq.(9) aswe assume the dot separation to be much shorter thanthe spin-orbit length in GaAs, making spin-orbit neg-ligible compared to other effects. The eigenstates ofEq.(9) can be easily determined numerically. In Fig. 2we plot the corresponding eigenenergies as a functionof the detuning ε for the case of large tunnel coupling t c = 150 µ eV. This figure also includes a schematic di-agram of the information-transfer process described inEq.(7). The system is photo-excited at negative detun-ing, where excitonic states are energetically favourable(bright spots in Fig. 2). The photo-excited electron isthen transferred to the GDQD by adiabatically increas-ing ε to the regime of separated states. The colour coderepresents the BC of each eigenstate, which clearly de-pends on the detuning. B. Error sources
In practice, the viability of the protocol sketched inEq.(7) depends on several factors and error sources.First, both selected excitonic states need to show suffi-cient optical coupling. The BC determines how rapidlya photon can be absorbed/reemitted from a certain state,and it will therefore determine the photon pulse-lengthneeded for optimal absorption. State-dependent photonpulse-shaping might be therefore required to compensatethe different BC of the excitonic states. Furthermore,ideally each exciton should couple to a single photonmode, in order to reduce dissipative losses and to ensure -1 -0.5 0 0.5 1 (meV)-0.6-0.4-0.200.20.40.60.8 E n e r g y ( m e V ) c ε H ex Electron tunnels S e p a r a t e d s t a t e s E x c i t o n s t a t e s ~2t c |↓◦⇓⟩ x |↓◦⇑⟩ x |↑◦⇓⟩ x |↑◦⇑⟩ x ↑⇑ FIG. 2. (color online). Schematic diagram of the information-transfer process from an OAQD to a single spin qubit. Shownare the eigenenergies of the Hamiltonian Eq. (9) as a func-tion of the detuning ε for the case of strong tunnel coupling( t c = 150 µ eV). All remaining parameters are given in Tab. I.The colour code indicates the bright-state contribution (BC)of each state, see Eq. (5). The bright-spots indicate the detun-ing at which photo-excitation occurs and the branches chosenas basis-states. Once an exciton is created in the OAQD,the photo-excited electron is transferred into the GDQD byadiabatically increasing the detuning ε . that the photon emission/absorption process is a unitaryprocess. Bragg mirrors and solid immersion lenses mayhave to be employed to increase the collection efficiencyof the OAQD. These important, setup specific, engineer-ing problems go however beyond the scope of this paper.Here we will focus instead on the intrinsic error sourcesthat can affect the adiabatic transfer process, assumingthat optimal mode engineering ensures a unitary map-ping between the photon state and the exciton state.The first source of errors in the adiabatic transfer pro-cess are non-adiabatic transitions to other states, whichcan occur if the detuning ε is increased too quickly. Givena time-dependent Hamiltonian H ( t ), with instantaneouseigenstates | m ( t ) (cid:105) (i.e. H ( t ) | m ( t ) (cid:105) = E m ( t ) | m ( t ) (cid:105) ), anecessary condition for adiabatic evolution is (cid:88) m (cid:54) = n (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) m ( t ) | ˙ n ( t ) (cid:105) ω mn ( t ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) (cid:126) ω mn ( t ) = E m ( t ) − E n ( t ). If this criterion is vi-olated, transitions between different eigenstates are ex-pected. For the simple case of a two-level system, theprobability of transitions between the two levels whensweeping through the avoided crossing is given by thewell known Landau-Zener formula P LZ = exp (cid:18) − π (∆ E AC / (cid:126) v ε (cid:19) , (11)where ∆ E AC is the energy splitting at the avoided cross-ing and v ε is the sweep speed. This formula allows tocalculate the highest possible sweep speed for a given∆ E AC and a targeted maximum transition probability(e.g., P LZ = 1%). To obtain a similar bound on thesweep speed for the eight-level system that we are con-sidering, we notice that the exponent in Eq.(11) is closelyrelated to the quantity on the left hand side of Eq. (10),being (cid:88) m (cid:54) = n (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) m | ˙ n (cid:105) ω mn (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:126) (cid:104) | ˙2 (cid:105) ∆ E AC (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v ε (cid:126) E AC ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12)for the case of a two-level system. This motivates us totake 1 v ε = − P LZ ) π (cid:88) m (cid:54) = n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) m | ∂n∂ε (cid:105) ω mn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (13)as a bound for the maximal sweep speed that is allowedin order to have a transition probability between differenteigenstates smaller or equal to P LZ . In the following wewill require P LZ = 1%, i.e. a 99% success probability forthe adiabatic transfer.The sweep speed v ε and the sweep range ∆ ε determinethe time T tr on which the transfer can be completed.On the timescale of T tr various factors that hinder thetransfer process are at play. First of all, excitons candecay due to radiative recombination. We estimate theprobability of recombination in a certain time t as follows P rec ( t ) = 1 − exp (cid:18) − (cid:90) t Γ( t (cid:48) ) dt (cid:48) (cid:19) , (14)where Γ( t ) = BC(Ψ( t )) /τ is the instantaneous decay rateof the state | Ψ( t ) (cid:105) , with τ the characteristic lifetime ofa bright exciton. Here | Ψ( t ) (cid:105) stands generically for theinstantaneous state of the system at time t . Radiativerecombination reduces the efficiency of the information-transfer process, but it does not introduces errors in theencoding of the information.Other factors hindering the transfer process are chargeand nuclear-spin noise, which are well known sources ofdephasing, causing random fluctuations of the rela-tive phase accumulated between two states | Ψ (cid:105) and | Ψ (cid:105) ϕ ( t ) = (cid:90) t ∆ E ( t (cid:48) ) (cid:126) dt (cid:48) (15)accumulated between two states | Ψ (cid:105) and | Ψ (cid:105) , where∆ E is the energy difference between the states. Chargenoise affects ∆ E (and therefore ϕ ( t )) by causing ran-dom fluctuations of the detuning ε . Nuclear spins in the TABLE I. Set of realistic parameters for GaAs based devices.Parameter[Source] Symbol ValueDark-bright splitting ∆ µ eVBright state splitting ∆ µ eVDark state splitting ∆ µ eVMagnetic field B g e , ˜ g e − . g h . t c − µ eVExciton recombination time τ ε rms µ V /L Fast uncorrelated charge noise S ε × − V Hz − /L Gate lever-arm L e − Nuclear spin noise SAQD B (rms)OF
50 mTNuclear spin noise GDQD ˜ B (rms)OF η < . t DD µ eVCoulomb repulsion U V + . µ eVCoulomb energy triplet V − µ eV host material affect ∆ E by creating a randomly fluctu-ating magnetic field, the Overhauser field (OF). Here weconsider both quasi-static and fast uncorrelated chargenoise, as well as nuclear-spin noise. Quasi-static noiseis due to fluctuations that occur on time scales muchlonger than the transfer time T tr , corresponding to aspectral density centered around zero frequency. We de-note the root-mean-squared (rms) charge noise amplitudeby ε rms . Nuclear-spin fluctuations are to a good approx-imation quasi-static and quantified by their rms. am-plitude, B (rms)OF . On the contrary, fast uncorrelated noisehas equal weight S ε at all frequencies (white noise). Typ-ical values for ε rms , S ε and B (rms)OF in GaAs-based devicesare ε rms = 8 µ V /L , S ε = 5 × − V Hz − /L , and B (rms)OF = 5 −
50 mT, where the first value is typicalfor GDQDs and the second for SAQDs. L denotesthe lever arm converting voltages on gates to detuningvariations. For each source of noise we evaluate the quan-tity (cid:104) δϕ (cid:105) describing the dephasing due to that particularnoise source, as detailed in Appendix A. Assuming thatall noise sources are uncorrelated, the total dephasing isgiven by (cid:104) δϕ (cid:105) tot = (cid:104) δϕ (cid:105) ch − qs + (cid:104) δϕ (cid:105) ch − fast + (cid:104) δϕ (cid:105) spins . In order to better compare the effect of dephasing toother mechanisms that lead to failure of the transfer pro-cess, we introduce the failure probability due to dephas-ing , which we define as the probability of the depolar-izing channel with the same average gate fidelity as thedephasing channel (see Appendix A) P deph − fail = (cid:104) δϕ (cid:105) tot / . (16)In the following, we discuss results obtained for the setof realistic parameters presented in Tab. I for the casewhere the OAQD is an InAs self-assembled quantum dot.For simplicity we assume ∆ = ∆ = 0 µ eV, as thesesplittings are typically small compared to ∆ , and donot lead to qualitative changes. For the case of largetunnel coupling considered in Fig. 2 ( t c = 150 µ eV) , wefind that the maximal sweep velocity with P LZ = 1 % is v ε = 14 . / ns, so that a sweep over the whole dis-played detuning range can be completed in T tr = 0 .
14 ns.On this time scale, the probability of radiative recombi-nation is P rec = 2% for the state with parallel spins and5 . P deph − fail = 0 . P fail − tot = P LZ + P rec + P deph − fail ≤ . t c = 50 µ eV), the eigen-states of Eq.(9) exhibit a series of crossing and anti-crossing, see Fig. 3. In Fig. 3a the colour code representsagain the bright-state contributions of the various eigen-states, while in Fig. 3b red and blue indicate eigenstateswith parallel or anti-parallel spins, respectively. Sincethese form two separate subspaces, crossings can occurbetween them. The labels | Ψ (cid:105) and | Ψ (cid:105) indicate thestates involved in the adiabatic-transfer process sketchedin Eq. (7), i.e. | Ψ (cid:105) ≈ |◦ ↑⇑(cid:105) , | Ψ (cid:105) ≈ |◦ ↓⇑(cid:105) for large,negative detuning and | Ψ (cid:105) ≈ |↑ ◦ ⇑(cid:105) , | Ψ (cid:105) ≈ |↓ ◦ ⇑(cid:105) for large, positive detuning. Fig. 3c represents the in-verse of the maximal sweep velocity 1 /v ε that allowsadiabatic evolution along the states | Ψ (cid:105) (red) and | Ψ (cid:105) (blue), calculated according to Eq. (13) for P LZ = 1%.As expected, 1 /v ε shows maxima in correspondence ofthe anticrossings. Knowing v ε , we can calculate the to-tal time required for the adiabatic transfer. For sim-plicity we assume that the transfer occurs with a con-stant speed, equal to the smaller possible maximal speed1 /v ε = 0 .
70 ns / meV (see Fig. 3c). Because of this lowsweep speed, it is convenient to photo-excite the sys-tem not in the strongly detuned regime, but close to ε = 0 meV, to limit the time spent in a hybridizedcharge state and exposed to the strong nuclear spin fieldin the SAQD, as well as to minimize the probability ofradiative recombination. Specifically, we choose photo-excitation to occur at excitation point ε EP = − .
035 meV(see dashed-line in Fig. 3). At this point, the two states | Ψ (cid:105) and | Ψ (cid:105) have the same bright-states contributionBC = 20 . ε EP to the final value ε = 0 .
25 meV at the constant speed1 /v ε = 0 .
70 ns / meV is T tr = 0 .
20 ns. On this timescale,radiative recombination only marginally limits the prob-ability of a successful transfer, being P rec = 1 . | Ψ (cid:105) and 0 .
8% for the state | Ψ (cid:105) (see Fig. 3d). -0.2 -0.1 0 0.1 0.200.30.6-0.200.2-0.200.2 BC |↓◦⇓⟩ x |↓◦⇑⟩ x |↑◦⇓⟩ x |↑◦⇑⟩ x E n e r g y ( m e V ) v ε ( n s / m e V ) ε EP AdiabaticConditionEigenenergiesSubspaces ε (meV)abc │ Ψ 〉│ Ψ 〉│ Ψ 〉│ Ψ 〉 - P r ec t (ns) ɛ (meV) Recombination
Quasi static noiseFast white noiseNuclear spin noiseTotal dephasing
Dephasing - P d e ph - f a il de FIG. 3. (color online). Performance of the protocol for trans-ferring information to a single spin qubits. All plots are for theparameters given in Tab. I with t c = 50 µ eV. a) Eigenenergiesof the coupled GDQD-exciton system, Eq. (9). The color scaleindicates the BC of each state. b) Same as in a), but now thecolor scale represents the two independent subspaces of theHilbert space. The bright spots indicate the detuning ε EP atwhich optical excitation occurs, as well as the branches cho-sen as basis-states for the information transfer-process (alsolabeled as | Ψ (cid:105) and | Ψ (cid:105) ). c) Plot of the inverse maximalsweep-speed 1 /v ε that allows adiabatic evolution along thebranches | Ψ (cid:105) (red) and | Ψ (cid:105) (blue). d) Probability thatno optical-recombination occurs during the adiabatic transferalong the branches | Ψ (cid:105) (red) and | Ψ (cid:105) (blue). e) Probabil-ity of completing the adiabatic transfer without dephasing.Different noise sources are separately accounted. The finalvalues are displayed on the right. t is the time elapsed fromthe beginning of the protocol. The probability of concluding the transfer without de-phasing is shown in Fig. 3e. The results of Fig. 3e cor-respond to a probability of failure due to dephasing of P deph − fail = 0 .
25 %, and a total failure probability forthe transfer process of P fail − tot ≤ .
75 %.
III. INFORMATION TRANSFER TO ASINGLET-TRIPLET QUBITA. Transfer protocol
We consider now a different system, namely, instead ofthe coupling to a single-spin qubit, we consider the casewhere the electronic state in the OAQD is tunnel coupledto a gate-defined double quantum-dot (DD), see Fig. 1c.One of the advantages of this setup is that a DD can beused to encode a singlet-triplet qubit, which allows highmanipulation fidelities in systems with large hyperfineinteraction such as GaAs. As in Sec. II, we assume herethat the state of a photon is mapped onto an excitonin the OAQD in the presence of an in-plane magneticfield (cid:126)B = B(cid:126)e x , and discuss under which conditions thephoto-excited electron can be coherently transfered to aneighbouring DD. Furthermore, we assume that beforethe optical excitation, an electron is initialized in theleft side of the DD (i.e. in the one more far away fromthe OAQD, see Fig. 1c) by an appropriate choice of thedetuning ε DD .The Hamiltonian of the coupled DD-exciton sys-tem can be divided into two different subspaces: oneformed by states where the both photo-excited elec-tron and hole are localized on the OAQD, whichwe call excitonic states (ES), and the other formedby states where the photo-excited electron has beentransferred to the DD, which we call separates states(SS). The subset of excitonic states is spannedby the basis {|↑ ◦(cid:105) , |↓ ◦(cid:105)} ⊗ {|↓⇑(cid:105) , |↑⇓(cid:105) , |↑⇑(cid:105) , |↓⇓(cid:105)} ),while the subset of separated states is spanned by {| S(0 , (cid:105) | S(2 , (cid:105) , |↑↓(cid:105) , |↓↑(cid:105) , |↑↑(cid:105) , |↓↓(cid:105)} ⊗ {|◦ ⇑(cid:105) , |◦ ⇓(cid:105)} ,where the spin quantization axis is taken along the di-rection of the applied magnetic field. In this notation,the kets on the left represent the state of the DD, with | S(0 , (cid:105) ( | S(2 , (cid:105) ) representing the singlet state with thetwo electrons in the right (left) side of the DD, and |↑↓(cid:105) , |↓↑(cid:105) , . . . representing states with one electron oneach side of the DD. The two subsets of excitonic andseparated states are connected by a spin-conserving tun-nel Hamiltonian, T , with coupling strength t c . The totalHamiltonian of the DD-exciton system then reads H = (cid:18) ε · + H ES TT † H SS (cid:19) , (17)where H ES and H SS are the Hamiltonians acting on theES and SS subspaces, respectively, and ε is the energydetuning between the two subspaces. The expressionsfor H ES and H SS are given in Appendix B.When occupied by two electrons, a DD can be op-erated as s singlet-triplet (ST) qubit, using the singlet | S (cid:105) = √ ( |↑↓(cid:105) − |↓↑(cid:105) ) and triplet | T (cid:105) = √ ( |↑↓(cid:105) + |↓↑(cid:105) )as states of the computational basis. The remainingtriplet states | T + (cid:105) = |↑↑(cid:105) and | T − (cid:105) = |↓↓(cid:105) are split off inenergy by the external magnetic field. The most relevantenergy scale for the operation of a singlet-triplet qubit isthe energy splitting J = E S − E T between the singlet | S (cid:105) and the triplet | T (cid:105) . It depends on the inter-dot tunnelcoupling t DD (see Fig. 1c), as well as the inter-dot detun-ing ε DD , which is an easily accessible parameter. A straightforward extension of the information-transfer process described in Sec. II to the case of a DDwould work as follows: |↑ ◦(cid:105) | ω , V (cid:105) photo-excit. −→ |↑ ◦(cid:105) |↓⇑(cid:105) adiabatic transf. −→ |↑↓(cid:105) |◦ ⇑(cid:105) , |↑ ◦(cid:105) | ω , H (cid:105) photo-excit. −→ |↑ ◦(cid:105) |↑⇑(cid:105) adiabatic transf. −→ |↑↑(cid:105) |◦ ⇑(cid:105) , representing only the evolution of the basis states). Thestate |↑↓(cid:105) |◦ ⇑(cid:105) can then be easily mapped onto the state | T (cid:105) |◦ ⇑(cid:105) by adiabatically increasing the exchange in-teraction in the DD (i.e. by increasing J ). How-ever, mapping |↑↑(cid:105) |◦ ⇑(cid:105) onto | S (cid:105) |◦ ⇑(cid:105) , which is the otherstate of the computational basis, requires some spin-non-conserving mechanism such as, for example, the Over-hauser field or the spin-orbit interaction. These effectsintroduce an anti-crossing between the states | S (cid:105) and | T + (cid:105) = |↑↑(cid:105) in the regime where the exchange splitting J approximately equals the Zeeman splitting. This an-ticrossing can be used to transform |↑↑(cid:105) into | S (cid:105) , howeverthis approach would suffer from strong charge dephasing,since T ∗ ∝ ( dJ/dε DD ) − and dJ/dε DD is fairly large atthe S-T + transition. Furthermore, the phase acquiredduring the process would also depend strongly on thehyperfine field.For this reason we take a different approach, which isbased on exploiting the exchange interaction between theelectron initialized in the left part of the DD and the holein the OAQD. This interaction stems from the combina-tion of t c , the exciton coupling in the OAQD, and theexchange interaction J in the DD, according the schemeschematically represented in the following diagram: |↓ ◦(cid:105) |↓⇓(cid:105) t c ←→ |↓↓(cid:105) |◦ ⇓(cid:105) ∆ (cid:48)(cid:48) (cid:108)|↓ ◦(cid:105) |↑⇑(cid:105) t c ←→ |↓↑(cid:105) |◦ ⇑(cid:105)(cid:108) J |↑ ◦(cid:105) |↓⇑(cid:105) t c ←→ |↑↓(cid:105) |◦ ⇑(cid:105) ∆ (cid:48) (cid:108)|↑ ◦(cid:105) |↑⇓(cid:105) t c ←→ |↑↑(cid:105) |◦ ⇓(cid:105) (18)for one of the two independent subspaces that forms theHilbert space of the system (see Appendix B). Here, eacharrow reflects a coupling term, with ∆ (cid:48) = − + ∆ − ∆ , ∆ (cid:48)(cid:48) = − − ∆ +∆ . The energy eigenstates corre-sponding to this subspace are plotted in Fig. 4a. This ex-change interaction between the electron initialized in theleft side of the DD and the hole in OAQD creates an indi-rect coupling between the states |↓ ◦(cid:105) |↓⇑(cid:105) and |↑ ◦(cid:105) |↓⇓(cid:105) and more generally, between the T + -like and the S-likebranches in Fig. 4a. Exploiting this coupling, it is possi-ble to induce transitions between these two branches byapplying a suitable ac-modulation of the detuning ε . -1 -0.5 0 0.5 1 1.52.251.50.750-0.3-0.2-0.10-0.3-0.2-0.100.10.2 BC VP E n e r g y ( m e V ) T i m e ( n s ) E n e r g y ( m e V ) ε EP ε * ε (meV) ε ε DD abc |T − ⟩|◦⇓⟩|S(2,0)⟩|◦⇑⟩|T ⟩|◦⇑⟩|S ⟩|◦⇑⟩|T + ⟩|◦⇓⟩ | ↓◦ ⟩ | ↓ ⇓ ⟩ | ↑◦ ⟩ | ↓ ⇑ ⟩ | ↓◦ ⟩ | ↑ ⇑ ⟩ | ↑◦ ⟩ | ↑ ⇓ ⟩ Rabidriving
FIG. 4. (color online). Schematic of the protocol fortransferring information to a singlet-triplet qubit. a) En-ergy levels of the subspace sketched in Eq. (18), as a func-tion of the detuning for the parameter set ( t c , t DD , ε DD ) =(150 µ eV , µ eV , − .
03 meV). The remaining parameters aregiven in Tab. I. The colorscale represents the bright-state con-tribution of the various branches. The labels close to eachbranch indicate the main contribution to the eigenstates in thecorresponding regime. The bright spots indicate the photo-excited branches. b) Same as above, with the color scalenow representing the vertical-polarization contribution of therelevant protocol branches. c) Sketch of the pulse sequenceinvolved in the protocol. First the system is detuned to thevalue ε = ε EP , where the optical excitation occurs. Then ε is adiabatically swept to the driving point ε ∗ , where a Rabi π -pulse is applied. At the end of the pulse, ε is further in-creased to large positive values, into the regime of separatedstates. The inter-dot detuning of the double dot ε DD is keptnegative until the end of the Rabi pulse to confine the elec-tron initialised in the DD on the left dot. At the end of thepulse, ε DD is adiabatically swept to zero. This fact can be used to transfer information encodedinto photons with different energy but the same polar- ization according to the following scheme: |↑ ◦(cid:105) | ω , V (cid:105) photo − excit . → |↑ ◦(cid:105) |↓⇑(cid:105) adiabatic transf. → | T (cid:105) |◦ ⇑(cid:105) , |↑ ◦(cid:105) | ω , V (cid:105) photo − excit . → |↑ ◦(cid:105) |↑⇓(cid:105) Rabi + ad. transf. → | S (cid:105) |◦ ⇑(cid:105) , (19)where again we only represent the evolution of the basisstates. The idea is the following. First, the system is op-tically excited at a certain value of the detuning ε = ε EP ,transferring the state of the photon into the sub-space ofexcitons with anti-parallel spins. Then ε is swept to thedriving point ε ∗ , where a Rabi pulse is applied to drivethe transition between the T + -like branch and the S-likebranch. During the whole procedure the double-dot de-tuning ε DD is set to finite negative values to provide alarge enough J and to prevent tunnelling of the electroninitialized in the left part of the DD to the right part.After the Rabi pulse, ε DD is swept to zero and ε is tunedto the regime of separated states, thus mapping the stateinto the subspace spanned by the computational basis {| S (cid:105) , | T (cid:105)} ⊗ |◦ ⇑(cid:105) . The pulse scheme for ε and ε DD re-quired for such a protocol is sketched in Fig. 4c. In thediscussion above we assumed that the DD is initializedin the |↑ ◦(cid:105) state (which can be achieved with standardprocedures), however the protocol can be easily adaptedto match the cases where the DD is initially in the state |↓ ◦(cid:105) and/or to the case where the photon has horizontalpolarization. For the sake of clarity, in the following wefocus only on the case described above. B. Feasibility of the transfer protocol
The transfer protocol described above depends on thechoice of a number of parameters. The excitation point ε EP has to be chosen in such a way that the photon polar-ization prevents the excitation of states other than thoseconsidered in Eq. (19). To do so, we consider the degreeof vertical polarization of each state, i.e. the projectionon the sub-space with excitonic states with antiparallelspin: VP(Ψ) = |(cid:104) Ψ | ↑⇓(cid:105) x | + |(cid:104) Ψ | ↓⇑(cid:105) x | . This quantityis plotted in Fig. 4b for the the relevant branches. Wechose ε EP by requiring VP = 20% for the S-like branchat the excitation point, to limit direct excitation of thisbranch in combination with energy selectivity, which willhelp to achieve a high fidelity.The next parameter to be fixed is the position of thedriving point ε ∗ , which has to be chosen such as to al-low an efficient Rabi π -pulse, i.e. a pulse that drivesthe transition between the T + -like branch and the S-like branch in the the shortest possible time. For a two-level system driven by a rectangular pulse of the form ε ( t ) = ε ∗ + ∆ ε ( t ) cos( ω d t ), with ∆ ε ( t ) = ∆ ε for time t ∈ [ t ∗ , t ∗ + T Rabi ] and zero otherwise, the probabilityof transition | a (cid:105) → | b (cid:105) is given by the well-known Rabi -0.5 0 0.5 10120.751.50.050.1-0.200.2 E n e r g y ( m e V ) v ɛ ( n s / m e V ) T R a b i ( n s ) a b ε EP ε * T + T ST − S(2,0)
TransitionelementsEnergylevelsRabi pulsedurationAdiabaticcondition ε (meV)abcd
1% 5%2%0.5%0.25%
FIG. 5. (color online). Analysis of the protocol pre-sented in Fig. 4. The parameter for these plots aregiven in Tab. I, with the additional choice ( t c , t DD , ε DD ) =(150 µ eV , µ eV , − .
03 meV). a) Energy levels of the sub-space sketched in Eq. (18). Different branches are indicatedby different colours. The left dashed-line marks the excitationpoint ε EP = 0 .
09 meV and the right line the driving point ε ∗ = 0 .
17 meV. b) Amplitude of the transition-matrix ele-ments λ ab , from the black (T + ) branch to the other branches.The colour code is the one defined in panel a. c) In this panel,the red curve represents the condition T Rabi = π (cid:126) / (∆ ελ T + S ),with ∆ ε chosen such that the coupling λ T + S does not varymore than 50% in the range [ ε − ∆ ε, ε + ∆ ε ]. The thin greycurves represent selected levels of constant leakage probabil-ity, P Rabi − leak = P Ψ T+ → Ψ T0 = 5% down to 0 . + -like branch (black) and alongthe S- and the T -like branches (red-blue), calculated accord-ing to Eq. (13). formula P a → b ( δ ab , Ω ab ) = Ω ab Ω ab + δ ab sin (cid:18)(cid:113) Ω ab + δ ab T Rabi (cid:19) , (20)where δ ab = ω d − ω ab , with (cid:126) ω ab = E a − E b , is thedetuning of the driving and (cid:126) Ω ab = ∆ ελ ab the Rabifrequency, with λ ab = (cid:10) a (cid:12)(cid:12) ∂H∂ε (cid:12)(cid:12) b (cid:11)(cid:12)(cid:12) ε ∗ the transition ma-trix element between the states. The conditions for a π -pulse are therefore δ ab = 0 (resonant driving) and T Rabi = π/ Ω ab = π (cid:126) / (∆ ελ ab ). For the case of the tran-sition between the T + - and the S-like branches, the cou-pling element λ T + S depends on ε as shown by the red- curve in Fig. 5b. For each value of ε , we fix the drivingamplitude ∆ ε by requiring that λ T + S does not vary morethan 50% in the detuning range [ ε − ∆ ε, ε + ∆ ε ]. Thecorresponding time required for a Rabi π -pulse is plottedas a red curve in Fig. 5c. The minimum of this curvegives the optimal point (cid:15) ∗ to apply the Rabi pulse. Forthe case of Fig. 5, it is ε ∗ = 0 . T Rabi = 0 . ω d = 2 π · . -like branch, which is energetically very close to theS-like one, and has similar coupling matrix elements. Toestimate this leakage, we consider the three-level systemformed by the states | Ψ S (cid:105) , | Ψ T + (cid:105) and | Ψ T (cid:105) , which rep-resent the S-like, the T + -like and the T -like branchesat the excitation point (cid:15) ∗ , and evaluate the transitionprobability P i → j ( T Rabi ) = (cid:12)(cid:12)(cid:12) (cid:104) j | e − i (cid:126) (cid:82) T Rabi0 H RWA ( t ) dt | i (cid:105) (cid:12)(cid:12)(cid:12) . (21)Here, H RWA is the the Hamiltonian of the system in therotating frame with respect to the drive with the rotatingwave approximation H RWA ( t ) = (cid:126) ω ST + − ω d ) Ω ST + ( t )Ω ST + ( t ) 0 Ω T + T ( t )0 Ω T + T ( t ) 2( ω T T + + ω d ) . The grey curves in in Fig. 5c show selected levels ofconstant leakage, P Rabi − leak = P Ψ T+ → Ψ T0 = 5% downto 0 . P Rabi − leak = 0 . P LZ = 1%. The (inverse) maxi-mal sweep-speed, 1 /v ε , is plotted in Fig. 5d. In the fol-lowing we assume for simplicity a constant sweep-speedfrom the excitation point ε EP = 0 .
09 meV to the drivingpoint ε ∗ = 0 .
17 meV, and from here to the final detun-ing ε = 1 . /v ε = 1 .
45 ns / meV , which then correspond to a trans-fer time of 0.12 ns from the excitation point ε EP to thedriving point ε ∗ , and of 1.93 ns from the driving point tothe final detuning ε = 1 . ε DD isswept to zero at the end of the Rabi-pulse (see Fig. 4),separating the S(2 , -like and the T + -like bands. In the ideal case of adia-batic evolution during the sweeps and of a perfect Rabipulse, the basis states | Ψ (cid:105) , | Ψ (cid:105) (where we use the no-tation of App. A) evolve as follows: | Ψ ( t ) (cid:105) = | Ψ T ( t ) (cid:105) - P r ec E n e r g y ( m e V ) - P d e ph - f a il Dephasing
Quasi static noiseFast white noiseNuclear spin noiseTotal dephasing
RecombinationEnergy levels t * t * + T Rabi abc t (ns) T + T ST − S(2,0)S(2,0)
Rabi drive
FIG. 6. (color online). Performance analysis of the protocolpresented in Fig. 4. a) Schematic evolution of the energies ofthe relevant protocol branches as a function of time. Dashedlines represent unpopulated branches, while full-lines popu-lated ones. The color code is the same as in Fig. 5a. b)Probability of completing the transfer without recombinationfor the S-like branch (red) and the T -like branch (blue). c)Failure probability due to dephasing due to different noisesources. for any time, and | Ψ ( t ) (cid:105) = | Ψ T + ( t ) (cid:105) for 0 < t < t ∗ , | Ψ ( t ) (cid:105) = e iω d t cos (cid:18) Ω T + S t (cid:19) | Ψ T + (cid:105) ε ∗ +sin (cid:18) Ω T + S t (cid:19) | Ψ S (cid:105) ε ∗ for t ∗ < t < t ∗ + T Rabi and, finally, | Ψ ( t ) (cid:105) = | Ψ S ( t ) (cid:105) for t > t ∗ + T Rabi , where again | Ψ T (cid:105) , | Ψ T + (cid:105) and | Ψ S (cid:105) , repre-sent the T -like, the T + -like and the the S-like branches,respectively. Assuming this time evolution, we evaluatethe probability of recombination, Eq. (14), as well as thefailure probability due to dephasing, Eq. (16), during theprotocol. The results are plotted in Fig. 6b-c. The proba-bility of recombination lies between P rec = 7 . − . P deph − fail = 1 .
1% and it is dominated by nuclear-spinnoise.Finally, we take into account that the performanceof the Rabi pulses is also affected by charge and spinnoise, as they both affect the resonance condition, aswell as the Rabi frequency. We implement this by nu-merically calculating the average transition probability.Assuming that all noise sources are quasi-static and un-
TABLE II. Performance of the presented protocols.Single-spin qubit singlet-tripletFig. 2 Fig. 3 Fig. 6 P LZ
1% 1% 1% P rec − . . − . . − . P deph − fail .
4% 0 .
3% 1 . P Rabi − leak - - 0 . P Rabi − fail - - 0 . P success > . > . > . correlated, we calculate the failure probability due toeach source of noise separately. For example, fluctu-ations of the detuning ε cause the failure probability P Rabi − fail ,δε = 1 −(cid:104) P Ψ T+ → Ψ S ( δ δε , Ω δε ) (cid:105) δε , with P Ψ T+ → Ψ S as given by Eq.(20) and δ δε = ∂ω ST + ∂ε δε, (cid:126) Ω δε = ∆ ε (cid:28) Ψ S (cid:12)(cid:12)(cid:12)(cid:12) ∂H∂ε (cid:12)(cid:12)(cid:12)(cid:12) Ψ T + (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ε ∗ + δε . The average (cid:104)·(cid:105) δε is calculated assuming zero-mean Gaus-sian fluctuations of δε . In a similar way we calculate thefailure rate of the Rabi pulse due to fluctuations in theinter-dot detuning δε DD , as well as in the Overhauserfield in the different dots δ ˜ B (L)OF , δ ˜ B (R)OF , and δB OF . Fi-nally, we sum over all these failure probabilitys to es-timate the reliability of the Rabi pulse. For the set ofparameters used in Fig. 6 we obtain P Rabi − fail = 0 . P success > . IV. CONCLUSIONS
We presented a feasibility analysis of two protocols fortransferring information from a photonic qubit to qubitsrealized in GDQDs, considering both the cases of single-spin qubits, and of singlet-triplet qubits. The protocolsare based on using an OAQD as interface between thephotonic and the spin qubit. Our analysis is based oneffective Hamiltonian models for describing the hybridsystems formed by a bound exciton in the OAQD that istunnel coupled to a single or to a double GDQD. We fo-cus in particular on the error sources that can affect thetransfer process. Specifically, we take into account theinfluence of the adiabatic transfer conditions, the recom-bination of the exciton, the decoherence due to chargeand nuclear-spin noise, as well as the inaccuracy of theRabi pulse needed for the case of information transferto a singlet-triplet qubit. We use simple noise models,1which are expected to account for the most importanteffects.As a concrete example, we consider the case wherethe OAQD is realized by a InAs SAQD. We find thatfor the realistic set of parameters summarized in Tab. I,the single-spin protocol can be completed within thecoherence time with a success probability in the range > . P success > . as samples withbetter performances have been reported. Furthermore,the OF field fluctuations can be further reduced usingdynamic nuclear polarization with feedback.
Wedid not address specifically the important but setupspecific issue of how to ensure optimal optical couplingbetween the photonic qubit and the OAQD. We also didnot consider the additional constrains that might occurin an experimental implementation of our protocols(e.g. achievable sweep-speed, idle and pulse rise times),though the values obtained are compatible with high-endequipment. On the other hand our estimates are conser-vative and leave substantial room for improvement (e.g.nonlinear sweeps and pulse shaping). We thus expectthat the the proposed schemes could be implementedwith reasonably high success probabilities.
V. ACKNOWLEDGMENTS
This work was supported by the Alfried Krupp vonBohlen und Halbach Foundation and the European Re-search Council (ERC) under the European Union’s Hori-zon 2020 research and innovation programme (grantagreement No. 679342). P.C. acknowledges support byDeutsche Telekom Stiftung.
Appendix A: Dephasing noise
The relative phase ϕ accumulated in a certain time t between two states | Ψ (cid:105) and | Ψ (cid:105) is given in Eq.(15),where ∆ E is the energy difference between the twostates. Any physical mechanism that causes random fluc-tuations of ∆ E leads to random fluctuations δϕ of therelative phase,inducing dephasing in a superposition of | Ψ (cid:105) and | Ψ (cid:105) . For the case of zero-average Gaussiannoise, the induced dephasing is quantified with e −(cid:104) δϕ (cid:105) / (A1)where (cid:104) δϕ (cid:105) is the variance of the phase fluctuations.
1. Charge noise
Charge noise introduces stochastic fluctuations of thedetuning, ε (cid:55)→ ε + δε , where δε is a randomly fluctuatingquantity, and therefore of the energy difference∆ E ( ε + δε ) ≈ ∆ E ( ε ) + ∂ ∆ E ( ε ) ∂ε δε, (A2)and in the accumulated phase δϕ = (cid:90) t δε ( t (cid:48) ) (cid:126) χ ( t (cid:48) ) dt (cid:48) , (A3)with χ ( t ) = ∂ ∆ E ( t ) /∂ε . The variance of the phasefluctuations induced by charge noise can be written as (cid:104) δϕ (cid:105) = (cid:90) t dt (cid:90) t dt (cid:104) δε ( t ) δε ( t ) (cid:105) (cid:126) χ ( t ) χ ( t ) , (A4)where the angle-bracket represents the statistical aver-age over all realizations of δε . The correlation function (cid:104) δε ( t ) δε ( t ) (cid:105) is nothing but the Fourier transform of thenoise spectral density S ε ( ω ): (cid:104) δε ( t ) δε ( t ) (cid:105) = 12 π (cid:90) ∞−∞ dω e − iω ( t − t ) S ε ( ω )2 , (A5)where the additional factor takes into account that weuse the one-sided spectral density.Here we consider two types of charge noise: quasi-staticcharge noise and fast, uncorrelated charge noise. Thefirst type represents charge fluctuations that occur ontime scales much longer than the transfer time T tr , sothat the charge background is essentially static duringeach transfer. In this case S ε ( ω ) = 4 πε δ ( ω ) , where ε rms is the root-mean-squared fluctuation in ε , which gives for the variance of phase fluctuations (cid:104) δϕ (cid:105) ch − qs = ε (cid:126) (cid:18)(cid:90) t χ ( t (cid:48) ) dt (cid:48) (cid:19) . (A6)Vice versa, fast uncorrelated noise has equal contribu-tions at all frequencies, i.e. S ε ( ω ) = S ε = const. (whitenoise). In this case the variance of phase fluctuations isgiven by (cid:104) δϕ (cid:105) ch − fast = S ε (cid:126) (cid:90) t χ ( t (cid:48) ) dt (cid:48) . (A7)The quantity χ ( t ) can be evaluated as follows. Takinginto account that the detuning ε enters in the Hamilto-nian Eq. (9) as: H ε = ε (cid:88) i | ES i (cid:105) (cid:104) ES i | − ε (cid:88) i | SS i (cid:105) (cid:104) SS i | , (A8)2where | ES i (cid:105) ∈ {|◦ ↓⇑(cid:105) x , |◦ ↑⇓(cid:105) x , |◦ ↑⇑(cid:105) x , |◦ ↓⇓(cid:105) x } and | SS i (cid:105) ∈ {|↓ ◦ ⇑(cid:105) x , |↑ ◦ ⇓(cid:105) x , |↑ ◦ ⇑(cid:105) x , |↓ ◦ ⇓(cid:105) x } , the quan-tity χ ( t ) becomes χ ( t ) = ∂ ∆ E ( t ) ∂ε = (cid:104) Ψ ( t ) | ∂H ε ∂ε | Ψ ( t ) (cid:105) − (cid:104) Ψ ( t ) | ∂H ε ∂ε | Ψ ( t ) (cid:105) = (cid:88) i |(cid:104) ES i | Ψ ( t ) (cid:105)| − |(cid:104) ES i | Ψ ( t ) (cid:105)| . (A9)For the case of the singlet-triplet qubit, one has alsoto take into account fluctuations in the double-dot de-tuning ε DD . Assuming the fluctuations in ε and ε DD to be uncorrelated, the total variance of phase fluctu-ations due to charge noise is given by (cid:104) δϕ (cid:105) charge = (cid:104) δϕ (cid:105) ch − qs + (cid:104) δϕ (cid:105) ch − fast + (cid:104) δϕ (cid:105) ε DD − qs + (cid:104) δϕ (cid:105) ε DD − fast .Here, (cid:104) δϕ (cid:105) ε DD − qs and (cid:104) δϕ (cid:105) ε DD − fast have the same struc-ture as Eq.(A6) and Eq.(A7), but with χ ( t ) replaced by χ ε DD ( t ) = ∂ ∆ E ( t ) ∂ε DD = (cid:12)(cid:12)(cid:12) (cid:104) ˜S(2 , | Ψ (cid:105) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (cid:104) ˜S(2 , | Ψ (cid:105) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) (cid:104) ˜S(0 , | Ψ (cid:105) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) (cid:104) ˜S(0 , | Ψ (cid:105) (cid:12)(cid:12)(cid:12) , where | ˜S(2 , (cid:105) = | S(2 , (cid:105) ⊗ ( |◦ ⇑(cid:105) + |◦ ⇓(cid:105) ), and similarlyfor | ˜S(0 , (cid:105) .Finally we note that, in principle, one should alsomodel the effects of 1 /f -noise. However, these canalso be taken into account with reasonable accuracy bychoosing the white noise level S ε such that it correspondsto the 1 /f -noise level in the frequency range relevant for the experiment.
2. Nuclear-spin noise
GaAs is a III-V semiconductor which exhibits a nu-clear spin field that interacts with the spin of the chargecarriers via hyperfine interaction. Since the number ofnuclear spins interacting with an electron (or a hole)is very large (typically 10 to 10 ), their effect can beconveniently described in terms of an effective magneticfield, the Overhauser field (OF). Here we consideronly the OF component parallel to the external magneticfield (i.e. along the x axis), as this is the one that moststrongly affects the dynamics of the spin of the chargecarriers (transverse OF components give only higher or-der contributions). With respect to the basis Eq.(8),the effective Hamiltonian of the hyperfine interaction canthen be written as H OF = µ B (cid:18) B OF ( g e S x + ηg h J x ) 00 ˜ B OF ˜ g e S x + ηB OF g h J x (cid:19) . (A10)Here, S x and J x are spin operators for electrons andheavy holes, with eigenvalues + ( − ) for states withspin-up (spin-down). Terms with a tilde ( ˜ B OF , ˜ g e ) takein to account the fact that an electron can experienceboth different g -factor and different Overhauser field inthe OAQD and in the GDQD. The factor η accountsfor the different OF experienced by the electron and thehole in the OAQD. The block-diagonal structure re-flects again the separation between excitonic states andseparated-states.As the OF fluctuates randomly in time, it causes fluc-tuation in the energy difference ∆ E . To lowest orderin the fluctuations it is∆ E ( B OF + δB OF , ˜ B OF + δ ˜ B OF ) ≈ ∆ E ( B OF , ˜ B OF ) + ∂ ∆ E ∂B OF δB OF + ∂ ∆ E ∂ ˜ B OF δ ˜ B OF . (A11)Spin fluctuations can be considered as quasi-static noise. Assuming the fluctuations δB OF and δ ˜ B OF to be uncorre-lated, we get for the variance of the phase fluctuations induced by spin noise the following result: (cid:104) δϕ (cid:105) spins = µ (cid:104) δB (cid:105) (cid:126) (cid:18) (cid:90) t ∂ ∆ E ( t ) ∂B OF (cid:19) + µ (cid:104) δ ˜ B (cid:105) (cid:126) (cid:18) (cid:90) t ∂ ∆ E ( t ) ∂ ˜ B OF (cid:19) . (A12)The root-mean-square fluctuations in the nuclear spinfield are referred to as B (rms)OF = (cid:112) (cid:104) δB (cid:105) and ˜ B (rms)OF = (cid:113) (cid:104) δ ˜ B (cid:105) in Tab. I.In the singlet-triplet qubit case we proceed analo-gously, but we have to account for the OF in the two parts of the DD, i.e. we replace the term ˜ B OF S x in Eq. (A10)by ˜ B (L)OF S (L) x + ˜ B (R)OF S (R) x . We furthermore assume the fluc-tuations in B OF , ˜ B (L)OF and ˜ B (R)OF to be all independent.3
3. Failure probability due to dephasing
In order to compare the effects of dephasing to othereffects that lead to the failure of the transfer process, weintroduce the failure probability due to dephasing , whichwe define as the probability of the depolarizing channelwith the same average gate fidelity.The average gate fidelity for a two-level system can becalculated as F ( E , U ) = 12 + 112 (cid:88) k =1 tr (cid:0) U σ k U † E ( σ k ) (cid:1) , (A13)where σ k are the Pauli matrices, U is a quantum gate,and E is a trace preserving quantum operation that ap-proximate U . If F ( E , U ) = 1, then E implements U perfectly, while F ( E , U ) < E is a noisy(or otherwise imperfect) implementation of U . Withthis definition, the infidelity between the phase gate U δϕ = exp( − i δϕ σ z ) and the identity operation becomes1 − F ( U δϕ , ) = 13 −
13 cos( δϕ ) . (A14)If δϕ is a normal distributed random variable with zeromean, the expectation value of the infidelity is (cid:104) − F ( U ( δϕ ) , ) (cid:105) = 13 − (cid:104)
13 cos( δϕ ) (cid:105) ≈ (cid:104) δϕ (cid:105) . (A15)A depolarizing channel is defined by E ( ρ, P ) = P − P ) ρ, (A16)where P is the depolarization probability. The infidelityof the depolarizing channel is simply1 − F ( E ( ρ, P ) , ) = P/ . (A17) Comparing this result with (A15), we define the failureprobability due to dephasing as P deph − fail = (cid:104) δϕ (cid:105) . (A18)Here and above δϕ represents the phase fluctuationdue to all different noise sources. Since we assume thelatter to be uncorrelated, it is (cid:104) δϕ (cid:105) = (cid:104) δϕ (cid:105) ch − qs + (cid:104) δϕ (cid:105) ch − fast + (cid:104) δϕ (cid:105) spins . Appendix B: Hamiltonians H ES and H SS Here we give explicit expressions for the Hamil-tonians H ES and H SS entering, Eq.(17). The firstterm, H ES , represents the projection of the Hamilto-nian of the coupled DD-exciton system on the sub-space of excitonic states spanned by {|↑ ◦(cid:105) x , |↓ ◦(cid:105) x } ⊗{|↓⇑(cid:105) x , |↑⇓(cid:105) x , |↑⇑(cid:105) x , |↓⇓(cid:105) x } ). It is given by H ES = H ⊗ ex + ⊗ H ex (B1)where H = µ B B x (cid:18) ˜ g e − ˜ g e (cid:19) (B2)represents the Zeeman Hamiltonian of the single electroninitialized in the DD, and H ex is given in Eq.(2).Similarly, H SS represents the projection of theHamiltonian of the coupled exciton-DD systemon the subspace of separated states spanned by {| S(0 , (cid:105) x | S(2 , (cid:105) x , |↑↓(cid:105) x , |↓↑(cid:105) x , |↑↑(cid:105) x , |↓↓(cid:105) x } ⊗{|◦ ⇑(cid:105) x , |◦ ⇓(cid:105) x } , and it is given by H SS = H ⊗ + ⊗ H . (B3)Here H = − ε DD + U − t DD t DD ε DD + U − t DD t DD − t DD − t DD V − + V + V − − V + t DD t DD V − − V + V − + V + V − + ˜ g e µ B B x
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