Multi-spin errors in the optical control of a spin quantum memory
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Multi-spin errors in the optical control of a spin quantum memory
Michal Grochol and Carlo Piermarocchi
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824 USA (Dated: December 3, 2018)We study a quantum memory composed of an array of charged quantum dots embedded in a planarcavity. Optically excited polaritons, i.e. exciton-cavity mixed states, interact with the electron spinsin the dots. Linearly polarized excitation induces two-spin and multi-spin interactions. We discusshow the multi-spin interaction terms, which represent a source of errors for two-qubit quantumgates, can be suppressed using local control of the exciton energy. We show that using detuningconditional phase shift gates with high fidelity can be obtained. The cavity provides long-range spincoupling and the resulting gate operation time is shorter than the spin decoherence time.
PACS numbers: 03.67.Lx, 71.36.+c, 78.20.Bh, 78.67.Hc a. Introduction
In the last few years there have beengreat advances towards quantum information processingin the solid state. Yet, there are many theoretical andpractical problems that remain to be addressed. In par-ticular, there is not yet a solid state systems for which allthe feasibility criteria for quantum computing (i.e. deco-herence, reliable one- and two-qubit operations, scalablequbit, initialization and read-out ) have been simultane-ously demonstrated. Lately, electron spins in semicon-ductors, localized either in low-dimensional nanostruc-tures, i.e. QDs or in impurities, are increasingly receiv-ing attention as qubits due to their very long decoherencetime, which is typically of the order of T = 3 µ s. The long coherence time of the electron spin is due toits weak interaction with the environment, which on theother hand makes its control more demanding. In thisframework, optical techniques are very promising sincein this case the control is realized using an optically ac-tive ancillary excited state, e.g. a trion state in quan-tum dots, leading to a control that can be obtainedin picoseconds. Optical initialization, single qubitmeasurement, and selective one-qubit control of QD’sspin have been already demonstrated. Similar exper-iments on impurity states have also been carried out.
The two-qubit control represents a more challenging task.Optically mediated long range spin-spin interaction in acavity system has been explored theoreticaly only for twoQDs.
In this paper, we show that an array of charged QDsembedded in a planar cavity (see Fig. 1) is a good candi-date for a controlable quantum memory. We extend theprevious works on polariton mediated spin coupling tothe case of many dots, which leads to the appeareance of multi-spin Ising-like coupling terms. We consider a sys-tem in which the energy of the ancillary states on eachdot can be controlled independently, for instance usinggates on each dot. We calculate the fidelity of the phasegate of two spins being in resonance with the cavity modeand show that by controlling the detuning of the remain-ing dots, gates with very small error can be obtained. Er-rors due to multi-spin terms in the case of quantum dotdirectly coupled by wavefunction overlap have also beenstudied recently. The model of multi-spin coupling is
FIG. 1: (a) Energy diagram with cavity photon dispersion,and detunig ∆ P and ∆ X discussed in the text. (b) Schemeof the quantum memory composed by charged quantum dotsin a planar cavity. Two dots brought into resonance with thecavity are highlighted. (c) Diagram of allowed spin configura-tions for a charged dot excited by circulartly polarized light.The distance between the trion energy in the two configura-tion defines an antiferromagnetic spin coupling between theelectron spin and the exciton spin (polarization). also applicable to similar systems like e.g. superconduct-ing qubits embedded in a cavity, for which the two-qubitcontrol has been demonstrated in a recent experiment. b. Polariton-Spin Hamiltonian Our assumptions forthe system studied are the following: (i) the trion energy∆
X,j of each dot can be independently controlled e.g.by applying a local voltage, (ii) the quantum dots arewell separated so there is not direct overlap of the trionwavefunction, (iii) each dot can be occupied only by oneadditional exciton, (iv) the heavy-hole light-hole splittingis large enough that only the heavy-hole exciton is takeninto account, and (v) the cavity is ideal. The role of thecavity is to enhance the range of the interaction betweendots and their spins. The Hamiltonian descibing thememeory can therefore be written as (¯ h = 1 throughoutthe paper)ˆ H g = X α n − X j ∆ X,j C † jα C jα + X qj ( g qj a qα C † jα + h.c. ) + X q ω q a † αq a αq o (1)+ X j J S S jz P jz + ˆ H L , where C † jα ( C jα ) is the creation (annihilation) opera-tor of exciton on the j th dot at position R j with po-larization α , a † αq ( a αq ) is the creation (annihilation) op-erator of the photon with two-dimensional momentum q , g qj = g e − q β e iqR j is the dot-photon coupling con-stant with β being the effective dot size, J S is the energydifference between trion states with parallel and anti-parallel spins as schematically shown in Fig. 1c, S jz isthe z -component of the electron spin in the j th QD, and P jz = C † j ↑ C j ↑ − C † j ↓ C j ↓ is the operator corresponding tothe z component of the exciton polarization. A σ + ( σ − )polarized photon creates a bright exciton with ↓ ( ↑ ) elec-tron spin in the growth ( z ) direction. For excitons in III-V confinded systems the possible values of the electronspin are σ ez = ± and the heavy hole spin are σ hhz = ∓ .We assume troughout the paper that the light is lin-early polarized. This choice simplifies considerably themulti-spin problems since it makes all multi-spin termsof odd order identically zero. The coupling of the cavityto the external electromagnetic field is described usingthe quasi-mode model ˆ H L = P αq ( V αq e iω L t a αq + h.c. ),where V αq is the laser-cavity coupling constant propor-tional to the cavity area ∼ √ A . c. Multi-Spin Hamiltonian The effective spinHamiltonian can be calculated introducing the level shiftoperator R ( ω L ) as ˆ H s = P R ( ω L ) P = P ˆ H L Q ω L − Q ˆ H Q ˆ H L P , (2)where P = P λ | λ ih λ |⊗ | ih | h Q = 1 − P = P λβ | λ ih λ |⊗| β ih β | i is the projection operator on the subspace of allspin states λ and zero [one] excitation. Assuming the ro-tating wave approximation and linearly polarized laserlight propagating perpendicularly to the cavity plane( q = 0) then ˆ H L = V ↓ a ↓ + V ↑ a ↑ + h.c. .By solving first the polariton problem for q = 0 andboth polarizations, we can write ˆ H P | α ↑ ( ↓ ) i = ω α | α ↑ ( ↓ ) i and | α ↑ ( ↓ ) i = (cid:0) P j u αj C † j ↑ ( ↓ ) + P Q v α Q a † ↑ ( ↓ ) (cid:1) | i where Q is a reciprocal lattice vector of the dot lattice.Then the matrix element between the spin state reads R λλ ′ = X αβ v α v ∗ β X γ = ↑ , ↓ V γ h αγ |h λ | ( ω L − ˆ H ) − | λ ′ i| βγ i . (3)The off-diagonal terms h λ | ( ω L − ˆ H ) − | λ ′ i are zero sinceall spin dependent terms are proportional to S z . This FIG. 2: Diagram illustrating multiple scattering events thatlead to a multi-spin coupling J (4)8 , , , , as derived in Eq. (6). allows us to calculate the energies in Eq. (3) exactly.Perturbation theory can also be applied and, assuminglinearly polarized light, only even contributions ( ∼ J (2 n ) S )are nonzero givingˆ H T = X i>j ˜ J (2) ij S iz S jz + X i>j>k>l ˜ J (4) ijkl S iz S jz S kz S lz + . . . , (4)where the coupling constants are renormalized to takeinto account multiple scattering, e.g.˜ J (2)12 = J (2)12 + J (2)21 + X i P J (4) P (12 ii ) + X ij P J (6) P (12 iijj ) + · · · , (5)where P indicates a permutation of all the indices. The z -coupling constants can be explicitely expressed as J ( n ) i ...i n = J nS V LP ( C − i ) ∗ T i i · · · T i n − i n C + i n (6)in terms of the photon-exciton coupling function and ex-citon inter-dot transfer probability (see scheme in Fig.2) C +( − ) i = X α v α u ∗ αi ω L − ω α ± iη , T ij = X α u αi u ∗ αj ω L − ω α + iη , (7)where η is the exciton and photon homogeneous broaden-ing, assumed identical for simplicity, and V LP = V ↑ + V ↓ is the effective light-polariton coupling constant.Let us now consider two dots labeled by { , } with asmall detuning with respect to the lowest cavity mode,i.e. ∆ X, = ∆ P . The remaining quantum dots aredetuned by a lager amount: ∆ X,j =1 , > ∆ P , as schemat-ically plotted in Fig. 1 (a). Dots shifted off-resonance bya DC Stark shift will also have a weaker light-dot cou-pling g due to the decrease of the electron-hole overlap.However, in order to have a conservative estimate of theerror we neglect this effect.We have used the following parameters: β = 35 nm, g = 70 µ eV, η = 50 µ eV, V ↑ = V ↓ = 0 . ω L = ω q =0 , J S = 0 .
39 meV, and exciton detuning up l og ( | J n | ) FIG. 3: (Color online) The logarithmic plot of J ( n ) R (solid)and J ( n ) O (dashed) [see text for details] as a function of thedetuning ∆ X in a 3 × P =1 meV and the lattice constants a = 100 nm for n = 2 (black), n = 4 (red), and n = 6 (blue) are shown. to ∆ X = 20 meV, which is about the upper limit for aStark shift that can be obtained in current experiments.In the numerical calculation we consider a finite systemwith 9 dots and we used periodic boudary conditions inorder to match the excitonic states in the dots with thecontinuous two dimensional photon modes.The dependence of different multi-spin terms on thedetuning is shown in Fig. 3 where we separate the termsthat involve the two dots nearly resonant with the cav-ity from the others. We plot J + J (solid black) and P ij / ∈{ , } | J ij | (black dashed) for n = 2 spin terms. Thecontributions, that renormalize the effective coupling be-tween 1 and 2 ( J ( n ) R ) from contributions that involve onlythe dots strongly detuned from the cavity ( J ( n ) O ), are sep-arated for multi-spin terms ( n = 4, n = 6). For instance,for n = 4 the resonant (off-resonant) terms are defined as J (4) R = P i P | J (4) P (12 ii ) | ( J (4) O = P ijkl | J (4) ijkl | − | J (4) R | ). Thisdefinition enables us to better estimate the contributionof the off-resonant terms. In fact, even if the magnitudeof the indivual terms J ( n ) i ..i n is very small (e.g. 10 − for J (6) ijklmn ) we get a sizeable effect due to the large num-ber of n -dot combinations ( ∼ (cid:0) nN D (cid:1) ). Note that there isalmost no dependence on the detuning for the resonantterms and a strong decrease for the off-resonant terms( J ( n ) ∼ ∆ − ( n − X ) as expected from the form of the cou-pling in Eq. (6). d. Fidelity of a conditional phase shift gate We willnow explicitely estimate the error in the implementa-tion of a conditional phase shift gate due to multi-spininteraction terms. The conditional phase gate (PG),is a universal two-qubit gate, i.e. can realize univer-sal quantum computation when combined with singlequbit operations. In the computational basis {| ↓↑i , | ↓↑i , | ↑↓i , | ↑↑i} the PG can be written as diagonal ma-trix with elements U P G = { , , , − } . Assuming theIsing-like interaction between two spins ∼ S z S z , the La tt i c e c on s t an t ( n m ) −4−3.5−3−2.5−2−1.5−1−0.5 FIG. 4: (Color online) Logarithmic plot of the error E as afunction of the detuning ∆ X and lattice constant a in a PhaseGate between two most distant dots in a 3 × P = 1 meV. following sequence gives the PG U P G = e iπ/ [ S z + S z ][ − S z S z ] with [ P ] = e iπ/ P . A quantitative mea-sure of the gate quality can be been given using the gate fidelity defined as F = |h Ψ | U † I U R | Ψ i| , where U I isthe ideal gate matrix, and U R is the real gate matrix,i.e. the one that includes the effects of multi-spin terms.Ψ is an arbitrary initial pure state, and |h Ψ | . | Ψ i| in-dicates averaging over all pure states. Working in thebasis of the full spin-Hamiltonian eigenstates { φ i } (with2 N D states), we can define an eigenvector fidelity as F i = h φ i | U † I U R | φ i i . Since the total Hamiltonian doesnot allow for spin flip processes, the fidelity can the beexpressed as F = | N D P i F i | . In order to calculate thefidelity, we calculate the dynamics exp( − iH R t C ), wherethe time t C is optimized so to obtain maximal fidelity.The gate can be described as follows: (i) two selecteddots { , } are brought adiabatically into resonace withthe cavity by controlling the exciton energy with localelectric field, (ii) the laser is switched on for a time t C ,and (iii) dots are brought back into the off-resonant state.The calculated error E = 1 − F as a function of thedetuning and lattice constant of the dot array is shownin Fig. 4. The fidelity F increases at larger detuning∆ X since only the two selected dots { , } remain in res-onance with the cavity and the multi-spin coupling withthe other dots is suppressed. On the other hand, in-creasing the lattice constant decreases the fidelity sincethe exciton transfer, even if considerably enhanced by thecavity, decreases with distance. The strong dependenceof the fidelity on the detuning reflects the competitionbetween the resonant and off-resonant terms as shown inFig. 3. Furthermore, note that the maximal value of theindividual dot detuning is limited by the inter dot sepa-ration a . Then the fidelity function F (∆ , a ) can be usedto select an optimal lattice constant.Another important characteristic of the PG is its op-eration time, i.e. the time during which the spin-interaction is switched on. The operation time increaseswith increasing detuning since the spin-spin coupling ∼ J decreases. Note that the time t C grows like ∆ P ,following the dependence of the resonant terms in Eqs.(6) and (7). Typical values of the operation times are t C = 100 ps ( t C = 450 ps) for a = 100 nm ( a = 1300 nm)[∆ X = 20 meV]. These characteristic gate times areshorter than the spin decoherence time T , which is oforders of at least µs . e. Conclusions We have studied an array of chargedquantum dots embedded in a planar cavity as a candi-date for the realization of a spin quantum memory. Wehave shown that optical excitation can be used to controlthe spins and implement quantum gates. The optical ex-citation couples many dots in the quantum memory, and multi-spin interaction terms beyond the ideal two-spininteraction are generated. We have shown that the multi-spin terms can induce errors in the gate operation evenif their value is small, due to their multiplicity. Theseerror can be corrected by a local control of the excitonicresonance on each dot. In the control scheme we alsoinclude a planar cavity that modifies the photon densityof states by providing a spectral region where dots donot couple to radiation. 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