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Single-qubit operations in the double-donor structure driven by optical and voltage pulses

We study theoretically the quantum dynamics of an electron in the singlyionized double-donor structure in the semiconductor host under the influence of laser pulses whose frequencies are close to structure resonant frequencies. This system can be used as a charge qubit where the logical states are defined by the lowest two energy states of the remaining valence electron localized around one or another donor. The quantum operations are performed via resonant or Raman-like transitions between the localized (qubit) states and the excited states delocalized over the structure, combined with phase shifts between qubit states generated by voltage pulses. The possibility of realization of arbitrary single-qubit rotations is demonstrated.


I. INTRODUCTION
In view of recent progress in the development of the controlled-donor implantation techniques [1], Kane's paradigm of the solid-state quantum computations [2] has gained new insights. The alternative schemes using the orbital [3,4] or spin [5,6] degrees of freedom of the donor-bounded electrons to encode the quantum information instead of the nuclear donor spins, have been proposed. Besides, several refinements of the original proposal concerning the initialization [7] and read-out [8,9,10,11] as well as the information transfer through the quantum networks [12,13,14] have been made.
In particular, the pair of donors sharing an electron has been considered as very promising candidate for the solid-state qubit embodiment [3]. The qubit is presented by the electron orbital states positioned at the different donors. There are two main driving mechanisms for the coherent electron evolution defining a quantum operation on such a qubit. First uses the electric fields through an application of the adiabatically switched voltages to the surface gates placed above the donor structure to modify the confinement potential, thus varying the electron tunneling rates between neighboring donors [3,15,16]. The desired state of the qubit is realized by an appropriate choice of the voltage parameters. The second scheme relies upon the optical dipole transitions between the size-quantized one-electron levels induced by the resonant pulses [17,18]. This technique requires one to irradiate the donors encapsulated into near-surface semiconductor layers by laser pulses with frequencies lying in terahertz range. Such kind of radiation sources is currently under extensive exploration. To all appearances, the quantum cascade lasers [19] in which photon generation takes place via electronic intersubband transitions in semiconductor heterostructures, will be able in near future to cover this frequency range. As it was shown, an arbitrary single-qubit operation can be achieved with two simultaneously switched pulses connecting the qubit states via the excitation of the state delocalized over the structure. The latter scheme is likely to be more preferable than the former due to its higher selectivity and lower field intensity.
The successful implementation of the quantum operations, however, requires one to provide the high precision in the durations, the frequencies, the polarizations and the strengths of the pulses together with reliable control over the delay time and the phase difference between the pulses. Besides, the use of the intermediate state lying in the neighborhood of the continuum introduces the decoherence caused by the ionization and the spontaneous emission from this state.
Here we propose the way to overcome the difficulties inherent to the electron charge manipulations by optical means. We show that the off-resonant laser pulses combined with electrostatic fields, can be used to generate an arbitrary rotation of the qubit-state vector as well as to drive an electron between the remote donors.
It is essential that this method is based on the Raman-like transitions between the localized electron states of an effective molecular ion where the excited (delocalized over the double-donor system) states irrespective of their number are used as the transport channels. As we shall see, the coherent electron dynamics is described by the simple analytical model. In order to check analytical results, we have performed numerical simulations on single-electron dynamics in the effective hydrogen molecular ion. This system is very popular object for modelling electronic properties of real donor structures (see, e.g., Refs. [18] and [20]). The eigenenergies and eigenfunctions of molecular ion were found from the stationary Schrödinger equation and then used to calculate the set of eigenfrequencies and the matrix elements of optical dipole transition between eigenstates of hydrogen molecular ion that define the field-structure interaction strengths. The time-dependent Schrödinger equation generating the coherent electron evolution in the external fields was integrated numerically for different values of structure and field parameters. By comparison of analytical expressions of the Rabi frequencies for both resonant and off-resonant cases, we found them to be in good agreement with those extracted from numerical solutions. This result confirms the validity of approximations made during the analytical treatment. The pulse and structure parameters needed for those types of quantum evolution may be evaluated from the results obtained in this study.
The paper is organized as follows. In Section II we describe general model of coherent electron dynamics in double-donor structure under the influence of nearresonant electromagnetic pulse. The Raman-like off-resonant electronic transitions are studied analytically in Section III. Besides, in this Section the possible realization of several basic single-qubit quantum operations is proposed. The results of the numerical study of electron dynamics in hydrogen molecular ion are given in Section IV. The advantages of the off-resonant driving scheme in comparison with the resonant scheme proposed earlier as well as some questions concerned with further exploration of optically-driven donor-based charge qubits, are discussed through the Section V. We conclude our paper by Section VI.

II. MODEL
We begin with the description of the one-electron double-donor (DD) structure ( Fig. 1 (a)). Let the donors A and B be placed on the axis z from each other at the distance R (hereafter -internuclear distance) large enough to consider their ground orbital states to be isolated. Due to this fact those states may be used as the qubit states |0 and |1 (for example, if an electron is localized on the donor A(B), the qubit is in the state |0 (|1 )). The energy difference ∆ = ε 1 − ε 0 may be introduced due to the structural asymmetry caused by the fabrication process and/or via the bias voltages generated by the surface gate V a . The coupling between the excited states of individual donors through the electron tunneling gives rise to the forming of hybridized states delocalized over the DD structure. However, if R ≫ a * B (a * B is the effective Bohr radius of host material) the low-lying excited states are hybridized weakly and do not participate the two-donor dynamics. We shall be interested therefore only in the excited states of individual donors whose wave functions considerably overlap. The single-electron spectrum of the DD structure is presented by the sequence of the states {|k } exc which for ∆ = 0 are the doublets composed of the symmetric and antisymmetric superpositions of isolated donor states. If ∆ = 0, the spectrum is assumed to be much more complex, as one can see from Fig. 1 (b). Taking into account the hydrogen-like spectrum of the isolated donors we expect the excited states close to the edge of the potential barrier separating the donors to have the quasi-continuous energy distribution.
In what follows we shall study the one-electron quantum dynamics involving the localized (qubit) states, |0 and |1 , and the states {|k } exc delocalized over the structure. Our aim is to choose the field and structure parameters so that to drive In the absence of an external field the DD structure is characterized by the stationary Hamiltonian H 0 with the eigenstates {|k } and the eigenenergies {ε k }: The eigenstates {|k } form the complete orthonormal set so that k |k k| = 1.
In the presence of the electromagnetic field the system Hamiltonian reads where e is the electron charge, E (t) is the field strength, r is the radius-vector of an electron. With the help of Eqs. (1) and (2) we rewrite the Eq. (3) in terms of the projection operators: where d km = k| − er |m is the matrix element of optical dipole transition between the states |k and |m . The state vector of the system may be presented in the form and is governed by the non-stationary Schrödinger equation with the initial condition |Ψ (t 0 ) = α 0 |0 + β 0 |1 (hereafterh ≡ 1).
Let the electromagnetic field imposed on the structure to have (in the dipole approximation) the form of two phase-locked pulses where the pulse envelopes E 0 (t) = E 0 f 0 (t) , E 1 (t) = E 1 f 1 (t) are the slowly-varying (compared to optical frequencies) time-dependent functions, ω 0,1 are the pulse fre-quencies, and ϕ 0,1 are the pulse phases. We require both pulses to be in the twophoton resonance with the DD structure, i.e., ε 0 + ω 0 = ε 1 + ω 1 or, alternatively, where δ 0(1)k = ω 0(1) − ω 0(1)k is the detuning of the pulse frequency ω 0(1) from the resonant frequency ω 0(1)k . Note, that we use two pulses only if ∆ = 0. In the symmetric structure, where ∆ = 0, the single pulse Making use of the rotating-wave approximation (that implies ∆ ≪ ω 0(1) and |δ k | ≪ ω 0(1) ), we obtain from Eq. (7) the following set: Eq. (10) describes the dynamical process involving many three-level excitation schemes that act in parallel. Each of them is characterized by the set of parameters ∆, λ 0(1)k , µ 0(1)k , and δ k , where k ∈ {k} exc . We shall suppose the values of λ 0(1)k and µ 0(1)k to be of the same order. Depending on the ratios between these parameters, k-th excitation scheme may be classified in the following way. First we consider the case of small detunings. If the coupling coefficients of the optical dipole transitions λ 0(1)k and the detunings δ k satisfy the inequality the three-level scheme works in the resonant symmetric regime. Instead, the applicability of the resonant asymmetric scheme [17] is provided by the condition We see that the asymmetry/symmetry of the structure isn't defined by the presence/absence of the energy difference ∆ only but by the ratio between ∆ and the coupling coefficients |λ 0k | , |λ 1k | as well. In other words, a driven DD structure can be treated (relative to the k-th transition scheme) as symmetric if the influence of the parameter ∆, introducing a "static" asymmetry, is compensated by an appropriate value of the field strength defined from (11). In this case only one external pulse is sufficient to excite both transitions [21].
Next we shall examine the opposite case where the states |0 and |1 are connected through the off-resonant transitions involving a collection of the excited states lying at the edge of the potential barrier. Two situations are possible again: and The first of these inequalities corresponds to the off-resonant symmetric excitation scheme. This situation was studied in Refs. [22,23] for the double-dot structures with three and four levels. As it was shown, the set of single-qubit operations produced by such one-electron dynamics is incomplete, since in order to realize an arbitrary rotation of the qubit-state vector, the structure symmetry must be broken.
In what follows, our attention will be focused on both the symmetric case and the off-resonant asymmetric case for which the conditions (14) are satisfied and each pulse drives the transitions between only one of localized state and the transport states {|k } exc . This implies also that the values of ∆ and δ k must be rather different from each other for all k to prevent the single-donor resonant dynamics.

III. THE OFF-RESONANT DYNAMICS
In this Section we consider two cases of the off-resonant electronic quantum dynamics in DD structure, i.e., the off-resonant symmetric dynamics and the offresonant asymmetric dynamics. We begin with the second one, initially specifying the conditions imposed on pulse and structure parameters: (The case of large ∆ is out of scope of this study; see Sec. IV D for the reasons).
Strictly speaking, the rotating-wave approximation used in derivation of Eq. (10) and based on the averaging of Eq. (7) over time interval [τ − π/ω 0 , τ + π/ω 0 ] (RWA 1) [24], cannot be applied to high-lying states for which the detunings δ k become comparable to the pulse frequencies. Let us define effective "maximal" detuning δ max ≤ ω 0 that corresponds to the upper bound of energy range, where RWA 1 is still valid, and average Eq. (10) over time interval [τ − π/δ max , τ + π/δ max ] (RWA 2). This step is justified by first of the inequalities (15). After some manipulations, we obtain two equations for time-averaged functions c 0 and c 1 : Again, careful consideration of averaging procedure requires us to handle only with the detunings that are sufficiently small in comparison with δ max . It is easy to see that the equations (16) do not contain the fast-oscillating terms c k =0,1 . Therefore, these equations describe effective two-state dynamics involving only the localized states |0 and |1 . From third equation of the set (10) we conclude that the functions c k =0,1 have the amplitudes ∼ |λ 0k /δ k | << 1 and oscillate at the frequencies ∼ |δ k |.
The reduction of set like Eq. (10) to the two equations for slowly-varying probability amplitudes c 0 and c 1 is known in atomic optics as the adiabatic elimination procedure [25] - [27]. To proceed further, let us average the equations (16) over the time interval [τ − π/∆, τ + π/∆] (RWA 3) on which all of the time-dependent functions except exp (±i∆t) may be replaced by their mean values so that after integration we arrive at the following set of equations: Now we return to the symmetric driving scheme that is simpler than the asymmetric one. By setting ∆ = 0 and E 1 = 0, we transform the set of Eq. (10) where b k = c k exp (iδ k t) , k ∈ {k} exc and we redefine the coupling coefficients as The inequalities (13) allow one to apply the adiabatic elimination procedure to the intermediate levels The equations for two remaining probability amplitudes c 0 and c 1 in the matrix form for both asymmetric and symmetric dynamics read where for symmetric case one has Λ 0 (t) whereas for asymmetric case the functions entering into Eq. (20) are given by expres- It allows us to treat both cases in the same manner, however, keeping in mind that for symmetric structure Λ 0 (t) = Λ 1 (t), since λ 0k = ±λ 1k for the state |k whose electronic wave function possesses even/odd parity relative to the center of DD structure, that does not take place for asymmetric structure. Below we present the general solution of Eq. (20) that will describe both types of off-resonant electron dynamics.
The eigenstates and the eigenenergies of the matrix in right-hand side of Eq. (20) may be written as and respectively, where and is the instantaneous Rabi frequency. Using the unitary transformation we represent the state vector in the instantaneous basis {|+ , |− } as and rewrite Eq. (20) in the new basis as Here we restrict our interest by the diagonal evolution followed from the choice of the system parameters for whichΘ (t) ≪ E ± (t). In this case the solution of Eq.
(28) is straightforward: With the help of equations (26), (27), and (30) we may write down the expression for the evolution matrix of the qubit-state vector |Ψ (t) in the laboratory frame: where The expressions (31) -(33) describe the effective two-level dynamics that corresponds to the continuous evolution of the qubit state vector on the Bloch sphere. In the remainder of this section we show how to choose the pulse and structure parameters in order to realize the most important single-qubit gates. We illustrate the qubit state engineering by considering a particular case of the driving pulses sharing the same time dependence, i.e., f 0 (t) = f 1 (t) ≡ f (t). The conditionΘ = 0 is then satisfied and the components of the evolution matrix (31) take the form can be realized (up to the common phase) given the following choices of the pulse -structure parameters: and respectively. Here k, l, m, and n are the integers and T is the pulse duration. Of course, this is not a unique parameter choice to attain the above quantum operations.
Careful estimation of the operation times requires the detailed knowledge of the energy spectrum of DD structure and the values of λ 0(1)k .

IV. A HYDROGEN MOLECULAR ION: NUMERICAL STUDY
We kg is the free electron mass). In practice, the single-valley approximation was applied to study the phonon-induced decoherence of electron in Si:P + 2 structure [20] and the effect of surface gate on single phosphorous donor in the silicon [28], as well as to calculate the Rabi frequency of resonant electron transfer between localized states in Si:P + 2 structure in three-level approximation [18]. The comparison of the results of calculations, where intervalley interference is taken into account, with those neglecting such effects, demonstrates qualitative agreement between them (especially in presence of an external field, see [16]). In our study, we are mostly interested in validation of the approximations, made during the paper to reveal the specific features of resonant and off-resonant electron dynamics in a multilevel structure.
We expect, that the dynamical analysis developed above and confirmed numerically below for effective single-valley-approximated structure, can be applied to real solidstate DD structures as well.
In what follows we show how to find the eigenenergies and eigenfunctions of hydrogen molecular ion (HMI) without LCAO approximation. We calculate the matrix elements of optical dipole transition and study their behavior under axial electric field. We shall work with atomic units (a.u.), keeping for energy 1 a.u. = 2Ry * = E D (E D is the effective ionization energy) and for distance 1 a.u. = a * B .

A. Eigenenergies and eigenfunctions of HMI at zero bias field
There exist several ways how to compute the eigenenergies and eigenfunctions of HMI. We consider one of them using variable separation in the time-independent Schrödinger equation followed by representation of solution in new variables via appropriate series expansions (see the work [29] and references therein). It is well known that the Schrödinger equation for HMI where r A(B) denotes the distance between an electron and atom A(B), is sepa- is in a series of associated Legendre polynomials, whereas for Ξ (ξ) the power series expansion is applied. Substituting these functions into corresponding wave equa-tions and requiring them to be minimal solutions of these equations, we obtain the expansion coefficients and separation constants C ξ and C η that at fixed m depend continuously on the energy parameter E and on the internuclear distance R. To obtain electronic eigenenergies we now have to find the set of separation constants common for both radial and angular solutions for given m and R. Graphically, these can be sought as intersection points of curves from sets {C ξ (E)} and {C η (E)} and corresponding values of parameter E will be the eigenenergies of HMI (for more details, see the Ref. [29]).
The HMI eigenstates is completely specified by the set of quantum numbers that is enough for this study, but the described algorithm enables one to calculate them with absolute accuracy 10 −20 a.u. or even higher [29]. From Fig. 2 (a) it is seen that two low-lying states of HMI, i.e., |1sσ g and |2pσ u , are almost degenerate, with R. The dependence of the electronic tunneling time τ 1sσg −2pσu =h ∆ 1sσg−2pσu (hereh = 6.582 × 10 −16 eV× s) between these states on R can be seen at The collection of low-lying electronic excited states falls into the subbands characterized at R >>1 by the principal quantum number n of isolated hydrogen atom.
The sets of HMI eigenstates, whose energies are pictured at Fig. 2 (b), namely, {|7jσ u , |6hσ g , ..} (not labelled), correspond to n =2, n =3, and n =4, respectively. In their turn, these subbands can be subdivided further into the doublets ; and so on. The degree of hybridization/localization can be evaluated from state quantum numbers.
For example, the pairs of states with l = s, p become degenerate at comparatively small internuclear distances. The doublets of states with l = d, f are well resolved up to R ≈30 and collapse quickly for larger R. We observe almost complete dissociation of HMI states, pertaining to the subband with n =2, into two pairs of states 2s A(B) = (|2sσ g ± |3pσ u + |3dσ g ± |4f σ u )/2 and 2pσ A(B) = (|2sσ g ± |3pσ u − |3dσ g ∓ |4f σ u )/2 of isolated hydrogen atoms. Taking the average over the energies of states pertaining to the subbbands with n = 1 and n = 2 and adding the energy of internuclear repulsion ε nucl = 1/R, we arrive at isolated hydrogen atom energies ε(n) = 1/2n 2 . On the other hand, the states with higher values of l remain non-degenerate on the whole interval of R under consideration. One can therefore treat those states as reliable transport channels to drive an electron between logical states. As we shall see below, the states |5gσ g and |6hσ u (whose energies are plotted as thick red lines at Fig. 2 (b)) meet the conditions imposed on the choice of transport states by the high excitation selectivity and appropriate dipole coupling strength requirements. The states, whose energies are plotted as thick blue lines, may be exploited as transport ones either at R < 20 (|3dσ g and |4f σ u ) or at R > 30 (|6gσ g , |7hσ u ), |7iσ g ).

B. Matrix elements of optical dipole transition at zero bias field
The numerical solution of Eq. (6) implies the knowledge of matrix elements {d km } of optical dipole transition (ODT) between all states entering into the Eq. (4).
In our simulation we restrict ourselves by consideration of electric fields polarized along z axis, so that the field-structure interaction Hamiltonian in Eq. (3) reads −eE (t) z. In this case, the matrix elements of ODT between states |k and |m can be calculated in spheroidal coordinates through the expression where J (ξ, η) is the Jacobian of the transformation from cartesian frame to spheroidal frame, and the relation z = (R/2) ξη is used. (The wave functions H k(m) (η) and Ξ k(m) of discrete part of HMI energy spectrum are real functions, hence the ODT matrix elements will be real as well). According to the selection rule ∆m = 0, im- Using the equations (38) and (39), we have calculated the ODT matrix elements between all pairs of 64 low-lying σ-states of HMI as functions of internuclear distance

R.
In what follows, they will be used to define the right-hand side of Eq. (6). At the same time, in Eq. (21) we proceed with matrix elements between logical states |0 and |1 and excited ones. These matrix elements can be derived from expression  Let us introduce the dimensionless field energy ε f ield = eE 0 a * B /2Ry * and the dimensionless pulse duration T = 2Ry * t/h. According to the analysis given in Sec.
II, the coupling coefficients that in atomic units take the form λ 0(1)k = ε f ield d 0(1)k 2, have to satisfy the requirements imposed by a concrete optical excitation regime.
If one applies a resonant driving scheme, where HMI state |r is used as transport state, the detuning of pulse frequency ω 0 from resonant frequency ω 0r must be much smaller than the value of coupling coefficient |λ 0r |. In its turn, |λ 0r | must be much smaller than ω rr+1 and ω r−1r in order to minimize population leakage into the states nearest-in-energy to transport state. Instead, the exploitation of stronglydetuned pulses (Raman scheme) implies the values of coupling coefficients |λ 0k | for all excited states to be much smaller than corresponding detunings. It is easy to see that one could attain this condition taking pulse strength as small as possible and detunings as large as possible. However, this brings about considerable reduction in Rabi frequency. In so far, we shall be interested in determination of optimal pulse parameter set, that would amount to rapid and robust implementation of quantum operations.
We begin with resonant driving scheme that, being realized in symmetric structure, may be considered as auxiliary one, since it is only able to inverse the population of logical states at discrete set of pulse durations when the electron is concentrated into logical subspace. The electronic resonant population transfer in threelevel and in four-level structures was studied theoretically in Refs. [21] and [23], where the probabilities p 0 , p 1 , and p tr were found at exact resonance (zero detuning from transport level) to be and (34), in the symmetric structure Λ 0 − Λ 1 = 0, Θ 0 = π/2, and where Ω R = |Λ 2 | (see Eq. (25)). Relative to the transport states, we are only able to estimate under adiabatic elimination an order of magnitude of total probability as p tr ∼ (λ 0k /δ 0k ) 2 max << 1 and its oscillation frequency Ω tr ∼ (δ 2 0k /λ 0k ) max >> Ω R , so that p tr exhibits fast oscillations with small amplitude.
In order to compare the Rabi frequencies found numerically with those calculated within analytical framework, we plot both data types on Fig. 7  To find them, let us consider the sum Λ 2 in more details. As we have mentioned before, the excited states of HMI can be classified by tunnel coupling strength between symmetric and antisymmetric states pertaining to the same doublet. For weakly hybridized states |k 1 and |k 2 , provided that the value of tunnel splitting is much smaller than pulse detuning from one of these states, we can write δ k 1 ≈ δ k 2 .
We suppose, that the pulse energy bandwidth is much larger than |δ k 1 − δ k 2 | so we are disable to resolve the doublet states under consideration. Furthermore, in this case λ 0k 1 ≈ λ 0k 2 , λ 0k 1 ≈ λ 1k 1 , and λ 0k 2 ≈ −λ 1k 2 (see Sec. IV B). Therefore, the terms λ 0k λ 1k /δ k arising from these states have opposite signs and cancel each other, so that the contribution of the doublet {|k 1 , |k 2 } to Λ 2 is minimal. To estimate the contributions from higher states with energies ε ≥ −0.05, we should take into account two circumstances. i) The ODT matrix elements between logical states and excited states with equal l decrease with the energy growth whereas the detunings increase with the energy growth. ii) High-lying states are closely spaced to each other and the distance between two neighboring states (not necessarily from same doublet) decrease rapidly with the energy growth. Thus, one may expect that corresponding terms in Λ 2 will either cancel out each other, as it was for degenerate states, or have insignificant effect on the sum convergence because of their subsequent reduction versus state energy. We think that both issues are important and their cooperative effect takes place, establishing our observation on the oscillation type of p tr . Strictly speaking, for long times T , when Ω R T >> 1, this is not true, because an internal structure of the sum p tr , containing the oscillating terms of different but close frequencies, causes these oscillations to be averaged out and p tr tends to its average value ∼ max (p tr ) /2.
Utilizing similar arguments, we specify an algorithm of choice of the pulse frequency and amplitude in off-resonant driving regime. First of all, we determine the doublet whose states k 1 and k 2 are characterized by sufficiently large ODT matrix elements and high tunneling rates (large value of the energy splitting in comparison with the pulse energy bandwidth). Then, by adjusting the pulse frequency to the center of energy gap between doublet states, we obtain for detunings In this case, the terms λ 0k λ 1k /δ k enter into Λ 2 with the same sign resulting in amplification in the sum. Doing so, we should remember that the field energy ε f ield (i.e., pulse strength) must be much smaller than the doublet energy splitting. As it can be seen from Fig. 2 (b), upper bound value for ε f ield is to be set ∼ 0.005 at the whole considered range of R. Actually, it means that we can reliably operate with frequencies that fall within the interval containing the states from HMI subbands with n = 3 and n = 4. Moreover, the frequency choice just outlined cannot guarantee low population of excited states. A general way to minimize this population implies the use of moderate pulse strengths at fixed detunings. Note, that two-level Rabi oscillation regime with max(p tr ) < 0.01 can be achieved under this frequency choice, since marked points on Fig. 7 lie in the middles of energy gapes separating the doublet states |5gσ g -|6hσ u , |6gσ g -|7hσ u , and |7iσ g -|8jσ u . With that, the pulses have to be strong enough in order to perform qubit rotations in the times that are shorter than decoherence time. For the off-resonant driving scheme, where the decoherence effects due to the finite population of excited states (e.g., the electron-phonon relaxation) are reduced considerably, the dephasing of localized states is expected to be the main source of coherence losses. In our simulations we require that the operation times (T op ∼ 1/Ω R ) do not exceed 10 −8 s.
It gives a lower bound value on the pulse strength to be ε f ield ∼ 0.001. For Si:P + 2 structure parameters, the allowed range 0.001 < ε f ield < 0.005 corresponds to the pulse strengths of hundreds of V/cm that is by two orders smaller than the field strengths considered in the voltage-based driving schemes [12] - [16].
Provided that the pulse parameters are selected correctly, the probability amplitudes c 0 and c 1 (and, consequently, probabilities p 0 and p 1 ) evolve smoothly against time thus allowing one to continuously rotate qubit state vector along fixed meridian with ϕ = π/2 on the Bloch sphere, namely, to create the superposition of logical states of the form |Ψ (t) = p 0 (t) |0 − i p 1 (t) |1 , where p 0 (t) + p 1 (t) ≈ 1. Control over relative phase between logical states, needed for creation of an arbitrary superposition state of the qubit, requires the logical states to be addressed independently by two different pulses to maintain Λ 0 = Λ 1 in Eq. (20). To achieve this goal, one should break the central symmetry of HMI making use of external field, thus the important quantum operations given by Eqs. (34) -(36) will be realized in this case after application of bichromatic pulse. Besides, there exists a more familiar way utilizing the adiabatically varied bias field to produce a shift between the energies of logical states.

D. Phase and population dynamics in nonzero-bias case
As it was demonstrated in the works [3], [16], [20], and [30], an uniform electrostatic field, polarized along z axis, disturbs spatial symmetry of DD structure where R = 38, requires an application of bias field with the strength E 0 ∼ 10 V/cm (see Fig. 8 (a)). If ε st f ield > 0, the states, whose energies bring down (up) with the static field strength, are localized predominantly on "deep" donor B ("shallow" donor A). From Fig. 8 (b) one can observe a general tendency of the HMI spec-trum to become complicated with the field growth due to both the full lifting of Coulomb degeneracy (in contrast to the partial lifting in HMI spectrum without field) and the hybridization of states pertaining to different subbands (giving rise to crossing-anticrossing pattern on Fig. 8 (b)). Otherwise, the energies of delocalized states |5gσ and |6hσ (thick red curves on Fig. 8(b)) vary with static field is the tunnel splitting at zero bias field. It means that the tunneling rate for those states remains sufficiently high if ε st f ield ≤ 0.001.
Therefore, we shall continue to operate with those states exploiting them as the transport channels. The dependencies of the matrix elements on the field strength for transitions, connecting logical states and transport states |5gσ and |6hσ , are given on Fig. 9. The examination of the values of matrix elements confirms the fact that the maximum of electron density for the state |5gσ , originated from symmetric state |5gσ g , is displaced onto "deep" donor B, whereas for the state |6hσ , originated from asymmetric state |6hσ u , it goes onto "shallow" donor A. The deeps on the curves appear due to the electron density redistribution at crossing/anticrossing points, where transport states interact with localized states (see Fig. 8 (b)).
In Ref. [17], it was demonstrated that elementary single-qubit operations can be performed in asymmetric DD structure, driven by two resonant pulses. Here we check this proposal for zero-detuning case by the simulation of NOT operation. According to our previous results, the resonant asymmetric three-level scheme requires that |λ 0tr | = |λ 1tr |, or |E 0 /E 1 | = |d 1tr /d 0tr |. Since the ODT matrix elements for distinct arms of the excitation scheme differ from each other, we need to compensate the difference in coupling coefficients λ 0tr and λ 1tr by choosing the pulse strengths so that to fulfill above condition. Now we integrate Eq. (6)  Unlike in the case of symmetric HMI, here we were unable to find pulse parameters for asymmetric off-resonant driving scheme that would provide a robust and fast implementation of basic quantum operations. This lack takes place because the pulse with frequency ω 0 , being tuned slightly below the state |5gσ for the transition |0 ↔ |5gσ , addresses unwanted nearly-resonant transitions connecting the state |1 with the states localized on "shallow" donor. Thus, the selectivity requirement is violated. The attempts to achieve the reliable two-level oscillation picture have forced us both to reduce the pulse strengths and to vary the detuning in such a way that both pulses would drive only the transitions prescribed by ideal excitation scheme. The goal is likely to be attained by this method, but the characteristic times will be inappropriately long in comparison with decoherence time. In particular, the asymmetric off-resonant scheme, described in Sec. III, will function correctly if ∆ is much smaller than the energy spacings between the states lying around the energy ε 0 + ω 0 . At same time, the coupling coefficients must be much smaller than ∆, in order to address separately each arm of driving scheme by its own pulse. From the Fig. 8 (b) we obtain the estimations ε st f ield < 10 −4 and ∆ < 0.004 that, in its turn, implies |λ 0(1)k | to be less than 10 −3 . It tells us that under these conditions the off-resonant Rabi frequency Ω R is of order of 10 8 s −1 (or smaller) that is out of interest of this paper.
Actually, it means that we can only deal with symmetric DD structure where the possibility of realization of the off-resonant qubit-state σ X rotation has been already demonstrated. However, we know this type of quantum evolution is not sufficient for implementation of arbitrary quantum operation, since the phase control is also required. This difficulty may be overcome with the help of direct introduction of the phase difference between logical states by the voltage pulse that produces necessary energy shift. This technique is widely used in quantum dot structures to align the energy levels of different dots that is needed for experimental investigations of their transport and spectral properties. Here we consider a simplified model of voltage pulse action on the HMI replacing non-isotropic electric field, generated by the gate V a (Fig. 1 (a)), by uniform axial electric field, as it is demonstrated at Fig. 1 (b). The time dependency of voltage pulse is approximated by the step function.
In this case, elementary calculations predict the qubit phase oscillations with the frequency Ω phase = ∆/h. The general expression for qubit-state evolution operator in the two-level approximation, accounting for different voltage pulse shapes, can be found in Ref. [30]. We illustrate this type of quantum dynamics at Fig. 11, Summarizing the results obtained in this and previous subsections, we conclude the numerical study of electron dynamics by observation that the strategy, utilizing sequential implementation of σ X and σ Z rotations, appears to be more reliable than the strategy based on simultaneous action of voltage and optical pulses. With that, one may regard the formulas deduced in Sec. III for asymmetric off-resonant scheme as describing non-ideal nearly-symmetric scheme driven by single pulse, where other external fields (e.g., acting on nearest qubits) slightly disturb the spatial symmetry of structure in uncontrollable way. In this case, small differences in frequencies of the arms quantified by the asymmetry parameter ∆, will amount to quantum error.
Of course, we should take into account that the driving scheme of Eqs. (34) -(36), whose efficiency has not been confirmed above for effective HMI, can describe the qubit dynamics for other models of DD structure (for example, that of heteropolar molecular ions [16]). The main issue, that makes the verification of proposed algorithm of quantum control difficult, is concerned with necessity to know in details the energy spectrum and the ODT matrix elements for each structure under consideration.

V. DISCUSSION
To provide more clarity in the understanding of the advantages of the charge qubit-state engineering presented above, let us compare the resonant and off-resonant excitation schemes. The resonant optical driving of the DD structure modelled by single three-level scheme has been studied in Refs. [17,18]. In the case considered here a more complex dynamics can take place involving more than one three-level scheme. For example, if we tune the lasers on resonance with the transition between the logical subspace {|0 , |1 } and a state |r , r ∈ {k} exc located near the top of the barrier, a number of states with the energies close to ε r will be excited as well.
This picture is quite expected in the hydrogen-like molecular ions for the high-lying states which energies are within the interval ∆ε r ≤ λ 0(1)r . However, for the symmetric structure it means that there will be no transitions between the qubit states.
It is because the exited states belonging to the same doublet are presented by the symmetric and antisymmetric superpositions of the excited states of isolated donors which, being excited simultaneously, interfere constructively on one donor and destructively on another one. As a consequence, the donors are excited independently.
This effect becomes more significant as the interdonor distance R increases and the tunnel coupling between the donors decreases. We have observed such type of electronic dynamics when pulse was tuned into resonance with low-lying excited states with l = s, p, d, f . When the energy splitting of the maximally-resolved doublet becomes comparable with the coupling coefficients of the optical dipole transitions, the process of the electron transfer between the donors is terminated. Note that the optically driven DD structure will demonstrate the similar behavior if one of the pulses is short enough so that its duration is T ≤ 1/∆ε r and thus it contains harmonics in the frequency range δω ∼ 1/T ≥ ∆ε r . Again, the states with the energies belonging to the interval ∆ε r will be excited simultaneously giving rise to the electron transfer blockage just outlined. In our simulations, we have arrived at this regime via application to HMI a short and intense (ε f ield ∼ 0.1) pulse.
The reliable resonant scheme thus deals with single transport state (for HMI, the states 5gσ (g) and 6hσ (u) ) and is very sensitive to the pulse detuning from the resonance with that state. For example, the non-zero detuning always produces an amplitude error in NOT gate because of incomplete depopulation of the initial state when the pulse is off [21]. On the contrary, the use of the off-resonant pulses enables one to exploit the whole number of excited states (from which only several ones participate substantially in electronic dynamics). Moreover, we don't need to control the pulse frequencies with high accuracy since a small variation in the pulse detunings brings about an insignificant change in the Rabi frequency (see Fig.   7). The computational errors originated from the frequency renormalization can be corrected by the corresponding change in the pulse duration due to the smooth time dependencies of the probability amplitudes, as it was shown at Fig. 6. The only requirement that must be followed closely for successful electron state manipulations is the Raman two-photon resonant condition (9).
The selectivity of the electron resonant transfer requires also a strict control over the pulse polarizations. The transport states in the molecular ion are formed through the hybridization of those individual donor states whose wave functions are extended along the axis z that coincides with the interdonor direction. Other states (e.g., πstates) are hybridized weakly and cannot assist efficiently in the electron dynamics.
Their excitations are due to the pulse components polarized along the axes x and y. It amounts to the population leakage into the non-hybridized single-donor states with the energies lying in the close proximity to the energy of the transport state.
Let us define the small angles γ n x and γ n y that characterize the deviations of the n-th pulse polarization from the axis z: E n = E n z + E n x cos (π/2 + γ n x ) + E n y cos (π/2 + γ n y ) , |γ n x | , |γ n y | ≪ 1, n = 0, 1, Note that the complete population transfer between the qubit states, or NOT operation, requires that Λ 0 = Λ 1 . This is naturally met for nearly symmetric DD structures where ∆ ≈ 0 and |d 0k | ≈ |d 1k |. In general, however, one should keep in mind that |d 0k | = |d 1k | that makes the performing of the condition Λ 0 = Λ 1 very problematic. It seems then reasonable to point the other way for the population transfer based upon the pulse-shaped techniques. Such methods, e.g., the stimulated Raman adiabatic passage (STIRAP) [26], are very robust against the pulse/structure imperfections and would allow one to handle the quantum information carefully. The theory of the adiabatic population transfer via multiple intermediate states, including the off-resonant case, was presented in Ref. [27]. Note that for the pulses strongly concerning the arrangement of the pulse shapes in STIRAP can be found elsewhere [26].
The effect of the detuning δ two−ph from two-photon resonance (the spacing between dashed horizontal lines on Fig. 1 (b)) should be also taken into account at more profound level of investigations. In nearly-symmetric off-resonant case (|Λ 0 − Λ 1 | ≪ |Λ 2 |) the problem enables analytical solution. If electron was initially localized into the state |0 , the total excitation probability (including transfer probability into state |1 ) at large detunings (|Λ 2 | ≪ |δ two−ph |) is of order of (Λ 2 /δ two−ph ) 2 ≪ 1. Therefore, if the detunings from two-photon resonance are significant, electronic excitations from localized state |0 do not occur. In the opposite case of small detunings, i.e., when |δ two−ph | ≪ |Λ 2 |, the probability of successful electronic transfer is given by formula p 1 ≈ 1 − (δ two−ph /2Λ 2 ) 2 (provided that other parameters are chosen in optimal way).
Finally, in our analytical treatment we consider continuum states only as virtually excited transport channels, neglecting the possibility of two-photon resonant electron transitions from logical subspace into the continuum states, that reside within interval around the energy ε 2ω 0 = ε 0 + 2ω 0 , via intermediate low-lying excited states (not necessarily highly populated) with energies ∼ ε 0 + ω 0 . It is known that in isolated hydrogen atom the matrix elements for the transition |1s ↔ |2pσ and the matrix elements for transitions connecting the state |2pσ with s (or d) continuum waves with energies pertaining to the interval pointed above, have the same order of magnitude. Hence, one could expect the electron dynamics in DD structure induced by optical driving pulses to be much more complex. However, in experimental investigations of low-frequency excitations in atomic systems these processes do not manifest themselves at the level that would establish the necessity to include the continuum into theoretical model. Perhaps, it could be explained by the arguments we have used in Sec. IV C to account for our numerical results reflecting very small influence of high-lying states of discrete part of HMI spectrum on electron dynamics. In principle, the continuum states being addressed directly can also be used as transport channels, but, at the same time, can bring about additional decoherence (see, e.g., [32]).

VI. CONCLUSIONS
In this paper we have considered the one-electron double-donor structure sub- The information about the structure and pulse parameters is contained in the Rabi frequency of the two-level oscillations. This frequency can be defined experimentally for each set of the detunings, the strengths, and the durations of the pulses.
The results of those measurements could be used to reconstruct the features of the spectrum of the DD structure.
Note that the method of the electron-state manipulations by optical means can be applied also to the spin-based encoding schemes like that of Ref. [2]. The implementation of optically controlled effective electron spin exchange described in Ref. [33] for the two-electron double-dot structure, can be generalized on the two-electron DD structure. The use of the adiabatic schemes is of the particular interest.     The pulse is in exact resonance with transport state |6hσ u . Other parameters are given at the plot. HMI subband with n = 4 (red-squared point on Fig. 7 (b)). Other parameters are given at the plot.