Spin entangled state transfer in quantum dot arrays: Coherent adiabatic and speed-up protocols

Long-distance transfer of quantum states is an indispensable part of large-scale quantum information processing. We propose a novel scheme for the transfer of two-electron entangled states, from one edge of a quantum dot array to the other by coherent adiabatic passage. This protocol is mediated by pulsed tunneling barriers. In a second step, we seek for a speed up by shortcut to adiabaticity techniques. This significantly reduces the operation time and, thus, minimizes the impact of decoherence. For typical parameters of state-of-the-art solid state devices, the accelerated protocol has an operation time in the nanosecond range and terminates before a major coherence loss sets in. The scheme represents a promising candidate for entanglement transfer in solid state quantum information processing.

Semiconductor quantum dots (QDs) with long spin coherence times, are promising platforms for quantum information processing. Scaling up these devices requires techniques for long-range qubit control, for which several methods and techniques have been proposed and implemented. For example, spin circuit quantum electrodynamics architectures with spin qubits coherently coupled to microwave frequency photons allow photon-mediated long-distance spin entanglement [1][2][3][4][5]. Surface acoustic waves can capture and steer electrons [6] between distant QDs and, thus, transfer quantum information [7].
Already triple quantum dots (TQDs) in series allow to investigate transfer between sites that are not directly coupled. Experimental evidence of direct electron spin transfer between the edge dots [8,9] and of photo-assisted long range electron transfer [10][11][12][13] demonstrates new transfer mechanisms mediated by virtual transitions. Recent experiments with larger QD arrays [14][15][16][17][18] put the challenge of transferring quantum states over larger distances. Since such transfer obviously is more involved, proposals for fast and reliable protocols are particularly welcome.
Adiabatic transfer has been widely invoked for quantum information processing. A single electron in a TQD can be directly transferred between outer dots by adiabatic passage via a dark state (DS) without ever populating other instantaneous eigenstates of the system [19,20]. Such techniques, known as coherent transfer by adiabatic passage (CTAP), have been extended to more complicated architectures in solid state devices [21][22][23][24] as well as in cold atoms [25,26]. Other proposals consider the level detuning as a control parameter to transfer a qubit state [27]. Recent experiments enable coherent shuttling of electron spin entangled states in QD arrays [28,29], where one party of an entangled spin pair is adiabatically transferred to a long-distant site by detuning levels. Also shuttling in parallel two and three electrons at a time has been implemented in silicon QD arrays [30].
In this Letter, we propose a new protocol which allows for transferring two-particle entangled states by applying pulses in an adiabatic manner. We focus on the transfer of a singlet and a triplet from the first two dots of an array to the last two dots, see Fig. 1. Interestingly, in contrast to the case of a single particle [19,31], we will find that CTAP of an entangled pair can be achieved also in arrays with an even number of dots. Furthermore, we improve our protocol by means of shortcuts to adiabaticity (STA) [32], which results in a significant speed-up that reduces the impact of decoherence. The feasibility is underlined by a numerical study that considers charge noise as well as fluctuations of the pulse intensities.
Hamiltonian and Dark State.-We consider two electrons confined in an array of n QDs described by the Hubbard model, in the presence of an external magnetic field B ( = 1), H = H ε + H τ + H U + H B , where H ε = j,σ ε j n j,σ with n j,σ = c † jσ c jσ the occu- Direct coherent transfer of a singlet between outer dots in an array of n QDs. With the application of pulses τ12(t), . . . , τ (n−1)n (t), high-fidelity transfer of the singlet state |S12 to |S (n−1)n is achieved. The subscripts refer to the dots occupied by one electron.  pation of dot j with spin σ and onsite energy ε j and H τ = <i,j>,σ τ ij c † iσ c jσ + h.c., with tunnel barriers τ ij . Coulomb interaction H U = U 0 j n j,↑ n j,↓ and B yields H B = j,σ ∆ j S z j,σ with the Zeeman splittings ∆ j . For a sufficiently large B, the states with parallel spins are energetically far off and can be neglected. Then only the triplets with antiparallel spins and the singlets are relevant.
Owing to the Pauli principle, the triplets within the same QD can be formed only by electrons in different orbital levels and they are largely detuned. Thus, the triplet basis consists of N = 1 2 (n − 1)n states |T 0 ij = (|↑ i ↓ j + |↓ i ↑ j )/ √ 2, where the subscripts i, j ≤ n refer to the dots occupied. An important ingredient for CTAP is the emergence of a DS, i.e., an eigenstate of the Hamiltonian with zero energy at all times. In the triplet subspace, it reads [33] where ν 2 1 = n−1 j=1 τ 2 j(j+1) . Its interesting feature is that the occupation of a triplet |T 0 j(j+1) is fully determined by the values of τ 2 j(j+1) /ν 2 1 . Therefore, they can be controlled by adiabatically switching τ j(j+1) such that |φ T DS,1 turns from the initial state |T 0 12 to the desired final state |T 0 (n−1)n [19,34]. For the singlets, we have N single-occupancy states, but in an anti-symmetric spin state, namely In addition, there are n double occupied states (DOST), |S ii = |↑ i ↓ i with energy of the order U 0 . It can be expected that for U 0 ≫ |τ max ij | 2 , with τ max ij the maximal value of the pulses, both singlet subspaces are energetically separated such that no transition between them occurs. Then one finds again the DS in Eq. (1), but with T 0 replaced by S and accordingly denoted by |φ S DS,1 . CTAP Transfer of entangled states in a TQD.-To work out the principles of the transfer, we start with a TQD in which the initial and the final states are |T 0 12 and |T 0 23 , respectively, while the DS reads |φ T DS,1 = cos θ|T 0 12 − sin θ|T 0 23 , where tan θ = τ 23 /τ 12 . Setting ε j = 0 for all j, we employ pulses of the form with the pulse parameters b = t f /2 and c = t f /14 [see Fig. 2(a)]. These pulses are chosen such that the boundary conditions θ(0) = 0 and θ(t f ) = π/2 are fulfilled and, thus, the initial state |T 0 12 turns into the final state |T 0 23 . We choose the operation time t f = 50 (in units of 2π/τ max ij ), with the maximal intensity of the pulse τ max ij = 2τ 0 = 2π, such that the adiabaticity condition τ max ij t f = 100π ≫ 1 [19] is fulfilled. For QDs, a possible experimental value is τ max ij = 5µeV, such that t f = 260ns. Figure 2(b) illustrates how the state |T 0 12 is adiabatically transferred to |T 0 23 along the DS, while |T 0 13 remains unpopulated. The computed fidelity of this process for For small U 0 , we find that the fidelity F = | S 23 |Ψ(t f ) | 2 of the protocol oscillates heavily. This is characteristic for U 0 such that DOST are energetically close to the single occupied ones. Then the DS |φ S DS,1 is no longer an instantaneous eigenstate. For the more realistic U 0 1400π ≫ τ 0 (1400π corresponds to 3.5 meV), however, the energetic separation of the two singlet subspaces is sufficiently large such that the transfer fidelity for both singlets and triplets is similar. CTAP of entangled states in quadruple QDs.-The case of a quadruple QD (QQD) deserves special attention, because the traditional CTAP protocol for one electron requires an odd number of sites [19]. For twoelectron entanglement transfer, by contrast, we find that this restriction does not apply. Furthermore, we find out that there are two different DSs instead of one. Each of them allows for different transfer protocols. One of the two DS reads |φ T DS,1 = [τ 12 , 0, −τ 23 , 0, 0, τ 34 ] T /ν 1 for triplets, where τ 12 and τ 34 correspond to τ 12 and τ 23 for a TQD [Eq. (2)] and τ 23 is chosen as where σ = 3t f /16. Then the population evolves via the DS |φ T DS,1 from |T 0 12 to |T 0 34 , as shown in Fig. 3. Similarly, a singlet is transferred provided that . Long-range entanglement transfer can be adiabatically realized by coherently moving one electron along the DS |φ T DS,2 . Specifically, |S 12 can be transferred to |S 14 , as shown in Fig. 4(a), where the conditions τ 23 (0) = τ 34 (0) = 0, τ 12 (t f ) = 0 and τ 23 (t f ) ≫ τ 34 (t f ) are satisfied. To fulfill the above requirements, we choose τ 12 , τ 34 in the same form of τ 12 and τ 23 from Eq. (2), respectively, where τ 0 = π/2, b = t f /7, and where b 2 = 3t f /5, a 0 = 20 ( Fig. 4(b)). High fidelity transfer can be obtained by tuning the ratio |τ 23 (t f )/τ 34 (t f )|. Increasing τ 34 (t f ) by adjusting a 0 , can further improve the fidelity. However, the pulse intensities should be sufficiently small, as otherwise we witness leakage to higher states. For instance, for the amplitude τ 23 (t f ) = 20π ∼ 12GHz, which is experimentally feasible [17], F = 0.998 is reached (see Fig. 4 (c)) with U 0 = 1200π ∼ 3meV. It approaches 1 as U 0 increases. Transfer of entangled states in longer arrays.-We propose another protocol to transfer the entangled states in a QD array with arbitrary length n, which is based on the protocol for a TQD discussed above. |T 0 12 can be transferred to |T 0 (n−1)n with the application of n − 2 pulse sequences such that the total protocol has the duration (n − 2)t f and  Accelerating the transfer.-In order to speed up the transfer, we use reverse engineering, a technique of STA, which allows to design the pulses in order to reduce the transfer time. Let us consider first TQD. Since all onsite energies ε j = 0, we can employ the ansatz Ψ(t) = cos χ cos η|1 −i sin η|2 −sin χ cos η|3 , with χ(t) and η(t) to be determined. The conditions χ(0) = 0, χ(t f ) = π/2, η(0) = 0 and η(t f ) = 0 correspond to the initial and final state of the protocol, while a smooth onset of the pulses is ensured by the conditionsχ(0) =χ(t f ) =χ(0) = χ(t f ) = 0 andη(0) =η(t f ) = 0 [35]. To satisfy all the above conditions, we introduce the scaled time s = t/t f and choose χ(t) = π 2 s − 15 64 sin(2πs) − 1 192 sin(6πs). Besides the term linear in t, χ(t) consists of only the lowest odd Fourier components, which is a common choice ("Gutman 1-3 trajectory") for obtaining smooth pulses [36]. In addition, the function η(t) = arctan(χ/α) is used, where the tunable parameter α allows one to re-duce the maximal occupation of the intermediate state to the value P max 2 = sin 2 η(t 0 ) =χ 2 (t 0 )/[χ 2 (t 0 ) + α 2 ] where t 0 = t f /2. Inserting the ansatz into the time-dependent Schrödinger equation, we obtain τ STA 12 (t) =η cos χ +χ cot η sin χ, τ STA 23 (t) = −η sin χ +χ cot η cos χ, for reversely engineering the pulses. Figures 2 compare the CTAP and STA protocols for a triplet transfer. The operation time required is t f = 1, i.e. 50 times shorter than the one obtained with CTAP, which corresponds to t f = 5.2ns when τ max ij = 5µeV. The designed pulses in Eq. (7) enable to speed-up the singlet transfer as well, but requires that DOST are energetically distant from the single occupied ones. For U 0 = 1400π ∼ 3.5 meV, the transfer from |S 12 to |S 23 occurs with F > 0.999, while the occupation of |S 13 stays below 1% during the whole process. A more detailed discussion concerning interaction can be found in [33].
For the extension to an arbitrarily long QD array, we combine the ideas of CTAP in longer arrays and the present STA pulses, i.e., we replace on the righthand side of Eq. (6) the pulses in Eq. (7). In a QQD, the dynamics for the resulting STA protocol is shown in Fig. 5 for the singlet transfer. Our benchmark is again the fidelity which should lie above the threshold F > 0.999. This can be achieved with the pulses as in Eq. 6, where τ CTAP ij are substituted by τ STA ij , and the interaction U 0 = 1200π ∼ 3meV.
Robustness with respect to dephasing.-So far, we have considered purely coherent quantum dynamics with perfect pulses. Experiments may operate under less ideal conditions. We therefore extend our numerical studies to the presence of decoherence stemming from substrate phonons and fluctuations of the pulse strength.
Let us assume that each QD is coupled to a separate environment that creates quantum noise coupled to the onsite energy. Then our Hamiltonian must be extended by macroscopic number of bosonic modes and a coupling Hamiltonian H c = j,λ (n j,↑ + n j,↓ )(g jλ a † jλ + g * jλ a jλ ) where a jλ is the annihilation operator of mode λ coupled to QD j with strength g jλ . By standard techniques, one eliminates the bath within second order perturbation theory and for an Ohmic spectral density of the bath modes [37]. After a Born-Markov and a rotatingwave approximation, one finds the master equationρ = −i[H(t), ρ] − j Γ j [n j↑ + n j↓ , [n j↑ + n j↓ , ρ]] /2, with the dephasing rates Γ j which in our numerical studies assume to be all equal. Figures 6(a,b) show the lack of fidelity 1 − F of the protocol for the triplet and the singlet. It turns out that it is proportional to the dephasing rate, 1 − F ∝ Γ. For the singlet case, it is slightly larger, which can be explained by the additional decay channel to DOST. In both cases, the STA pulses perform significantly better than the CTAP pulses. Regarding pulse amplitudes fluctuations, we consider τ 12 = τ 12 +ǫ 12 ,τ 23 = τ 23 +ǫ 23 , where ǫ 12 and ǫ 23 are independent fluctuations. We model them as Gaussian pulses with strength ǫ 0 at times t l and sign σ l = ±1, such that their average vanishes.
The effects coming from the noise are mainly related with the density of t l points, d = N/t f , and the width of distribution ξ, where N are the total number of t l points. For the same value of fluctuation density d, shorter t f results in less points of fluctuations. On the other hand, a narrow width ξ leads to more stable pulses. We compare the dependence of F on the amplitude fluctuations by using CTAP and STA protocols in a TQD, respectively. As shown in Figs. 6(c,d), the transfer fidelities for both triplet and singlet states are higher by using STA. A further noise source may be the hyperfine interaction with the nuclear spins, as discussed in [33].
Conclusions.-We have proposed a novel CTAP scheme for the long-range transfer of spin entangled states in QDs and derived for it a speed-up via STA. The protocol works with rather large fidelity for both spin singlets and triplets. Importantly, simultaneous transport of two particle spins from one edge to the other one of the atomic array is achieved while preserving their entanglement. While CTAP is designed for slow operations, the operation of the STA is significantly faster, in our numerical studies typically by a factor 50.
When the coherent time evolution is affected by the interaction with an environment, the STA protocol provides clear advantages, because the reduced operation time make it less sensitive to decoherence. Fluctuations of the pulse strength lead to similar conclusions. For CTAP, already fluctuations of the pulse strength below 1% are noticeable, while the STA scheme is fault tolerant towards much larger imperfections.
The precise control of electric pulses in the present experiments will allow the implementation of the protocols presented here. In fact, the transfer of quantum states between distant sites with high fidelity, could be experimentally implemented not only in QDs but also in different physical systems such as cold atoms or photonic crystals, which are of significance for large-scale quantum information processing. This work is supported by the Spanish Ministry of Science, Innovation, and Universities via Grant No. MAT2017-86717-P, NSFC (11474193), SMSTC (18010500400 and 18ZR1415500), and the Program for Eastern Scholar. Y.B. thanks the Juan de la Cierva Program of the Spanish MINECO.
Supplemental Material for "Spin entangled state transfer in quantum dot arrays: Coherent adiabatic and speed-up protocols" In this section, we derive the Hamiltonian for twoelectron entangled states transfer in a TQD. Our aim is to transfer the triplet state |T 0 or the singlet state |S from one edge to the other one of a TQD. The triplet basis consists of three states where the subscripts label the dots occupied by electrons.
In contrast, the Hilbert space for the singlets consists of three single occupied and three double occupied states: The Hamiltonian expanded in the basis of the triplet states reads Accordingly, the Hamiltonian in the basis of the singlet states becomes By using the adiabatic elimination for H S , i.e., we assume that the first time derivative of the wavefunction of double occupied states is zero, we can find the effective Hamiltonian of single occupied states including implicitly the effects of the double occupied states For U 0 ≫ |τ max ij | 2 , this Hamiltonian becomes equal to the one in Eq. (3) when setting all ε j = 0, such that in this limit, the CTAP and STA pulses for the singlet states can also be employed for the triplet states, as discussed in the main text.
With the same coupling pulses of τ 12 and τ 23 for the triplet transfer, the singlet state can be transferred with high fidelity from |S 1,2 to |S 2,3 , for U 0 ≫ |τ max ij | 2 . We calculate the fidelity of the singlet transfer F = | S 23 |Ψ(t f ) | 2 for different values of the intradot Coulomb interaction U 0 for both CTAP Fig. S1 (a) and STA S1 (b). Both of them indicate that the fidelity of the singlet transfer reaches values F = 0.999 for U 0 = 1400π ∼ 3.5 meV which corresponds to Coulomb interaction values within the experimental ones for semiconductor QDs. In order to see the effects of double occupied states during the transfer in a quadruple QD, we also do the adiabatic elimination for H S , and find the Hamiltonian with zero detuning One of the main spin decoherence mechanisms in GaAs quantum dots is hyperfine interaction. Here, we include in our analysis the effect of hyperfine interaction between electron and nuclear spins. The effective magnetic field B N j resulting from a random configuration of many nu-clear spins localized in each quantum dot, affects the spin of the electrons, and also causes transitions between singlet and triplet states. Now, we consider a Hamiltonian with both singlets and triplets where we add also the parallel spin states |T + ij = |T ↑↑ ij and |T − ij = |T ↓↓ ij . Here, we consider zero detuning and an external magnetic field of several hundreds of mT, i.e., much larger than the Overhauser field B N which is of the order of 5mT. We consider a phenomenological model in GaAs QDs where we include in the time evolution of the density matrix equation a phenomenological spin-flip rate. As the difference between the |S and |T 0 energy levels is much smaller than the one between |S and |T ± , and the one between |T 0 and |T ± , transitions occur mainly between |S ij and |T 0 ij . Thus, we consider the Hilbert space which includes both subspaces and we consider a spin relaxation time of T 1 = 50µs and a spin decoherence time of T 2 = 1µs, (which are typical experimental values for GaAs QDs). Then, we solve the reduced density matrix by using the pulses designed by CTAP and STA. We obtain the transfer fidelity for the singlet and triplet states, as shown in Table I.
Using the strategy of STA, we obtain high fidelity, even in the presence of hyperfine interaction, for the transfer of both triplet and singlet states. The reason is that decoherence is avoided to large extent by shortening the operation time from t f = 50 (260ns) into t f = 1 (5.2ns), which is much smaller than the spin decoherence time considered in our calculation T 2 = 1µs.