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

TL;DR

This paper introduces a novel method for transferring two-electron entangled states across quantum dot arrays using adiabatic passage and shortcut techniques, significantly reducing operation time and decoherence effects.

Contribution

It presents a new protocol combining coherent adiabatic passage with shortcut techniques for fast, reliable entanglement transfer in quantum dot arrays.

Findings

Operation time reduced to nanoseconds with shortcut techniques.

Protocol effectively minimizes decoherence impact.

Suitable for solid-state quantum information processing.

Abstract

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.

Equations

$$|{\varphi }_{DS,1}^T⟩=\frac{1}{|{\nu }_1|}\sum _{j=1}^{n-1}(-1{)}^{j+1}{\tau }_{j(j+1)}|T_{j(j+1)}^0⟩,$$

$${\tau }_{12}(t)=-{\tau }_0\left[\tanh \left(\frac{t-b}{c}\right)-1\right]=2{\tau }_0-{\tau }_{23}(t)$$

$${\tau }_{23}=\frac{3{\tau }_0}{2}\exp \left[-\frac{(t-t_f/2{)}^2}{{\sigma }^2}\right],$$

$$|{\varphi }_{DS,2}^T⟩=\frac{1}{{\nu }_2}{\left[{\tau }_{12}^2-{\tau }_{34}^2,0,-{\tau }_{12}{\tau }_{23},{\tau }_{23}{\tau }_{34},0,0\right]}^T,$$

$${\tau }_{23}$$

$${\tau }_{k-2,k-1}^{(n)}$$

$${\tau }_{k-1,k}^{(n)}$$

$${\tau }_{12}^{\text{STA}}(t)$$

$${\tau }_{23}^{\text{STA}}(t)$$

Full text

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 cavity-spin1 ; cavity-spin2 ; cavity-spin3 ; cavity-spin4 ; david . Surface acoustic waves can capture and steer electrons SAW1 between distant QDs and, thus, transfer quantum information SAW2 .

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 quantum-coherence1 ; quantum-coherence2 and of photo-assisted long range electron transfer photo-cotunneling1 ; photo-cotunneling2 ; photo-cotunneling3 ; photo-cotunneling4 demonstrates new transfer mechanisms mediated by virtual transitions. Recent experiments with larger QD arrays van4 ; tar4 ; tar5 ; PuddyAPL2015 ; 12dots-Petta 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 CTAP-QD ; sigmund . Such techniques, known as coherent transfer by adiabatic passage (CTAP), have been extended to more complicated architectures in solid state devices CTAP-QD-chain1 ; CTAP-QD-chain2 ; CTAP-spin1 ; CTAP-exchange-spin-qubit as well as in cold atoms CTAP-atom-transport ; CTAP-atom-entanglement . Other proposals consider the level detuning as a control parameter to transfer a qubit state ST-detuning-TQD . Recent experiments enable coherent shuttling of electron spin entangled states in QD arrays Vandersypen-2017 ; Tarucha-2018 , 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 petta .

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 CTAP-QD ; QDarrays-STA , 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) STA , 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 $B$ ($\hbar=1$), $H=H_{\varepsilon}+H_{\tau}+H_{U}+H_{B}$, where $H_{\varepsilon}=\sum_{j,\sigma}\varepsilon_{j}n_{j,\sigma}$ with $n_{j,\sigma}=c^{\dagger}_{j\sigma}c_{j\sigma}$ the occupation of dot $j$ with spin $\sigma$ and onsite energy $\varepsilon_{j}$ and $H_{\tau}=\sum_{<i,j>,\sigma}\tau_{ij}c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.,$ with tunnel barriers $\tau_{ij}$. Coulomb interaction $H_{U}=U_{0}\sum_{j}n_{j,\uparrow}n_{j,\downarrow}$ and $B$ yields $H_{B}=\sum_{j,\sigma}\Delta_{j}S^{z}_{j,\sigma}$ with the Zeeman splittings $\Delta_{j}$. 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 $N=\frac{1}{2}(n-1)n$ states $|T^{0}_{ij}\rangle=(|{\uparrow}_{i}{\downarrow}_{j}\rangle+|{\downarrow}_{i}{\uparrow}_{j}\rangle)/\sqrt{2}$, where the subscripts $i,j\leq n$ 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 supple

[TABLE]

where $\nu_{1}^{2}=\sum_{j=1}^{n-1}\tau^{2}_{j(j+1)}$. Its interesting feature is that the occupation of a triplet $|T^{0}_{j(j+1)}\rangle$ is fully determined by the values of $\tau^{2}_{j(j+1)}/\nu_{1}^{2}$. Therefore, they can be controlled by adiabatically switching $\tau_{j(j+1)}$ such that $|\phi^{T}_{DS,1}\rangle$ turns from the initial state $|T^{0}_{12}\rangle$ to the desired final state $|T^{0}_{(n-1)n}\rangle$ CTAP-QD ; MenchonEnrich2016a .

For the singlets, we have $N$ single-occupancy states, but in an anti-symmetric spin state, namely $|S_{ij}\rangle=(|{\uparrow}_{i}{\downarrow}_{j}\rangle-|{\downarrow}_{i}{\uparrow}_{j}\rangle)/\sqrt{2}$. In addition, there are $n$ double occupied states (DOST), $|S_{ii}\rangle=|{\uparrow}_{i}{\downarrow}_{i}\rangle$ with energy of the order $U_{0}$. It can be expected that for $U_{0}\gg|\tau^{\text{max}}_{ij}|^{2}$, with $\tau^{\text{max}}_{ij}$ 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 $|\phi^{S}_{DS,1}\rangle$.

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 $|T^{0}_{12}\rangle$ and $|T^{0}_{23}\rangle$, respectively, while the DS reads $|\phi^{T}_{DS,1}\rangle=\cos\theta|T^{0}_{12}\rangle-\sin\theta|T^{0}_{23}\rangle$, where $\tan\theta=\tau_{23}/\tau_{12}$. Setting $\varepsilon_{j}=0$ for all $j$, we employ pulses of the form

[TABLE]

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 $\theta(0)=0$ and $\theta(t_{f})=\pi/2$ are fulfilled and, thus, the initial state $|T^{0}_{12}\rangle$ turns into the final state $|T^{0}_{23}\rangle$. We choose the operation time $t_{f}=50$ (in units of $2\pi/\tau_{ij}^{\text{max}}$), with the maximal intensity of the pulse $\tau^{\text{max}}_{ij}=2\tau_{0}=2\pi$, such that the adiabaticity condition $\tau^{\text{max}}_{ij}t_{f}=100\pi\gg 1$ CTAP-QD is fulfilled. For QDs, a possible experimental value is $\tau^{\text{max}}_{ij}=5\mu$eV, such that $t_{f}=260$ns. Figure 2(b) illustrates how the state $|T^{0}_{12}\rangle$ is adiabatically transferred to $|T^{0}_{23}\rangle$ along the DS, while $|T^{0}_{13}\rangle$ remains unpopulated. The computed fidelity of this process for $t_{f}=50$ is $F=|\langle T^{0}_{23}|\Psi(t_{f})\rangle|^{2}\gtrsim 0.999$.

For the singlet case, we have to include DOST. With the same coupling pulses $\tau_{12}$ and $\tau_{23}$ [Eq. (2)], the singlet state can be transferred with high fidelity from $|S_{12}\rangle$ to $|S_{23}\rangle$, for $U_{0}\gg|\tau_{ij}^{\textrm{max}}|^{2}$. In this case, adiabatic elimination of DOST in the Hamiltonian $H_{S}$ can be applied supple . For small $U_{0}$, we find that the fidelity $F=|\langle S_{23}|\Psi(t_{f})\rangle|^{2}$ 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 $|\phi^{S}_{DS,1}\rangle$ is no longer an instantaneous eigenstate. For the more realistic $U_{0}\gtrsim 1400\pi\gg\tau_{0}$ ($1400\pi$ 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 CTAP-QD . For two-electron 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 $|\phi^{T}_{DS,1}\rangle=\left[\tau_{12},0,-\tau_{23},0,0,\tau_{34}\right]^{T}/\nu_{1}$ for triplets, where $\tau_{12}$ and $\tau_{34}$ correspond to $\tau_{12}$ and $\tau_{23}$ for a TQD [Eq. (2)] and $\tau_{23}$ is chosen as

[TABLE]

where $\sigma=3t_{f}/16$. Then the population evolves via the DS $|\phi^{T}_{DS,1}\rangle$ from $|T^{0}_{12}\rangle$ to $|T^{0}_{34}\rangle$, as shown in Fig. 3. Similarly, a singlet is transferred provided that $U_{0}\gg|\tau^{\text{max}}_{ij}|^{2}$. The other DS in a QQD is

[TABLE]

where $\nu_{2}=\sqrt{(\tau_{12}^{2}-\tau_{34}^{2})^{2}+\tau_{12}^{2}\tau_{23}^{2}+\tau_{23}^{2}\tau_{34}^{2}}$. Long-range entanglement transfer can be adiabatically realized by coherently moving one electron along the DS $|\phi^{T}_{DS,2}\rangle$. Specifically, $|S_{12}\rangle$ can be transferred to $|S_{14}\rangle$, as shown in Fig. 4(a), where the conditions $\tau_{23}(0)=\tau_{34}(0)=0$, $\tau_{12}(t_{f})=0$ and $\tau_{23}(t_{f})\gg\tau_{34}(t_{f})$ are satisfied. To fulfill the above requirements, we choose $\tau_{12}$, $\tau_{34}$ in the same form of $\tau_{12}$ and $\tau_{23}$ from Eq. (2), respectively, where $\tau_{0}=\pi/2$, $b=t_{f}/7$, and

[TABLE]

where $b_{2}=3t_{f}/5$, $a_{0}=20$ (Fig. 4(b)). High fidelity transfer can be obtained by tuning the ratio $|\tau_{23}(t_{f})/\tau_{34}(t_{f})|$. Increasing $\tau_{34}(t_{f})$ 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 $\tau_{23}(t_{f})=20\pi\sim 12$GHz, which is experimentally feasible 12dots-Petta , $F=0.998$ is reached (see Fig. 4 (c)) with $U_{0}=1200\pi\sim 3$meV. 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. $|T^{0}_{12}\rangle$ can be transferred to $|T^{0}_{(n-1)n}\rangle$ with the application of $n-2$ pulse sequences such that the total protocol has the duration $(n-2)t_{f}$ and

[TABLE]

where $3\leq k\leq n$. The pulses $\tau_{12}^{\text{CTAP}}$ and $\tau_{23}^{\text{CTAP}}$ refer to the ones for TQD defined in Eq. (2).

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 $\varepsilon_{j}=0$, we can employ the ansatz $\Psi(t)=\cos\chi\cos\eta|1\rangle-i\sin\eta|2\rangle-\sin\chi\cos\eta|3\rangle$, with $\chi(t)$ and $\eta(t)$ to be determined. The conditions $\chi(0)=0$, $\chi(t_{f})=\pi/2$, $\eta(0)=0$ and $\eta(t_{f})=0$ correspond to the initial and final state of the protocol, while a smooth onset of the pulses is ensured by the conditions $\dot{\chi}(0)=\dot{\chi}(t_{f})=\ddot{\chi}(0)=\ddot{\chi}(t_{f})=0$ and $\dot{\eta}(0)=\dot{\eta}(t_{f})=0$ three-level-I . To satisfy all the above conditions, we introduce the scaled time $s=t/t_{f}$ and choose $\chi(t)=\frac{\pi}{2}s-\frac{15}{64}\sin(2\pi s)-\frac{1}{192}\sin(6\pi s)$. Besides the term linear in $t$, $\chi(t)$ consists of only the lowest odd Fourier components, which is a common choice (“Gutman 1-3 trajectory”) for obtaining smooth pulses Booktrajectory . In addition, the function $\eta(t)=\arctan(\dot{\chi}/\alpha)$ is used, where the tunable parameter $\alpha$ allows one to reduce the maximal occupation of the intermediate state to the value $P_{2}^{\text{max}}=\sin^{2}\eta(t_{0})=\dot{\chi}^{2}(t_{0})/[\dot{\chi}^{2}(t_{0})+\alpha^{2}]$ where $t_{0}=t_{f}/2$. Inserting the ansatz into the time-dependent Schrödinger equation, we obtain

[TABLE]

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 $t_{f}=5.2$ns when $\tau^{\text{max}}_{ij}=5\mu$eV. The designed pulses in Eq. (Spin entangled state transfer in quantum dot arrays: Coherent adiabatic and speed-up protocols) enable to speed-up the singlet transfer as well, but requires that DOST are energetically distant from the single occupied ones. For $U_{0}=1400\pi\sim 3.5$ meV, the transfer from $|S_{12}\rangle$ to $|S_{23}\rangle$ occurs with $F>0.999$, while the occupation of $|S_{13}\rangle$ stays below $1\%$ during the whole process. A more detailed discussion concerning interaction can be found in supple .

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 right-hand side of Eq. (6) the pulses in Eq. (Spin entangled state transfer in quantum dot arrays: Coherent adiabatic and speed-up protocols). 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 $\tau^{\textrm{CTAP}}_{ij}$ are substituted by $\tau^{\textrm{STA}}_{ij}$, and the interaction $U_{0}=1200\pi\sim 3$meV.

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 $H_{\textrm{c}}=\sum_{j,\lambda}(n_{j,\uparrow}+n_{j,\downarrow})(g_{j\lambda}a^{{\dagger}}_{j\lambda}+g_{j\lambda}^{*}a_{j\lambda})$ where $a_{j\lambda}$ is the annihilation operator of mode $\lambda$ coupled to QD $j$ with strength $g_{j\lambda}$. By standard techniques, one eliminates the bath within second order perturbation theory and for an Ohmic spectral density of the bath modes Breuer-book . After a Born-Markov and a rotating-wave approximation, one finds the master equation $\dot{\rho}=-i[H(t),\rho]-\sum_{j}\Gamma_{j}\left[n_{j\uparrow}+n_{j\downarrow},\left[n_{j\uparrow}+n_{j\downarrow},\rho\right]\right]/2,$ with the dephasing rates $\Gamma_{j}$ 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, $1-F\propto\Gamma$. 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 $\tilde{\tau}_{12}=\tau_{12}+\epsilon_{12}$, $\tilde{\tau}_{23}=\tau_{23}+\epsilon_{23}$, where $\epsilon_{12}$ and $\epsilon_{23}$ are independent fluctuations. We model them as Gaussian pulses with strength $\epsilon_{0}$ at times $t_{l}$ and sign $\sigma_{l}=\pm 1$, such that their average vanishes. Thus, $\epsilon=\epsilon_{0}\sum_{l=1}^{N}s_{l}f(t-t_{l})$ and $f(t-t_{l})=\exp[-(t-t_{l})^{2}/\xi^{2}]$. 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 $\xi$, 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 $\xi$ 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 supple .

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.