Transfer of a quantum state from a photonic qubit to a gate-defined quantum dot

TL;DR

This paper proposes two protocols for transferring quantum information from photonic qubits to gate-defined quantum dot spin qubits, using excitons as intermediaries, with high success probabilities under realistic conditions.

Contribution

It introduces novel protocols for quantum state transfer between photonic and solid-state qubits via excitons in quantum dots, supported by effective models and noise analysis.

Findings

Protocols achieve 85-97% success probability

Effective Hamiltonian models describe hybrid systems

Realistic noise models validate protocol viability

Abstract

Interconnecting well-functioning, scalable stationary qubits and photonic qubits could substantially advance quantum communication applications and serve to link future quantum processors. Here, we present two protocols for transferring the state of a photonic qubit to a single-spin and to a two-spin qubit hosted in gate-defined quantum dots (GDQD). Both protocols are based on using a localized exciton as intermediary between the photonic and the spin qubit. We use effective Hamiltonian models to describe the hybrid systems formed by the the exciton and the GDQDs and apply simple but realistic noise models to analyze the viability of the proposed protocols. Using realistic parameters, we find that the protocols can be completed with a success probability ranging between 85-97%.

Figures

Tables

Table 1. Table 1: Set of realistic parameters for GaAs based devices.

Parameter[Source]	Symbol	Value
Dark-bright splitting(Bayer et al., 2002)	${\Delta }_0$	$100\mu \mathrm{eV}$
Bright state splitting(Bayer et al., 2002)	${\Delta }_1$	$0\mu \mathrm{eV}$
Dark state splitting(Bayer et al., 2002)	${\Delta }_2$	$0\mu \mathrm{eV}$
Magnetic field	$B$	$5T$
Electron g-factor	$g_{\text{e}},{\tilde{g}}_{\text{e}}$	$-0.44$
Hole g-factor	$g_{\text{h}}$	$0.2$
Tunnel-coupling	$t_{\text{c}}$	$50-150\mu \mathrm{eV}$
Exciton recombination time	$\tau$	$1\mathrm{ns}$
Quasi static charge noise(Dial et al., 2013)	${\epsilon }_{\text{rms}}$	$8\mu V/L$
Fast uncorrelated charge noise(Dial et al., 2013; Cerfontaine et al., 2014)	$S_{\epsilon }$	$5\text{×}{10}^{-20}{V}^2{\mathrm{Hz}}^{-1}/L^2$
Gate lever-arm	$L$	$10e^{-1}$
Nuclear spin noise SAQD(Taylor et al., 2007; Prechtel et al., 2016)	$B_{\text{OF}}^{\text{(rms)}}$	$50\mathrm{mT}$
Nuclear spin noise GDQD(Taylor et al., 2007)	${\tilde{B}}_{\text{OF}}^{\text{(rms)}}$	$5\mathrm{mT}$
Transverse e-h hyperfine ratio(Prechtel et al., 2016)	$\eta$
DD Parameter
DD tunnel-coupling	$t_{\text{DD}}$	$50\mu \mathrm{eV}$
Coulomb repulsion(Cerfontaine et al., 2014)	$U$	$2\mathrm{meV}$
Coulomb energy singlet(Cerfontaine et al., 2014)	$V_{+}$	$0.8\mu \mathrm{eV}$
Coulomb energy triplet(Cerfontaine et al., 2014)	$V_{-}$	$0\mu \mathrm{eV}$

Table 2. Table 2: Performance of the presented protocols.

	Single-spin qubit		singlet-triplet
	Fig. 2	Fig. 3	Fig. 6
$P_{\mathrm{LZ}}$	$1\%$	$1\%$	$1\%$
$P_{\mathrm{rec}}$	$2-5.4\%$	$0.8-1.5\%$	$7.9-12.0\%$
$P_{\mathrm{deph}-\mathrm{fail}}$	$0.4\%$	$0.3\%$	$1.1\%$
$P_{\mathrm{Rabi}-\mathrm{leak}}$	-	-	$0.6\%$
$P_{\mathrm{Rabi}-\mathrm{fail}}$	-	-	$0.3\%$
$P_{\mathrm{success}}$	$>93.2\%$	$>97.2\%$	$>85.0\%$

Equations

$${\mathbf{H}}_{0,z}=\frac{1}{2}\left(\begin{array}{cccc}{\Delta }_0& {\Delta }_1& 0& 0\\ {\Delta }_1& {\Delta }_0& 0& 0\\ 0& 0& -{\Delta }_0& {\Delta }_2\\ 0& 0& {\Delta }_2& -{\Delta }_0\end{array}\right),$$

$${\mathbf{H}}_{\text{ex}}={\mathbf{H}}_{0,x}+{\mathbf{H}}_{\text{Ze},x},$$

$${\mathbf{H}}_{0,x}=\frac{1}{4}\left(\begin{array}{cccc}-{\Delta }_1-{\Delta }_2& -2{\Delta }_0+{\Delta }_1-{\Delta }_2& 0& 0\\ -2{\Delta }_0+{\Delta }_1-{\Delta }_2& -{\Delta }_1-{\Delta }_2& 0& 0\\ 0& 0& {\Delta }_1+{\Delta }_2& -2{\Delta }_0-{\Delta }_1+{\Delta }_2\\ 0& 0& -2{\Delta }_0-{\Delta }_1+{\Delta }_2& {\Delta }_1+{\Delta }_2\end{array}\right),$$

$${\mathbf{H}}_{\text{Ze},x}=\frac{{\mu }_{\text{B}}B}{2}\left(\begin{array}{cccc}-g_{\text{e}}-g_{\text{h}}& 0& 0& 0\\ 0& g_{\text{e}}+g_{\text{h}}& 0& 0\\ 0& 0& g_{\text{e}}-g_{\text{h}}& 0\\ 0& 0& 0& -g_{\text{e}}+g_{\text{h}}\end{array}\right),$$

$$\mathrm{BC}(\Psi )={\left|⟨\Psi ⟩{\uparrow \Downarrow }_z\right|}^2+{\left|⟨\Psi ⟩{\downarrow \Uparrow }_z\right|}^2.$$

$$\alpha |{\omega }_1,\mathrm{H}⟩+\beta |{\omega }_2,\mathrm{V}⟩\to \alpha {|\widetilde{\uparrow \Uparrow }⟩}_x+\beta {|\widetilde{\downarrow \Uparrow }⟩}_x$$

$$\alpha |{\omega }_1,\mathrm{H}⟩+\beta |{\omega }_2,\mathrm{V}⟩\stackrel{\mathrm{photo}-\mathrm{excitation}}{⟶}\alpha {|\circ \widetilde{\uparrow \Uparrow }⟩}_x+\beta {|\circ \widetilde{\downarrow \Uparrow }⟩}_x\stackrel{\text{ adiabatic transfer}}{⟶}\alpha {|\uparrow \circ \Uparrow ⟩}_x+\beta {|\downarrow \circ \Uparrow ⟩}_x,$$

$$\begin{array}{rl}& \{{|\circ \downarrow \Uparrow ⟩}_x,{|\circ \uparrow \Downarrow ⟩}_x,{|\circ \uparrow \Uparrow ⟩}_x,{|\circ \downarrow \Downarrow ⟩}_x,\\ & {|\downarrow \circ \Uparrow ⟩}_x,{|\uparrow \circ \Downarrow ⟩}_x,{|\uparrow \circ \Uparrow ⟩}_x,{|\downarrow \circ \Downarrow ⟩}_x\},\end{array}$$

$$\mathbf{H}=\left(\begin{array}{cc}{\mathbf{H}}_{\text{0},x}+{\mathbf{H}}_{\text{Ze},x}+\frac{\epsilon }{2}1& t_{\text{c}}1\\ t_{\text{c}}1& {\tilde{\mathbf{H}}}_{\text{Ze},x}-\frac{\epsilon }{2}1\end{array}\right),$$

$$\sum _{m\ne n}\left|\frac{⟨m(t)⟩\dot{n}(t)}{{\omega }_{mn}(t)}\right|\ll 1$$

$$P_{\text{LZ}}=\exp \left(-2\pi \frac{(\Delta E_{\text{AC}}/2{)}^2}{\hslash v_{\epsilon }}\right),$$

$$\sum _{m\ne n}\left|\frac{⟨m⟩\dot{n}}{{\omega }_{mn}}\right|=\left|\frac{\hslash ⟨1⟩\dot{2}}{\Delta E_{\text{AC}}}\right|=\left|\frac{v_{\epsilon }\hslash }{2{\left(\Delta E_{\text{AC}}\right)}^2}\right|$$

$$\frac{1}{v_{\epsilon }}=-\frac{4\ln (P_{\text{LZ}})}{\pi }\sum _{m\ne n}\left|\frac{⟨m⟩\frac{\partial n}{\partial \epsilon }}{{\omega }_{mn}}\right|$$

$$P_{\text{rec}}(t)=1-\exp \left(-{\int }_0^t\Gamma (t^{\prime })d{t}^{\prime }\right),$$

$$\phi (t)={\int }_0^t\frac{\Delta E_{12}(t^{\prime })}{\hslash }d{t}^{\prime }$$

$$⟨\delta {\phi }^2{⟩}_{\mathrm{tot}}=⟨\delta {\phi }^2{⟩}_{\mathrm{ch}-\mathrm{qs}}+⟨\delta {\phi }^2{⟩}_{\mathrm{ch}-\mathrm{fast}}+⟨\delta {\phi }^2{⟩}_{\mathrm{spins}}.$$

$$P_{\mathrm{deph}-\mathrm{fail}}=⟨\delta {\phi }^2{⟩}_{\mathrm{tot}}/3.$$

$$\mathbf{H}=\left(\begin{array}{cc}\epsilon \cdot 1+{\mathbf{H}}_{\text{ES}}& \mathbf{T}\\ {\mathbf{T}}^{†}& {\mathbf{H}}_{\text{SS}}\end{array}\right),$$

$$|\uparrow \circ ⟩|{\omega }_1,\text{V}⟩\stackrel{\text{photo-excit.}}{⟶}|\uparrow \circ ⟩|\downarrow \Uparrow ⟩\stackrel{\text{adiabatic transf.}}{⟶}|\uparrow \downarrow ⟩|\circ \Uparrow ⟩,$$

$$|\uparrow \circ ⟩|{\omega }_2,\text{H}⟩\stackrel{\text{photo-excit.}}{⟶}|\uparrow \circ ⟩|\uparrow \Uparrow ⟩\stackrel{\text{adiabatic transf.}}{⟶}|\uparrow \uparrow ⟩|\circ \Uparrow ⟩,$$

$$\begin{array}{r}|\downarrow \circ ⟩|\downarrow \Downarrow ⟩\stackrel{t_{\text{c}}}{⟷}|\downarrow \downarrow ⟩|\circ \Downarrow ⟩\\ {\Delta }_0^{\prime \prime }\updownarrow \\ |\downarrow \circ ⟩|\uparrow \Uparrow ⟩\stackrel{t_{\text{c}}}{⟷}|\downarrow \uparrow ⟩|\circ \Uparrow ⟩\\ \updownarrow J\\ |\uparrow \circ ⟩|\downarrow \Uparrow ⟩\stackrel{t_{\text{c}}}{⟷}|\uparrow \downarrow ⟩|\circ \Uparrow ⟩\\ {\Delta }_0^{\prime }\updownarrow \\ |\uparrow \circ ⟩|\uparrow \Downarrow ⟩\stackrel{t_{\text{c}}}{⟷}|\uparrow \uparrow ⟩|\circ \Downarrow ⟩\end{array}$$

$$|\uparrow \circ ⟩|{\omega }_1,\text{V}⟩\stackrel{\mathrm{photo}-excit.}{\to }|\uparrow \circ ⟩|\downarrow \Uparrow ⟩\stackrel{\text{adiabatic transf.}}{\to }|{\text{T}}_0⟩|\circ \Uparrow ⟩,$$

$$|\uparrow \circ ⟩|{\omega }_2,\text{V}⟩\stackrel{\mathrm{photo}-excit.}{\to }|\uparrow \circ ⟩|\uparrow \Downarrow ⟩\stackrel{\text{Rabi + ad. transf.}}{\to }|\text{S}⟩|\circ \Uparrow ⟩,$$

$$P_{a\to b}({\delta }_{ab},{\Omega }_{ab})=\frac{{\Omega }_{ab}^2}{{\Omega }_{ab}^2+{\delta }_{ab}^2}{\sin }^2\left(\sqrt{{\Omega }_{ab}^2+{\delta }_{ab}^2}\frac{T_{\mathrm{Rabi}}}{2}\right),$$

$$P_{i\to j}(T_{\mathrm{Rabi}})={\left|⟨j|e^{-\frac{i}{\hslash }{\int }_0^{T_{\mathrm{Rabi}}}{\mathbf{H}}_{\mathrm{RWA}}(t)dt}|i⟩\right|}^2.$$

$${\mathbf{H}}_{\mathrm{RWA}}(t)=\frac{\hslash }{2}\left(\begin{array}{ccc}2({\omega }_{\mathrm{S}{\mathrm{T}}_{+}}-{\omega }_{\mathrm{d}})& {\Omega }_{\mathrm{S}{\mathrm{T}}_{+}}(t)& \\ {\Omega }_{\mathrm{S}{\mathrm{T}}_{+}}(t)& 0& {\Omega }_{{\mathrm{T}}_{+}{\mathrm{T}}_0}(t)\\ 0& {\Omega }_{{\mathrm{T}}_{+}{\mathrm{T}}_0}(t)& 2({\omega }_{{\mathrm{T}}_0{\mathrm{T}}_{+}}+{\omega }_{\mathrm{d}})\end{array}\right).$$

$$|{\Psi }_2(t)⟩=e^{i{\omega }_{\mathrm{d}}t}\cos \left(\frac{{\Omega }_{{\mathrm{T}}_{+}\mathrm{S}}}{2}t\right){|{\Psi }_{{\mathrm{T}}_{+}}⟩}_{{\epsilon }^{\ast }}+\sin \left(\frac{{\Omega }_{{\mathrm{T}}_{+}\mathrm{S}}}{2}t\right){|{\Psi }_{\mathrm{S}}⟩}_{{\epsilon }^{\ast }}$$

$${\delta }_{\delta \epsilon }=\frac{\partial {\omega }_{\mathrm{S}{\mathrm{T}}_{+}}}{\partial \epsilon }\delta \epsilon ,$$

$$\hslash {\Omega }_{\delta \epsilon }=\Delta \epsilon {⟨{\Psi }_{\text{S}}⟩\frac{\partial H}{\partial \epsilon }{\Psi }_{{\text{T}}_{+}}|}_{{\epsilon }^{\ast }+\delta \epsilon }.$$

$$e^{-⟨\delta {\phi }^2⟩/2}$$

$$\Delta E_{12}(\epsilon +\delta \epsilon )\approx \Delta E_{12}(\epsilon )+\frac{\partial \Delta E_{12}(\epsilon )}{\partial \epsilon }\delta \epsilon ,$$

$$\delta \phi$$

Full text

I Intoduction

Semiconductor quantum-dot devices have demonstrated considerable potential for quantum information applications. A prominent example are gate-defined quantum dots (GDQD), i.e. quantum dots realized in semiconductor heterostructures in which individual electrons are confined by an electrostatic trapping potential. Spin qubits based on GDQD in GaAs/$\textrm{Al}_{x}\textrm{Ga}_{x-1}\textrm{As}$ heterostructures have demonstrated all key requirements for quantum information processing, such as qubit initialization, readout,(Elzerman et al., 2004; Barthel et al., 2009) coherent control(Koppens et al., 2006; Foletti et al., 2009) with high fidelity(Yoneda et al., 2014; Cerfontaine et al., 2016) and two-qubit gates.(Nowack et al., 2011; Shulman et al., 2012) Moreover, thanks to their similarity to the transistors used in modern computer chips, these top-down fabricated quantum dots have good prospects for realizing large scale quantum processing nodes. However, unlike self-assembled quantum dots, where excellent optical control and information transfer has been demonstrated,(Press et al., 2008; De Greve et al., 2012; Gao et al., 2012) GDQDs pose a number of challenges when it comes to couple them coherently with light. The problems come from the lack of exciton confinement: while the electron states are confined, the hole states are not. Since in the creation of an exciton the spin of the photo-excited electron is always entangled with the one of the hole, discarding the hole-spin inevitably leads to decoherence of the electron spin. This limits considerably the possibility of optically controlling and manipulating spins in GDQDs, and it hinders their applicability in quantum communications.

Despite these difficulties, first steps towards the goal of coherently coupling photons and electron spins in GDQDs have already been made, by trapping and detecting photo-generated carriers in GDQDs,Fujita et al. (2013) and by proving transfer of angular momentum between photons and electrons.(Fujita et al., 2015) Much of this effort is motivated by the fact that robust spin-photon entanglement is a key requirement for quantum repeaters for long-distance quantum communications,(Dür et al., 1999) as well as for distributed quantum computing, where different computing nodes based on GDQD are connected by optical channels.(Van Meter et al., 2006) One strategy to avoid entanglement between the spins of the electron and the hole is to use g-factor engineering to obtain a much smaller g-factor for the electrons than for holes.(Kosaka et al., 2008; Kuwahara et al., 2010; Kosaka, 2011)

Here we propose a different strategy, which relies on a localized exciton in an optically active quantum dot (OAQD) as interface between a photonic qubit and a spin qubit in a GDQD. The OAQD could be a self-assembled quantum dot (SAQD) – as also proposed by Engel and coworkers(Engel et al., 2006) – an impurity,(Fu et al., 2008) or a bound exciton localized with local electric-gates by exploiting the quantum Stark effect.(Schinner et al., 2013) Using effective Hamiltonian models to describe the hybrid system formed by a bound exciton tunnel coupled to a GDQD, we analyse the feasibility of two different information transfer protocols. First, we consider the case where the quantum state of the photon is mapped onto the state of a single-spin qubit, and then the case where the mapping is to a singlet-triplet qubit in a double GDQD. In both cases, the first step of the transfer process is the photo-excitation of an exciton in the OAQD in the Voigt configuration, i.e. in the presence of a strong in-plane magnetic field and normal incident light beam. We focus in particular on the effects that can hinder the coherent transfer of the photo-excited electron from the OAQD to the GDQD, assuming a unitary mapping between the photon state and the exciton state. Throughout the paper we use a InAs SAQD as a concrete example of OAQD (see Fig. 1). The described protocols can however be straightforwardly extended to other OAQD with appropriate tunnel coupling to the GDQD. We estimate the performance of the protocols using a realistic set of parameters for InAs SAQDs. According to these estimates, the proposed protocols could be completed with a success probability of approximately 85% for the case of the singlet-triplet qubit, and up to 97% probability for the single-spin qubit.

The paper is organized as follows. In Section II, we discuss in detail the protocol for transferring information to a single-spin qubit, including the possible error sources (section IIB). Section III is dedicated to the protocol for transferring information to a singlet-triplet qubit encoded in a double dot. Details on how we deal with the different noise sources and on the model Hamiltonian employed in Sec. III are given in Appendix A and B, respectively.

II Information transfer to a single spin-qubit

II.1 Transfer protocol

Transferring the information encoded in the polarization of one photon to the spin of one electron in a GDQD using an OAQD as intermediary requires two steps: (i) the creation of a bound exciton in the OAQD by absorption of the incident photon; (ii) the adiabatic transfer of the photo-excited electron into the GDQD. Here, we do not model explicitly the absorption process. Rather, we assume that the process is coherent and the photo-generated exciton in the OAQD reflects the state of the absorbed photon, and discuss under which conditions the photo-excited electron can be coherently transferred to the GDQD.

In OAQDs embedded in GaAs, the light- and the heavy-hole bands are split in energy by several tens of meV due to strainBayer et al. (2002) or confinement.(El Khalifi et al., 1989) In the following we will use the notation $\ket{\downarrow\Uparrow}_{z},\ket{\uparrow\Downarrow}_{z},\dots$ to indicate the electron-hole states, where the regular arrow represents the projection of the electron spin along the growth-direction $z$ ($S^{(\rm e)}_{z}=\pm 1/2\hbar$), and double arrows the projection of the heavy-hole spin ($S^{(\rm h)}_{z}=\pm 3/2\hbar$). In this system, electron and hole states with antiparallel spins, $\{\ket{\downarrow\Uparrow}_{z},\ket{\uparrow\Downarrow}_{z}\}$, have angular momentum $\pm\hbar$, and are optically addressable with circularly polarized light. Hence, they are referred to as bright states. States with parallel spin, $\{\ket{\uparrow\Uparrow}_{z},\ket{\downarrow\Downarrow}_{z}\}$, are optically inactive and are referred to as dark states. The Hamiltonian of the electron-hole exchange interaction takes the block-diagonal form (Bayer et al., 2002; Pikus and Bir, 1971; Van Kesteren et al., 1990; Ivchenko et al., 1993; Blackwood et al., 1994)

[TABLE]

with respect to the basis $\{\ket{\downarrow\Uparrow}_{z},\ket{\uparrow\Downarrow}_{z},\ket{\uparrow\Uparrow}_{z},\ket{\downarrow\Downarrow}_{z}\}$. Here, $\Delta_{0}$ is the energy splitting between the dark and the bright states originating from the electron-hole exchange interaction. The off-diagonal terms $\Delta_{1}$ and $\Delta_{2}$ are responsible for the energy splitting of the bright and dark excitons, respectively.

The excitation of the bright-states by photo-absorption induces entanglement between the spins of the electron and that of the hole as follows: $\alpha\ket{\sigma^{+}}+\beta\ket{\sigma^{-}}\to\alpha\ket{\downarrow\Uparrow}_{z}+\beta\ket{\uparrow\Downarrow}_{z}$, where $\alpha$ and $\beta$ are complex numbers and $\ket{\sigma^{+}}$ and $\ket{\sigma^{+}}$ represent left and right circularly polarized photons, respectively. This poses a fundamental problem if we want to map the state of the photon onto the spin of the electron only, and discard the hole. To avoid this problem, it is necessary to eliminate the entanglement between the spins of the electron and of the hole. One way to achieve this is via g-factor engineering.(Kosaka et al., 2008; Kuwahara et al., 2010; Kosaka, 2011) However, one difficulty with this approach is that the resulting small Zeeman splitting between the electron spin-states makes the system very susceptible to nuclear spin fluctuations.(Reilly et al., 2010) Furthermore, this approach is strictly limited to (Al,Ga)As-based systems and cannot be extended to other material system (e.g. II/VI)(Sanaka et al., 2009; Pawlis et al., 2011). Also, it cannot be simply extended to two-electron spin qubits with singlet-triplet encoding, which have the advantage of full electrical control.(Foletti et al., 2009)

Here, we investigate a different approach, which is based on applying a strong in-plane magnetic field that mixes bright and and dark states making all of them optically accessible(Bayer et al., 2002), as described below. We assume the in-plane magnetic field to be along the $x$-direction. Taking this as as the spin-quantization axis, the exciton Hamiltonian takes the form

[TABLE]

where

[TABLE]

represents the electron-hole exchange interaction with respect to the basis $\{\ket{\downarrow\Uparrow}_{x},\ket{\uparrow\Downarrow}_{x},\ket{\uparrow\Uparrow}_{x},\ket{\downarrow\Downarrow}_{x}\}$, and

[TABLE]

accounts for the Zeeman splitting induced by the in-plane magnetic field $\vec{B}=B\hat{e}_{x}$. Here, $g_{\rm e}$ and $g_{\rm h}$ are the g-factors for the electron and for the hole, respectively (with the sign convention that $g_{\rm e}$ is negative and $g_{\rm h}$ is positive). In the limit of large magnetic field ($|g_{\textrm{e}}\pm g_{\textrm{h}}|\mu_{\textrm{B}}B\gg\Delta_{0},\Delta_{1},\Delta_{2}$), the eigenstates of $\mathbf{H}_{\textrm{ex}}$ almost coincide with the basis kets $\{\ket{\downarrow\Uparrow}_{x},\ket{\uparrow\Downarrow}_{x},\ket{\uparrow\Uparrow}_{x},\ket{\downarrow\Downarrow}_{x}\}$. We will therefore denote them by their dominant basis-state contribution, e.g. $\ket{\widetilde{\uparrow\Uparrow}}_{x}=\sqrt{1-\delta}\ket{\uparrow\Uparrow}_{x}+\sqrt{\delta}\ket{\downarrow\Downarrow}_{x}$, where $\delta$ is small for large $B$. All eigenstates have a contribution from the bright-states, i.e. they are all optically active.

In general, the *bright-state contribution * (BC) of a state $\ket{\Psi}$ can be quantified as follow:

[TABLE]

The BC is a factor determining how fast a photon can be absorbed (or reemitted). BC substantially smaller than one are not fundamentally problematic, as they can be compensated by longer photon wave-packets. Eigenstates with parallel-spins $\left(\ket{\widetilde{\uparrow\Uparrow}}_{x},\ket{\widetilde{\downarrow\Downarrow}}_{x}\right)$ can be addressed only with horizontally polarized photons, while eigenstates with antiparallel-spins $\left(\ket{\widetilde{\downarrow\Uparrow}}_{x},\ket{\widetilde{\uparrow\Downarrow}}_{x}\right)$ require vertically polarized photons. The idea is now to use the pair $\{\ket{\widetilde{\uparrow\Uparrow}}_{x},\ket{\widetilde{\downarrow\Uparrow}}_{x}\}$ to map a photon state as follow:

[TABLE]

(or, alternatively, $\alpha\ket{\omega_{1}^{\prime},{\rm V}}+\beta\ket{\omega_{2}^{\prime},{\rm H}}\to\alpha\ket{\widetilde{\uparrow\Downarrow}}_{x}+\beta\ket{\widetilde{\downarrow\Downarrow}}_{x}$), where $\alpha$ and $\beta$ are complex numbers and $\ket{\omega,{\rm H(V)}}$ represents a photon state with energy $\omega$ and horizontal (vertical) polarization. In this kind of mapping the whole information on the state of the photon is entirely mapped on the spin of the electron alone, since the excitonic states $\ket{\widetilde{\uparrow\Uparrow}}_{x}$ and $\ket{\widetilde{\downarrow\Uparrow}}_{x}$ have the same spin projection for the hole.

The next step of the protocol – and the main subject of our analysis – is the coherent transfer of the photo-excited electron from the OAQD to a GDQD. If the OAQD and the GDQD are tunnel coupled, the coherent transfer between the two can be achieved by adiabatically increasing the detuning $\varepsilon$ between the electronic levels in the two system (see Fig. 1b). Ideally, the whole transfer protocol will then work as follows:

[TABLE]

where now $\ket{\circ\widetilde{\uparrow\Uparrow}}_{x}$ represents the state where the GDQD is empty and there is an exciton with the parallel spins in the OAQD (see schematic in Fig.1b), while $\ket{\uparrow\circ\Uparrow}_{x}$ represents the state where the electron has been transferred into the GDQD, leaving a hole alone in the OAQD (and similarly for the other states).

We model the GDQD as a single electronic level and use the basis

[TABLE]

to represent the states of the coupled exciton-GDQD system. With respect to this basis the Hamiltonian of the coupled exciton-GDQD system reads

[TABLE]

where $t_{c}$ is a spin-conservingFujita et al. (2013) tunneling matrix element, and $\varepsilon$ is the gate-dependent energy detuning between the OAQD and the GDQD, see Fig. 1b. We call the states where the electron and the hole are both on the OAQD excitonic states, and those where the electron has been transferred to the GDQD separated states. The excitonic exchange-interaction, $\mathbf{H}_{{\rm 0},x}$, has non-vanishing matrix elements only between excitonic states. $\tilde{\mathbf{H}}_{\textrm{Ze},x}$ has the same structure as in Eq.(4), but it can differ numerically from $\mathbf{H}_{\textrm{Ze},x}$ because of a different Zeeman splitting in the GDQD in the OAQD (e.g., because of a different $g$-factor, $\tilde{g}_{\rm e}$, in the GDQD). Spin-orbit effects, which in principle can lead to spin-flip processes during the adiabatic transfer,Schreiber et al. (2011) are not included in Eq.(9) as we assume the dot separation to be much shorter than the spin-orbit length in GaAs,111Typical values for the spin-orbit length in GaAs are in the range $1-$10\text{,}\mathrm{\SIUnitSymbolMicro m}$$(Petta et al., 2010; Nichol et al., 2015). making spin-orbit negligible compared to other effects.

The eigenstates of Eq.(9) can be easily determined numerically. In Fig. 2 we plot the corresponding eigenenergies as a function of the detuning $\varepsilon$ for the case of large tunnel coupling $t_{\textrm{c}}=$150\text{,}\mathrm{\SIUnitSymbolMicro eV}$$. This figure also includes a schematic diagram of the information-transfer process described in Eq.(7). The system is photo-excited at negative detuning, where excitonic states are energetically favourable (bright spots in Fig. 2). The photo-excited electron is then transferred to the GDQD by adiabatically increasing $\varepsilon$ to the regime of separated states. The colour code represents the BC of each eigenstate, which clearly depends on the detuning.

II.2 Error sources

In practice, the viability of the protocol sketched in Eq.(7) depends on several factors and error sources. First, both selected excitonic states need to show sufficient optical coupling. The BC determines how rapidly a photon can be absorbed/reemitted from a certain state, and it will therefore determine the photon pulse-length needed for optimal absorption. State-dependent photon pulse-shaping might be therefore required to compensate the different BC of the excitonic states. Furthermore, ideally each exciton should couple to a single photon mode, in order to reduce dissipative losses and to ensure that the photon emission/absorption process is a unitary process. Bragg mirrors and solid immersion lenses may have to be employed to increase the collection efficiency of the OAQD. These important, setup specific, engineering problems go however beyond the scope of this paper. Here we will focus instead on the intrinsic error sources that can affect the adiabatic transfer process, assuming that optimal mode engineering ensures a unitary mapping between the photon state and the exciton state.

The first source of errors in the adiabatic transfer process are non-adiabatic transitions to other states, which can occur if the detuning $\varepsilon$ is increased too quickly. Given a time-dependent Hamiltonian $H(t)$, with instantaneous eigenstates $\ket{m(t)}$ (i.e. $H(t)\ket{m(t)}=E_{m}(t)\ket{m(t)}$), a necessary condition for adiabatic evolution is(Messiah, 1962; Tong, 2010)

[TABLE]

with $\hbar\omega_{mn}(t)=E_{m}(t)-E_{n}(t)$. If this criterion is violated, transitions between different eigenstates are expected. For the simple case of a two-level system, the probability of transitions between the two levels when sweeping through the avoided crossing is given by the well known Landau-Zener formulaZener (1932); Shevchenko et al. (2010)

[TABLE]

where $\Delta E_{\textrm{AC}}$ is the energy splitting at the avoided crossing and $v_{\varepsilon}$ is the sweep speed. This formula allows to calculate the highest possible sweep speed for a given $\Delta E_{\textrm{AC}}$ and a targeted maximum transition probability (e.g., $P_{\rm LZ}=1\%$). To obtain a similar bound on the sweep speed for the eight-level system that we are considering, we notice that the exponent in Eq.(11) is closely related to the quantity on the left hand side of Eq. (10), being

[TABLE]

for the case of a two-level system. This motivates us to take

[TABLE]

as a bound for the maximal sweep speed that is allowed in order to have a transition probability between different eigenstates smaller or equal to $P_{\rm LZ}$. In the following we will require $P_{\rm LZ}=1\%$, i.e. a $99\%$ success probability for the adiabatic transfer.

The sweep speed $v_{\varepsilon}$ and the sweep range $\Delta\varepsilon$ determine the time $T_{\rm tr}$ on which the transfer can be completed. On the timescale of $T_{\rm tr}$ various factors that hinder the transfer process are at play. First of all, excitons can decay due to radiative recombination. We estimate the probability of recombination in a certain time $t$ as follows

[TABLE]

where $\Gamma(t)={\rm BC}(\Psi(t))/\tau$ is the instantaneous decay rate of the state $\ket{\Psi(t)}$, with $\tau$ the characteristic lifetime of a bright exciton. Here $\ket{\Psi(t)}$ stands generically for the instantaneous state of the system at time $t$. Radiative recombination reduces the efficiency of the information-transfer process, but it does not introduces errors in the encoding of the information.

Other factors hindering the transfer process are charge and nuclear-spin noise, which are well known sources of dephasing,Chirolli and Burkard (2008); Chekhovich et al. (2013) causing random fluctuations of the relative phase accumulated between two states $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$

[TABLE]

accumulated between two states $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$, where $\Delta E_{12}$ is the energy difference between the states. Charge noise affects $\Delta E_{12}$ (and therefore $\varphi(t)$) by causing random fluctuations of the detuning $\varepsilon$. Nuclear spins in the host material affect $\Delta E_{12}$ by creating a randomly fluctuating magnetic field, the Overhauser field (OF).(Barthel et al., 2009) Here we consider both quasi-static and fast uncorrelated charge noise, as well as nuclear-spin noise. Quasi-static noise is due to fluctuations that occur on time scales much longer than the transfer time $T_{\rm tr}$, corresponding to a spectral density centered around zero frequency. We denote the root-mean-squared (rms) charge noise amplitude by $\varepsilon_{\rm rms}$. Nuclear-spin fluctuations are to a good approximation quasi-static and quantified by their rms. amplitude, $B_{\rm OF}^{(\rm rms)}$. On the contrary, fast uncorrelated noise has equal weight $S_{\varepsilon}$ at all frequencies (white noise). Typical values for $\varepsilon_{\rm rms}$, $S_{\varepsilon}$ and $B_{\rm OF}^{(\rm rms)}$ in GaAs-based devices are $\varepsilon_{\rm rms}=$8\text{,}\mathrm{\SIUnitSymbolMicro V}$/L$,Dial et al. (2013) $S_{\varepsilon}=$5\text{\times}{10}^{-20}\text{,}{\mathrm{V}}^{2}\text{,}{\mathrm{Hz}}^{-1}$/L^{2}$,(Dial et al., 2013)222The values given in Ref.Dial et al., 2013 refer to gate-equivalent voltage noises, and have to be scaled by a lever-arm of the order of 10\text{\,}{\mathrm{\text{ e }}}^{-1} to be converted into detuning energy noise. and $B_{\rm OF}^{(\rm rms)}=5-$50\text{,}\mathrm{mT}$$, where the first value is typical for GDQDs(Taylor et al., 2007) and the second for SAQDs.(Taylor et al., 2007; Prechtel et al., 2016) L denotes the lever arm converting voltages on gates to detuning variations.

For each source of noise we evaluate the quantity $\langle\delta\varphi^{2}\rangle$ describing the dephasing due to that particular noise source, as detailed in Appendix A. Assuming that all noise sources are uncorrelated, the total dephasing is given by

[TABLE]

In order to better compare the effect of dephasing to other mechanisms that lead to failure of the transfer process, we introduce the failure probability due to dephasing, which we define as the probability of the depolarizing channel with the same average gate fidelity as the dephasing channel (see Appendix A)

[TABLE]

In the following, we discuss results obtained for the set of realistic parameters presented in Tab. 1 for the case where the OAQD is an InAs self-assembled quantum dot. For simplicity we assume $\Delta_{1}=\Delta_{2}=$0\text{,}\mathrm{\SIUnitSymbolMicro eV}, as these splittings are typically small compared to $\Delta_{0}$,(Bayer *et al.*, [2002](#bib.bib22)) and do not lead to qualitative changes. For the case of large tunnel coupling considered in Fig. [2](#S2.F2) ($t_{\textrm{c}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro eV}) , we find that the maximal sweep velocity with $P_{\rm LZ}=$1\text{,}\mathrm{\char 37\relax} is $v_{\varepsilon}=14.8\,{\rm meV/ns}$, so that a sweep over the whole displayed detuning range can be completed in $T_{\rm tr}=$0.14\text{\,}\mathrm{ns}. On this time scale, the probability of radiative recombination is $P_{\rm rec}=2\%$ for the state with parallel spins and $5.5\text{\,}\mathrm{\char 37\relax}$ for the state with anti-parallel spins. At the same time, the effects of charge noise as well as of nuclear-spin noise are rather small, giving $P_{\rm deph-fail}=$0.4\text{,}\mathrm{\char 37\relax}$$. The total failure probability then is $P_{\rm fail-tot}=P_{\rm LZ}+P_{\rm rec}+P_{\rm deph-fail}\leq 6.8\%$. Thus, the state-transfer process will work rather reliably and fail mostly because of recombination.

For smaller tunnel coupling ($t_{\textrm{c}}=$50\text{,}\mathrm{\SIUnitSymbolMicro eV}), the eigenstates of Eq.([9](#S2.E9)) exhibit a series of crossing and anti-crossing, see Fig. [3](#S2.F3). In Fig. [3](#S2.F3)a the colour code represents again the bright-state contributions of the various eigenstates, while in Fig. [3](#S2.F3)b red and blue indicate eigenstates with parallel or anti-parallel spins, respectively. Since these form two separate subspaces, crossings can occur between them. The labels $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$ indicate the states involved in the adiabatic-transfer process sketched in Eq. ([7](#S2.E7)), i.e. $\ket{\Psi_{1}}\approx\ket{\circ\uparrow\Uparrow}$, $\ket{\Psi_{2}}\approx\ket{\circ\downarrow\Uparrow}$ for large, negative detuning and $\ket{\Psi_{1}}\approx\ket{\uparrow\circ\Uparrow}$, $\ket{\Psi_{2}}\approx\ket{\downarrow\circ\Uparrow}$ for large, positive detuning. Fig. [3](#S2.F3)c represents the inverse of the maximal sweep velocity $1/v_{\varepsilon}$ that allows adiabatic evolution along the states $\ket{\Psi_{1}}$ (red) and $\ket{\Psi_{2}}$ (blue), calculated according to Eq. ([13](#S2.E13)) for $P_{\rm LZ}=1\%$. As expected, $1/v_{\varepsilon}$ shows maxima in correspondence of the anticrossings. Knowing $v_{\varepsilon}$, we can calculate the total time required for the adiabatic transfer. For simplicity we assume that the transfer occurs with a constant speed, equal to the smaller possible maximal speed $1/v_{\varepsilon}=$0.70\text{\,}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{m}\mathrm{eV} (see Fig. 3c). Because of this low sweep speed, it is convenient to photo-excite the system not in the strongly detuned regime, but close to $\varepsilon=$0\text{,}\mathrm{meV}, to limit the time spent in a hybridized charge state and exposed to the strong nuclear spin field in the SAQD, as well as to minimize the probability of radiative recombination. Specifically, we choose photo-excitation to occur at excitation point $\varepsilon_{\rm EP}=$-0.035\text{\,}\mathrm{meV} (see dashed-line in Fig. 3). At this point, the two states $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$ have the same bright-states contribution ${\rm BC}=$20.3\text{,}\mathrm{\char 37\relax}. The transfer time required to sweep from $\varepsilon_{\rm EP}$ to the final value $\varepsilon=0.25$ meV at the constant speed $1/v_{\varepsilon}=$0.70\text{\,}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{m}\mathrm{eV} is $T_{\rm tr}=$0.20\text{,}\mathrm{ns}. On this timescale, radiative recombination only marginally limits the probability of a successful transfer, being $P_{\rm rec}=$1.5\text{\,}\mathrm{\char 37\relax} for the state $\ket{\Psi_{1}}$ and $0.8\%$ for the state $\ket{\Psi_{2}}$ (see Fig. 3d). The probability of concluding the transfer without dephasing is shown in Fig. 3e. The results of Fig. 3e correspond to a probability of failure due to dephasing of $P_{\rm deph-fail}=$0.25\text{,}\mathrm{\char 37\relax}, and a total failure probability for the transfer process of $P_{\rm fail-tot}\leq$2.75\text{\,}\mathrm{\char 37\relax}.

III Information transfer to a singlet-triplet qubit

III.1 Transfer protocol

We consider now a different system, namely, instead of the coupling to a single-spin qubit, we consider the case where the electronic state in the OAQD is tunnel coupled to a gate-defined double quantum-dot (DD), see Fig. 1c. One of the advantages of this setup is that a DD can be used to encode a singlet-triplet qubit, which allows high manipulation fidelities in systems with large hyperfine interaction such as GaAs.Cerfontaine et al. (2014) As in Sec. II, we assume here that the state of a photon is mapped onto an exciton in the OAQD in the presence of an in-plane magnetic field $\vec{B}=B\vec{e}_{x}$, and discuss under which conditions the photo-excited electron can be coherently transfered to a neighbouring DD. Furthermore, we assume that before the optical excitation, an electron is initialized in the left side of the DD (i.e. in the one more far away from the OAQD, see Fig. 1c) by an appropriate choice of the detuning $\varepsilon_{\textrm{DD}}$.

The Hamiltonian of the coupled DD-exciton system can be divided into two different subspaces: one formed by states where the both photo-excited electron and hole are localized on the OAQD, which we call excitonic states (ES), and the other formed by states where the photo-excited electron has been transferred to the DD, which we call separates states (SS). The subset of excitonic states is spanned by the basis $\left\{\ket{\uparrow\circ},\ket{\downarrow\circ}\right\}\otimes\left\{\ket{\downarrow\Uparrow},\ket{\uparrow\Downarrow},\ket{\uparrow\Uparrow},\ket{\downarrow\Downarrow}\right\}$), while the subset of separated states is spanned by $\{\ket{\rm S(0,2)}\ket{\rm S(2,0)},\ket{\uparrow\downarrow},\ket{\downarrow\uparrow},\ket{\uparrow\uparrow},\ket{\downarrow\downarrow}\}\otimes\{\ket{\circ\Uparrow},\ket{\circ\Downarrow}\}$, where the spin quantization axis is taken along the direction of the applied magnetic field. In this notation, the kets on the left represent the state of the DD, with $\ket{\rm S(0,2)}$ ($\ket{\rm S(2,0)}$) representing the singlet state with the two electrons in the right (left) side of the DD, and $\ket{\uparrow\downarrow},\ket{\downarrow\uparrow},\dots$ representing states with one electron on each side of the DD. The two subsets of excitonic and separated states are connected by a spin-conserving tunnel Hamiltonian, ${\bf T}$, with coupling strength $t_{c}$. The total Hamiltonian of the DD-exciton system then reads

[TABLE]

where $\mathbf{H}_{\textrm{ES}}$ and $\mathbf{H}_{\textrm{SS}}$ are the Hamiltonians acting on the ES and SS subspaces, respectively, and $\varepsilon$ is the energy detuning between the two subspaces. The expressions for $\mathbf{H}_{\textrm{ES}}$ and $\mathbf{H}_{\textrm{SS}}$ are given in Appendix B.

When occupied by two electrons, a DD can be operated as s singlet-triplet (ST) qubit, using the singlet $\ket{\textrm{S}}=\frac{1}{\sqrt{2}}\left(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow}\right)$ and triplet $\ket{\textrm{T}_{0}}=\frac{1}{\sqrt{2}}\left(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow}\right)$ as states of the computational basis.Burkard et al. (1999) The remaining triplet states $\ket{\textrm{T}_{+}}=\ket{\uparrow\uparrow}$ and $\ket{\textrm{T}_{-}}=\ket{\downarrow\downarrow}$ are split off in energy by the external magnetic field. The most relevant energy scale for the operation of a singlet-triplet qubit is the energy splitting $J=E_{\rm S}-E_{\rm T_{0}}$ between the singlet $\ket{\rm S}$ and the triplet $\ket{\rm T_{0}}$. It depends on the inter-dot tunnel coupling $t_{\rm DD}$ (see Fig. 1c), as well as the inter-dot detuning $\varepsilon_{\rm DD}$, which is an easily accessible parameter.Burkard et al. (1999); Stepanenko et al. (2012)

A straightforward extension of the information-transfer process described in Sec. II to the case of a DD would work as follows:

[TABLE]

representing only the evolution of the basis states). The state $\ket{\uparrow\downarrow}\ket{\circ\Uparrow}$ can then be easily mapped onto the state $\ket{\rm T_{0}}\ket{\circ\Uparrow}$ by adiabatically increasing the exchange interaction in the DD (i.e. by increasing $J$).333The state $\ket{\uparrow\downarrow}\ket{\circ\Uparrow}$ has an higher energy than $\ket{\downarrow\uparrow}\ket{\circ\Uparrow}$ because of the exchange interaction with the hole, and it therefore naturally evolves into the triplet state $\ket{\rm T_{0}}\ket{\circ\Uparrow}$ (and not into $\ket{\rm S}\ket{\circ\Uparrow}$) as the exchange interaction $J$ is adiabatically increased. The energy splitting between $\ket{\uparrow\downarrow}\ket{\circ\Uparrow}$ and $\ket{\downarrow\uparrow}\ket{\circ\Uparrow}$ can furthermore be adjusted by introducing a small magnetic field gradient $\Delta B_{z}$ between the two dots, by dynamical nuclear spin polarization. However, mapping $\ket{\uparrow\uparrow}\ket{\circ\Uparrow}$ onto $\ket{\rm S}\ket{\circ\Uparrow}$, which is the other state of the computational basis, requires some spin-non-conserving mechanism such as, for example, the Overhauser field or the spin-orbit interaction. These effects introduce an anti-crossing between the states $\ket{\rm S}$ and $\ket{\rm T_{+}}=\ket{\uparrow\uparrow}$ in the regime where the exchange splitting $J$ approximately equals the Zeeman splitting.Stepanenko et al. (2012); Cerfontaine et al. (2014) This anticrossing can be used to transform $\ket{\uparrow\uparrow}$ into $\ket{\rm S}$, however this approach would suffer from strong charge dephasing, since ${\textrm{T}_{2}^{*}}\propto\left(dJ/d\varepsilon_{\textrm{DD}}\right)^{-1}$ and $dJ/d\varepsilon_{\textrm{DD}}$ is fairly large at the S-$\textrm{T}_{+}$ transition.(Dial et al., 2013) Furthermore, the phase acquired during the process would also depend strongly on the hyperfine field.

For this reason we take a different approach, which is based on exploiting the exchange interaction between the electron initialized in the left part of the DD and the hole in the OAQD. This interaction stems from the combination of $t_{c}$, the exciton coupling in the OAQD, and the exchange interaction $J$ in the DD, according the scheme schematically represented in the following diagram:

[TABLE]

for one of the two independent subspaces that forms the Hilbert space of the system (see Appendix B). Here, each arrow reflects a coupling term, with $\Delta_{0}^{\prime}=-2\Delta_{0}+\Delta_{1}-\Delta_{2}$, $\Delta_{0}^{\prime\prime}=-2\Delta_{0}-\Delta_{1}+\Delta_{2}$. The energy eigenstates corresponding to this subspace are plotted in Fig. 4a. This exchange interaction between the electron initialized in the left side of the DD and the hole in OAQD creates an indirect coupling between the states $\ket{\downarrow\circ}\ket{\downarrow\Uparrow}$ and $\ket{\uparrow\circ}\ket{\downarrow\Downarrow}$ and more generally, between the T*+*-like and the S-like branches in Fig. 4a. Exploiting this coupling, it is possible to induce transitions between these two branches by applying a suitable ac-modulation of the detuning $\varepsilon$.

This fact can be used to transfer information encoded into photons with different energy but the same polarization according to the following scheme:

[TABLE]

where again we only represent the evolution of the basis states. The idea is the following. First, the system is optically excited at a certain value of the detuning $\varepsilon=\varepsilon_{\rm EP}$, transferring the state of the photon into the sub-space of excitons with anti-parallel spins. Then $\varepsilon$ is swept to the driving point $\varepsilon^{*}$, where a Rabi pulse is applied to drive the transition between the T*+*-like branch and the S-like branch. During the whole procedure the double-dot detuning $\varepsilon_{\rm DD}$ is set to finite negative values to provide a large enough $J$ and to prevent tunnelling of the electron initialized in the left part of the DD to the right part. After the Rabi pulse, $\varepsilon_{\rm DD}$ is swept to zero and $\varepsilon$ is tuned to the regime of separated states, thus mapping the state into the subspace spanned by the computational basis $\{\ket{\textrm{S}},\ket{\textrm{T}_{0}}\}\otimes\ket{\circ\Uparrow}$. The pulse scheme for $\varepsilon$ and $\varepsilon_{\rm DD}$ required for such a protocol is sketched in Fig. 4c. In the discussion above we assumed that the DD is initialized in the $\ket{\uparrow\circ}$ state (which can be achieved with standard procedures), however the protocol can be easily adapted to match the cases where the DD is initially in the state $\ket{\downarrow\circ}$ and/or to the case where the photon has horizontal polarization. For the sake of clarity, in the following we focus only on the case described above.

III.2 Feasibility of the transfer protocol

The transfer protocol described above depends on the choice of a number of parameters. The excitation point $\varepsilon_{\rm EP}$ has to be chosen in such a way that the photon polarization prevents the excitation of states other than those considered in Eq. (19). To do so, we consider the degree of vertical polarization of each state, i.e. the projection on the sub-space with excitonic states with antiparallel spin: ${\rm VP}(\Psi)=\left|\braket{\Psi}{\uparrow\Downarrow}_{x}\right|^{2}+\left|\braket{\Psi}{\downarrow\Uparrow}_{x}\right|^{2}.$ This quantity is plotted in Fig. 4b for the the relevant branches. We chose $\varepsilon_{\rm EP}$ by requiring ${\rm VP}=20\%$ for the S-like branch at the excitation point, to limit direct excitation of this branch in combination with energy selectivity, which will help to achieve a high fidelity.

The next parameter to be fixed is the position of the driving point $\varepsilon^{*}$, which has to be chosen such as to allow an efficient Rabi $\pi$-pulse, i.e. a pulse that drives the transition between the T*+*-like branch and the S-like branch in the the shortest possible time. For a two-level system driven by a rectangular pulse of the form $\varepsilon(t)=\varepsilon^{*}+\Delta\varepsilon(t)\cos(\omega_{\textrm{d}}t)$, with $\Delta\varepsilon(t)=\Delta\varepsilon$ for time $t\in[t^{*},t^{*}+T_{\rm Rabi}]$ and zero otherwise, the probability of transition $\ket{a}\to\ket{b}$ is given by the well-known Rabi formula (Rabi, 1937; Boradjiev and Vitanov, 2013; Batista, 2015)

[TABLE]

where $\delta_{ab}=\omega_{\rm d}-\omega_{ab}$, with $\hbar\omega_{ab}=E_{a}-E_{b}$, is the detuning of the driving and $\hbar\Omega_{ab}=\Delta\varepsilon\lambda_{ab}$ the Rabi frequency, with $\lambda_{ab}=\left.\Braket{a}{\frac{\partial H}{\partial\varepsilon}}{b}\right|_{\varepsilon^{*}}$ the transition matrix element between the states. The conditions for a $\pi$-pulse are therefore $\delta_{ab}=0$ (resonant driving) and $T_{\rm Rabi}=\pi/{\Omega_{ab}}={\pi\hbar}/(\Delta\varepsilon\lambda_{ab})$. For the case of the transition between the T*+*- and the S-like branches, the coupling element $\lambda_{\rm T_{+}S}$ depends on $\varepsilon$ as shown by the red-curve in Fig. 5b. For each value of $\varepsilon$, we fix the driving amplitude $\Delta\varepsilon$ by requiring that $\lambda_{\rm T_{+}S}$ does not vary more than 50% in the detuning range $[\varepsilon-\Delta\varepsilon,\varepsilon+\Delta\varepsilon]$. The corresponding time required for a Rabi $\pi$-pulse is plotted as a red curve in Fig. 5c. The minimum of this curve gives the optimal point $\epsilon^{*}$ to apply the Rabi pulse. For the case of Fig. 5, it is $\varepsilon^{*}=0.17$meV, corresponding to a pulse of duration $T_{\rm Rabi}=0.55$ns and a resonance frequency $\omega_{\rm d}=2\pi\cdot 14.5$ GHz.

An important point to take into account when considering the Rabi pulse is the possible unwanted leakage to the T0-like branch, which is energetically very close to the S-like one, and has similar coupling matrix elements. To estimate this leakage, we consider the three-level system formed by the states $\ket{\Psi_{\textrm{S}}}$, $\ket{\Psi_{\textrm{T}_{+}}}$ and $\ket{\Psi_{\textrm{T}_{0}}}$, which represent the S-like, the T*+-like and the T0*-like branches at the excitation point $\epsilon^{*}$, and evaluate the transition probability

[TABLE]

Here, ${\bf H}_{\rm RWA}$ is the the Hamiltonian of the system in the rotating frame with respect to the drive with the rotating wave approximation

[TABLE]

The grey curves in in Fig. 5c show selected levels of constant leakage, $P_{\rm Rabi-leak}=P_{\Psi_{\rm T_{+}}\to\Psi_{\rm T_{0}}}=5\%$ down to $0.25\%$. For the considered Rabi pulse given by the minimum of the red curve, the leakage probability is $P_{\rm Rabi-leak}=0.6\%$

Another possible source of errors during the protocol are non-adiabatic transitions between the branches caused by an excessive sweep-speed. As in Sec. II, we use Eq. (13) to get a bound on the maximally allowed sweep-speed, requiring $P_{\rm LZ}=1\%$. The (inverse) maximal sweep-speed, $1/v_{\varepsilon}$, is plotted in Fig. 5d. In the following we assume for simplicity a constant sweep-speed from the excitation point $\varepsilon_{\rm EP}=0.09$ meV to the driving point $\varepsilon^{*}=0.17$ meV, and from here to the final detuning $\varepsilon=1.5$ meV. The results of Fig. 5d indicate that in this range of detuning the largest allowed sweep-speed is $1/v_{\varepsilon}=$1.45\text{,}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{m}\mathrm{eV}$$ , which then correspond to a transfer time of 0.12 ns from the excitation point $\varepsilon_{\rm EP}$ to the driving point $\varepsilon^{*}$, and of 1.93 ns from the driving point to the final detuning $\varepsilon=1.5$ meV.

The energies of the relevant protocol branches as a function of time are sketched in Fig. 6a, where we also take into account that the inter-dot detuning $\varepsilon_{\rm DD}$ is swept to zero at the end of the Rabi-pulse (see Fig. 4), separating the S$(2,0)$-like branch from the basis states. The bright spots represents the photo-excitation of the T0-like and the T*+*-like bands. In the ideal case of adiabatic evolution during the sweeps and of a perfect Rabi pulse, the basis states $\ket{\Psi_{1}}$, $\ket{\Psi_{2}}$ (where we use the notation of App. A) evolve as follows: $\ket{\Psi_{1}(t)}=\ket{\Psi_{\rm T_{0}}(t)}$ for any time, and $\ket{\Psi_{2}(t)}=\ket{\Psi_{\rm T_{+}}(t)}$ for $0<t<t^{*}$,

[TABLE]

for $t^{*}<t<t^{*}+T_{\rm Rabi}$ and, finally, $\ket{\Psi_{2}(t)}=\ket{\Psi_{\rm S}(t)}$ for $t>t^{*}+T_{\rm Rabi}$, where again $\ket{\Psi_{\textrm{T}_{0}}}$, $\ket{\Psi_{\textrm{T}_{+}}}$ and $\ket{\Psi_{\textrm{S}}}$, represent the T0-like, the T*+*-like and the the S-like branches, respectively. Assuming this time evolution, we evaluate the probability of recombination, Eq. (14), as well as the failure probability due to dephasing, Eq. (16), during the protocol. The results are plotted in Fig. 6b-c. The probability of recombination lies between $P_{\rm rec}=7.9\%-12.0\%$, while the failure probability due to dephasing amount to $P_{\rm deph-fail}=1.1\%$ and it is dominated by nuclear-spin noise.

Finally, we take into account that the performance of the Rabi pulses is also affected by charge and spin noise, as they both affect the resonance condition, as well as the Rabi frequency. We implement this by numerically calculating the average transition probability. Assuming that all noise sources are quasi-static and uncorrelated, we calculate the failure probability due to each source of noise separately. For example, fluctuations of the detuning $\varepsilon$ cause the failure probability $P_{{\rm Rabi-fail},\delta\varepsilon}=1-\braket{P_{\Psi_{\rm T_{+}}\rightarrow\Psi_{\rm S}}(\delta_{\delta\varepsilon},\Omega_{\delta\varepsilon})}_{\delta\varepsilon}$, with $P_{\Psi_{\rm T_{+}}\rightarrow\Psi_{\rm S}}$ as given by Eq.(20) and

[TABLE]

The average $\langle{\cdot}\rangle_{\delta\varepsilon}$ is calculated assuming zero-mean Gaussian fluctuations of $\delta\varepsilon$. In a similar way we calculate the failure rate of the Rabi pulse due to fluctuations in the inter-dot detuning $\delta\varepsilon_{\textrm{DD}}$, as well as in the Overhauser field in the different dots $\delta\tilde{B}^{\rm(L)}_{\textrm{OF}}$, $\delta\tilde{B}^{\rm(R)}_{\textrm{OF}}$, and $\delta B_{\textrm{OF}}$. Finally, we sum over all these failure probabilitys to estimate the reliability of the Rabi pulse. For the set of parameters used in Fig. 6 we obtain $P_{\rm Rabi-fail}=0.3\%$.

The failure probabilities due to the different error sources are summarized in Tab. 2. Adding all failure probabilities, we conclude that the transfer protocol can be completed with a success probability of $P_{\rm success}>85.0\%$.

IV Conclusions

We presented a feasibility analysis of two protocols for transferring information from a photonic qubit to qubits realized in GDQDs, considering both the cases of single-spin qubits, and of singlet-triplet qubits. The protocols are based on using an OAQD as interface between the photonic and the spin qubit. Our analysis is based on effective Hamiltonian models for describing the hybrid systems formed by a bound exciton in the OAQD that is tunnel coupled to a single or to a double GDQD. We focus in particular on the error sources that can affect the transfer process. Specifically, we take into account the influence of the adiabatic transfer conditions, the recombination of the exciton, the decoherence due to charge and nuclear-spin noise, as well as the inaccuracy of the Rabi pulse needed for the case of information transfer to a singlet-triplet qubit. We use simple noise models, which are expected to account for the most important effects.

As a concrete example, we consider the case where the OAQD is realized by a InAs SAQD. We find that for the realistic set of parameters summarized in Tab. 1, the single-spin protocol can be completed within the coherence time with a success probability in the range $>93.2\%$, depending on the strength of tunnel coupling between SAQD and GDQD. The presented version of the singlet-triplet protocol can be complete with a success probability above $P_{\rm success}>85.0\%$. These results are based on rather conservative estimates of charge fluctuation amplitudes,(Dial et al., 2013; Cerfontaine et al., 2014) as samples with better performances have been reported.(Kuhlmann et al., 2013) Furthermore, the OF field fluctuations can be further reduced using dynamic nuclear polarization with feedback.(Bluhm et al., 2010; Latta et al., 2009; Xu et al., 2009) We did not address specifically the important but setup specific issue of how to ensure optimal optical coupling between the photonic qubit and the OAQD. We also did not consider the additional constrains that might occur in an experimental implementation of our protocols (e.g. achievable sweep-speed, idle and pulse rise times), though the values obtained are compatible with high-end equipment. On the other hand our estimates are conservative and leave substantial room for improvement (e.g. nonlinear sweeps and pulse shaping). We thus expect that the the proposed schemes could be implemented with reasonably high success probabilities.

V Acknowledgments

This work was supported by the Alfried Krupp von Bohlen und Halbach Foundation and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 679342). P.C. acknowledges support by Deutsche Telekom Stiftung.

Appendix A Dephasing noise

The relative phase $\varphi$ accumulated in a certain time $t$ between two states $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$ is given in Eq.(15), where $\Delta E_{12}$ is the energy difference between the two states. Any physical mechanism that causes random fluctuations of $\Delta E_{12}$ leads to random fluctuations $\delta\varphi$ of the relative phase,inducing dephasing in a superposition of $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$. For the case of zero-average Gaussian noise, the induced dephasing is quantified with

[TABLE]

where $\langle\delta\varphi^{2}\rangle$ is the variance of the phase fluctuations.

A.1 Charge noise

Charge noise introduces stochastic fluctuations of the detuning, $\varepsilon\mapsto\varepsilon+\delta\varepsilon$, where $\delta\varepsilon$ is a randomly fluctuating quantity, and therefore of the energy difference

[TABLE]

and in the accumulated phase

[TABLE]

with $\chi(t)=\partial\Delta E_{12}(t)/\partial\varepsilon$. The variance of the phase fluctuations induced by charge noise can be written as

[TABLE]

where the angle-bracket represents the statistical average over all realizations of $\delta\varepsilon$. The correlation function $\braket{\delta\varepsilon(t_{1})\delta\varepsilon(t_{2})}$ is nothing but the Fourier transform of the noise spectral density $S_{\varepsilon}(\omega)$:(Cywiński et al., 2008)

[TABLE]

where the additional factor $\frac{1}{2}$ takes into account that we use the one-sided spectral density.

Here we consider two types of charge noise: quasi-static charge noise and fast, uncorrelated charge noise. The first type represents charge fluctuations that occur on time scales much longer than the transfer time $T_{\rm tr}$, so that the charge background is essentially static during each transfer. In this case

[TABLE]

where $\varepsilon_{\textrm{rms}}$ is the root-mean-squared fluctuation in $\varepsilon$,(Dial et al., 2013) which gives for the variance of phase fluctuations

[TABLE]

Vice versa, fast uncorrelated noise has equal contributions at all frequencies, i.e. $S_{\varepsilon}(\omega)=S_{\varepsilon}=\textrm{const.}$ (white noise). In this case the variance of phase fluctuations is given by

[TABLE]

The quantity $\chi(t)$ can be evaluated as follows. Taking into account that the detuning $\varepsilon$ enters in the Hamiltonian Eq. (9) as:

[TABLE]

where $\ket{\textrm{ES}_{i}}\in\{\ket{\circ\downarrow\Uparrow}_{x},\ket{\circ\uparrow\Downarrow}_{x},\ket{\circ\uparrow\Uparrow}_{x},\ket{\circ\downarrow\Downarrow}_{x}\}$ and $\ket{\textrm{SS}_{i}}\in\{\ket{\downarrow\circ\Uparrow}_{x},\ket{\uparrow\circ\Downarrow}_{x},\ket{\uparrow\circ\Uparrow}_{x},\ket{\downarrow\circ\Downarrow}_{x}\}$, the quantity $\chi(t)$ becomes

[TABLE]

For the case of the singlet-triplet qubit, one has also to take into account fluctuations in the double-dot detuning $\varepsilon_{\rm DD}$. Assuming the fluctuations in $\varepsilon$ and $\varepsilon_{\rm DD}$ to be uncorrelated, the total variance of phase fluctuations due to charge noise is given by $\langle\delta\varphi^{2}\rangle_{\rm charge}=\langle\delta\varphi^{2}\rangle_{\rm ch-qs}+\langle\delta\varphi^{2}\rangle_{\rm ch-fast}+\langle\delta\varphi^{2}\rangle_{\varepsilon_{\rm DD}-\rm qs}+\langle\delta\varphi^{2}\rangle_{\varepsilon_{\rm DD}-\rm fast}$. Here, $\langle\delta\varphi^{2}\rangle_{\varepsilon_{\rm DD}-\rm qs}$ and $\langle\delta\varphi^{2}\rangle_{\varepsilon_{\rm DD}-\rm fast}$ have the same structure as Eq.(27) and Eq.(28), but with $\chi(t)$ replaced by

[TABLE]

where $\ket{\tilde{\textrm{S}}(2,0)}=\ket{\textrm{S}(2,0)}\otimes(\ket{\circ\Uparrow}+\ket{\circ\Downarrow})$, and similarly for $\ket{\tilde{\textrm{S}}(0,2)}$.

Finally we note that, in principle, one should also model the effects of $1/f$-noise. However, these can also be taken into account with reasonable accuracy by choosing the white noise level $S_{\varepsilon}$ such that it corresponds to the $1/f$-noise level in the frequency range relevant for the experiment.

A.2 Nuclear-spin noise

GaAs is a III-V semiconductor which exhibits a nuclear spin field that interacts with the spin of the charge carriers via hyperfine interaction. Since the number of nuclear spins interacting with an electron (or a hole) is very large (typically $10^{4}$ to $10^{6}$), their effect can be conveniently described in terms of an effective magnetic field, the Overhauser field (OF).(Chirolli and Burkard, 2008; Bluhm et al., 2011) Here we consider only the OF component parallel to the external magnetic field (i.e. along the $x$ axis), as this is the one that most strongly affects the dynamics of the spin of the charge carriers (transverse OF components give only higher order contributions).(Bluhm et al., 2011; Neder et al., 2011; Botzem et al., 2016) With respect to the basis Eq.(8), the effective Hamiltonian of the hyperfine interaction can then be written as

[TABLE]

Here, ${\bf S}_{x}$ and $\mathbf{J}_{x}$ are spin operators for electrons and heavy holes, with eigenvalues $+\frac{1}{2}$ ($-\frac{1}{2}$) for states with spin-up (spin-down). Terms with a tilde ($\tilde{B}_{\rm OF}$, $\tilde{g}_{\textrm{e}}$) take in to account the fact that an electron can experience both different $g$-factor and different Overhauser field in the OAQD and in the GDQD. The factor $\eta$ accounts for the different OF experienced by the electron and the hole in the OAQD.(Prechtel et al., 2016) The block-diagonal structure reflects again the separation between excitonic states and separated-states.

As the OF fluctuates randomly in time, it causes fluctuation in the energy difference $\Delta E_{12}$. To lowest order in the fluctuations it is

[TABLE]

Spin fluctuations can be considered as quasi-static noise.(Barthel et al., 2009) Assuming the fluctuations $\delta B_{\textrm{OF}}$ and $\delta\tilde{B}_{\textrm{OF}}$ to be uncorrelated, we get for the variance of the phase fluctuations induced by spin noise the following result:

[TABLE]

The root-mean-square fluctuations in the nuclear spin field are referred to as $B_{\textrm{OF}}^{\textrm{(rms)}}=\sqrt{\braket{\delta B_{\textrm{OF}}^{2}}}$ and $\tilde{B}_{\textrm{OF}}^{\textrm{(rms)}}=\sqrt{\braket{\delta\tilde{B}_{\textrm{OF}}^{2}}}$ in Tab. 1.

In the singlet-triplet qubit case we proceed analogously, but we have to account for the OF in the two parts of the DD, i.e. we replace the term $\tilde{B}_{\rm OF}\mathbf{S}_{x}$ in Eq. (31) by $\tilde{B}_{\rm OF}^{\rm(L)}\mathbf{S}_{x}^{\rm(L)}+\tilde{B}_{\rm OF}^{\rm(R)}\mathbf{S}_{x}^{\rm(R)}$. We furthermore assume the fluctuations in ${B}_{\rm OF}$, $\tilde{B}_{\rm OF}^{\rm(L)}$ and $\tilde{B}_{\rm OF}^{\rm(R)}$ to be all independent.

A.3 Failure probability due to dephasing

In order to compare the effects of dephasing to other effects that lead to the failure of the transfer process, we introduce the failure probability due to dephasing, which we define as the probability of the depolarizing channel with the same average gate fidelity.

The average gate fidelity for a two-level system can be calculated as

[TABLE]

where $\sigma_{k}$ are the Pauli matrices, $U$ is a quantum gate, and $\mathcal{E}$ is a trace preserving quantum operation that approximate $U$.(Bowdrey et al., 2002) If $\mathcal{F}(\mathcal{E},U)=1$, then $\mathcal{E}$ implements $U$ perfectly, while $\mathcal{F}(\mathcal{E},U)<1$ indicates that $\mathcal{E}$ is a noisy (or otherwise imperfect) implementation of $U$. With this definition, the infidelity between the phase gate $\mathcal{U}_{\delta\varphi}=\exp(-i\frac{\delta\varphi}{2}\sigma_{z})$ and the identity operation becomes

[TABLE]

If $\delta\varphi$ is a normal distributed random variable with zero mean, the expectation value of the infidelity is

[TABLE]

A depolarizing channel is defined by

[TABLE]

where $P$ is the depolarization probability. The infidelity of the depolarizing channel is simply

[TABLE]

Comparing this result with (36), we define the failure probability due to dephasing as

[TABLE]

Here and above $\delta\varphi$ represents the phase fluctuation due to all different noise sources. Since we assume the latter to be uncorrelated, it is $\langle\delta\varphi^{2}\rangle=\langle\delta\varphi^{2}\rangle_{\rm ch-qs}+\langle\delta\varphi^{2}\rangle_{\rm ch-fast}+\langle\delta\varphi^{2}\rangle_{\rm spins}.$

Appendix B Hamiltonians $\mathbf{H}_{\rm ES}$ and $\mathbf{H}_{\rm SS}$

Here we give explicit expressions for the Hamiltonians $\mathbf{H}_{\rm ES}$ and $\mathbf{H}_{\rm SS}$ entering, Eq.(17). The first term, $\mathbf{H}_{\rm ES}$, represents the projection of the Hamiltonian of the coupled DD-exciton system on the subspace of excitonic states spanned by $\left\{\ket{\uparrow\circ}_{x},\ket{\downarrow\circ}_{x}\right\}\otimes\left\{\ket{\downarrow\Uparrow}_{x},\ket{\uparrow\Downarrow}_{x},\ket{\uparrow\Uparrow}_{x},\ket{\downarrow\Downarrow}_{x}\right\}$). It is given by

[TABLE]

where

[TABLE]

represents the Zeeman Hamiltonian of the single electron initialized in the DD, and $\mathbf{H}_{\rm ex}$ is given in Eq.(2).

Similarly, $\mathbf{H}_{\rm SS}$ represents the projection of the Hamiltonian of the coupled exciton-DD system on the subspace of separated states spanned by $\{\ket{\rm S(0,2)}_{x}\ket{\rm S(2,0)}_{x},\ket{\uparrow\downarrow}_{x},\ket{\downarrow\uparrow}_{x},\ket{\uparrow\uparrow}_{x},\ket{\downarrow\downarrow}_{x}\}\otimes\{\ket{\circ\Uparrow}_{x},\ket{\circ\Downarrow}_{x}\}$, and it is given by

[TABLE]

Here

[TABLE]

represents the Hamiltonian of the doubly occupied DD in the Hund-Mulliken approximation,(Burkard et al., 1999; Coish and Loss, 2005; Stepanenko et al., 2012) with $+(-)\varepsilon_{\textrm{DD}}$ representing the detuning energy of an electron located in the left or right side of the DD , and $t_{\textrm{DD}}$ the tunnel coupling between these two sides. $U$ is the Coulomb repulsion for two electrons located in the same dot, while $V_{+}$ and $V_{-}$ denote the Coulomb energy of the singlet and triplet state with one electron located in each dot, respectively. (Stepanenko et al., 2012) The Zeeman Hamiltonian of the hole remaining in the OAQD is given by $\mathbf{H}_{\rm 1h}$, which has the same structure of $\mathbf{H}_{\rm 1e}$, but with $\tilde{g}_{\rm e}$ replaced by $-{g}_{\rm h}$.

The Hilbert space of the coupled OAQD-DD system can be divided into two separate sub-spaces, spanned respectively by the basis states $\{\ket{\downarrow\circ}\ket{\downarrow\Downarrow}$, $\ket{\downarrow\downarrow}\ket{\circ\Downarrow}$, $\ket{\downarrow\circ}\ket{\uparrow\Uparrow}$, $\ket{\downarrow\uparrow}\ket{\circ\Uparrow}$, $\ket{\uparrow\circ}\ket{\downarrow\Uparrow}$, $\ket{\uparrow\downarrow}\ket{\circ\Uparrow}$, $\ket{\uparrow\circ}\ket{\uparrow\Downarrow}$, $\ket{\uparrow\uparrow}\ket{\circ\Downarrow}$, $\ket{S(2,0)}\ket{\circ\Uparrow}$, $\ket{S(0,2)}\ket{\circ\Uparrow}\}$ and by $\{\ket{\uparrow\circ}\ket{\downarrow\Downarrow}$, $\ket{\uparrow\downarrow}\ket{\circ\Downarrow}$, $\ket{\uparrow\circ}\ket{\uparrow\Uparrow}$, $\ket{\uparrow\uparrow}\ket{\circ\Uparrow}$, $\ket{\downarrow\circ}\ket{\downarrow\Uparrow}$, $\ket{\downarrow\downarrow}\ket{\circ\Uparrow}$, $\ket{\downarrow\circ}\ket{\uparrow\Downarrow}$, $\ket{\downarrow\uparrow}\ket{\circ\Downarrow}$, $\ket{S(2,0)}\ket{\circ\Downarrow}$, $\ket{S(0,2)}\ket{\circ\Downarrow}\}$.