Robust quantum state transfer via topologically protected edge channels in dipolar arrays

The qubits $\alpha =1,..,{N}_{{\rm{q}}}$ forming the quantum memory layer, with ground state $| g\rangle$ and excited state $| e\rangle$, are located at positions $({x}_{\alpha },{y}_{\alpha },{z}_{\alpha })$. The qubit Hamiltonian is given by

$$\begin{eqnarray}&&{H}_{{\rm{q}}}={{\rm{\Delta }}}_{{\rm{q}}}\displaystyle \sum _{\alpha =1}^{{N}_{{\rm{q}}}}{\sigma }_{\alpha }^{+}{\sigma }_{\alpha }^{-},\end{eqnarray} \tag{ 2 }$$

with ${\sigma }_{\alpha }^{-}=| g{\rangle }_{\alpha }\langle e|$, ${\sigma }_{\alpha }^{+}={({\sigma }_{\alpha }^{-})}^{\dagger }$ and energy offsets ${{\rm{\Delta }}}_{{\rm{q}}}$. We note that in this model the qubits are not directly coupled, i.e. they can only interact via the topological spin system.

The topological spin system is represented by V-type three-level systems ${\bf{j}}=({j}_{x},{j}_{y})$ with ${j}_{x}=1,..,{N}_{{\rm{X}}},{j}_{y}=1,..,{N}_{{\rm{Y}}}$, where ${| 0\rangle }_{{\bf{j}}}$ denotes the ground state and ${| 1\rangle }_{{\bf{j}}}{| 2\rangle }_{{\bf{j}}}$ the two excitation states of atom ${\bf{j}}$, which are placed at fixed positions $({x}_{{\bf{j}}},{y}_{{\bf{j}}},{z}_{{\bf{j}}}=0)$ in the $X,Y$ plane according to a square lattice of lattice spacing a. Similar to the Harper Hofstadter Hamiltonian [45, 46], which has been recently realised in cold atom experiments [23–26], a topological band structure is obtained by allowing the spin excitations ($| 1\rangle$, $| 2\rangle$), also known as magnons, to acquire a phase when hopping around a closed loop in the lattice. The corresponding Hamiltonian is given by

$$\begin{eqnarray}&&{H}_{{\rm{T}}}=\displaystyle \sum _{{\bf{j}}}\displaystyle \sum _{\nu =1,2}{\delta }_{\nu }{b}_{{\bf{j}},\nu }^{\dagger }{b}_{{\bf{j}},\nu }+\displaystyle \sum _{{\bf{j}}\ne {\bf{l}}}{{\bf{b}}}_{{\bf{j}}}^{\dagger }h({{\bf{r}}}_{{\bf{j}},{\bf{l}}}){{\bf{b}}}_{{\bf{l}}}.\end{eqnarray} \tag{ 3 }$$

Here, we map the spin excitations to hard-core boson particles where the two-element vector ${{\bf{b}}}_{{\boldsymbol{j}}}=[{b}_{{\bf{j}},1},{b}_{{\bf{j}},2}]$, with ${b}_{{\bf{j}},\nu }=| 0{\rangle }_{{\bf{j}}}\langle \nu |$, represents the two (hard-core boson) annihilation operators of the excitations at site ${\bf{j}}$ and ${\delta }_{\nu }$ is the corresponding energy offset. The matrix $h({{\bf{r}}}_{{\bf{j}},{\bf{l}}})$, with ${{\bf{r}}}_{{\bf{j}},{\bf{l}}}=({x}_{{\bf{j}}}-{x}_{{\bf{l}}}){\bf{X}}+({y}_{{\bf{j}}}-{y}_{{\bf{l}}}){\bf{Y}}$, describes the transfer of an excitation between sites ${\bf{j}}$ and ${\bf{l}}$ and can be written as

$$\begin{eqnarray}h({{\bf{r}}}_{{\bf{j}},{\bf{l}}})=\left[\begin{array}{cc}{t}_{1}({{\bf{r}}}_{{\bf{j}},{\bf{l}}}) & w({{\bf{r}}}_{{\bf{j}},{\bf{l}}}){e}^{-i\phi ({{\bf{r}}}_{{\bf{j}},{\bf{l}}})}\\ w({{\bf{r}}}_{{\bf{j}},{\bf{l}}}){e}^{i\phi ({{\bf{r}}}_{{\bf{j}},{\bf{l}}})} & {t}_{2}({{\bf{r}}}_{{\bf{j}},{\bf{l}}})\end{array}\right],\end{eqnarray} \tag{ 4 }$$

where the phases $\phi ({{\bf{r}}}_{{\bf{j}},{\bf{l}}})$ are responsible for the existence of Quantum Hall topological bands characterised by non-vanishing Chern numbers [37, 38] and thus chiral edge states [21, 22, 38] (c.f. figure 2(a)).

By considering for convenience the limit of an infinite number of atoms in one direction, ${N}_{{\rm{Y}}}\to \infty$, we obtain the dispersion relation of bulk and edge modes by Fourier transforming along the Y axis and diagonalising the TSS Hamiltonian (c.f. appendix A),

$$\begin{eqnarray}&&{H}_{{\rm{T}}}=\displaystyle \sum _{m}{\int }_{-\pi /a}^{\pi /a}{\rm{d}}k\,{\omega }_{m}(k){b}_{k,m}^{\dagger }{b}_{k,m},\end{eqnarray} \tag{ 5 }$$

where ${\omega }_{m}(k)$ denotes the dispersion relation corresponding to the TSS eigenmode ${b}_{k,m}$ ⁴ . Figure 2(b) shows a typical example of dispersion relation curves ${\omega }_{m}(k)$. The edge-state dispersion relation corresponding to $m={\rm{a}}{\omega }_{{\rm{a}}}(k)$ (blue line) represents localised modes propagating with group velocity ${v}_{{\rm{a}}}(k)\equiv \partial {\omega }_{{\rm{a}}}(k)/\partial k$. Their chirality originates from the fact that the modes propagating with positive velocity are located on the left edge, while the modes moving in the other direction are located on the right edge (c.f. figure 2). At this point, we want to emphasise that the absence of counter-propagating modes (for example with a negative velocity on the left edge) guarantees the absence of reflections originated from local defects. We analyse in more detail the robustness of the edge-state channels in sections 4 and 5.

The last part of the Hamiltonian H of our model corresponds to the coupling between the qubits and the topological spin system. To achieve the quantum state transfer, we are interested in coupling the qubits predominantly to the edge modes. This is achieved by positioning the qubits in the vicinity of the edge of the topological spin system⁵ and choosing the qubit transition frequency ${{\rm{\Delta }}}_{{\rm{q}}}$ to match with the energy of an edge state ${b}_{{\bar{k}}_{{\rm{a}}},{\rm{a}}}$, where the resonant wave vector ${\bar{k}}_{{\rm{a}}}$ is defined via ${\omega }_{{\rm{a}}}({\bar{k}}_{{\rm{a}}})={{\rm{\Delta }}}_{{\rm{q}}}$. The coupling Hamiltonian is written as

$$\begin{eqnarray}&&{H}_{\mathrm{qT}}=\displaystyle \sum _{{\bf{j}},\alpha ,\nu }{g}_{\nu }({{\bf{r}}}_{{\bf{j}},\alpha },t){b}_{{\bf{j}},\nu }^{\dagger }{\sigma }_{\alpha }^{-}+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ 6 }$$

where ${g}_{\nu }({{\bf{r}}}_{{\bf{j}},\alpha },t)$ represents the coupling of an excitation ${| e\rangle }_{\alpha }$ of the qubit α to a TSS excitation at site ${\bf{j}}$, encoded in one of the two levels $\nu =1,2$ and depends on the relative vector ${{\bf{r}}}_{{\bf{j}},\alpha }=({x}_{{\bf{j}}}-{x}_{\alpha }){\bf{X}}+({y}_{{\bf{j}}}-{y}_{\alpha }){\bf{Y}}-{z}_{\alpha }{\bf{Z}}$. Moreover, we consider the coupling terms ${g}_{\nu }({{\bf{r}}}_{{\bf{j}},\alpha },t)$ to be time-dependent, which allows to form wave packets of edge modes propagating with a well-defined shape [1, 47]. In the basis of the eigenmodes ${b}_{k,m}$, the coupling Hamiltonian takes the form

$$\begin{eqnarray}&&{H}_{\mathrm{qT}}=\displaystyle \sum _{m,\alpha }\int {\rm{d}}k\,{g}_{k,m}^{(\alpha )}(t){e}^{-{{iky}}_{\alpha }}{b}_{k,m}^{\dagger }{\sigma }_{\alpha }^{-}+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ 7 }$$

with the coupling strength ${g}_{k,m}^{(\alpha )}(t)=(1/\sqrt{2\pi }){\sum }_{{\bf{j}},\nu }{e}^{-{ik}({y}_{{\bf{j}}}-{y}_{\alpha })}{g}_{\nu }({{\bf{r}}}_{{\boldsymbol{j}},\alpha },t){c}_{{x}_{{\bf{j}}},\nu }^{(k,m)}$, where the coefficients ${c}_{{x}_{{\bf{j}}},\nu }^{(k,m)}$ describe the spatial properties of the eigenmodes and are given in appendix A.

Let us now apply our model to realise a quantum state transfer protocol [1] using chiral edge states [19]. The formal process we want to achieve is the transfer of any superposition state $| s\rangle =A| g\rangle +B| e\rangle$ of a qubit $\alpha =1$ to a second qubit $\beta =2$ mediated by a wave packet propagating in the quantum channel (c.f. figures 1(a), (c))

$$\begin{eqnarray}&&[A{| g\rangle }_{1}+B{| e\rangle }_{1}]{| 0\rangle }_{{\rm{T}}}{| g\rangle }_{2}\\ &&\quad \to {| g\rangle }_{1}\left[A{| 0\rangle }_{{\rm{T}}}+B\displaystyle \int {\rm{d}}k\,{c}_{k,{\rm{a}}}(t){e}^{-i{\omega }_{m}(k)t}{b}_{k,{\rm{a}}}^{\dagger }{| 0\rangle }_{{\rm{T}}}\right]{| g\rangle }_{2}\\ &&\quad \to {| g\rangle }_{1}{| 0\rangle }_{{\rm{T}}}[A{| g\rangle }_{2}+B{| e\rangle }_{2}],\end{eqnarray} \tag{ 8 }$$

with ${| 0\rangle }_{{\rm{T}}}={\prod }_{{\bf{j}}}{| 0\rangle }_{{\bf{j}}}$ the excitation vacuum of the topological spin system. To derive the form of the coupling ${g}_{k,m}^{(\alpha )}(t)$, which achieves the quantum state transfer, we write the general wave function

$$\begin{eqnarray}&&| \psi (t)\rangle =\left(\displaystyle \sum _{\alpha }{c}_{e,\alpha }(t){e}^{-i{{\rm{\Delta }}}_{{\rm{q}}}t}{\sigma }_{\alpha }^{+}+\displaystyle \sum _{m}\int {\rm{d}}k\,{c}_{k,m}(t){e}^{-i{\omega }_{m}(k)t}{b}_{k,m}^{\dagger }\right)| V\rangle ,\end{eqnarray} \tag{ 9 }$$

describing the propagation of a single excitation in the total system, with $| V\rangle \equiv {| 0\rangle }_{{\rm{T}}}\otimes {\prod }_{\alpha }{| g\rangle }_{\alpha }$. As presented in appendix B, the Wigner-Weisskopf treatment, valid in the weak coupling regime $h({{\bf{r}}}_{{\bf{j}},{\bf{l}}})\gg {g}_{\nu }({{\bf{r}}}_{{\bf{j}},\alpha })$, eliminates the contribution of the TSS, resulting in the following set of equations for the qubit amplitudes:

$$\begin{eqnarray}{\dot{c}}_{e,1}(t) & = & -\,\displaystyle \frac{1}{2}{\gamma }_{{\rm{a}},1}(t){c}_{e,1}(t)\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\dot{c}}_{e,2}(t)=-\,\displaystyle \frac{1}{2}{\gamma }_{{\rm{a}},2}(t){c}_{e,2}(t)\\ &&-\sqrt{{\gamma }_{{\rm{a}},1}(t-{\tau }_{{\rm{a}}}){\gamma }_{{\rm{a}},2}(t)}{e}^{i({\bar{k}}_{{\rm{a}}}d-({\eta }_{2}-{\eta }_{1}))}{c}_{e,1}(t-{\tau }_{{\rm{a}}}),\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}&&{\gamma }_{{\rm{a}},\alpha }(t)=2\pi \displaystyle \frac{| {g}_{{\bar{k}}_{{\rm{a}}},{\rm{a}}}^{(\alpha )}(t){| }^{2}}{| {v}_{{\rm{a}}}({\bar{k}}_{{\rm{a}}})| }\end{eqnarray} \tag{ 12 }$$

denotes the coupling of the qubit α to the edge mode $m={\rm{a}}$, ${g}_{k,{\rm{a}}}^{(\alpha )}(t)=| {g}_{k,{\rm{a}}}^{(\alpha )}(t)| {e}^{i{\eta }_{\alpha }}$, $d={y}_{2}-{y}_{1}$ is the distance between the two qubits along the Y axis and ${\tau }_{{\rm{a}}}=d/| {v}_{{\rm{a}}}({\bar{k}}_{{\rm{a}}})|$ represents the time delay between the qubits. We emphasise that equations (10) and (11) illustrate the unidirectional coupling between the two qubits. Similar to the original proposal [1], the quantum state transfer protocol can be then realised with the time-dependent coupling [47, 48]

$$\begin{eqnarray}&&{\gamma }_{{\rm{a}},1}(t)=\displaystyle \frac{2\sqrt{c}{\gamma }_{0}{e}^{-c{(t-{t}_{0})}^{2}}}{2\sqrt{c}-\sqrt{\pi }{\gamma }_{0}\mathrm{erf}(\sqrt{c}(t-{t}_{0}))},\end{eqnarray} \tag{ 13 }$$

which generates a Gaussian edge-state wave packet of temporal width $\sqrt{c}$. This symmetric pulse can be then reabsorbed by the second qubit via a time-reversed pulse ${\gamma }_{{\rm{a}},2}(t)={\gamma }_{{\rm{a}},1}({\tau }_{{\rm{a}}}-t)$.

The TSS Hamiltonian ${H}_{{\rm{T}},\mathrm{imp}}={H}_{\mathrm{at}}+{H}_{\mathrm{int}}$ has two contributions. The first term H_at, representing the energies of the levels and the laser excitation, can be written in a rotating frame determined by the laser frequencies

$$\begin{eqnarray}{H}_{\mathrm{at}} & = & -\displaystyle \sum _{{\bf{j}}}\displaystyle \sum _{3\leqslant i\leqslant 7}{{\rm{\Delta }}}_{i}| i{\rangle }_{{\bf{j}}}\langle i| \\ & & +\displaystyle \sum _{{\bf{j}}}\left\{{{\rm{\Omega }}}_{0}| 3{\rangle }_{{\bf{j}}}\langle 0| +{{\rm{\Omega }}}_{1}| 6{\rangle }_{{\bf{j}}}\langle 1| +{{\rm{\Omega }}}_{2}\left(| 5{\rangle }_{{\bf{j}}}\langle 2| +\displaystyle \frac{\sqrt{3}}{2}| 7{\rangle }_{{\bf{j}}}\langle 1| \right)+{\rm{h}}.{\rm{c}}.\right\},\end{eqnarray} \tag{ 14 }$$

with the Rabi frequencies $\{{{\rm{\Omega }}}_{0},{{\rm{\Omega }}}_{1},{{\rm{\Omega }}}_{2}\}$, the laser detunings ${{\rm{\Delta }}}_{i}$ and ${| i\rangle }_{{\bf{j}}}$ the state i of atom ${\bf{j}}$. For simplicity, we consider in the following ${{\rm{\Delta }}}_{i}={\rm{\Delta }}$.

As shown in appendix C, the second part of the Hamiltonian H_int representing the dipole-dipole interactions between Rydberg-excited states can be written as

$$\begin{eqnarray}&&{H}_{\mathrm{int}}=\displaystyle \sum _{{\bf{j}}\ne {\bf{l}}}\displaystyle \frac{{C}_{3}}{{r}_{{\bf{j}},{\bf{l}}}^{3}}{c}_{{\bf{j}}}^{\dagger }{h}_{\mathrm{dd}}({\theta }_{{\bf{j}},{\bf{l}}},{\phi }_{{\bf{j}},{\bf{l}}}){c}_{{\bf{l}}}\end{eqnarray} \tag{ 15 }$$

with the shorthand notation ${c}_{{\bf{j}}}=[| 5{\rangle }_{{\bf{j}}}\langle 3| ,| 6{\rangle }_{{\bf{j}}}\langle 3| ,| 5{\rangle }_{{\bf{j}}}\langle 4| ,| 6{\rangle }_{{\bf{j}}}\langle 4| ]$ and ${h}_{\mathrm{dd}}({\theta }_{{\bf{j}},{\bf{l}}},{\phi }_{{\bf{j}},{\bf{l}}})$, a dimensionless 4 × 4 matrix that depends on the spherical angles ${\theta }_{{\bf{j}},{\bf{l}}}$, and ${\phi }_{{\bf{j}},{\bf{l}}}$ of the relative vector ${{\bf{r}}}_{{\bf{j}},{\bf{l}}}$ with respect to ${\bf{z}}$. The radial coefficient ${C}_{3}={\left(\int {{drR}}_{S}(r){r}^{3}{R}_{P}(r)\right)}^{2}$ is a function of R_S (R_P), the radial wave function associated with the ${S}_{1/2}$ (${P}_{1/2}$) Rydberg states [43]. We emphasise that we consider the large distance limit ${C}_{3}/{r}_{{\bf{j}},{\bf{l}}}^{3}\ll {\rm{\Delta }}$ where only the resonant flip-flop processes of the type ${P}_{1/2}{S}_{1/2}\to {S}_{1/2}{P}_{1/2}$ contribute.

In the perturbative (or dressing) regime, ${{\rm{\Omega }}}_{\mathrm{0,1,2}}\ll {\rm{\Delta }}$, the Rydberg states can be eliminated in perturbation theory, leading to an effective Hamiltonian governing the dynamics of the ground-state levels [39–41]. To do so, we apply the Van Vleck formalism [54] reducing the TSS Hamiltonian ${H}_{{\rm{T}},\mathrm{imp}}$ to the form of (3) with

$$\begin{eqnarray}&&{t}_{\nu =\mathrm{1,2}}({{\bf{r}}}_{{\bf{j}},{\bf{l}}})={(-1)}^{\nu }\displaystyle \frac{{C}_{3}}{{r}_{{\bf{j}},{\bf{l}}}^{3}}\displaystyle \frac{{{\rm{\Omega }}}_{0}^{2}{{\rm{\Omega }}}_{\nu }^{2}}{9{{\rm{\Delta }}}^{4}}(1-3{\cos }^{2}{\theta }_{{\bf{j}},{\bf{l}}})\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&w({{\bf{r}}}_{{\bf{j}},{\bf{l}}})=\displaystyle \frac{{C}_{3}}{{r}_{{\bf{j}},{\bf{l}}}^{3}}\displaystyle \frac{{{\rm{\Omega }}}_{0}^{2}{{\rm{\Omega }}}_{1}{{\rm{\Omega }}}_{2}}{6{{\rm{\Delta }}}^{4}}\sin (2{\theta }_{{\bf{j}},{\bf{l}}}),\end{eqnarray} \tag{ 17 }$$

and the second and fourth-order AC-Stark shifts

$$\begin{eqnarray}{\delta }_{1} & = & \displaystyle \frac{3{{\rm{\Omega }}}_{2}^{2}}{4{\rm{\Delta }}}-\displaystyle \frac{9{{\rm{\Omega }}}_{2}^{4}}{16{{\rm{\Delta }}}^{3}}+\displaystyle \frac{{{\rm{\Omega }}}_{1}^{2}}{{\rm{\Delta }}}-\displaystyle \frac{{{\rm{\Omega }}}_{1}^{4}}{{{\rm{\Delta }}}^{3}}-\displaystyle \frac{3{{\rm{\Omega }}}_{1}^{2}{{\rm{\Omega }}}_{2}^{2}}{2{{\rm{\Delta }}}^{3}}-\displaystyle \frac{{{\rm{\Omega }}}_{0}^{2}}{{\rm{\Delta }}}+\displaystyle \frac{{{\rm{\Omega }}}_{0}^{4}}{{{\rm{\Delta }}}^{3}}\\ {\delta }_{2} & = & \displaystyle \frac{{{\rm{\Omega }}}_{2}^{2}}{{\rm{\Delta }}}-\displaystyle \frac{{{\rm{\Omega }}}_{2}^{4}}{{{\rm{\Delta }}}^{3}}-\displaystyle \frac{{{\rm{\Omega }}}_{0}^{2}}{{\rm{\Delta }}}+\displaystyle \frac{{{\rm{\Omega }}}_{0}^{4}}{{{\rm{\Delta }}}^{3}},\end{eqnarray} \tag{ 18 }$$

where we set ${\delta }_{0}=0$. To obtain a topological phase [38], it is important that the energy-splitting ${\delta }_{1}-{\delta }_{2}$ does not dominate over the flip-flop term (17). This condition can be achieved for example by choosing the ratio ${{\rm{\Omega }}}_{2}/{{\rm{\Omega }}}_{0}$, ${{\rm{\Omega }}}_{1}/{{\rm{\Omega }}}_{0}$ according to (18) in order to obtain ${\delta }_{1}={\delta }_{2}=0$.

Finally, we emphasise that in our dressing implementation, the local and time-dependent control of the laser intensities ${{\rm{\Omega }}}_{\mathrm{0,1,2}}$ allows to effectively disconnect atoms from the rest of the topological spin system and therefore to dynamically reshape the edges (c.f. figure 1(d)).

The implementation of the qubits is similar to the one of the TSS atoms, with the difference that the level $| 2\rangle$ is energetically excluded from the dynamics (for instance via the second-order AC-Stark shifts). Following the same procedure as for the derivation of the TSS Hamiltonian, we obtain the coupling Hamiltonian in the form of (6) with

$$\begin{eqnarray}&&{g}_{1}({{\bf{r}}}_{{\bf{j}},\alpha },t)=\displaystyle \frac{{C}_{3}}{{r}_{{\bf{j}},\alpha }^{3}}\displaystyle \frac{{\tilde{{\rm{\Omega }}}}_{0,\alpha }(t){\tilde{{\rm{\Omega }}}}_{1,\alpha }(t){{\rm{\Omega }}}_{0}{{\rm{\Omega }}}_{1}}{9{{\rm{\Delta }}}^{4}}(3{\cos }^{2}{\theta }_{{\bf{j}},\alpha }-1)\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&{g}_{2}({{\bf{r}}}_{{\bf{j}},\alpha },t)=\displaystyle \frac{{C}_{3}}{{r}_{{\bf{j}},\alpha }^{3}}\displaystyle \frac{{\tilde{{\rm{\Omega }}}}_{0,\alpha }(t){\tilde{{\rm{\Omega }}}}_{1,\alpha }(t){{\rm{\Omega }}}_{0}{{\rm{\Omega }}}_{2}}{6{{\rm{\Delta }}}^{4}}\sin (2{\theta }_{{\bf{j}},\alpha }){e}^{i{\phi }_{{\bf{j}},\alpha }},\end{eqnarray} \tag{ 20 }$$

where ${\tilde{{\rm{\Omega }}}}_{0,\alpha }(t),{\tilde{{\rm{\Omega }}}}_{1,\alpha }(t)$ denote the local Rabi frequencies addressing the qubits. Finally, considering as in the case of the TSS atoms, that the AC-Stark shifts can be compensated, for simplicity we set the qubit Hamiltonian (2) to zero. Moreover, we consider the distance between the qubits to be sufficiently large to neglect the direct dipole-dipole interactions between them. Consequently, the interactions between the qubits are exclusively mediated by the TSS atoms, as presented in section 2.

We conclude this section by assessing the regime of validity of our implementation calculating the relevant time scales. For a lattice spacing $a=15\,\mu {\rm{m}}$ and TSS Rabi frequencies ${{\rm{\Omega }}}_{0}={{\rm{\Omega }}}_{2}=2\pi \times 8\,\mathrm{MHz}$, ${{\rm{\Omega }}}_{1}=2\pi \times 4\,\mathrm{MHz}$, detuning ${\rm{\Delta }}=2\pi \times 25\,\mathrm{MHz}$ exciting the TSS atoms to Rydberg states with principal quantum number n = 65, the value of the dipole-dipole coefficient ${C}_{3}=2\pi \times 19\ {\hslash }$ GHz $\mu {{\rm{m}}}^{3}$ leads to flip-flop interactions of the order of 1 kHz, which is larger than the effective decoherence rate ${\rm{\Gamma }}\sim {({\rm{\Omega }}/{\rm{\Delta }})}^{2}{{\rm{\Gamma }}}_{{\rm{r}}}\sim 2\pi \times 0.3\,\mathrm{kHz}$, induced by the Rydberg state admixture [39–41] of the ground-state atoms (${{\rm{\Gamma }}}_{{\rm{r}}}$ is the typical decay rate of the S and P Rydberg states [55]). Note that for these specific parameters, the quantum state transfer cannot be realised for very large distances $d\gg a$ as the time needed to realise the experiment, which should be smaller than the decoherence time $1/{\rm{\Gamma }}$, increases linearly with the distance d. However, this limitation can be simply overcome by reducing the lattice spacing, leading to much faster time scales ⁶ . Finally, we emphasise that we consider ground-state atoms, which in contrast to Rydberg atoms, remain trapped while experiencing the dipole-dipole interactions and therefore decoherence effects arising from the motion of the atoms are negligible [56].

As shown in Peter et al [38] in the context of polar molecules, the TSS Hamiltonian exhibits a quantum Hall topological band structure phase associated with the existence of two edge modes $m={\rm{a}},{\rm{b}}$. In figure 4 we show the dispersion relation ${\omega }_{m}(k)$ and the edge states localisation and spin properties, obtained by numerically diagonalising the TSS Hamiltonian using the dimension reduction analysis along the Y axis (c.f. appendix A) and for the numbers given in section 3.3. The two edge-state channels have the same chirality, i.e. they propagate in the same direction along each edge. The existence of two edge modes offers in principle the advantage of performing multiplexing protocols but also gate operations based on spin-spin collisions (see [16] in the context of spin chains). However, in the following we consider that one edge-state mode $m={\rm{a}}$ is predominantly excited, which can be achieved by exploiting the spin and localisation properties of the edge states (c.f. figure 4(b)).

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The quantum state transfer protocol relies on effective time-dependent couplings of the form of (13), which in the context of our implementation can be achieved by dynamically varying the qubit Rabi frequencies ${\tilde{{\rm{\Omega }}}}_{0,\alpha }(t),{\tilde{{\rm{\Omega }}}}_{1,\alpha }(t)$ according to (12), (19) and (20). We study the performance of the protocol by numerically calculating the corresponding dynamics governed by (1) with the initial condition $| {\rm{\Psi }}(t=0)\rangle ={\sigma }_{1}^{+}| V\rangle$. As an example, we choose the two qubits to be separated by a distance $d={y}_{2}-{y}_{1}$ and position them with respect to the closest TSS atom ${\bf{j}}[\alpha ]$ (with ${x}_{{\bf{j}}[\alpha ]}=0$) via ${{\bf{r}}}_{{\bf{j}}[\alpha ],\alpha }=b({\bf{Z}}-{\bf{X}})$, with $b=8.1\,\mu {\rm{m}}$. For this geometric configuration, we obtain a dominant coupling ${\gamma }_{{\rm{a}}}$ to the edge state indicated in red in figure 4.

The results of the quantum state transfer protocol are shown in figure 5, where we represent the fidelity of the quantum state transfer ${ \mathcal F }=| {c}_{e,2}({t}_{f}){| }^{2}$ for a final time t_f = 8 ms. In panel (a), we study the effect of the finite size of the topological spin system by representing the fidelity ${ \mathcal F }$ for different TSS sizes $({N}_{{\rm{X}}},{N}_{{\rm{Y}}})$, a distance $d=6a$ and ${\gamma }_{0}=2\pi \times 0.64\,\mathrm{kHz}$. For large TSS sizes ${N}_{{\rm{X}}},{N}_{{\rm{Y}}}\gt 10$, the fidelity converges towards a constant value ${ \mathcal F }\approx 0.95$, showing that the dispersion relation that we derived under the assumption of a semi-infinite topological spin system is relevant to describe the quantum state transfer in large but finite systems. To explain the cause of the deviation from the ideal fidelity ${ \mathcal F }=1$, we represent ${ \mathcal F }$ as a function of d and ${\gamma }_{0}$ in panel (b). At short distances and small coupling strengths, we observe state transfer with a small error that we attribute to the influence of the second edge-state channel $m={\rm{b}}$ and to the off-resonant contribution of the bulk modes $m\ne {\rm{a}},{\rm{b}}$. At large distances, the fidelity is a decreasing function of d and of the coupling strength ${\gamma }_{0}$ indicating the onset of dispersion effects [16] that distort the edge-state wave packet while propagating.

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Furthermore, we study the influence of disorder, which in cold atom experiments typically manifests by an on-site probability ${ \mathcal P }$ for TSS atom vacancies. The fidelity ${ \mathcal F }$ as a function of ${ \mathcal P }$ is shown in figure 5(c) for a fixed distance $d=26a$ and coupling ${\gamma }_{0}=2\pi \times 0.64\,\mathrm{kHz}$. Despite the absence of back-scattering in our topological quantum channel, the presence of disorder distorts wave packets of edge states, affecting the fidelity of the quantum state transfer. Moreover, considering the individual disorder realisations shown as red circles, we notice that the robustness of the protocol crucially depends on the position of the vacancies.

To summarise this section, we demonstrated the potential of using a Rydberg-dressing implementation of chiral channels for quantum state transfer. We studied the role of dispersion and disorder as sources of imperfection, the latter having a crucial influence on the fidelity despite the topological character of the edge modes. In the next section, we show that these limitations can be overcome by performing a protocol based on the spectroscopy of the transmission channel.

We now present the different steps of the spectroscopy of the quantum channel. As depicted in figure 6, we consider the first qubit $\alpha =1$ to be initially excited and to emit for times $t\lt 0$ a wave packet with a well-defined shape which is then distorted while propagating. At time t = 0, the wave packet reaches the second qubit, whose dynamics, following appendix B, are given by

$$\begin{eqnarray}&&{\dot{c}}_{e,2}(t)=-\left[\displaystyle \frac{{\gamma }_{{\rm{a}},2}(t)}{2}+i\delta (t)\right]{c}_{e,2}(t)+\sqrt{{\gamma }_{{\rm{a}},2}}f(t),\end{eqnarray} \tag{ 21 }$$

with $f(t)=-{{iv}}_{{\rm{a}}}({\bar{k}}_{{\rm{a}}}){e}^{i{\bar{k}}_{{\rm{a}}}{v}_{{\rm{a}}}({\bar{k}}_{{\rm{a}}})t-i{\eta }_{2}}{c}_{{y}_{2}-{v}_{{\rm{a}}}({\bar{k}}_{{\rm{a}}})t}(0)$ where ${c}_{y}(t)=\int {\rm{d}}{k}\,{e}^{{iky}}{c}_{k,{\rm{a}}}(t)$ represents the wave packet amplitude in real space and we assume ${c}_{e,1}(0)\approx 0$. In contrast to the standard state transfer protocol that supposes that a wave packet propagates without deformation [1], we assume here that the second qubit receives an unknown wave packet f(t) whose amplitude $| f(t)|$ and phase ${\rm{\Phi }}(t)\equiv \arg (f(t))$ ⁸ are both unknown⁹ . The time-dependent quantity $\delta (t)$ (c.f. figure 6) allows to change the transition frequency of the second qubit dynamically, which is assumed not to modify the coupling ${\gamma }_{{\rm{a}},2}(t)$. As explained in section 5.2 the chirp $\delta (t)$ is a crucial ingredient to realise the quantum state transfer.

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We now show how to measure the wave packet f(t). Using the ansatz $d(t)={e}^{{\gamma }_{{\rm{a}},2}t/2}{c}_{e,2}(t)$, (21) becomes

$$\begin{eqnarray}&&\dot{d}=-i\delta d+\sqrt{{\gamma }_{{\rm{a}},2}}{e}^{{\gamma }_{{\rm{a}},2}t/2}f(t).\end{eqnarray} \tag{ 22 }$$

In a typical experimental setup, we only have access to the population of the qubit, and thus to the modulus $r\equiv | d|$, while the phase $\theta \equiv \arg (d)$ cannot be directly measured. Therefore, it is not possible to extract the function f(t) via (22) with a single measurement of the qubit population. However, the measurement can be repeated for different (time-independent) δ leading to the response d(t) and consequently f(t). A simple option is to realise two measurements of the qubit population: (i) the first measurement gives the function r₀ for $\delta =0$ and (ii) the second measurement, performed at a small constant value $\delta (t)=\delta$, leads to the knowledge of ${r}_{0}^{\prime }\equiv {(\partial r/\partial \delta )}_{\delta =0}$ via finite differentiation. We can then eliminate f(t) using (22) and obtain

$$\begin{eqnarray}&&{\dot{r}}_{0}^{\prime }-{\dot{\theta }}_{0}u=0\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\dot{\theta }}_{0}{r}_{0}^{\prime }+\dot{u}+{r}_{0}=0,\end{eqnarray} \tag{ 24 }$$

with ${\theta }_{0}^{\prime }\equiv {(\partial \theta /\partial \delta )}_{\delta =0}$, $u={r}_{0}{\theta }_{0}^{\prime }$. Knowing r₀ and ${r}_{0}^{\prime }$, (23), (24) can be integrated to find ${\theta }_{0}$ and finally f(t) using (22). As we are interested here in studying deviations from the ideal case where the function f(t) is real, we solve the differential equations (23), (24) iteratively in ${\dot{\theta }}_{0}$, the zeroth order ${\dot{\theta }}_{0}(t)=-{\dot{r}}_{0}^{\prime }(t)/\int {\rm{d}}{t}^{\prime} \;{r}_{0}(t^{\prime} )$ in the examples below will be a good approximation. This concludes the spectroscopy of the quantum channel: comparing f(t) with the initial pulse sent by the first qubit indicates how a wave packet is transmitted towards the second qubit including the effects of dispersion and disorder. We now use this information to realise a robust state transfer protocol.

Under the assumption of static quantum channel imperfections, the knowledge of the wave packet f(t), which reaches the second qubit, can be used to realise a robust state transfer protocol. To do so, we realise, after the first two measurements related to the quantum channel spectroscopy, a third experiment with time-dependent couplings ${\gamma }_{{\rm{a}},2}(t)$ and chirp $\delta (t)$. For a perfect absorption of the entire wave packet f(t) with the evolution ${c}_{e,2}(t)={e}^{i{\rm{\Phi }}(t)}\sqrt{{\int }_{0}^{t}{\rm{d}}{t}^{\prime} \;| f(t^{\prime} ){| }^{2}}$, we obtain the conditions

$$\begin{eqnarray}&&{\gamma }_{{\rm{a}},2}(t)=| f(t){| }^{2}/{\int }_{0}^{t}{\rm{d}}{t}^{\prime} \;| f(t^{\prime} ){| }^{2}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&\delta (t)=-\dot{{\rm{\Phi }}}(t),\end{eqnarray} \tag{ 26 }$$

where we used (21). The first condition (25) resembles the typical pulse shape obtained in the standard state transfer protocol [1, 47, 48] with a real envelop f(t) (c.f. (11) in the Gaussian case). The second 'phase-matching' condition allows to compensate for the existence of the phase Φ by a 'chirped' frequency $\delta (t)$. In this way, the frequency of the second qubit is dynamically synchronised with the evolution of the phase ${\rm{\Phi }}(t)$.

We now apply our protocol based on the spectroscopy of the channel, using two toy models. First, in section 5.3.1, we consider a 1D quantum channel subject to dispersive effects while we present in section 5.3.2 the results obtained in the case of a disordered topological spin system. In both models, we assess the efficiency of the protocol by numerically simulating the dynamics of the combined system formed by the qubits and the quantum channel, similar to the study presented in section 4.

5.3.1. Compensation of dispersion effects in structured 1D waveguides

We consider a unidimensional waveguide (${N}_{{\rm{X}}}=1$), where the excitations are encoded in a single excited state $| 1\rangle$. The matrix $h({{\bf{r}}}_{{\bf{j}},{\bf{l}}})$ (c.f. (3)) is a simple scalar with nearest neighbour interactions $h({{\bf{r}}}_{{\bf{j}},{\bf{l}}})=-J{\delta }_{{j}_{y},{l}_{y}\pm 1}$. This model was studied in detail in [16], showing dispersive effects similar to the ones presented here in section 4. In this case, the dispersion relation has an analytical expression ${\omega }_{{\rm{a}}}(k)=-2J\cos ({ka})+{\delta }_{\nu }$ where we choose ${\delta }_{\nu }=0.5J$, ${{\rm{\Delta }}}_{{\rm{q}}}=0$. The fidelity of the quantum state transfer protocol is shown in figure 7(a). The blue curve corresponds to the standard state transfer protocol, showing how dispersion affects the quantum state transfer at large distances d/a. The green curve represents the case where the coupling of the second qubit has been adapted according to (25), whereas the red curve also includes the 'chirp' condition (26). The second condition is the crucial requirement to obtain a robust state transfer protocol. For a distance $d=400a$, we obtain a fidelity of 96%, corresponding to a reduction of the error of 90% compared with the standard state transfer protocol. The robustness of the protocol simply comes from the fact that the chirp compensates for the phases accumulated by the Fourier components of the propagating wave packet.

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5.3.2. Robustness against disorder in a topological spin system

Finally, we study the robustness of the protocol in the case of a disordered topological spin system. For simplicity, we consider a model which, in contrast to the Rydberg implementation in section 3, includes a single edge-state channel [57]. The Hamiltonian is commonly written in terms of Pauli matrices

$$\begin{eqnarray}{H}_{{\rm{T}}} & = & \displaystyle \sum _{{\bf{j}}}J\left({b}_{{j}_{x},{j}_{y}}^{\dagger }\displaystyle \frac{{\sigma }_{z}-i{\sigma }_{x}}{2}{b}_{{j}_{x}+1,{j}_{y}}+{b}_{{j}_{x},{j}_{y}}^{\dagger }\displaystyle \frac{{\sigma }_{z}-i{\sigma }_{y}}{2}{b}_{{j}_{x},{j}_{y}+1}+{\rm{h}}.{\rm{c}}.\right)\\ & & +m\displaystyle \sum _{{\bf{j}}}{b}_{{\bf{j}}}^{\dagger }{\sigma }_{z}{b}_{{\boldsymbol{j}}},\end{eqnarray} \tag{ 27 }$$

where we fix in the following $m=-1$. Moreover, the two qubits are coupled to a single TSS atom ${\bf{j}}[\alpha ]$ (with ${y}_{{\bf{j}}[\alpha ]}=0$) according to (6) with ${g}_{\nu }{({{\bf{r}}}_{{\bf{j}},\alpha },t)=-\tilde{J}(t)(-1)}^{\nu }/\sqrt{2}{\delta }_{{\bf{j}},{\bf{j}}[\alpha ]}$.

The results of the protocol are shown in figure 7(b) with the same graphical conventions as in figure 7(a). The comparison between the different curves shows that adapting the coupling pulse according to (25) increases substantially the fidelity ${ \mathcal F }$ of the quantum state transfer, while adding a chirp term (c.f. (26)) is in this case not required. Our interpretation is that in contrast to dispersion effects, the effect of missing atoms mainly leads to a time-delayed absorption of the wave packet, corresponding to the time needed by the wave packet to 'avoid' the defects. Finally, we note that our protocol is not robust when the coupling between the qubits and the edge-state channel is affected by the disorder, which in this example occurs if one site ${\bf{j}}[\alpha ]$ is vacant. This is illustrated in figure 7(b) by the orange line, which represents the averaged fidelity for disorder realisations where the two sites ${\bf{j}}[1],{\bf{j}}[2]$ are both occupied and corresponds to a much higher fidelity compared to the red curve.