Experimental simulation and limitations of quantum walks with trapped ions

For the experimental implementation we confine a single ²⁵Mg⁺ ion in a linear Paul trap [41]. The motional frequency related to the confinement in the axial direction of the trap is set to ω_z = 2π × 2.13 MHz and in the radial directions to ω_x ≈ ω_y ≈ 2π × 5 MHz. We define two out of 12 electronic states of the hyperfine ground state manifolds [38] (figure 4)

$$\begin{equation} \begin{array}{@{}l@{}} \vert H\rangle\equiv\vert ^2S_{1/2}, F=2, m_F=2\rangle,\\ \vert T\rangle\equiv\vert ^2S_{1/2}, F=3, m_F=3\rangle \end{array} \end{equation} \tag{ 13 }$$

as the coin states. Further, we will use the state

$$\begin{equation} \vert A\rangle\equiv\vert ^2S_{1/2}, F=2, m_F=-2\rangle \end{equation} \tag{ 14 }$$

in the detection procedure. To lift the degeneracy within each hyperfine manifold, we apply a magnetic field inducing a Zeeman shift with an energy separation related to ω_Zm ≈ 2π·3 MHz between neighboring states. The energy difference (including the hyperfine splitting) between |H〉 and |T〉 amounts to a frequency of ω_coin = 2π × 1.77 GHz.

Our realization of the QW consists in the application of a sequence of laser and radiofrequency (RF) pulses to (1) initialize the ion's electronic and motional state, (2) to implement the QW and (3) to read out the final state via photon scattering. The experiments are repeated of the order of 1000 times for each set of parameters to obtain the required statistical relevance. A concise discussion of these tools in a generic context can be found in [42] and [43].

At the beginning of each experiment the ion is prepared in the coin state |T〉 with a fidelity ⩾0.99 by optical pumping [44], while the axial mode of motion is cooled close to the ground state by Doppler cooling ($\overline {n} \approx 10$) [45] and subsequent sideband cooling ($\overline {n} < 0.03$) [44]. The phase space of the axial mode of motion is used to encode the position of the walker. The radial modes are Doppler cooled ($\overline {n} \approx 4$), which enables a sufficient decoupling from the axial degree of freedom.

We drive coherent transitions between the coin states by applying an RF field for a duration t with frequency ω_coin (figures 3 and 4) [42]. This implements the operator $R \left (\theta , \phi \right)$ (6) with θ = Ω·t and Ω = 2π × 100 kHz, the Rabi frequency of the transition. The phase ϕ for the first pulse of each experimental cycle is arbitrary. For every following pulse the phase ϕ is set identically with respect to the first pulse. The duration of one R(π,ϕ)-pulse amounts to T_π = 5 μs. The coherence time (drop of oscillation contrast below 50%) for the RF field exceeds several tens of milliseconds [38]. The duration of a single experimental cycle (without initialization and detection) of the QW with three steps amounts to 150 μs (figure 10). Therefore the dephasing of the coin remains small. Spin-echo sequences [42], which are included in the QW pulse sequence (figure 10), further reduce the dephasing.

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Ideally, we encode the positions of the QW into coherent motional states |α_k〉 of the ion's axial harmonic motion. We manipulate the motion by implementing the shift operator S (7) via the application of a coin-state-dependent optical dipole force (see figure 3(b)). In the following we describe this method and its limitations.

3.4.1. Experimental tools

The initial motional state after sideband cooling is close to the ground state |n = 0〉 ($\overline {n}<0.03$). We apply a two-photon-stimulated Raman transition between the coin states by applying two laser beams (R1, R2) (figures 3(b) and 4), at a detuning of Δ = 2π × 80 GHz from the P_3/2 state manifold and a fixed phase relation [42]. The frequency difference between R1 and R2 amounts to

$$\begin{equation} \omega_L = \omega_1 - \omega_2 = \omega_z - \delta, \end{equation}\noindent \tag{ 15 }$$

with δ = 2π × 100 kHz. The effective wave vector is k₁ − k₂ = ke_z, pointing into the axial direction (figure 3(b)). This allows for two-photon stimulated Raman transitions $\vert T\rangle\kern-0.5pt \vert n\rangle \leftrightarrow \vert T\rangle\kern-0.5pt \vert n+1\rangle$ and $\vert H\rangle\kern-0.5pt \vert n\rangle \leftrightarrow \vert H\rangle\kern-0.5pt \vert n+1\rangle$ (∀n). In a simplified picture the two laser beams provide a walking standing wave causing a state dependent ac-Stark shift. This yields a coin-state-dependent force (F_T , F_H), proportional to the spacial gradient of the walking wave and oscillating with frequency ω_L. The ratio of the forces acting on the coin states amounts to F_H/F_T  ≈ −2/3. The polarizations and intensities of the laser beams are adjusted such that the time-averaged ac-Stark shift for pulse durations T ≫ 0.5 μs is negligible [46]. Thus application of the dipole force does not change the relation between the relative phase of the coin states and the phase of the RF, which implements the coin operator. The effective wavelength of the walking wave amounts to λ ≈ 200 nm. With the width of the axial ground-state wave function of z₀ ≈ 10 nm this results in a Lamb–Dicke parameter of η = z₀ × 2π/λ = 0.31 [42].

3.4.2. Description of the dynamics

We consider the following Hamiltonian describing a two-level system coupled to a harmonic oscillator and interacting with a classical light field [42]:

$$\begin{eqnarray} &&\fl \mathcal{H} = \mathcal{H}_{\mathrm{coin}} + \mathcal{H}_{\mathrm{motion}} + \mathcal{H}_{\mathrm{interaction}}\nonumber\\ &&\fl \hphantom{\mathcal{H}}= \frac{\hbar}{2}\omega_{\mathrm{coin}}\sigma_z + \hbar \omega_z \left(a^{\dagger}a + {\textstyle\frac{1}{2}}\right)+\hbar\underline{\Omega}_D\cos(k\cdot z-\omega_Lt+\phi_0)\nonumber\\ &&\fl \hphantom{\mathcal{H}}= \frac{\hbar}{2}\omega_{\mathrm{coin}}\sigma_z + \hbar \omega_z \left(a^{\dagger}a + {\textstyle\frac{1}{2}}\right)+ \frac{\hbar}{2}\underline{\Omega}_D (\mathrm{e}^{\mathrm{i}(\eta (a + a^{\dagger})-\omega_Lt+\phi_0)}+\mathrm{h.c}.), \end{eqnarray} \tag{ 16 }$$

where $\underline {\Omega }_D=\Omega _D(\vert T\rangle \langle T\vert -\frac {2}{3}\vert H\rangle \langle H\vert)$ with Ω_D being the coupling factor and σ_z the Pauli z-matrix. The dynamics have been investigated for many applications in QIP [47, 48] (in the LDA, see below), for the simulation of nonlinear optics [49] and in the context of mesoscopic entanglement [36, 37]. In the interaction picture, with the free Hamiltonian being $\mathcal {H}_{\mathrm {coin}} + \mathcal {H}_{\mathrm {motion}}$, the interaction Hamiltonian can be written as

$$\begin{eqnarray} &&\fl \mathcal{H}_I(t) =\frac{\hbar}{2} \underline{\Omega}_D \otimes \displaystyle\sum^{\infty}_{m=0} \sum^{\infty}_{n=0} \ \ \ \vert m\rangle\langle m\vert \mathrm{e}^{\mathrm{i} \eta (a + a^{\dagger})} \vert n\rangle\langle n\vert\nonumber\\ &&\fl\hspace{6pc}\times\left(\mathrm{e}^{\mathrm{i}((m-n) \omega_z - \omega_L) t+\mathrm{i}\phi_0} + (-1)^{\vert m-n \vert} \,\mathrm{e}^{\mathrm{i}((m-n) \omega_z + \omega_L) t-i\phi_0}\right). \end{eqnarray}\noindent \tag{ 17 }$$

We apply the dipole force with a small detuning of δ = ω_z − ω_L = 2π × 100 kHz, such that the terms corresponding to first-sideband transitions, $\vert n\rangle \leftrightarrow \vert n+1\rangle$, rotate slowest and thus dominate. However, we also consider contributions up to the third sideband, $\vert m-n\vert =3$, an approximation we refer to as 3SB (figures 6, 8 and 12).

When considering only the slowest rotating terms, i.e. applying the usual rotating-wave approximation (RWA), the interaction Hamiltonian is reduced to

$$\begin{eqnarray}&&\fl \mathcal{H}^{\mathrm{RWA}}_I(t) = \frac{\hbar}{2} \underline{\Omega}_D \otimes \!\sum^{\infty}_{n=0} \left(\langle n+1\vert \mathrm{e}^{\mathrm{i} \eta (a + a^{\dagger})} \vert n\rangle \mathrm{e}^{\mathrm{i}(\delta t+\phi_0)} \vert n+1\rangle\langle n\vert\right.\nonumber\\ &&\fl\hphantom{\mathcal{H}^{\mathrm{RWA}}_I(t) =} - \left.\langle n\vert \mathrm{e}^{\mathrm{i} \eta (a + a^{\dagger})} \vert n+1\rangle \mathrm{e}^{-\mathrm{i} (\delta t+\phi_0)} \vert n\rangle\langle n+1\vert\right). \end{eqnarray} \tag{ 18 }$$

For states with $k\sqrt {\langle {z^2}\rangle}=\eta \sqrt {\langle {(a+a^\dagger)^2}\rangle}\ll 1$, we can approximate $\langle n+1\vert \mathrm {e}^{\mathrm {i}\eta (a+a^\dagger)}\vert n\rangle \approx \mathrm {i}\eta \sqrt {n+1}$ [43]. That is, the potential providing the dipole force changes linearly over the extension of the wave function. We can then simplify the interaction Hamiltonian to

$$\begin{equation} \mathcal{H}^{\mathrm{LD}}_{\mathrm{I}}(t) = \frac{\hbar}{2} \underline{\Omega}_D \otimes \mathrm{i} \eta \left(a^{\dagger}\,\mathrm{e}^{\mathrm{i}(\delta t+\phi_0)} - a\,\mathrm{e}^{-\mathrm{i}(\delta t+\phi_0)}\right)\!. \end{equation} \tag{ 19 }$$

This is the Lamb–Dicke approximation (LDA) [42] or the linear approximation, respectively. In the following, we set ℏ = 1 and ϕ₀ = 0, as ϕ₀ represents the initial phase relation between the ion motion and the dipole force, which does not influence the results of the QW (see section 3.4.3).

Within the LDA, the time evolution operator is given by [50]

$$\begin{equation} \fl U (t) = \vert T\rangle\langle T\vert \otimes D(\alpha(t))\cdot \mathrm{e}^{\mathrm{i} \Phi\left(\alpha(t), t\right)}+\vert H\rangle\langle H\vert \otimes D\left(-\frac{2}{3}\alpha(t)\right)\cdot \mathrm{e}^{\mathrm{i} \Phi\left(-\frac{2}{3}\alpha(t), t\right)}, \end{equation} \tag{ 20 }$$

which is a displacement operator D(α(t)) with a phase factor, where the phase amounts to

$$\begin{equation} \Phi \left(\alpha(t), t\right) = \mathrm{Im}\left(\int^t_0 \mathrm{d}\tau \: \alpha^{\ast}(\tau) \frac{\mathrm{d}\alpha(\tau)}{\mathrm{d}\tau}\right). \end{equation} \tag{ 21 }$$

The factor 2/3 in the displacement operator for the Heads part results from the difference of the state-dependent dipole force, F_H/F_T  = −2/3 (see figure 3). The complex parameter appearing in the displacement operator and in the phase amounts to

$$\begin{equation} \alpha(t)=\frac{\eta\Omega_D}{2}\cdot \int^t_0\mathrm{e}^{\mathrm{i}\delta t}\,\mathrm{d}t = -\mathrm{i}\frac{\eta\Omega_D}{2\delta} \cdot (\mathrm{e}^{\mathrm{i}\delta t}-1) \end{equation} \tag{ 22 }$$

and corresponds to a circular trajectory in a co-rotating phase space, given by the interaction picture as

$$\begin{equation} \left(\begin{array}{@{}c@{}}\mathrm{Re}(\alpha(t))\\ [6pt] \mathrm{Im}(\alpha(t))\end{array}\right) = \left(\begin{array}{@{}cc@{}}\cos(\omega_z t) & \sin(\omega_z t) \\ -\sin(\omega_z t) & \cos(\omega_z t) \end{array}\right) \left(\begin{array}{@{}c@{}} \frac{1}{2z_0}\langle{z}\rangle(t) \\ z_0 \langle{p}\rangle(t) \end{array}\right), \end{equation} \tag{ 23 }$$

with 〈z〉(t) and 〈p〉(t) being the expectation values of position and momentum.

For each coin state, the motional wave function is coherently displaced along a circular trajectory in the co-rotating phase space (figure 6). The circular shape of the trajectory is caused by the dipole force being applied with a detuning δ relative to the oscillator frequency. Thus the relative phase between dipole force and oscillation of the ion evolves in time as ϕ_D(t) = δ·t. After a duration of T_π = π/δ of driving the motional state and increasing its amplitude, the relative phase amounts to ϕ_D(T_π) = π and therefore the dipole force starts to decelerate the oscillation of the ion. After a duration of T_2π = 2π/δ the coherent state returns to its initial location in the co-rotating phase space. The total acquired phase of the motional state, which equals the enclosed area of the trajectory [47], amounts to

$$\begin{equation} \Phi_T=\pi\left(\frac{\eta\Omega_D}{2\delta}\right)^2 \end{equation} \tag{ 24 }$$

for |T〉 and Φ_H = (4/9)·Φ_T for |H〉.

The nonlinearity of the potential causing spatial variations of the dipole force can be described by the absolute values of the transition matrix elements, i.e. $\Omega _{n+1,n} = \langle n+1\vert \exp (\mathrm {i} \eta (a + a^{\dagger }))\vert n\rangle$ (figure 5). For small Fock state numbers n they remain close to the approximative results within the LDA (where the potential is linear in z). Close to n = 8 ≡ g₁ (for η = 0.31) they start to significantly deviate from that approximation. In particular, the transition matrix elements feature a maximum value at g₁. The excitation of motion via the dipole force ceases at n = 37 ≡ g₂, due to $\vert \Omega _{38,37}\vert \approx 0$. It therefore represents an upper bound for the motional excitation with a dipole force applied close to resonance. (Applying the dipole force with a frequency ω_L = 2·ω_z would allow populating higher Fock states, but since the overall time evolution is then described by a squeezing operator, it cannot be used for the implementation of a QW based on coherent displacements, following [31].)

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Figure 6 presents the results of our numerical simulations of the time evolution, comparing the three different approximations 3SB, RWA and LDA. Starting in the motional ground state, the trajectory in the co-rotating phase space first follows the circular evolution, as long as the amplitude remains small, i.e. 〈n〉 < g₁. This regime can be well described by the LDA (with the driving potential being linear in z). As the amplitude of the motional state approaches g₁, the driving potential becomes sufficiently nonlinear to severely affect the subsequent evolution. The amount of displacement per time interval is substantially reduced, as the transition rates $\vert \Omega _{n+1,n}\vert$ decrease for n ⩾ g₁, such that the trajectory remains in the vicinity of g₁. At this point motional squeezing occurs. The probability distribution of the wave function in the Fock state basis becomes narrower than Poissonian, which results in a squeezed shape of the corresponding Wigner function. The relative phase between the dipole force and the oscillation of the ion changes faster than in the linear case. Thus the squeezed wavefunction reaches the origin of the phase space after a time significantly shorter than 2π/δ. The dependence of the return time and the amount of squeezing on the maximal motional amplitude severely affect an implementation of a QW with position states outside the LDR, following the scheme described in [31]. However, in section 6 we propose an alternative protocol for the implementation of the shift operator that circumvents this restriction. The faster rotating terms, which are taken into account in the 3SB approximation, cause additional modulations of the trajectory with low amplitudes and high frequencies (2ω_z + δ) and (3ω_z + δ), respectively.

Further motional excitation (up to g₂) can be realized either by applying the dipole force resonantly (figure 7) [49] or by the repeated off-resonant application of a weak dipole force with duration π/δ and constant time delays between the pulses (figure 8). However, in either way severe motional squeezing occurs as the amplitude approaches and becomes larger than g₁.

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3.4.3. Implementation of the shift operator

We implement several shifts into a certain direction in the co-rotating phase space by a synchronized application of dipole force pulses. Switching the dipole force is realized by acousto-optical modulators (AOMs), refracting the laser beams R1 and R2 into the Paul trap [42]. As the laser and the electronic oscillator driving the AOMs are continuously on during the whole experiment, the phase relation between the dipole force and the motion of the ion continuously evolves in time as ϕ_D(t) = δ·t even when the optical dipole force is switched off. Therefore, after the first pulse of the dipole force, the relative phase ϕ_D(t) and thus the direction of the displacement caused by the following dipole force pulse depend on the intermitted delay. We apply pulses of the duration T^QW_D ≈ π/δ = 5 μs and mutual delays of even multiples of T^QW_D (see timing in figure 10) such that the phase difference evolves by multiples of 2π and the displacements are concatenated along a line in the co-rotating phase space (figure 8).

However, the motional states corresponding to |H〉 and |T〉 acquire different phase factors Φ_H, Φ_T  (24) during each displacement. To compensate for these coin-state-dependent phases, we implement the shift operation of the QW as a combined pulse, which consists of two dipole force pulses, each followed by an R(π,0)-pulse (figure 9). In this scheme each coin state acquires the sum of the phase factors, Φ_T  + Φ_H, which therefore turns into a global phase factor not affecting the QW. A schematic diagram of the overall pulse sequence for the QW is depicted in figure 10.

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For the implementation of the QW the following is crucial. After a step fulfilling the operation $\vert T\rangle\kern-0.5pt \vert \widetilde {\alpha }_{k}\rangle \rightarrow \vert T\rangle\kern-0.5pt \vert \widetilde {\alpha }_{k+1}\rangle$, the subsequent coin toss and shift operation have to ensure the operation $\vert H\rangle\kern-0.5pt \vert \widetilde {\alpha }_{k+1}\rangle \rightarrow \vert H\rangle\kern-0.5pt \vert \widetilde {\alpha }_{k}\rangle$, i.e. the motional state $\vert \widetilde {\alpha }_{k+1}\rangle$ must be transferred back to the previous state $\vert \widetilde {\alpha }_{k}\rangle$. This must be fulfilled for all k simultaneously (see figure 1). In order to reach the state $\vert \widetilde {\alpha }_{k}\rangle$, the duration T_D and detuning δ of the dipole force have to be adjusted properly (in the LDR to T_D = π/δ), implementing semi-circular trajectories in the co-rotating phase space. But since the return time is reduced outside the LDR (figure 6), the shift operation implements the transition from $\vert \widetilde {\alpha }_{k+1}\rangle$ to some other state $\vert \widetilde {\alpha }_{k_2}\rangle \neq \vert \widetilde {\alpha }_{k}\rangle$. The reduced overlap $\langle {\widetilde {\alpha }_{k_2}\vert \widetilde {\alpha }_{k}}\rangle<1$ leads to reduced interference during the succeeding coin toss (figure 1).

In principle, the shift operator can alternatively be implemented by the dipole force on resonance (δ = 0). However, in that case, small variations of ω_L and ω_z have a much stronger influence than in the detuned case. This can be seen by comparing the difference in displacement, i.e. $\vert \alpha (\delta ,t)-\alpha (\delta +\epsilon ,t)\vert$, for a fixed duration t and a fixed difference in the detuning, , using equation (22) for different values of δ, where α(δ,t) fulfills equation (22) for the detuning δ.

The state after three steps of the QW is

$$\begin{equation} \vert \widetilde{\psi}_3\rangle=\frac{1}{\sqrt{2}}(\vert T\rangle\vert M_T\rangle+\vert H\rangle\vert M_H\rangle), \end{equation} \tag{ 25 }$$

with $\vert M_T\rangle =\sum ^3_{k=-3} c^T_k \vert \widetilde {\alpha }_{k}\rangle$ and $\vert M_H\rangle =\sum ^3_{k=-3} c^H_k \vert \widetilde {\alpha }_{k}\rangle$ (cf equation (11)). The basics for the readout are state-of-the-art techniques in quantum information processing with trapped ions [42, 51].

We read out the coin state by driving the cycling transition |T〉|n〉 → |2P_3/2,F = 4,m_F  = 4〉|n〉 (figure 4) for a duration of 20 μs and detect scattered photons with a photomultiplier. This transition is in good approximation independent of the motional state. The average number of detected photons is proportional to the probability $P_T(\widetilde {\psi }_3)$, related to the coin state |T〉 [42]. The probability $P_H(\widetilde {\psi }_3)$ is accessible via the application of an $R\left (\pi ,0\right)$-pulse before the coin-state detection.

To characterize the motional states of $\vert \widetilde {\psi }_3\rangle$ we determine the position-state probabilities $\vert c^T_k \vert ^2$ and $\vert c^H_k\vert ^2$. To analyze |T〉|M_T 〉, we isolate the other part, |H〉|M_H〉, by transfering |H〉|M_H〉 → |A〉|M_H〉 using appropriate RF pulses (figure 4). The part |A〉|M_H〉 of the ion's state is not affected by the subsequent operations. We then apply a two-photon stimulated Raman transition $\vert T\rangle\kern-0.5pt \vert n\rangle \leftrightarrow \vert H\rangle\kern-0.5pt \vert n+1\rangle$ (∀n) using the lasers BR and R2 (figure 4), with a frequency difference of ω_L = ω_coin + ω_z (BSB) for a variable duration t_BSB [42]. The corresponding Rabi frequency for each n is proportional to Ω_n+1,n (figure 5). Finally, we apply the coin-state detection, as described above. The average number of detected photons is proportional to the probability [42]

$$\begin{equation} P_T(t_{\mathrm{BSB}}) = \frac{1}{2}\left(1+\sum^\infty_{n=0} \vert a^T_n\vert^2 \cdot \cos(\Omega_{n+1,n}\cdot t_{\mathrm{BSB}})\mathrm{ e}^{-\gamma \cdot t_{\mathrm{BSB}}}\right), \end{equation} \tag{ 26 }$$

with a^T _n = 〈n|M_T 〉 being the coefficients of |M_T 〉 in the Fock-state basis. The damping factor γ accounts for decohering effects, mainly of the BSB operation [42]. A discrete Fourier transform of the function P_T (t_BSB) allows one to access the Fock-state probabilities $\vert a^T_n\vert ^2 = \vert\! \sum _k c_k^T \langle {n\vert \widetilde {\alpha}_k}\rangle\vert ^2$.

We numerically simulate the QW and generate a corresponding function P^S_T (t_BSB) (where the index ^S denotes the result of the simulation) and optimize the parameters of the simulation in order to fit P^S_T (t_BSB) onto the experimental data, P^E_T (t_BSB) (indicated by the index ^E). The state $\vert \widetilde {\psi }_3^S\rangle$, generated by the simulation, respectively the Fock-state probabilities of its motional parts, are fed into the algorithm for the discrete Fourier transform of P^E_T (t_BSB) to improve the convergence of the results. The Fock-state probabilities $p_n\left (\widetilde {\alpha }_k\right)=\vert \langle {n\vert \widetilde {\alpha }_k}\rangle\vert ^2$ of the position states $\vert \widetilde {\alpha }_{k}\rangle$ are determined separately (figure 11), using the same method.

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To identify the position state probabilities $\vert c^T_k\vert ^2$ from the Fock-state probabilities of |M_T 〉, in particular to distinguish between the coefficients related to $p_n\left (\widetilde {\alpha }_k\right)$ and $p_n\left (\widetilde {\alpha }_{-k}\right)$, we additionally apply the motional-state readout to the shifted states $\vert \widetilde {\psi }^+_3\rangle = S\vert \widetilde {\psi }_3\rangle$ and $\vert \widetilde {\psi }^-_3\rangle = S^{-1}\vert \widetilde {\psi }_3\rangle$ (see equation (7) and figure 14), where S⁻¹ is (up to a global phase) implemented by a proper timing of the corresponding dipole force pulses.

Starting from the initial state |ψ₀〉 we apply 0–4 shift operations S with a dipole force duration of $T_D\approx \frac {\pi }{\delta }$, exciting the motion of the ion to one of the position states $\vert \widetilde {\alpha }_{k}\rangle$. Then we read out the motional state to determine its Fock-state probabilities $p_n\left (\widetilde {\alpha }_k\right)$ and the expectation of the number operator $\langle {n}\rangle \equiv \vert \widetilde {\alpha }_k\vert ^2$. Small deviations of the dipole force duration (see the next subsection) do not influence the probabilities significantly. We adjust the amplitude of the dipole force by adjusting the corresponding laser beam intensities to approximately meet the conditions $\vert \Delta \alpha \vert \geqslant 1$ and $\vert \widetilde {\alpha }_3\vert ^2\equiv \langle \widetilde {\alpha }_3\vert n\vert \widetilde {\alpha }_3\rangle \leqslant 9$ (three well-distinguishable position states within or close to the LDR). The Fock-state expectation values of the position states amount to $\langle {n}\rangle _0=\langle \widetilde {\alpha }_0\vert n\vert \widetilde {\alpha }_0\rangle =0$, 〈n〉₁ = 1.33, 〈n〉₂ = 4.71, 〈n〉₃ = 9.08 and 〈n〉₄ = 13.50. The outer position states $\vert \widetilde {\alpha }_{\pm 3}\rangle$ and $\vert \widetilde {\alpha }_{\pm 4}\rangle$ are not within the LDR. The step sizes therefore differ, reaching from $\vert \widetilde {\alpha }_1\vert -\vert \widetilde {\alpha }_0\vert =1.15$ to $\vert \widetilde {\alpha }_4\vert -\vert \widetilde {\alpha }_3\vert =0.66$. However, due to the motional squeezing the overlaps of all neighboring states amount to $\vert\kern-1pt\langle\widetilde{\alpha}_k\vert \widetilde {\alpha}_{k+1}\rangle\kern-1pt\vert ^2\approx 0.24<1/e$, which corresponds to the overlap of coherent states with a step size of $\vert \Delta \alpha \vert >1$ as assumed in the theory (see section 2). With δ = 2π × 100 kHz and η = 0.31 the corresponding coupling strength of the dipole force amounts to Ω_D = 2π × 0.24 MHz. The amplitude of the corresponding dipole force amounts to F_T  = (ℏηΩ_D)/(2z₀) = 2.54 × 10⁻²¹ N inside the LDR.

To estimate the amount of motional squeezing (not affecting the fidelity of our results on the QW), we compute coherent states |α_k〉 according to equation (3) with $\alpha _k\equiv \widetilde {\alpha }_k$. We then compute their overlaps $F_k=\vert\kern-1pt \langle\widetilde {\alpha }_k\vert \alpha _k\rangle\kern-1pt\vert ^2$, which amount to F₀ = 1.00, F₁ = 1.00, F₂ = 0.97, F₃ = 0.90 and F₄ = 0.78.

The duration T_D of the dipole force pulses (and the related mutual delays, see figure 10) is a very sensitive parameter for the implementation of the QW. For a given detuning δ, T_D determines the relative direction of subsequent shifts in the co-rotating phase space. By altering T_D we can control and maximize the overlap of the interfering parts of the wave function in the QW (figure 1). We repeat the QW pulse sequence with increasing values of T_D (figure 10) and acquire the coin-state probabilities P_T and P_H via the coin-state detection (figure 12). The ratio P_T /P_H is an indicator of the amount of interference. If no interference occurs, it amounts to P_T /P_H = 1. We maximize the ratio by iterating to the optimal dipole force duration, denoted by T^QW_D ≈ π/δ. The nonlinearity of the dipole force, in particular the reduced return time discussed in section 3.4.2, leads to a deviation of T^QW_D from the duration π/δ, optimal within the LDA only. Additionally, this method is suitable for implicitly determining the detuning δ to the required precision.

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As illustrated in figure 12, the maximum ratio of the measured probabilities amounts to P_T /P_H ≈ 3, with the related dipole force duration being defined as T^QW_D. In a separate precision measurement, averaged over 60 000 measurements the coin state probabilities at this point amount to P_T  = 0.741 ± 0.002 and P_H = 0.259 ± 0.001, which is close to the theoretical predictions 0.75 and 0.25 (figure 1). At slightly different values of the dipole force duration, T_D = T^QW_D × (1 ± 0.02) the coin state probabilities are approximately equal (P_T /P_H ≈ 1), indicating that the overlaps and hence the interference of different parts of the wave function vanish.

The results of a numerical simulation of this procedure within 3SB are in good agreement with the experimental data (figure 12). Additionally, the simulation shows similar splittings of the coin state probabilities at the dipole force durations T_D = 4.6 μs and T_D = 5.4 μs, for δ = 2π × 100 kHz (figure 13). These are QWs in which the step directions of |T〉 and |H〉 are exchanged at each step.

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The simulation also shows a high frequent modulation of the coin state probabilities as a function of T_D. This is due to the fast modulations of the trajectories in the rotating phase space, mainly caused by $\vert n+2\rangle \leftrightarrow \vert n\rangle$-contributions of the dipole force (see figure 6).

In the following, we propose the implementation of the shift operator with photon kicks [39, 40], which is substantially less dependent on the motional state and allows for the implementation of QWs with many steps. The principle of a photon kick is to apply a π-pulse on the coin states which is sufficiently short such that the free harmonic motion of the ion during the pulse itself is negligible. It was shown that the change of the momentum of the ion during such a pulse can be described by a displacement operator, allowing us to propose its application as a building block for the shift operator of a QW. In the original protocol [39], which has been realized recently [52], however, the influence of the motional state on the performance of the photon kicks has not been considered, since the amplitudes of the motional states were assumed to remain small. For the implementation of a QW with many steps, we have to consider (coherent) motional states with a very large amplitude and thus have to re-assess the validity of the above-mentioned approximation. We find that, for a given fidelity, the upper bound for the pulse duration scales inversely with the motional amplitude, and additionally, for coherent motional states, depends on the phase of their harmonic oscillation at the moment when the pulse is applied. In the following, we derive an analytic bound for general states and present the results of a numerical study for coherent motional states. With the latter we show that QWs with up to 100 steps for a step size of $\vert \Delta \alpha \vert =2$ should be possible with state-of-the-art technology.

Referring to [39], we start our analysis with the Hamiltonian

$$\begin{equation} \begin{array}{@{}l@{}} \mathcal{H} = \mathcal{H}_0+\mathcal{H}_1\\ \hphantom{\mathcal{H}}= \displaystyle\frac{\Omega}{2} \left(\sigma_+ \otimes \mathrm{e}^{\mathrm{i}\eta(a^\dagger+a)}+\sigma_- \otimes \mathrm{e}^{-\mathrm{i}\eta(a^\dagger+a)}\right)+\omega_z a^\dagger a. \end{array} \end{equation} \tag{ 27 }$$

This Hamiltonian can be implemented in various ways, e.g. via direct dipole coupling, two-photon stimulated Raman transitions or stimulated Raman adiabatic passage [39]. Each implementation imposes different constraints on pulse duration, laser intensities, etc. In the following, we will focus on the implementation with a two-photon stimulated Raman transition and consider the energy levels of ²⁵Mg⁺.

In this configuration, two laser beams (R_a,R_b) resonantly drive two-photon transitions between the coin states via a virtual state detuned from the P_3/2 manifold by Δ_R. Each laser beam drives only one of the two Raman branches, due to their different polarizations. In a RWA, terms varying at optical frequencies are neglected. This is valid in our case for pulse durations well above 1/10⁻¹⁵ Hz = 1 fs. Finally, an adiabatic elimination of the P_3/2 states requires $\vert \Omega /\Delta _R\vert \ll 1$. The pulse duration T_p in our case must therefore be sufficiently longer than 5 ps for Δ_R ≈ 2π × 10¹¹ Hz and T_pΩ = π (see below). The effective wave vector of the two-photon transition is k = k_a − k_b.

Hamiltonian (27) implements the desired displacement operator for a pulse duration of T_p = π/Ω, if we neglect the perturbation $\mathcal {H}_1$. The time evolution operator then reads

$$\begin{eqnarray}&&\fl U_0(T_{\mathrm{p}})=\mathrm{e}^{-\mathrm{i}\mathcal{H}_0T_{\mathrm{p}}}\nonumber\\ &&\fl \hphantom{U_0(T_{\mathrm{p}})}=\cos\left(\frac{\Omega T_{\mathrm{p}}}{2}\right)\cdot \mathbbm{1}_{\mathrm{coin}}\otimes\mathbbm{1}_{\mathrm{motion}}-\mathrm{i}\sin\left(\frac{\Omega T_{\mathrm{p}}}{2}\right)\cdot \left(\sigma_+\otimes D\left(\mathrm{i}\eta\right)+\sigma_-\otimes D\left(-\mathrm{i}\eta\right)\right)\nonumber\\ &&\fl \hphantom{U_0(T_{\mathrm{p}})}=-\mathrm{i}\left(\sigma_+\otimes D\left(\mathrm{i}\eta\right)+\sigma_-\otimes D\left(-\mathrm{i}\eta\right)\right), \end{eqnarray} \tag{ 28 }$$

which is obtained by expanding the exponential function, splitting the series into odd and even parts and using the properties of the Pauli matrices and displacement operators.

The shift operator itself, implementing the desired step size (i.e. $\vert \Delta \alpha \vert =2$, see figure 2), can be realized by the subsequent application of 2/η kicks in such a way that the displacements D(iη) of several π-pulses add up to D(Δα = iη·2/η). This can be achieved by changing the direction of the effective wave vector by 180° for each photon kick. In practice one can either switch between two Raman beam configurations with opposite effective wave vectors, or implement every second π-pulse by an RF transition for which the momentum transfer is negligible. Notably, with this protocol the step sizes for both directions of the QW are equal, in contrast to the method of optical dipole forces used in our current experiment.

In the following, we derive a conservative estimate for the deviation from a coherent-state displacement induced by $\mathcal {H}_1$. The total time evolution is

$$\begin{equation} U(t)=\mathrm{e}^{-\mathrm{i}\mathcal{H}t}= U_0(t)\cdot V(t) \end{equation} \tag{ 29 }$$

with $V(t)= \mathrm {e}^{\mathrm {i}\mathcal {H}_0t}\,\mathrm {e}^{-\mathrm {i}\mathcal {H}t}$. Note that V (t) can be differentiated

$$\begin{equation} \dot{V}(t)=-\mathrm{i}\,\mathrm{e}^{\mathrm{i}\mathcal{H}_0t}\mathcal{H}_1\,\mathrm{e}^{-\mathrm{i}\mathcal{H}_0t}\cdot V(t), \end{equation} \tag{ 30 }$$

leading to an equation that is formally solved by the integral equation

$$\begin{equation} V(t)=\mathbbm{1}-\mathrm{i}\int^t_0{\rm d} s \, \mathrm{e}^{\mathrm{i}\mathcal{H}_0s}\mathcal{H}_1\mathrm{e}^{-\mathrm{i}\mathcal{H}_0s}\cdot V(s), \end{equation} \tag{ 31 }$$

using $V(0)=\mathbbm{1}$. Now we define as the distance between the evolved state according to the full Hamiltonian and the desired evolved state according to $\mathcal {H}_0$:

$$\begin{eqnarray} &&\epsilon \equiv\parallel \left(U(t)-U_0(t)\right)\vert \psi\rangle\!\parallel \nonumber\\ &&\hphantom{\epsilon}=\parallel(V(t)-\mathbbm{1})\vert \psi\rangle\parallel \nonumber\\ &&\hphantom{\epsilon}= \parallel\int^t_0{\rm d} s \;\, \mathrm{e}^{\mathrm{i}\mathcal{H}_0s}\mathcal{H}_1\,\mathrm{e}^{-\mathrm{i}\mathcal{H}_0s} V(s) \vert \psi\rangle\!\parallel . \end{eqnarray} \tag{ 32 }$$

Approximating the last expression by the largest term of the first-order Dyson series, and considering a motional state |ψ〉 = |H〉|α〉, gives the following error estimate for a pulse with duration T_p [39]:

$$\begin{equation} \epsilon\approx\parallel \int_0^{T_{\mathrm{p}}}{\rm d} s \;\, \omega_z\mathbbm{1}\otimes a^\dagger a \vert \psi\rangle\parallel = T_{\mathrm{p}}\omega_z\vert\alpha\vert^2. \end{equation} \tag{ 33 }$$

Thus, for an initial state |H〉|α〉 (the coin state can be chosen arbitrarily), the pulse duration T_p necessary to implement the displacement operator with an error smaller than must fulfill

$$\begin{equation} T_{\mathrm{p}}\leqslant\frac{\epsilon}{\omega_z\cdot \vert\alpha\vert^2}. \end{equation} \tag{ 34 }$$

The scaling with $\vert \alpha \vert ^{-2}$ is, however, a rather rough estimate. This is shown by a numerical simulation of this process, in particular considering the application of photon kicks to (superpositions of) coherent motional states. We compute the fidelity $f=\vert \langle H\vert \langle \alpha \vert U_0^\dagger (T_{\mathrm {p}})U(T_{\mathrm {p}})\vert H\rangle\kern-0.5pt \vert \alpha \rangle\kern-0.5pt \vert ^2$ with the initial state |H〉|α〉, a pulse duration T_p and Ω = π/T_p, where the time evolution is implemented using a Runge–Kutta method. The results show that the fidelity strongly depends on the phase of ion oscillation at the moment of the photon kick.

Demanding a fidelity of f ⩾ 0.99 and for imaginary α, i.e. at the moment of the photon kick the ion is at the center of the harmonic potential and thus fastest, for our experimental parameters we find that⁶

$$\begin{equation} T^{\Im}_{f=0.99}(\vert\alpha\vert)=\exp(-17.55-0.63\ln(\vert\alpha\vert)-0.05\left(\ln\left(\vert\alpha\vert\right)\right)^2). \end{equation} \tag{ 35 }$$

For $\vert \alpha \vert =200$, an amplitude reached after the 100th step of a QW with $\vert \Delta \alpha \vert =2$, the pulse duration must be shorter than T^ℑ_f=0.99(200) = 0.21 ns.

However, applying the photon kick when the ion is at its turning point, i.e. the ion is slowest and α is real, the scaling is less demanding. We find that

$$\begin{equation} T^{\Re}_{f=0.99}(\vert\alpha\vert)=\exp(-17.03-0.02\ln(\vert\alpha\vert)-0.1(\ln(\vert\alpha\vert))^2). \end{equation} \tag{ 36 }$$

Most importantly, the prefactor of the term linear in $\ln (\vert \alpha \vert)$ is much smaller than for an imaginary α. For the 100th step, the pulse duration therefore only has to be shorter than T^ℜ_f=0.99(200) = 2.18 ns, which is within the specifications of a fast-switching electro-optic modulator and our current continuous-wave laser system. Timing the application of the photon kick to the (spatial) turning points of all the coherent oscillations occurring during the QW is possible, because we start the QW in the motional ground state and the position states are aligned along a line in the co-rotating phase space. Thus, coherent states of different $\vert \Delta \alpha \vert$ reach their turning points simultaneously.

Given our experimental parameters, in particular the width of the ground state wave function z₀ = 10 nm, the coherent motional state of maximal amplitude, $\vert \alpha _{\max }=200\rangle$, would have a real-space amplitude of $\langle \alpha _{\max }\vert z\vert \alpha _{\max }\rangle =4\,{\rm \mu}\mathrm {m}$. At such high motional amplitudes anharmonicities of the trapping potential must be considered. These depend on the design of the electrodes and could be eliminated, for example by designing the Paul trap electrodes in a hyperbolic shape [53]. Additionally, micromotion might increase the deviation from the ideal walk, for example by reducing the overlap of the additionally oscillating motional wave functions. However, it will remain negligible when the QW is implemented in the axial degree of freedom of an ion in a linear Paul trap.

A major challenge, but also an interesting research topic on its own, is the consideration of decoherence processes, in particular the heating and dephasing of the motional state and especially at the desired motional amplitude. Several studies—theoretical [54] and experimental [48, 55]—have been accomplished in the past. To estimate the amount of decoherence one can consider a Ramsey-interferometry experiment incorporating a Schrödinger-cat state, consisting of the two outermost states |α₁₀₀〉 and |α₋₁₀₀〉, and estimate the expected decay of the Ramsey fringes, which has experimentally been found to scale as exp(−d²λt) [48]. There $d=\vert \alpha _{100} - \alpha _{-100}\vert$, the phase-space distance of the coherent state, t being the duration between the creation of both states and λ representing the scaling parameter, incorporating heating and dephasing. Using a Paul trap with hyperbolically shaped electrodes to ensure a harmonic confinement implies a large electrode–ion distance of the order of millimeters. This reduces heating by several orders of magnitude compared, for example, to the setup used in [55], where the dependence of heating on the electrode–ion distance has been investigated. Additionally, cooling the electrodes to cryogenic temperatures, which further reduces the heating rate [55], is currently becoming state-of-the-art technology in trapped-ion experiments. Dephasing, on the other hand, depends mainly on the stability of the trap frequency ω_z against fluctuations. We believe that much improvement is possible there by extensive use of stabilization electronics, which has not yet been brought to the edge of the technically possible, since large motional amplitudes have not been a major issue in most experiments. In fact, our experiment of the three-step QW required an improved frequency stabilization, which was achieved to a sufficient degree by a simple electronic circuit. Still to be considered is the duration t, which, using the fast pulse protocol, is about one order of magnitude longer compared to [48]. However, the duration of the coin toss, being the constituent of longest duration in the protocol, can be reduced to a pulse duration of less than a microsecond, in analogy to [48].

A QW in two or three dimensions is possible by additionally considering the motion in the radial direction. The pulse sequence for a step of a QW is then the subsequent application of the shift operator in each direction where each operation is preceded by a coin toss.

More possibilities and reduced technical requirements might be achieved by trapping more than one ion and considering the collective degrees of motion in one direction. The work [56] describes the scheme with two ions, creating a four-sided coin, where two coin states affect the walk in the center-of-mass motional mode and the other two in the stretch mode of motion. In particular, possibilities with the coin being initialized in an entangled state are investigated.

A photon kick, as described above, induces motion in all motional modes in the direction of the effective wave vector k, according to the respective coin states. That is, with N ions, one step of the QW consists of a coin operation on the 2^N-sided coin and a single shift operation, which displaces the part of the motional wave function related to each coin state into the opposite direction in phase space of one motional mode. Particularly difficult is to assign the 2^N−1 pairs of coin states to N different axial motional modes such that for each coin state the corresponding state-dependent force induces motion in one direction in a certain mode [57]. One possible way to obtain the required number of motional modes is by adding ions that do not contain a transition corresponding to the coin states and are therefore not affected by the photon kicks. That would be in our case ²⁴Mg ions without hyperfine structure and therefore no coin states and no corresponding transition. With this, the implementation of a QW in four dimensions is possible using three ²⁵Mg ions and one ²⁴Mg ion. For more dimensions the issue arises that the ²⁴Mg and ²⁵Mg ions have to be arranged in such a way that for each coin state the corresponding state dependent force induces motion in a certain mode, which has not yet been clarified.