Quantum Walks with Nonorthogonal Position States

Quantum walks have by now been realized in a large variety of different physical settings. In some of these, particularly with trapped ions, the walk is implemented in phase space, where the corresponding position states are not orthogonal. We develop a general description of such a quantum walk and show how to map it into a standard one with orthogonal states, thereby making available all the tools developed for the latter. This enables a variety of experiments, which can be implemented with smaller step sizes and more steps. Tuning the nonorthogonality allows for an easy preparation of extended states such as momentum eigenstates, which travel at a well-defined speed with low dispersion. We introduce a method to adjust their velocity by momentum shifts, which allows us to experimentally probe the dispersion relation, providing a benchmarking tool for the quantum walk, and to investigate intriguing effects such as the analog of Bloch oscillations.

Figure 1

nQW implemented in a harmonic-oscillator phase space. The positions are coherent states, illustrated by their Husimi functions, $P(\alpha )=|⟨\alpha |{\alpha }_x⟩{|}^2$, for each $|{\alpha }_x⟩$ separately. The inlay illustrates their orientation in phase space, implementing a nQW along a line. Since the position states $|{\alpha }_x⟩$ are coherent states, they are not orthogonal. The step size $\Delta \alpha =|{\alpha }_1-{\alpha }_0|=\sqrt{2}/\sigma$ [cf. Eq. (2)] of the nQW determines their overlap. QWs of this type have been implemented with trapped ions [16, 17, 18].

Figure 2

Position probability distribution $P_t(x)$ (3) after $t=150$ steps of a nQW with overlap function $g(k)$ (2) and (a) the experimental coin $C_E$ and (b) the Hadamard coin $C_H$ for $\sigma =\{0,1.5,8\}$ [green (light grey), blue (dark grey), red (medium grey)]. [For the green (light grey) curve, only points $x$ with $P(x)\ne 0$ are connected.] In the orthogonal case [green (light grey)] the probability distributions of both types of walks are equal; however, for large overlaps they differ significantly [blue (dark grey), red (medium grey)]. In case (a), the probability distribution approaches a Gaussian shape centered at the origin of the walk, as $\sigma$ is increased. The spreading, which is still linear in the step number $t$, is vastly reduced [17]. In the Hadamard case (b), the probability distribution approaches a shape consisting of two Gaussian peaks centered around $\pm t/\sqrt{2}$. Thus, the (linear) spreading is increased, as the probabilities between the peaks vanish. The initial state is ${\rho }_0=|{\alpha }_0⟩⟨{\alpha }_0|\otimes |c_{+}⟩⟨c_{+}|$.

Figure 3

(a) Group velocities $v_k(p)$ (grey, dotted/dashed) of the Hadamard-walk and $\Vert {\tilde{\rho }}_0\Vert (p)/(2\pi )$ of the initial states localized at $p=0$ [blue (dark solid line)] and $p=\pi /2$ [red (light solid line), corresponds to the experimental walk] with $\sigma =4$. (b) Asymptotic position probability distribution $P_{\infty }(q)$ for each initial state [blue (dark solid line), red (light solid line)]. For each initial state only the group velocities around their points of localization determine the position probability distribution. That is, since the blue initial state (a) is localized at $p=0$, where $v_k(0)\approx \pm 1/\sqrt{2}$, $P_{\infty }(q)$ consists of two peaks moving away from the origin with that velocity (b). In contrast, the red initial state (a) is centered around $v_k(\pi /2)\approx 0$, which leads to a localized asymptotic position probability distribution $P_{\infty }(q)$. The coin part of the initial states is ${\rho }_{00}=|c_{+}⟩⟨c_{+}|$.

Figure 4

Probability density (grey scale with black: $P_t(x)=1$) of a Hadamard-nQW ($\sigma =14$) with Bloch oscillations. After 20 steps of the nQW [cf. Fig. 2b], the Bloch oscillations are switched on with $\Delta \Theta =\pi /10$, such that the positions of the peaks oscillate with a period of 20 steps and an amplitude of 5 positions. The Bloch oscillations are switched off after 65 steps, a point where the group velocity is zero [Fig. 3a], such that the peaks remain at their position during the remaining nQW. The expectation values of the harmonic-oscillator occupation-number operator $N$ range during the oscillations from $⟨N_{\min }⟩=1.3$ to $⟨N_{\max }⟩=2.7$, which is detectable with state-of-the-art trapped-ion technology [39].