Qudit encoding in Rydberg blockaded arrays of atoms

Abstract

We propose a protocol to realize arbitrary state synthesis and unitary operations on a qudit encoded in the collective dressed states of a Rydberg blockaded array of three-level atoms. This system is isomorphic to the Jaynes-Cummings model and acts as a multi-level Rydberg-superatom whose nonlinear spectrum can be precisely controlled through the parameters of the laser driving the intermediate-to-Rydberg transition. Control of the qudit state is possible through pulse sequences of the laser driving the ground-to-intermediate transition. The dimension of the qudit Hilbert space is scalable by adjusting the number of atoms involved in the Rydberg blockaded array. We estimate the fidelity of our protocol for realizing arbitrary unitaries and discuss the influence of the finite lifetime of the Rydberg state. Our work paves the way for processing quantum information with Rydberg blockaded arrays of atoms as an alternative to atom qubit arrays.

Figures (4)

• Figure 1: Rydberg blockaded array of single atoms with its collective dressed state level scheme. (a)$N$ individual three-level atoms are positioned regularly using either an array of optical tweezers or by loading the sites of an optical lattice with a tweezer. The atoms feature a ladder atomic-level structure where the upper level corresponds to a Rydberg state. The interatomic distance $a$ is chosen such that we can neglect light assisted collisions ($a > \lambda$, where $\lambda$ is the wavelength of the light driving either the upper or the lower transition -- if the frequency of the lower transition lies within the optical domain) and that all atoms are located inside a blockaded volume ($N^{1/d} a < R_b$, where $d$ is the dimensionality of the system). (b) Energy diagrams of the dressed states revealing the nonlinear Jaynes--Cummings ladder. Arrows represent the coupling terms of the Hamiltonian in the dressed-state basis.
• Figure 2: Realization of a qudit phase gate.(a) Pulse sequence for generating a phase gate with rotation angle $\pi/2$ on the target state $|\psi_{\text{target}}\rangle = \frac{1}{\sqrt{2N}}\sum_\pm \sum_{q=1}^{N} |\pm,q\rangle$ for $N=7$ atoms ($14$-level qudit). (b) Time evolution of the initial state $|\psi_{\text{ini}}\rangle = e^{-i \pi/4} |\psi_{\text{target}}\rangle$ for the pulse sequence of (a) (we added a global phase of $-\pi/4$ to improve the visual rendering of the figure). The resulting final state is $|\psi_{\text{final}}\rangle = e^{i \pi/2} |\psi_{\text{in}}\rangle = e^{i \pi/4} |\psi_{\text{target}}\rangle$. We clearly notice the three phases during the dynamics (i) the progressive mapping of the wavefunction to $|-,1\rangle$ (ii) in the middle of the pulse sequence, the $z$-rotation of angle $\pi/2$ acting on the subspace $\left\{|g,0\rangle, |-,1\rangle \right\}$ that temporarily populates $|g,0\rangle$ and (iii) mapping back the wavefunction to $|\psi_{\text{final}}\rangle$. (c) The phase gate leaves the states orthogonal to $|\psi_{\text{target}}\rangle$ unchanged. To illustrate this point, we show the dynamics of $|\psi_{\text{ini}}\rangle = \frac{1}{\sqrt{2}} \left( e^{i \pi/2} |+,7\rangle + e^{-i \pi/2} |-,7\rangle \right)$ that is unaffected by the pulse sequence i.e.$|\psi_{\text{final}}\rangle = |\psi_{\text{ini}}\rangle$. Note that in (b) and (d) we subtracted the trivial phase accumulation due to the diagonal terms of the Hamiltonian (which is anyway compensated at the end of the pulse sequence by the protocol). To improve the readability of the figure, we have set the phases to zero when the corresponding amplitudes are below $10^{-3}$.
• Figure 3: Matrix representation of the generalized phase gate. By simulating the time evolution of the pulse sequence of Fig. \ref{['Fig2']} (a) for each of the qudit basis vectors $\left\{ |\pm,q \rangle \right\}$ as initial state, we obtain the matrix representation of the generalized phase gate introduced in Fig. \ref{['Fig2']}. (a) Amplitude of the matrix. (b) Phase of the matrix. By comparing the gate $\hat{U}$ resulting from the simulation of the dynamics to the expected gate $\hat{U}_{\text{target}} = e^{i \pi/2} |\psi_{\text{target}}\rangle \langle\psi_{\text{target}}| + \left( \hat{I} - |\psi_{\text{target}}\rangle \langle\psi_{\text{target}}|\right)$, we extract a gate infidelity of is $9 \, 10^{-5}$ for $\Omega_{01} / \Omega_{1r} = 10^{-3}$ for $N=7$ ($14$-level qudit) where the gate infidelity is defined as $\epsilon = 1 - \frac{1}{(2N)^2} \left| \text{Tr} \left( \hat{U}_{\text{target}}^\dagger \hat{U} \right) \right|^2$.
• Figure 4: Generalized Hadamard gate.(a) Pulse sequence to generate the generalized Hadamard gate for $N=7$ atoms ($14$-level qudit) and $\Omega_{01}/\Omega_{1r} = 0.004$. It consists of repeatedly applying a phase gate with angle $\alpha_i$ to each of the $2N$ eigenvectors $|\psi_{\alpha_i}\rangle$ of the generalized Hadamard gate with eigenvalues $e^{i \alpha_i}$. (b) Left column: Amplitude and phase of the matrix representation of the target Hadamard gate for $N=7$ atoms. Right column: Matrix representation of the gate resulting from numerical time evolution of the pulse sequence given in (a) for each of the qudit basis vectors $\left\{ |\pm,q \rangle \right\}$ as initial state. For the sake of illustration, we have chosen $\Omega_{01}/\Omega_{1r} = 0.004$ so that the imperfections of the gate remain visible on the figure. For these parameters, the infidelity of the gate is $3\, 10^{-2}$. (c) Numerical simulations of the infidelities of the generalized Hadamard gate as a function of $\Omega_{01}/\Omega_{1r}$ for different values of the number of atoms $N$ (the dimension of the qudit Hilbert space is $2N$). The results confirm the scaling of the infidelities $\sim N^3 \Omega_{01}^2 / \Omega_{1r}^2$. The shaded areas depict the regions where the probability of relaxation of the Rydberg state $P_{\text{relax}}$ exceeds the gate infidelity $\epsilon$ for different values of $N$ with the same color coding as the legend of the simulation data for $\Gamma_r^{-1} = 100 \, \mu \text{s}$ and $\Omega_{1r}/(2\pi) = 25 \, \text{MHz}$. It thus makes it possible to realize experimentally a generalized Hadamard gate with up to a $14$-level qudit ($N=7$) before being hindered by the lifetime of the Rydberg state.