Phys. Rev. A 89, 032306 (2014) - One- and two-qubit quantum gates using superimposed optical-lattice potentials

One- and two-qubit quantum gates using superimposed optical-lattice potentials

Nils B. Jørgensen, Mark G. Bason, and Jacob F. Sherson

Phys. Rev. A 89, 032306 – Published 6 March 2014

Abstract

Steps towards implementing a collision-based two-qubit gate in optical lattices have previously been realized by the parallel merging of all pairs of atoms in a periodicity two superlattice. In contrast, we propose an architecture which allows for the merger of a selected qubit pair in a long-periodicity superlattice structure consisting of two optical lattices with close-lying periodicity. We numerically optimize the gate time and fidelity, including the effects on neighboring atoms and in the presence of experimental sources of error. Furthermore, the superlattice architecture induces a differential hyperfine shift, allowing for single-qubit gates. The fastest possible single-qubit gate times, given a maximal tolerable rotation error on the remaining atoms at various values of the lattice wavelengths, are identified. We find that robust single- and two-qubit gates with gate times of a few hundred microseconds and with error probabilities $\sim$ $10^{- 3}$ are possible.

Received 20 December 2013

DOI:https://doi.org/10.1103/PhysRevA.89.032306

Authors & Affiliations

Nils B. Jørgensen, Mark G. Bason, and Jacob F. Sherson^*

Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark

^*sherson@phys.au.dk

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Vol. 89, Iss. 3 — March 2014

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Figure 1
Overview of the superlattice and the single- and two-qubit spin-state gates. (a) Two optical-lattice potentials are superimposed to create a long-period superlattice. (b) The varying well depth throughout the superlattice results in a varying spin transition frequency for each atom. A microwave tuned to the transition of one atom (red) only partially switches another atom (green). If the partially switched population is kept sufficiently low, a single-qubit gate is realized. (c) When a lattice potential of longer wavelength is added to a lattice potential of shorter wavelength, two wells, each holding an atom, can be merged. Through control of phase and well depth, atoms are sent into the vibrational ground and first excited state, where they interact for an arbitrary amount of time, before reversing the process. The interaction causes a spin state exchange resulting in a two-qubit gate.
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Figure 2
A schematic overview of the mechanism of the single-qubit gate and how the detuning $| Δ_{R}^{i j} |$ is calculated. (a) A plot of an entire SLP, with both hyperfine levels as the blue line and the light blue line (left axis) and an enlargement of the hyperfine splitting as the dashed black line (right axis). The section marked by the thin black dashed line is enlarged in panels (b) and (c). The values used to plot are $A = 0.28$ , $λ_{1} = 1064$ nm, $λ_{2} = 4 / 5 λ_{1} = 851.2$ nm, and $P_{1} = P_{2} = 1$ . (b) When exposed to precisely controlled microwave radiation the target atom is switched while keeping the switched population of neighboring atoms under a threshold $P_{t}$ . The positions of the atoms are calculated by taking the potential minima. (c) The positions of the atoms are used to calculate the hyperfine splitting of both atoms, and the detuning is the difference in this splitting, which in this example is $ℏ | Δ_{R}^{i j} | / η E_{r} = 0.016$ .
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Figure 3
Results for single-qubit gate calculations. (a) The absolute detuning $ℏ | Δ_{R}^{i j} | / η E_{r}$ calculated for an array of different wavelengths and secondary lattice depths. The contour lines represent logarithmic scaling. The addressing time can be calculated through the detuning via $t_{a} = π / \sqrt{P_{t}} | Δ_{R}^{i j} |$ , with $E_{r} / h \approx 2$ kHz. (b) The corresponding probabilities of an operation without scattering the target atom. There is a maximum of high probability 0.9995 and the thick and thin contour lines represent steps of 0.001 and 0.0002. The black area represents probabilities beneath 0.99. Calculated with $λ_{1} = 1064$ nm and $P_{1} = P_{2} = 1$ .
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Figure 4
An example of the operation required to realize a two-qubit gate. (a) The time evolution of the potential leading to two atoms being sent into one well. (b) and (c) Optimal control pulses for the amplitude and phase of the primary lattice leading to the potential deformation seen in panel (a). Density profiles of the two atoms as a function of time illustrating the mapping of one atom into the excited state of a neighboring well (e), while the other atom ends in the ground state (d).
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Figure 5
Fidelity, $F$ , for the simulations of the superlattice two-qubit gate. (a) Different fidelities obtained at different operation times with $n = 5$ , with two results of interest highlighted as I and II. The optimized merging scheme at II corresponds to the example shown in Fig. 4. (b) The change in fidelity for II as a function of error also illustrating how $F_{error}$ is calculated, i.e., by including all fidelities within the dashed lines and assuming the worst. The contour lines represent steps of $10^{- 4}$ . (c) Results for the lattice configuration $n = 10$ with one more point of interest highlighted as III. The lattice beam wavelength $λ_{2}$ is fixed at 1064 nm. Further results for I–III are given in Table 1.
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Physical Review A

covering atomic, molecular, and optical physics and quantum science

One- and two-qubit quantum gates using superimposed optical-lattice potentials

Nils B. Jørgensen, Mark G. Bason, and Jacob F. Sherson

Phys. Rev. A 89, 032306 – Published 6 March 2014

Abstract

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Physical Review A

covering atomic, molecular, and optical physics and quantum science

One- and two-qubit quantum gates using superimposed optical-lattice potentials

Nils B. Jørgensen, Mark G. Bason, and Jacob F. Sherson

Phys. Rev. A 89, 032306 – Published 6 March 2014

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