Coherent control of trapped ions using off-resonant lasers

In this paper we develop a unified framework to study the coherent control of trapped ions subject to state-dependent forces. Taking different limits in our theory, we can reproduce previous designs of quantum gates and propose a different design of fast gates based on continuous laser beams. We demonstrate how to simulate Ising Hamiltonians in a many ions setup, and how to create highly entangled states and induce squeezing. Finally, in a detailed analysis we identify the physical limits of this technique and study the dependence of errors on the temperature.

Figure 1

Trajectories on phase space of a coherent wave packet subject to a single forced harmonic oscillator, $H=\omega a^{†}a+F\sin \left(2t\right)(a+a^{†})$. In (a) we plot the usual phase-space trajectories, $⟨a⟩=⟨X+iP⟩∕\sqrt{2}$, without forcing (dashed) and with $F=0.1$ (solid), for two initial conditions (empty circles). In (b) we plot the same, but on a rotating frame of reference, $⟨a⟩=⟨e^{-i\omega t}(X_r+i{P}_r)⟩∕\sqrt{2}$.

Figure 2

(a) Trajectory in phase space of the center-of-mass state of the ions $({\tilde{Q}}_c,{\tilde{P}}_c)$ during the two-qubit gate (solid line), connecting a possible initial state (black filled circle) to its final state (grey filled circle), for the protocol designed in Sec. 3B. Part (b) shows how the laser pulses (bars) distribute in time for this scheme. Below we plot (c) the number of pairs of pulses, and (d) maximum displacement in phase space required to produce a phase gate using this scheme.

Figure 3

(a) Optimal forcing for a gating time $T\nu =2.1$ (solid), 1.5 (dashed), and 0.9 (dash-dot), where $\omega =2\pi \times \nu$ is the frequency of the ion trap. (b) Intensity of the force vs duration of the gate (solid) and visual aid ${\left(T\nu \right)}^{-3∕2}$ (dashed). All magnitudes have been adimensionalized following the text.

Figure 4

Generation of a (a),(c) cluster state of $N=10$ ions and of (b),(d) a GHZ state of $N=20$ ions, using common forcing for a time $T=1.1∕\omega$. (a),(b) Time dependence of the forces, $g\left(t\right)$. (c),(d) Fidelity $F$, with respect to the target state as a function of time.