Colloquium: Trapped ions as quantum bits: Essential numerical tools

Trapped laser-cooled atoms and ions are quantum systems which can be experimentally controlled with an as yet unmatched degree of precision. Due to the control of the motion and the internal degrees of freedom, these quantum systems can be adequately described by a well-known Hamiltonian. In this colloquium, powerful numerical tools for the optimization of the external control of the motional and internal states of trapped neutral atoms, explicitly applied to the case of trapped laser-cooled ions in a segmented ion-trap are presented. Inverse problems when optimizing trapping potentials for ions, are solved. The presentation is complemented by a quantum-mechanical treatment of the wave-packet dynamics of a trapped ion. Efficient numerical solvers for both time-independent and time-dependent problems are provided. Shaping the motional wave functions and optimizing a quantum gate is realized by the application of quantum optimal control techniques. The numerical methods presented can also be used to gain an intuitive understanding of quantum experiments with trapped ions by performing virtual simulated experiments on a personal computer. Code and executables are supplied as supplementary online material.

Figure 1

(Color online) Electrode geometries of ion traps: The rf electrodes (dark shading) and dc electrodes (bright shading), are depicted. (a) Typical electrode configuration for a 3D ring trap with dynamic rf confinement in all three dimensions. (b) Electrode arrangement for a linear Paul trap. The dc electrodes are divided into segments numbered from 1 to 5. For the numerical simulations we assume the following parameters: segments have a width of $2\mathrm{mm}$ and a radius of $0.5\mathrm{mm}$. The central dc electrode is centered at the position $x=0$. The minimum distance of the electrode surface to the trap axis is $1.5\mathrm{mm}$.

Figure 2

Trapping potentials. (a) Electrostatic potentials along the $x$ axis generated by each individual electrode from the linear Paul trap geometry of Fig. 1b. Each curve corresponds to the respective electrode biased to $-1\mathrm{V}$ and all others to $0\mathrm{V}$. (b) Equipotential lines of the pseudopotential in the radial plane ($U_{\mathrm{rf}}=200{\mathrm{V}}_{\mathrm{pp}}$ and ${\omega }_{\mathrm{rf}}=2\pi \times 20\mathrm{MHz}$). Potentials are obtained as described in Sec. 2.

Figure 3

(Color online) Overlapping basis functions from Eq. (15) $v_k\left(x\right)$ (dotted lines) for the finite element method providing linear interpolation (black dashed line) of an arbitrary function (black solid line).

Figure 4

Comparison of the Euler and Störmer-Verlet simulation methods. (a) Trajectories in the radial plane simulated with the explicit Euler method. It can be seen that the trajectories are unstable. (b) Simulation for the same parameters with the Störmer-Verlet method results in stable trajectories. The small oscillations are due to the micromotion caused by the rf drive. (c) Phase-space trajectories of the harmonic axial motion of an ion numerically integrated with the explicit Euler method. The simulation was performed for a set of three different initial phase-space coordinates, as indicated by the triangles. Energy and phase-space area conservations are violated. (d) Equivalent trajectories numerically integrated with the Störmer-Verlet method which respects energy and phase-space area conservation. The parameters are simulation time $80\mu \mathrm{s}$, number of simulation steps 4000, $U_{\mathrm{rf}}=400{\mathrm{V}}_{\mathrm{pp}}$, ${\omega }_{\mathrm{rf}}=2\pi \times 12\mathrm{MHz}$, and $U_{\mathrm{dc}}=(0,1,0,1,0)\mathrm{V}$ for the trap geometry shown in Fig. 1b.

Figure 5

(Color online) Illustration of the regularization technique: Suppression of the divergence at zero (dashed curve) of the $1∕s$ term. The solid curve shows the Tikhonov regularization term, the singular behavior of the $1∕s$ inverse is avoided and diverging values are replaced by values near zero. The dotted curve tends towards 1 for $s\to 0$ where $1∕s$ is diverging. It is used to force diverging inverses towards a given value (see text).

Figure 6

(Color online) Voltage configurations to put the minimum of a harmonic potential with fixed curvature into different positions. The insets show the resulting potentials obtained by linear superposition of the individual electrode potentials.

Figure 7

(Color online) Illustration of the Numerov algorithm for the numerical solution of the TISE. The dashed line shows the trapping potential for a particle. The black solid line shows the ground-state eigenfunction. Dotted and dash-dotted lines show the result of numerical integration starting from right and left if the energy does not correspond to an energy eigenvalue.

Figure 8

(Color online) Fidelity increase after iterative optimization steps. (a) Initial guess of the time-dependent voltages applied to the electrodes. (b) Resulting wave function at the final time $T$. The fidelity is only 0.3. (c) Voltage configurations obtained after 100 iterative calls of the Krotov optimal control method. (d) The final wave function obtained at time $T$ with optimized voltages agrees well with the target ground-state wave function. The fidelity is increased to 0.997.

Figure 9

CNOT gate. (a) Quantum circuit for a CNOT gate between two ions realized by a SWAP operation on the control ion, a CNOT gate between the motional state, and the target ion and a ${\mathrm{SWAP}}^{-1}$ operation. (b) Composite pulse sequence to realize the total CNOT gate between two ions (95; 96).

Figure 10

(Color online) Optimal control of a CNOT gate. (a) Increase in fidelity from 0.43 to 0.975 after 50 iterations (black). The composite pulse sequence realizes a fidelity of 0.994. (b) Time-dependent phase applied on the blue sideband. The dotted curve shows the initial guess $\varphi \left(t\right)=0$ and the black curve shows the result obtained after 50 iterations of the Krotov algorithm. Note the similarity in the obtained phases concerning amplitude and period when compared to the composite pulse sequence as used in 95 and 96) (dashed curve).