Atom-ion quantum gate

Ultracold collisions of ions with neutral atoms in traps are studied. Recently, ultracold atom-ion systems have become available in experimental setups, where their quantum states can be coherently controlled. This control allows for an implementation of quantum information processing, combining the advantages of charged and neutral particles. The state-dependent dynamics that is a necessary ingredient for quantum computation schemes is provided in this case by the short-range interaction forces that depend on the hyperfine states of both particles. In this work, a theoretical description of spin-state-dependent trapped atom-ion collisions is developed in the framework of a multichannel quantum-defect theory and an effective single-channel model is formulated that reduces the complexity of the problem. Based on this description, a two-qubit phase gate between a ${}^{135}\mathrm{Ba}$${}^{+}$ ion and a ${}^{87}\mathrm{Rb}$ atom is simulated using a realistic combination of the singlet and triplet scattering lengths. The gate process is optimized and accelerated with the help of optimal control techniques. The result is a gate fidelity of $1-{10}^{-3}$ within $350$ $\mu$s.

Figure 1

Concept for quantum computation with atoms and ions: Atoms are prepared in an optical lattice in a Mott insulator phase. A movable ion entangles the atoms and can also be used for sympathetic cooling.

Figure 2

The long-range part of the atom-ion interaction potential equals $-C_4/r^4$. At distances smaller than the potential minimum $r_{\min }$, repulsive terms start to dominate. Quantum defect theory replaces the actual potential $W(r)$ (solid line) with a reference potential $V(r)$ (dashed line) and includes the short-range effects using a quantum-defect parameter related to the short-range phase of the relative wave function. The characteristic range $R^{*}$ of the interaction is typically much larger than $r_{\min }$.

Figure 3

An example correlation diagram calculated for $R^{*}=3.68l_0$ and the short-range phase $\phi =0.74\pi$, showing the energy spectrum versus the trap separation. The partial wave number $l$ is given for the lowest states at $d=0$. As explained in the text, molecular-state energies have an approximately parabolic $d$ dependence, indicated by the dashed line.

Figure 4

Trap-induced shape resonance: At a certain trap separation $d=d_{\mathrm{res}}$, the energy of a molecular bound state degenerates to a trap vibrational energy. The adiabatic eigenenergies exhibit an avoided crossing at this position. The arrows indicate that the molecular energy is shifted if the relative trap position is changed.

Figure 5

Correlation diagrams for $a_s=0.90$ and $a_t=0.95$, where for each of the qubit pairs we subtract the threshold energy of the corresponding channel. The energy curve followed in the adiabatic process is marked with a thick line between ${\Psi }_{\mathrm{ini}}$ and ${\Psi }_{\mathrm{mol}}$ in part (a). We only show the complete diagram for the $|11\rangle$ channel. Small energy differences of the channels around $d=d_{\min }$ can be seen in the close-up (b). These differences are the basis of our proposal for realizing an atom-ion phase gate.

Figure 6

Specific choice of the qubit states out of the manifold of hyperfine spin states of a ${}^{87}\mathrm{Rb}$ atom and a ${}^{135}\mathrm{Ba}$${}^{+}$ ion.

Figure 7

(a) The function $d(t)$ starts at $d_{\max }=1.4R^{*}$ with a slope of $d_1^{\text{'}}=0.5R^{*}/(\hslash /E^{*})$ large enough to diabatically pass weaker resonances. The velocity is changed to around $d=0.95R^{*}$ to $d_2^{\text{'}}=0.1R^{*}/(\hslash /E^{*})$ to ensure an adiabatic traversal of a stronger resonance. The system is brought to the initial distance with the reversed pulse and the kinks are smoothed to avoid motional excitations. The characteristic unit of speed is $R^{*}/(\hslash /E^{*})=2.05$ mm/s. (b) Gate phase as a function of $d_{\min }$ using the described $d(t)$ shape. We find that with $d_{\min }=0.591$ a gate phase of $\varphi =1.009\pi$ is reached.

Figure 8

Energy difference as a function of trap separation. For $d\lesssim 0.3R^{*}$ the function is practically constant, which, for the phase gate process, means that bringing the traps closer than $d_{\min }~0.3R^{*}$ does not lead to a significant advantage.

Figure 9

Population of the most important adiabatic eigenstates for the qubit channel $|11\rangle$ during the adiabatic gate process (the remaining channels show similar behavior). States are labeled according to their energy in the correlation diagram in ascending order. The initial state at $t=0$ is the relative-motion ground state in the trapping potential for $d=d_{\max }$. The label of this state is set to $n=1$. The molecular state marked in Fig. 5a then bears the label $n=4$. Around a crossing, the labels of molecular and trap states switch because their energies change order. Our goal is to follow the energy curve marked between ${\Psi }_{\mathrm{ini}}$ and ${\Psi }_{\mathrm{mol}}$ in Fig. 5a. We observe that indeed two resonances are passed diabatically. At $t=T/2$ the state $n=4$ is reached with relatively high fidelity, which means that a molecular ion is formed here. At $d(T)=d_{\max }$ the initial state is regained (the state-dependent phase of the final state is not depicted here). To better show the features of the curves, the time axis is squeezed between $t_1=0.12$ ms and $t_2=1.18$ ms, where the velocity is lower (see Fig. 7).

Figure 10

(a) Optimized $d(t)$ function and initial guess (dashed) for the fast gate process. The initial state is the trap ground state at $d_{\max }$. The populations of the adiabatic eigenstates are changed by the optimization process. (b) After the initial guess pulse is applied, higher vibrational and also molecular states become populated. (c) These excitations are prevented or undone using the optimized pulse. In this case, at $d_{\min }$ the desired molecular state $n=4$ is reached almost perfectly. For simplicity, only the plots for the qubit channel $|11\rangle$ are shown here; the situation is very similar for the other channels. Note the different time scales used for the transport and phase-accumulation sequences, respectively.