Quantum computations with atoms in optical lattices: Marker qubits and molecular interactions

We develop a scheme for quantum computation with neutral atoms, based on the concept of “marker” atoms, i.e., auxiliary atoms that can be efficiently transported in state-independent periodic external traps to operate quantum gates between physically distant qubits. This allows for relaxing a number of experimental constraints for quantum computation with neutral atoms in microscopic potential, including single-atom laser addressability. We discuss the advantages of this approach in a concrete physical scenario involving molecular interactions.

Figure 1

Basic operations with a marker atom on a quantum register: (i) forward and backward transport steps, (ii) local interaction with register qubits.

Figure 2

Realizing an entangling operation between distant atoms based on the elementary steps described in Fig. 1: (a) swapping the first qubit onto the marker atom, (b) transporting the marker atom unto the second qubit and local interaction, (c) transport back to the first qubit and inverse swap.

Figure 3

Laser setup. The plane containing the wave vectors ${\vec{k}}_{1,2}$ of a laser pair $a$ intersects the plane containing the wave vectors ${\vec{k}}_{3,4}$ of a laser pair $b$ at the $x$ axis. The planes can have a finite angle between them. The angle between the lasers in each of the pairs are ${\theta }_a$ and ${\theta }_b$, respectively (see also Ref. [26]).

Figure 4

Potential $U(x,t)$ at two different times and $V=100E_r$ (with recoil energy $E_r={\hslash }^2{\kappa }^2∕2m$). At $t_0=0$ $u_1\left(0\right)=u_2\left(0\right)=0$ (dashed line) while at $t_1>0$ we set $u_1\left(t_1\right)=0.5,u_2\left(t_2\right)=0.2$, and $\sigma =1,l=0$ (solid line). In the latter case the difference between the minima is approximately $20E_r$ and the height of every second barrier is reduced by approximately $50\%$.

Figure 5

Illustration of steps (i)–(iv) of the transport process as described in the text. Also shown are the square of the absolute value of the single particle wave functions of the atom which is transported (i.e., the “marker atom”), ${\mid {\psi }^M\left(x\right)\mid }^2$ (solid lines), and of the register atom which is supposed to remain at its lattice site, ${\mid {\psi }^R\left(x\right)\mid }^2$ (dashed lines). The example shown requires $\sigma =1$ and $l=0$.

Figure 6

Band structures for two different values of $u_1,u_2$ and $V=100E_r,\sigma =1,l=0$. (a) Initial values of the lattice parameters $\left[u_1\left(0\right)=u_2\left(0\right)=0\right]$. (b) Band structure during the transport process $\left(u_1=0.99,u_2=0.13\right)$. The two figures correspond to the situations shown in Figs. 5a, 5c.

Figure 7

Adiabatic atom transfer between lattice sites: (a) Instantaneous eigenenergies $E_{k=0}^{\left(n\right)}\left(t\right)$. The upper solid line corresponds to the motional energy of the atom to be transported and the lower solid line corresponds to the register atom initially in the right well (cf. see Fig. 5). (b) Corresponding control parameters $u_{1,2}$. (c) Amplitudes of the lattice components. (d) Phase difference between lattice components. The time axis is given in units of the recoil frequency ${\omega }_r\equiv E_r∕\hslash$. The time interval $[t_1,t_3]$ is not of the same scale as $[0,t_1],[t_3,t_4]$.

Figure 8

Nonadiabatic atom transfer between lattice sites. (a) Control parameters $u_{1,2}$. (b) Corresponding amplitudes of the lattice components. (c) Phase difference between lattice components. The time axis is given in units of the recoil frequency ${\omega }_r\equiv E_r∕\hslash$. The time interval $[t_1,t_3]$ is not of the same scale as $[0,t_1],[t_3,t_4]$.

Figure 9

Interaction of two atoms in an external trap. The two curves schematically describe the Born-Oppenheimer potential for two scattering channels around a Feshbach resonance.

Figure 10

Adiabatic (solid) and diabatic (dashed) energies of a 6-level expansion (molecular resonance level plus five trap levels of an isotropic $100\mathrm{kHz}$ trap) with a linear ramp of magnetic field with slope $5\mathrm{G}∕\mathrm{ms}$. The adiabatic energy levels show a set of avoided crossings. We use the Feshbach resonance level near $100\mathrm{mT}$. The coupling matrix element $V_0$ is $0.884h\nu$ for this trap.

Figure 11

Internal level scheme for a single atom: qubit and auxiliary states.

Figure 12

Internal level scheme for two coupled atoms: levels involved in single- (left) and in two-qubit operations (right). The Raman transitions ${\Omega }_1$ $\left({\Omega }_2\right)$ used for single- (two-)qubit operations are shown.

Figure 13

Dependence of the scattering length of ${}^{87}\mathrm{Rb}$ on the external magnetic field for collisions in channels $\mid 00⟩$ and $\mid 0x⟩$.

Figure 14

Two-qubit ($C$-phase) gate operation: resonance energy ${\epsilon }_{00}$ (top; off-resonant before and after gate operation), overlap between evolved and initial state (center), accumulated phase (bottom).