Microwave potentials and optimal control for robust quantum gates on an atom chip

We propose a two-qubit collisional phase gate that can be implemented with available atom chip technology and present a detailed theoretical analysis of its performance. The gate is based on earlier phase gate schemes, but uses a qubit state pair with an experimentally demonstrated, very long coherence lifetime. Microwave near fields play a key role in our implementation as a means to realize the state-dependent potentials required for conditional dynamics. Quantum control algorithms are used to optimize gate performance. We employ circuit configurations that can be built with current fabrication processes and extensively discuss the impact of technical noise and imperfections that characterize an actual atom chip. We find an overall infidelity compatible with requirements for fault-tolerant quantum computation.

Figure 1

(Color online) State-selective potential, atomic wave functions, and principle of the gate operation. (a) The state-independent potential $u_c\left(x\right)$ along $x$ for $t<0$ and $t>{\tau }_g$, before and after the gate operation, when $\lambda \left(t\right)=0$. The initial-state wave function of the two atoms in this potential is shown. (b) The state-dependent potential $u_c\left(x\right)+u_i\left(x\right)$ [here $\lambda \left(t\right)=1$] for $0\le t\le {\tau }_g$, during the gate operation. The atomic wave functions after half an oscillation period are shown. The state-independent part $u_c\left(x\right)$ is shown for comparison.

Figure 2

(Color online) Energy-level diagram of the hyperfine structure of the ${}^{87}\mathrm{Rb}$ ground state in a combined static magnetic and microwave field. $V_Z^{F,m_F}$ indicates the energy shift due to the static Zeeman effect, which is (nearly) identical for $\mid 0⟩$ and $\mid 1⟩$. Due to the magnetic field of the microwave, the level $\mid F,m_F⟩$ is shifted in energy by $V_{\mathrm{mw}}^{F,m_F}$. This shift has opposite sign for $\mid 0⟩$ and $\mid 1⟩$, as indicated in the figure (the shift of the other levels is not shown). The microwave transitions contributing to the shift of $\mid 0⟩$ and $\mid 1⟩$ are shown for ${\Delta }_{1,m_1}^{2,m_2}>0$ (blue detuning).

Figure 3

(Color online) Chip layout for the microwave collisional phase gate. (a) Cut through substrate. (b) Top view of wire layout. Wire parameters: $w=0.8\mathrm{\mu }\mathrm{m}$, $s=0.1\mathrm{\mu }\mathrm{m}$, $t=0.2\mathrm{\mu }\mathrm{m}$, $d_T=4.2\mathrm{\mu }\mathrm{m}$, $w_T=1.5\mathrm{\mu }\mathrm{m}$, and $t_T=1.0\mathrm{\mu }\mathrm{m}$. The lower wire (current $I_T$) is fabricated into a groove which was etched into the substrate and covered by an insulating layer. The dc currents $I_T$, $I_C$, $I_L$, and $I_R$ and the orientation of the magnetic bias field $\mathbf{B}$ used to create the state-independent potential $u_c\left(\mathbf{r}\right)$ are shown. The atomic wave functions in this potential are indicated. The three upper wires form a microwave coplanar waveguide (CPW); compare Fig. 4.

Figure 4

(Color online) Transverse ($xz$-plane) magnetic and electric fields of the microwave on the coplanar waveguide. The fields were obtained by a quasistatic simulation including conductor-loss effects. The longitudinal ($y$-direction) electric field, which is several orders of magnitude smaller than the transverse electric field, is not shown in the plots. (a) Magnitude of the fields for $\lambda =1$ shown in a cross section of the CPW. (b) Transverse microwave field components as a function of $x$ at a distance $d=1.80\mathrm{\mu }\mathrm{m}$ from the wire surface, corresponding to the line $z=z_0=1.90\mathrm{\mu }\mathrm{m}$ in (a), which indicates the $z$ position of the static trap minimum. (c) Microwave voltage and current amplitudes on the CPW used in the simulation. $\sigma$ is the conductivity of the gold wires, ${ϵ}_r$ the dielectric constant of the Si substrate. For the size of the conductors, see Fig. 3.

Figure 5

(Color online) Dynamics during the gate operation, shown for $N=3$ oscillations. The gate time is ${\tau }_g=1.110\mathrm{ms}$. Both $\lambda \left(t\right)$ and ${\omega }_{\perp }\left(t\right)$ are used as control parameters. (a) Optimal control of the microwave power. Upper plot: initial trial function ${\lambda }_0\left(t\right)$ (dashed line) and optimized control parameter $\lambda \left(t\right)$ (solid line). Each triangular ramp of $\lambda \left(t\right)$ corresponds to a full oscillation of state $\mid 1⟩$. Lower plot: the difference $\delta \lambda \left(t\right)=\lambda \left(t\right)-{\lambda }_0\left(t\right)$ shows small modulations. (b) Optimal control of the effective one-dimensional interaction strength via modulation of the transverse trap frequency ${\omega }_{\perp }\left(t\right)$. (c) Evolution of the overlap fidelities during the gate operation: $F_{00}\left(t\right)$ (dashed line), $F_{01}\left(t\right)=F_{10}\left(t\right)$ (dotted line), and $F_{11}\left(t\right)$ (solid line). (d) Evolution of the gate phase ${\varphi }_g\left(t\right)$. The phase shift steps are due to the six collisions in state $\mid 11⟩$.

Figure 6

Gate infidelity $1-F\left(T\right)$ as a function of temperature for the gate with $N=3$.

Figure 7

Transverse excitation probability $p_{\perp }^c\left({\tau }_g\right)$ due to collisions of the atoms for $N=3$. $p_{\perp }^c\left({\tau }_g\right)$ is shown as a function of the kinetic energy $E_{\mathrm{kin}}$ of one atom in state $\mid 1⟩$ at the time of the collision.